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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:04:06 -0700 |
|---|---|---|
| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:04:06 -0700 |
| commit | eccd1b51821186504623f1895f13af611c14e4f1 (patch) | |
| tree | f9be81a27f658eba1d16f35fba5e82281feb016c | |
| -rw-r--r-- | .gitattributes | 3 | ||||
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| -rw-r--r-- | LICENSE.txt | 11 | ||||
| -rw-r--r-- | README.md | 2 |
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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/35588-pdf.pdf b/35588-pdf.pdf Binary files differnew file mode 100644 index 0000000..06c4807 --- /dev/null +++ b/35588-pdf.pdf diff --git a/35588-pdf.zip b/35588-pdf.zip Binary files differnew file mode 100644 index 0000000..ce64988 --- /dev/null +++ b/35588-pdf.zip diff --git a/35588-t.zip b/35588-t.zip Binary files differnew file mode 100644 index 0000000..ae3e149 --- /dev/null +++ b/35588-t.zip diff --git a/35588-t/35588-t.tex b/35588-t/35588-t.tex new file mode 100644 index 0000000..799610b --- /dev/null +++ b/35588-t/35588-t.tex @@ -0,0 +1,9853 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% The Project Gutenberg EBook of Scientific Papers by Sir George Howard % +% Darwin, by George Darwin % +% % +% This eBook is for the use of anyone anywhere at no cost and with % +% almost no restrictions whatsoever. You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: Scientific Papers by Sir George Howard Darwin % +% Volume V. Supplementary Volume % +% % +% Author: George Darwin % +% % +% Commentator: Francis Darwin % +% E. W. Brown % +% % +% Editor: F. J. M. Stratton % +% J. Jackson % +% % +% Release Date: March 16, 2011 [EBook #35588] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK SCIENTIFIC PAPERS *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{35588} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% %% +%% array: Enhanced tabular features. Required. %% +%% longtable: Multi-page tables. Required. %% +%% dcolumn: Customized table column types. Required. %% +%% %% +%% footmisc: Multiple footnote series, and separate footnote %% +%% manyfoot: numbering on each page. Both required. %% +%% %% +%% multicol: Two-column formatting, for index. Required. %% +%% index: Indexing. Required. %% +%% %% +%% graphicx: Graphics inclusion. Required. %% +%% wrapfig: Inset figures. Required. %% +%% caption: Caption customization. Required. %% +%% %% +%% indentfirst: Indent first word of each sectional unit. Required. %% +%% %% +%% calc: Length calculations. Required. %% +%% %% +%% fancyhdr: Enhanced running headers and footers. Required. %% +%% %% +%% yfonts: Gothic text on copyright pages. Optional. %% +%% xcolor: Color, for watermark. Optional. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% Changes are noted in this file in two ways. %% +%% 1. \DPtypo{}{} for typographical corrections, showing original %% +%% and replacement text side-by-side. %% +%% 2. [** TN: Note]s for other comments. %% +%% %% +%% Variant hyphenations clock-work, co-operation, co-ordinates, %% +%% and re-writing have been retained from the original. %% +%% %% +%% Compilation Flags: %% +%% %% +%% The following behavior may be controlled by boolean flags. %% +%% %% +%% ForPrinting (false by default): %% +%% Compile a screen-optimized PDF file. Set to true for print- %% +%% optimized file (wide margins, two-sided, black hyperlinks). %% +%% %% +%% WaterMark (true by default): %% +%% Print "The Project Gutenberg ebook #..." in the footer. %% +%% %% +%% ShowSeparators (false by default): %% +%% Display (unsightly) page separators from the original book. %% +%% %% +%% %% +%% PDF pages: 183 (if ForPrinting set to false) %% +%% PDF page size: 5.5 x 8" (non-standard) %% +%% PDF bookmarks: created, point to ToC entries %% +%% PDF document info: filled in %% +%% Images: 7 png files, 1 jpg %% +%% %% +%% Summary of log file: %% +%% * Three overfull hboxes (all < 5 pt wide). %% +%% * Two underfull hboxes, five underfull vboxes. %% +%% %% +%% %% +%% Compile History: %% +%% %% +%% March, 2011: adhere (Andrew D. Hwang) %% +%% texlive2007, GNU/Linux %% +%% %% +%% Command block: %% +%% %% +%% pdflatex x3 # Run pdflatex three times %% +%% makeindex -s darwin.rst # config file name specified below %% +%% pdflatex x2 %% +%% %% +%% %% +%% March 2011: pglatex. %% +%% Compile this project with: %% +%% pdflatex 35588-t.tex ..... THREE times %% +%% makeindex -s darwin.rst 35588-t.idx %% +%% pdflatex 35588-t.tex ..... TWO times %% +%% %% +%% pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\listfiles +\documentclass[12pt]{book}[2005/09/16] + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\usepackage[utf8]{inputenc}[2006/05/05] + +\usepackage{ifthen}[2001/05/26] %% Logical conditionals + +\usepackage{amsmath}[2000/07/18] %% Displayed equations +\usepackage{amssymb}[2002/01/22] %% and additional symbols + +\usepackage{alltt}[1997/06/16] %% boilerplate, credits, license + +\usepackage{array}[2005/08/23] %% extended array/tabular features +\usepackage{longtable}[2004/02/01] +\usepackage{dcolumn}[2001/05/28] + + %% extended footnote capabilities +\usepackage[multiple,symbol,perpage]{footmisc}[2005/03/17] +\usepackage[perpage]{manyfoot}[2005/09/11] + +\usepackage{multicol}[2006/05/18] +\usepackage{index}[2004/01/20] + +\usepackage{graphicx}[1999/02/16]%% For diagrams +\usepackage{wrapfig}[2003/01/31] %% and wrapping text around them +\usepackage{caption} + +\usepackage{indentfirst}[1995/11/23] + +\usepackage{calc}[2005/08/06] + +\IfFileExists{yfonts.sty}{% + \usepackage{yfonts}[2003/01/08]% fraktur font (for copyright pages) +}{% + \providecommand{\textgoth}[1]{\textbf{##1}}% +} % fallback if no yfonts + +\IfFileExists{xcolor.sty}{% + \usepackage{xcolor}[2007/01/21]% for watermark +}{% + \providecommand{\textcolor}[2]{#2}% +} + +% for running heads +\usepackage{fancyhdr} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% Interlude: Set up PRINTING (default) or SCREEN VIEWING %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% ForPrinting=true (default) false +% Asymmetric margins Symmetric margins +% Black hyperlinks Blue hyperlinks +% Start Preface, ToC, etc. recto No blank verso pages +% +% Chapter-like ``Sections'' start both recto and verso in the scanned +% book. This behavior has been retained. +\newboolean{ForPrinting} + +%% UNCOMMENT the next line for a PRINT-OPTIMIZED VERSION of the text %% +%\setboolean{ForPrinting}{true} + +% COMMENT the second line below to remove the watermarks +\newboolean{WaterMark} +\setboolean{WaterMark}{true} + +% UNCOMMENT the second line below for (unsightly) visible page separators +\newboolean{ShowSeparators} +%\setboolean{ShowSeparators}{true} + + +%% Initialize values to ForPrinting=false +\newcommand{\Margins}{hmarginratio=1:1} % Symmetric margins +\newcommand{\HLinkColor}{blue} % Hyperlink color +\newcommand{\PDFPageLayout}{SinglePage} +\newcommand{\TransNote}{Transcriber's Note} +\newcommand{\TransNoteCommon}{% + The original copy of this book was generously made available for + scanning by the Department of Mathematics at the University of + Glasgow. + \bigskip + + Minor typographical corrections and presentational changes have + been made without comment. + \bigskip +} + +\newcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is optimized for screen viewing, but may easily be + recompiled for printing. Please see the preamble of the \LaTeX\ + source file for instructions. +} +%% Re-set if ForPrinting=true +\ifthenelse{\boolean{ForPrinting}}{% + \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins + \renewcommand{\HLinkColor}{black} % Hyperlink color + \renewcommand{\PDFPageLayout}{TwoPageRight} + \renewcommand{\TransNote}{Transcriber's Note} + \renewcommand{\TransNoteText}{% + \TransNoteCommon + + This PDF file is optimized for printing, but may easily be + recompiled for screen viewing. Please see the preamble of the + \LaTeX\ source file for instructions. + } +}{% If ForPrinting=false, don't skip to recto + \renewcommand{\cleardoublepage}{\clearpage} +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%% End of PRINTING/SCREEN VIEWING code; back to packages %%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\ifthenelse{\boolean{ForPrinting}}{% + \setlength{\paperwidth}{8.5in}% + \setlength{\paperheight}{11in}% + \usepackage[body={5.25in,7in},\Margins]{geometry}[2002/07/08] +}{% + \setlength{\paperwidth}{5.5in}% + \setlength{\paperheight}{8in}% + \usepackage[body={5.25in,7in},\Margins,includeheadfoot]{geometry}[2002/07/08] +} + +\providecommand{\ebook}{00000} % Overridden during white-washing +\usepackage[pdftex, + hyperfootnotes=false, + pdftitle={The Project Gutenberg eBook \#\ebook: <TITLE>}, + pdfauthor={George Howard Darwin}, + pdfkeywords={Laura Wisewell, Chuck Greif, Andrew D. Hwang, + Project Gutenberg Online Distributed Proofreading Team, + University of Glasgow Department of Mathematics}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=1, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2007/02/07] + + +%%%% Fixed-width environment to format PG boilerplate %%%% +\newenvironment{PGtext}{% +\begin{alltt} +\fontsize{9.2}{10.5}\ttfamily\selectfont}% +{\end{alltt}} + +% Length register for scratch work +\newlength{\TmpLen} + +%% Index tweaks +\makeindex + +% Index of original prints "A" at start of A entries, etc. +% Write index style configuration code to .rst file. +\makeatletter +\immediate\openout\@partaux darwin.rst +\immediate\write\@partaux + {heading_prefix "{\string\\centering\string\\bfseries "^^J% + heading_suffix "\string\\endgraf}\string\\nopagebreak\string\n"^^J% + headings_flag 1 }% +\immediate\closeout\@partaux + +% Custom index format +\renewcommand{\@idxitem}{\par\hangindent 20\p@\global\let\idxbrk\nobreak} +\renewcommand\subitem{\idxbrk\@idxitem \hspace*{10\p@}\let\idxbrk\relax} +\renewcommand{\indexspace}{\par\penalty-3000 \vskip 10pt plus5pt minus3pt\relax} + +\renewenvironment{theindex} + {\setlength\columnseprule{0.5pt}\setlength\columnsep{18pt}% + \Chapter{Index to Volume V} + \begin{multicols}{2}% + \footnotesize% + \setlength\parindent{0pt}\setlength\parskip{0pt plus 0.3pt}% + \let\item\@idxitem\raggedright } + {\end{multicols}\fancyhf{}% + \normalsize} +\makeatother + +%% Page separators +\newcommand{\DPPageSep}[2]{% Temp + \ifthenelse{\boolean{ShowSeparators}}{% + \ifthenelse{\boolean{ForPrinting}}{% + \ifmmode {} \else\mbox{}\marginpar{\centering\scriptsize Page #2}\fi% + }{% + \hfill\break\mbox{---\ #2 (\texttt{#1.png})\ ---} \\ + }% + }{}% + \ignorespaces +} + +%% No hrule in page header +\renewcommand{\headrulewidth}{0pt} + +% Mark the footline +\newcommand{\watermark}{% + \ifthenelse{\boolean{WaterMark}}{% + \fancyfoot[C]{% + \textsc{\tiny\textcolor{lightgray}{The Project Gutenberg ebook \#\ebook}}% + }% + }{}% +} + +% Page style to handle otherwise-undecorated pages +\fancypagestyle{myplain}{\fancyhf{}\watermark} + +% Specify entire caption manually +\captionsetup{labelformat=empty,font=footnotesize} + +% Top-level footnote numbers restart on each page +\MakePerPage{footnote} +%[** TN: Footnotes in essays are Arabic, in Darwin's papers are symbols] +\DeclareNewFootnote{N} % Numbered notes + +% Running heads +\newcommand{\SetRunningHeads}[2]{% + \fancyhead{} + \setlength{\headheight}{15pt} + \thispagestyle{myplain} + \fancyhead[CE]{\MakeUppercase{\footnotesize #1}} + \fancyhead[CO]{\MakeUppercase{\footnotesize #2}} + + \ifthenelse{\boolean{ForPrinting}}{% + \fancyhead[RO,LE]{\thepage}% + }{% + \fancyhead[R]{\thepage}% + } + + \watermark% +} + +%%%% Sectional divisions %%%% +\newcounter{ChapNo} +\setcounter{ChapNo}{0} + +\newcounter{SectNo} +\setcounter{SectNo}{0} + +\newcounter{NoteNo} +\setcounter{NoteNo}{0} + +\newcounter{FigNo} +\setcounter{FigNo}{0} + +\newcommand{\Chapter}[2][]{% + \ifthenelse{\boolean{ForPrinting}}{% + \clearpage\fancyhf{}% + }{}% + \cleardoublepage + \phantomsection + \setcounter{SectNo}{0} + \setcounter{FigNo}{0} + \refstepcounter{ChapNo} + \label{chapter:\theChapNo} + \ifthenelse{\equal{#1}{}}{% + \SetRunningHeads{#2}{#2}% + \pdfbookmark[0]{#2}{#2}% + }{% + \SetRunningHeads{#1}{#1}% + \pdfbookmark[0]{#1}{#1}% + } + \section*{\normalfont\large\centering\MakeUppercase{#2}} + \pagestyle{fancy} + + % Reset the page numbering + \ifthenelse{\equal{#2}{Inaugural Lecture}}{% + \pagenumbering{arabic}% + }{}% +} + +% Used both as chapter subheads and as stand-alone units +\newcommand{\Heading}[1]{% + \subsection*{\normalfont\small\centering\textsc{#1}} +} + +\newcommand{\BY}[1]{\Heading{BY\protect\\ \MakeUppercase{#1}}} + +\newcommand{\Section}[2]{% + \subsection*{\normalfont\normalsize\centering #1\textit{#2}} + + \refstepcounter{SectNo}% + \Pagelabel{section:\theChapNo-\theSectNo}% +} + +\newcommand{\Subsection}[1]{\subsubsection*{\normalfont\textit{#1}}} + +\newcommand{\Note}[2]{% + \subsection*{\normalfont\normalsize\centering\textsc{Note #1} \textit{#2}} + \refstepcounter{NoteNo}% + \Pagelabel{note:\theNoteNo}% +} + +\newcommand{\Appendix}[1]{% + \clearpage + \subsection*{\normalfont\centering APPENDIX.} + \subsubsection*{\normalfont\centering\textit{#1}} +} + +% Only used once +\newcommand{\tb}{% + \begin{center} + *\qquad*\qquad*\qquad*\qquad*\qquad* +\end{center} +} + +%\begin{Letter}{Date}{Salutation} +\newenvironment{Letter}[2]{% + \par + \small + \ifthenelse{\not\equal{#1}{}}{\Date{#1}}{}% + \ifthenelse{\not\equal{#2}{}}{\Salutation{#2}}{\vspace{-\topsep}}% + \begin{list}{}{% + \setlength{\leftmargin}{\parindent}% + \setlength{\parskip}{0pt}% + \setlength{\listparindent}{\parindent}% + \setlength{\labelwidth}{0pt}% + \setlength{\itemindent}{2\parindent}% + } + \item +}{% + \end{list}% + \normalsize +} + +\newcommand{\Date}[1]{\vspace{-\topsep}\null\hfill#1\qquad\break} +\newcommand{\Salutation}[1]{\indent\indent#1\vspace{-\topsep}} + +\newcommand{\Signature}[2]{% + \nopagebreak\par% + \begin{minipage}{\textwidth-\parindent-4em} + \null\hfill#1\qquad\qquad\\ + \null\hfill\textsc{#2} + \end{minipage} + \pagebreak[1] +} + +\newenvironment{Quote}{% + \begin{list}{}{% + \setlength{\leftmargin}{\parindent}% + \small\setlength{\parskip}{0pt}% + \setlength{\listparindent}{\parindent}% + \setlength{\labelwidth}{0pt}% + \setlength{\itemindent}{\parindent}% + } + \item +}{% + \end{list}% +} + +%% Illustrations +\newcommand{\Input}[2][\textwidth]{% + \includegraphics[width=#1]{./images/#2.png}% +} + +%% Table of contents is formatted manually +\newcommand{\ToCFrontis}[1]{% + \settowidth{\TmpLen}{\footnotesize\textit{Frontispiece}\ }% + \noindent\parbox[b]{\textwidth - \TmpLen}{% + \footnotesize\scshape #1\dotfill% + }% + \makebox[\TmpLen][r]{\hyperref[frontis]{\footnotesize\textit{Frontispiece}}}% +} + +\newcommand{\ToCChap}[2]{% + \settowidth{\TmpLen}{xxxx}% + \noindent\parbox[b]{\textwidth - \TmpLen}{% + \strut\footnotesize\scshape\hangindent2em #1\dotfill% + }% + \makebox[\TmpLen][r]{\footnotesize\Pgref{#2}}% +} + +\newcommand{\ToCSec}[3]{% + \settowidth{\TmpLen}{xxxx}% + \noindent\hspace*{2em}\parbox[b]{\textwidth - \TmpLen - 2em}{% + \strut\footnotesize\raggedright\hangindent6em \ToCBox{Note 2.}{#1}#2\dotfill% + }% + \makebox[\TmpLen][r]{\footnotesize\Pgref{section:7-#3}}% +} + +\newcommand{\ToCNote}[3]{% + \settowidth{\TmpLen}{xxxx}% + \noindent\hspace*{2em}\parbox[b]{\textwidth - \TmpLen - 2em}{% + \strut\footnotesize\raggedright\hangindent4em \ToCBox{Note 2.\quad}{#1}#2\dotfill% + }% + \makebox[\TmpLen][r]{\footnotesize\Pgref{#3}}% +} + +\newcommand{\ToCMisc}[1]{% + \settowidth{\TmpLen}{xxxx}% + \noindent\hspace*{2em}\parbox[b]{\textwidth - \TmpLen - 2em}{% + \footnotesize\hangindent2em #1% + }% +} + +\newcommand{\ToCBox}[2]{% + \settowidth{\TmpLen}{\footnotesize #1}% + \makebox[\TmpLen][l]{\footnotesize#2}% +} +\newcommand{\ToCPAGE}{\noindent\makebox[\textwidth][r]{\scriptsize PAGE}} + + +%% Textual conveniences +\newcommand{\Vol}[1]{\textsc{\MakeLowercase{#1}}} + +\newcommand{\First}[1]{\textsc{#1}} + +\newcommand{\Ditto}[1][August ]{% + \settowidth{\TmpLen}{#1}% + \makebox[\TmpLen][c]{''}% +} + +\newcommand{\DPtypo}[2]{#2} +\newcommand{\DPnote}[1]{} +\newcommand{\Add}[1]{#1} + +%% Math conveniences +\DeclareMathSizes{12}{11}{8}{6} +% Spacing +\newcommand{\Z}{\phantom{0}} +\newcommand{\Neg}{\phantom{-}} +\newcommand{\Allowbreak}{\displaybreak[1] \\[2pt plus 12pt]} + +% Alignment tweaks for long tables in Appendix to "On Librating Planets..." +\newcommand{\ColHead}[1]{\multicolumn{1}{c}{\ensuremath{#1}}} +\newcolumntype{.}[1]{D{.}{.}{#1}} +\newcolumntype{,}[1]{D{,}{\ }{#1}} + +% Degrees and minutes column heading +\newcommand{\AngleHeading}{% + \smash{\raisebox{-12pt}{% + \ensuremath{\Z\Z\Z\Z\stackrel{°}{\Z\Z}\ \stackrel{'}{\Z\Z}} + } + } +} + +\newcommand{\UnderNote}[2]{% + \underset{\makebox[0pt][c]{\footnotesize #2}}{#1}} + +\newcommand{\dd}{\partial} +\DeclareMathOperator{\cosec}{cosec} + +% Original uses \surd sporadically; use \sqrt for consistency +\newcommand{\Surd}[1]{\sqrt{#1}} + +% Add space after \therefore +\let\oldthus\therefore +\renewcommand{\therefore}{\oldthus\ } + +% Mixed upright and italic "a" and "m" in math +\DeclareMathSymbol{a}{\mathalpha}{operators}{`a} +\DeclareMathSymbol{m}{\mathalpha}{operators}{`m} +\renewcommand{\a}{\mathit{a}} +\newcommand{\m}{\mathit{m}} + +\DeclareUnicodeCharacter{00A3}{\pounds} +\DeclareInputText{183}{\ifmmode\cdot\else{\ \textperiodcentered\ }\fi} + +% For condensed intertext in bracketed equation groups +\newcommand{\lintertext}[2]{% + \settowidth{\TmpLen}{$\displaystyle #2\Biggl\}$}% + \llap{\makebox[0.5\textwidth - 0.5\TmpLen][l]{#1}}#2 +} + +% Cross-referencing: anchors +\newcommand{\Pagelabel}[1] + {\phantomsection\label{#1}} + +\newcommand{\Figlabel}[1] + {\phantomsection\label{fig:\theChapNo-#1}} + +\newcommand{\Tag}[1]{% + \phantomsection + \label{eqn:\theChapNo-#1} + \tag*{\normalsize\ensuremath{#1}} +} + +% and links +\newcommand{\Pageref}[2][p.]{\hyperref[#2]{#1~\pageref{#2}}} +\newcommand{\Pgref}[1]{% + \hyperref[#1]{\pageref{#1}}% +} + +% \Eqref{equation}{(<number>)} +\newcommand{\Eqno}[1]{\normalsize\ensuremath{#1}} +\newcommand{\Eqref}[2][]{% + \ifthenelse{\equal{#1}{}}{% + \hyperref[eqn:\theChapNo-#2]{\Eqno{#2}}% + }{% + \hyperref[eqn:\theChapNo-#2]{#1~\Eqno{#2}}% + }% +} + +\newcommand{\SecRef}[1]{% + \hyperref[section:\theChapNo-#1]{{\upshape §~#1}}% +} + +\newcommand{\FigRef}[2][fig.]{% + \hyperref[fig:\theChapNo-#2]{#1~#2}% +} + +% Loosen the horizontal spacing +\setlength{\emergencystretch}{1.5em} +\newcommand{\stretchyspace}{\spaceskip 0.375em plus 0.375em minus 0.25em} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +\pagestyle{empty} +\pagenumbering{Alph} + +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of Scientific Papers by Sir George Howard +Darwin, by George Darwin + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Scientific Papers by Sir George Howard Darwin + Volume V. Supplementary Volume + +Author: George Darwin + +Commentator: Francis Darwin + E. W. Brown + +Editor: F. J. M. Stratton + J. Jackson + +Release Date: March 16, 2011 [EBook #35588] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK SCIENTIFIC PAPERS *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Laura Wisewell, Chuck Greif +and the Online Distributed Proofreading Team at +http://www.pgdp.net (The original copy of this book was +generously made available for scanning by the Department +of Mathematics at the University of Glasgow.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\phantomsection +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +\newpage +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\DPPageSep{001}{Unnumbered page} +\begin{center} +\null\vfill +\LARGE\textbf{SCIENTIFIC PAPERS} +\vfill +\end{center} +\newpage +\DPPageSep{002}{Unnumbered page} +\begin{center} +\null\vfill +\scriptsize\setlength{\TmpLen}{2pt}% +CAMBRIDGE UNIVERSITY PRESS \\[\TmpLen] +C. F. CLAY, \textsc{Manager} \\[\TmpLen] +\textgoth{London}: FETTER LANE, E.C. \\[\TmpLen] +\textgoth{Edinburgh}: 100 PRINCES STREET \\[\TmpLen] +\includegraphics[width=1in]{./images/cups.png} \\[\TmpLen] +\textgoth{New York}: G. P. PUTNAM'S SONS \\[\TmpLen] +\textgoth{Bombay, Calcutta and Madras}: MACMILLAN AND CO., \textsc{Ltd.} \\[\TmpLen] +\textgoth{Toronto}: J. M. DENT AND SONS, \textsc{Ltd.} \\[\TmpLen] +\textgoth{Tokyo}: THE MARUZEN-KABUSHIKI-KAISHA +\vfill +\textit{All rights reserved} +\end{center} +\frontmatter +\pagenumbering{roman} +\DPPageSep{003}{i} +%[Blank Page] +\DPPageSep{004}{ii} +\null\vfill +\begin{figure}[p!] + \centering + \Pagelabel{frontis} + \ifthenelse{\boolean{ForPrinting}}{% + \includegraphics[width=\textwidth]{./images/frontis.jpg} + }{% + \includegraphics[width=0.875\textwidth]{./images/frontis.jpg} + } +\iffalse +[Hand-written note: From a water-colour drawing +by his daughter +Mrs Jacques Raverat +G. H. Darwin] +\fi +\end{figure} +\vfill +\clearpage +\DPPageSep{005}{iii} +\begin{center} +\setlength{\TmpLen}{12pt}% +\textbf{\Huge SCIENTIFIC PAPERS} +\vfil +\footnotesize% +BY \\[\TmpLen] +{\normalsize SIR GEORGE HOWARD DARWIN} \\ +{\scriptsize K.C.B., F.R.S. \\ +FELLOW OF TRINITY COLLEGE \\ +PLUMIAN PROFESSOR IN THE UNIVERSITY OF CAMBRIDGE} +\vfil +VOLUME V \\ +SUPPLEMENTARY VOLUME \\[\TmpLen] + +{\scriptsize CONTAINING} \\ + +BIOGRAPHICAL MEMOIRS BY SIR FRANCIS DARWIN \\[2pt] +AND PROFESSOR E. W. BROWN, \\[2pt] +LECTURES ON HILL'S LUNAR THEORY, \textsc{etc.} +\vfil +EDITED BY \\ +F. J. M. STRATTON, M.A., \textsc{and} J. JACKSON, M.A., \textsc{B.Sc.} +\vfil\vfil +\normalsize +Cambridge: \\ +at the University Press \\ +1916 +\end{center} +\newpage +\DPPageSep{006}{iv} +\begin{center} +\null\vfill\scriptsize +\textgoth{Cambridge}: \\ +PRINTED BY JOHN CLAY, M.A. \\ +AT THE UNIVERSITY PRESS +\vfill +\end{center} +\newpage +\DPPageSep{007}{v} + + +\Chapter{Preface} + +\First{Before} his death Sir~George Darwin expressed the view that his +lectures on Hill's Lunar Theory should be published. He made no +claim to any originality in them, but he believed that a simple presentation +of Hill's method, in which the analysis was cut short while the fundamental +principles of the method were shewn, might be acceptable to students of +astronomy. In this belief we heartily agree. The lectures might also +with advantage engage the attention of other students of mathematics +who have not the time to enter into a completely elaborated lunar theory. +They explain the essential peculiarities of Hill's work and the method of +approximation used by him in the discussion of an actual problem of +nature of great interest. It is hoped that sufficient detail has been given +to reveal completely the underlying principles, and at the same time not +be too tedious for verification by the reader. + +During the later years of his life Sir~George Darwin collected his +principal works into four volumes. It has been considered desirable to +publish these lectures together with a few miscellaneous articles in a fifth +volume of his works. Only one series of lectures is here given, although +he lectured on a great variety of subjects connected with Dynamics, Cosmogony, +Geodesy, Tides, Theories of Gravitation,~etc. The substance of +many of these is to be found in his scientific papers published in the four +earlier volumes. The way in which in his lectures he attacked problems +of great complexity by means of simple analytical methods is well illustrated +in the series chosen for publication. + +Two addresses are included in this volume. The one gives a view of +the mathematical school at Cambridge about~1880, the other deals with +the mathematical outlook of~1912. +\DPPageSep{008}{vi} + +The previous volumes contain all the scientific papers by Sir~George +Darwin published before~1910 which he wished to see reproduced. They +do not include a large number of scientific reports on geodesy, the tides and +other subjects which had involved a great deal of labour. Although the +reports were of great value for the advancement and encouragement of +science, he did not think it desirable to reprint them. We have not +ventured to depart from his own considered decision; the collected lists +at the beginning of these volumes give the necessary references for such +papers as have been omitted. We are indebted to the Royal Astronomical +Society for permission to complete Sir~George Darwin's work on Periodic +Orbits by reproducing his last published paper. + +The opportunity has been taken of securing biographical memoirs of +Darwin from two different points of view. His brother, Sir~Francis Darwin, +writes of his life apart from his scientific work, while Professor E.~W.~Brown, +of Yale University, writes of Darwin the astronomer, mathematician and +teacher. + +\footnotesize +\settowidth{\TmpLen}{F. J. M. S.\quad}% +\null\hfill\parbox{\TmpLen}{F. J. M. S.\\ J. J.} + +\scriptsize +\textsc{Greenwich,} \\ +\indent\indent6 \textit{December} 1915. + +\normalsize +\newpage +\DPPageSep{009}{vii} +%[** TN: Table of Contents] + + +\Chapter{Contents} +\enlargethispage{36pt} +\ToCFrontis{Portrait of Sir George Darwin}%{Frontispiece} + +\ToCPAGE + +\ToCChap{Memoir of Sir George Darwin by his brother Sir Francis Darwin} +{chapter:3}%{ix} + +\ToCChap{The Scientific Work of Sir George Darwin by Professor E. W. +Brown}{chapter:4}%{xxxiv} + +\ToCChap{Inaugural lecture (Delivered at Cambridge, in 1883, on Election to +the Plumian Professorship)}{chapter:5}%{1} + +\ToCChap{Introduction to Dynamical Astronomy}{chapter:6}%{9} + +\ToCChap{Lectures on Hill's Lunar Theory}{chapter:7}%{16} + +\ToCSec{§ 1.}{Introduction}{1}%{16} + +\ToCSec{§ 2.}{Differential Equations of Motion and Jacobi's Integral} +{2}%{17} + +\ToCSec{§ 3.}{The Variational Curve}{3}%{22} + +\ToCSec{§ 4.}{Differential Equations for Small Displacements from the +Variational Curve}{4}%{26} + +\ToCSec{§ 5.}{Transformation of the Equations in § 4}{5}%{29} + +\ToCSec{§ 6.}{Integration of an important type of Differential Equation} +{6}%{36} + +\ToCSec{§ 7.}{Integration of the Equation for~$\delta p$}{7}%{39} + +\ToCSec{§ 8.}{Introduction of the Third Coordinate}{8}%{43} + +\ToCSec{§ 9.}{Results obtained}{9}%{45} + +\ToCSec{§ 10.}{General Equations of Motion and their solution} +{10}%{46} + +\ToCSec{§ 11.}{Compilation of Results}{11}%{52} + +\ToCNote{Note 1.}{On the Infinite Determinant of § 5}{note:1}%{53} + +\ToCNote{Note 2.}{On the periodicity of the integrals of the equation +\[ +\frac{d^{2}\, \delta p}{d\tau^{2}} + \Theta\, \delta p = 0, +\] +where $\Theta = \Theta_{0} + \Theta_{1} \cos 2\tau + + \Theta_{2} \cos 4\tau + \dots$.}{note:2}%{55} + +\ToCChap{On Librating Planets and on a New Family of Periodic Orbits} +{chapter:8}%{59} + +\ToCMisc{[\textit{Monthly Notices of the Royal Astronomical Society}, Vol.~72 (1912), pp.~642--658.]} + +\ToCChap{Address to the International Congress of Mathematicians at +Cambridge in 1912}{chapter:9}%{76} + +\ToCChap{Index}{indexpage}%{80} +\DPPageSep{010}{viii} +% [Blank Page] +\DPPageSep{011}{ix} + +\cleardoublepage +\phantomsection +\pdfbookmark[-1]{Main Matter}{Main Matter} + + +\Chapter{Memoir of Sir George Darwin} +\BY{His Brother Sir Francis Darwin} +\SetRunningHeads{Memoir of Sir George Darwin}{By Sir Francis Darwin} +\index{Darwin, Sir Francis, Memoir of Sir George Darwin by}% +\index{Darwin, Sir George, genealogy}% +\index{Galton, Sir Francis}% + +George Howard, the fifth\footnoteN + {The third of those who survived childhood.} +child of Charles and Emma Darwin, was +born at Down July~9th, 1845. Why he was christened\footnoteN + {At Maer, the Staffordshire home of his mother.} +George, I cannot +say. It was one of the facts on which we founded a theory that our parents +lost their presence of mind at the font and gave us names for which there +was neither the excuse of tradition nor of preference on their own part. +His second name, however, commemorates his great-grandmother, Mary +Howard, the first wife of Erasmus Darwin. It seems possible that George's +ill-health and that of his father were inherited from the Howards. This at +any rate was Francis Galton's view, who held that his own excellent health +was a heritage from Erasmus Darwin's second wife. George's second name, +Howard, has a certain appropriateness in his case for he was the genealogist +and herald of our family, and it is through Mary Howard that the +Darwins can, by an excessively devious route, claim descent from certain +eminent people, e.g.~John of~Gaunt. This is shown in the pedigrees which +George wrote out, and in the elaborate genealogical tree published in Professor +Pearson's \textit{Life of Francis Galton}. George's parents had moved to +Down in September~1842, and he was born to those quiet surroundings of +which Charles Darwin wrote ``My life goes on like clock-work\DPnote{[** TN: Hyphenated in original]} and I am +fixed on the spot where I shall end it.\footnotemarkN'' It would have been difficult to +\footnotetextN{\textit{Life and Letters of Charles Darwin}, vol.~\Vol{I.} p.~318.}% +find a more retired place so near London. In 1842 a coach drive of some +twenty miles was the only means of access to Down; and even now that +railways have crept closer to it, it is singularly out of the world, with little +to suggest the neighbourhood of London, unless it be the dull haze of smoke +that sometimes clouds the sky. In 1842 such a village, communicating with +the main lines of traffic only by stony tortuous lanes, may well have been +enabled to retain something of its primitive character. Nor is it hard to +believe in the smugglers and their strings of pack-horses making their way +up from the lawless old villages of the Weald, of which the memory then +still lingered. +\DPPageSep{012}{x} + +George retained throughout life his deep love for Down. For the lawn +\index{Darwin, Sir George, genealogy!boyhood}% +with its bright strip of flowers; and for the row of big lime trees that +bordered it. For the two yew trees between which we children had our +swing, and for many another characteristic which had become as dear and +as familiar to him as a human face. He retained his youthful love of +the ``Sand-walk,'' a little wood far enough from the house to have for us +a romantic character of its own. It was here that our father took his daily +exercise, and it has ever been haunted for us by the sound of his heavy +walking stick striking the ground as he walked. + +George loved the country round Down,---and all its dry chalky valleys +of ploughed land with ``shaws,'' i.e.~broad straggling hedges on their +crests, bordered by strips of flowery turf. The country is traversed by +many foot-paths, these George knew well and used skilfully in our walks, +in which he was generally the leader. His love for the house and the +neighbourhood was I think entangled with his deepest feelings. In later +years, his children came with their parents to Down, and they vividly +remember his excited happiness, and how he enjoyed showing them his +ancient haunts. + +In this retired region we lived, as children, a singularly quiet life +practically without friends and dependent on our brothers and sisters for +companionship. George's earliest recollection was of drumming with his +spoon and fork on the nursery table because dinner was late, while a +barrel-organ played outside. Other memories were less personal, for instance +the firing of guns when Sebastopol was supposed to have been taken. His +diary of~1852 shows a characteristic interest in current events and in the +picturesqueness of Natural History: +\begin{Quote} +\centering +The Duke is dead. Dodos are out of the world. +\end{Quote} +He perhaps carried rather far the good habit of re-reading one's\DPnote{[** TN: [sic]]} favourite +authors. He told his children that for a year or so he read through every +day the story of Jack the Giant Killer, in a little chap-book with coloured +pictures. He early showed signs of the energy which marked his character +in later life. I am glad to remember that I became his companion and +willing slave. There was much playing at soldiers, and I have a clear +remembrance of our marching with toy guns and knapsacks across the +field to the Sand-walk. There we made our bivouac with gingerbread, +and milk, warmed (and generally smoked) over a ``touch-wood'' fire. I was +a private while George was a sergeant, and it was part of my duty to stand +sentry at the far end of the kitchen-garden until released by a bugle-call +from the lawn. I have a vague remembrance of presenting my fixed bayonet +at my father to ward off a kiss which seemed to me inconsistent with my +military duties. Our imaginary names and heights were written up on the +wall of the cloak-room. George, with romantic exactitude, made a small +\DPPageSep{013}{xi} +foot rule of such a size that he could conscientiously record his height as +$6$~feet and mine as slightly less, in accordance with my age and station. + +Under my father's instruction George made spears with loaded heads +which he hurled with remarkable skill by means of an Australian throwing +stick. I used to skulk behind the big lime trees on the lawn in the character +of victim, and I still remember the look of the spears flying through the air +with a certain venomous waggle. Indoors, too, we threw at each other lead-weighted +javelins which we received on beautiful shields made by the village +carpenter and decorated with coats of arms. + +Heraldry was a serious pursuit of his for many years, and the London +\index{Darwin, Sir George, genealogy!interested in heraldry}% +Library copies of Guillim and Edmonson\footnoteN + {Guillim, John, \textit{A display of heraldry}, 6th~ed., folio~1724. Edmonson,~J., \textit{A complete body + of heraldry}, folio~1780.} +were generally at Down. He +retained a love of the science through life, and his copy of Percy's \textit{Reliques} +is decorated with coats of arms admirably drawn and painted. In later life +he showed a power of neat and accurate draughtsmanship, and some of the +illustrations in his father's books, e.g.~in \textit{Climbing Plants}, are by his hand. + +His early education was given by governesses: but the boys of the family +\index{Darwin, Sir George, genealogy!education}% +used to ride twice or thrice a week to be instructed in Latin by Mr~Reed, the +Rector of Hayes---the kindest of teachers. For myself, I chiefly remember +the cake we used to have at 11~o'clock and the occasional diversion of looking +at the pictures in the great Dutch bible. George must have impressed his +parents with his solidity and self-reliance, since he was more than once +allowed to undertake alone the $20$~mile ride to the house of a relative at +Hartfield in Sussex. For a boy of ten to bait his pony and order his +luncheon at the Edenbridge inn was probably more alarming than the +rest of the adventure. There is indeed a touch of David Copperfield in +his recollections, as preserved in family tradition. ``The waiter always said, +`What will you have for lunch, Sir?' to which he replied. `What is there?' +and the waiter said, `Eggs and bacon'; and, though he hated bacon more +than anything else in the world, he felt obliged to have it.'' + +On August~16th, 1856, George was sent to school. Our elder brother, +William, was at Rugby, and his parents felt his long absences from home +such an evil that they fixed on the Clapham Grammar School for their +younger sons. Besides its nearness to Down, Clapham had the merit of +giving more mathematics and science than could them be found in public +schools. It was kept by the Rev.~Charles Pritchard\footnotemarkN, a man of strong +\footnotetextN{Afterwards Savilian Professor of Astronomy at Oxford. Born~1808, died~1893.}% +character and with a gift for teaching mathematics by which George undoubtedly +profited. In (I think) 1861 Pritchard left Clapham and was +succeeded by the Rev.~Alfred Wrigley, a man of kindly mood but without +the force or vigour of Pritchard. As a mathematical instructor I imagine +\DPPageSep{014}{xii} +Wrigley was a good drill-master rather than an inspiring teacher. Under +him the place degenerated to some extent; it no longer sent so many boys +to the Universities, and became more like a ``crammer's'' and less like a public +school. My own recollections of George at Clapham are coloured by an abiding +gratitude for his kindly protection of me as a shrinking and very unhappy +``new boy'' in~1860. + +George records in his diary that in 1863 he tried in vain for a Minor +\index{Darwin, Sir George, genealogy!at Cambridge}% +Scholarship at St~John's College, Cambridge, and again failed to get one at +Trinity in~1864, though he became a Foundation Scholar in~1866. These +facts suggested to me that his capacity as a mathematician was the result of +slow growth. I accordingly applied to Lord Moulton, who was kind enough +to give me his impressions: +\begin{Quote} +My memories of your brother during his undergraduate career +correspond closely to your suggestion that his mathematical power +developed somewhat slowly and late. Throughout most if not the +whole of his undergraduate years he was in the same class as myself +and Christie, the ex-Astronomer Royal, at Routh's\footnotemarkN. We all recognised +\footnotetextN{The late Mr~Routh was the most celebrated Mathematical ``Coach'' of his +day.}% +him as one who was certain of being high in the Tripos, but he did not +display any of that colossal power of work and taking infinite trouble +that characterised him afterwards. On the contrary, he treated his +work rather jauntily. At that time his health was excellent and he +took his studies lightly so that they did not interfere with his enjoyment +of other things\footnotemarkN. I remember that as the time of the examination +\footnotetextN{Compare Charles Darwin's words: ``George has not slaved himself, which makes his + success the more satisfactory.'' (\textit{More Letters of C.~Darwin}, vol.~\Vol{II.} p.~287)}% +came near I used to tell him that he was unfairly handicapped in being +in such robust health and such excellent spirits. + +Even when he had taken his degree I do not think he realised his +innate mathematical power\ldots. It has been a standing wonder to me that +he developed the patience for making the laborious numerical calculations +on which so much of his most original work was necessarily +based. He certainly showed no tendency in that direction during his +undergraduate years. Indeed he told me more than once in later life +that he detested Arithmetic and that these calculations were as tedious +and painful to him as they would have been to any other man, but that +he realised that they must be done and that it was impossible to train +anyone else to do them. +\end{Quote} + +As a Freshman he ``kept'' (i.e.~lived) in~A\;6, the staircase at the N.W. +corner of the New Court, afterwards moving to~F\;3 in the Old Court, +pleasant rooms entered by a spiral staircase on the right of the Great Gate. +Below him, in the ground floor room, now used as the College offices, lived +Mr~Colvill, who remained a faithful but rarely seen friend as long as George +lived. + +Lord Moulton, who, as we have seen, was a fellow pupil of George's at +Routh's, was held even as a Freshman to be an assured Senior Wrangler, +\DPPageSep{015}{xiii} +a prophecy that he easily made good. The second place was held by George, +and was a much more glorious position than he had dared to hope for. In +those days the examiners read out the list in the Senate House, at an early +hour, 8~a.m.\ I think. George remained in bed and sent me to bring the +news. I remember charging out through the crowd the moment the magnificent +``Darwin of Trinity'' had followed the expected ``Moulton of St~John's.'' +I have a general impression of a cheerful crowd sitting on George's bed and +literally almost smothering him with congratulations. He received the +following characteristic letter from his father\footnotemarkN: +\footnotetextN{\textit{Emma Darwin, A Century of Family Letters}, vol.~\Vol{II.} p.~186.}% +\index{Darwin, Charles, ix; letters of}% +\begin{Letter} + {\textsc{Down}, \textit{Jan.}~24\textit{th} [1868].} + {My dear old fellow,} + + I am so pleased. I congratulate you with all my heart and soul. I + always said from your early days that such energy, perseverance and + talent as yours would be sure to succeed: but I never expected such + brilliant success as this. Again and again I congratulate you. But + you have made my hand tremble so I can hardly write. The telegram + came here at eleven. We have written to W.~and the boys. + + God bless you, my dear old fellow---may your life so continue. + + \Signature{Your affectionate Father,}{Ch.~Darwin.} +\end{Letter} + +In those days the Tripos examination was held in the winter, and the +successful candidates got their degrees early in the Lent Term; George +records in his diary that he took his~B.A. on January~25th, 1868: also +that he won the second of the two Smith's Prizes,---the first being the +natural heritage of the Senior Wrangler. There is little to record in this +year. He had a pleasant time in the summer coaching Clement Bunbury, +the nephew of Sir~Charles, at his beautiful place Barton Hall in Suffolk. +In the autumn he was elected a Fellow of Trinity, as he records, ``with +Galabin, young Niven, Clifford, [Sir~Frederick] Pollock, and [Sir~Sidney] +Colvin.'' W.~K.~Clifford was the well-known brilliant mathematician who +died comparatively early. + +Chief among his Cambridge friends were the brothers Arthur, Gerald +\index{Darwin, Sir George, genealogy!friendships}% +and Frank Balfour. The last-named was killed, aged~31, in a climbing +accident in~1882 on the Aiguille Blanche near Courmayeur. He was +remarkable both for his scientific work and for his striking and most lovable +personality. George's affection for him never faded. Madame Raverat remembers +her father (not long before his death) saving with emotion, ``I dreamed +Frank Balfour was alive.'' I imagine that tennis was the means of bringing +George into contact with Mr~Arthur Balfour. What began in this chance +way grew into an enduring friendship, and George's diary shows how much +kindness and hospitality he received from Mr~Balfour. George had also the +\DPPageSep{016}{xiv} +advantage of knowing Lord Rayleigh at Cambridge, and retained his friendship +through his life. + +In the spring of~1869 he was in Paris for two months working at French. +His teacher used to make him write original compositions, and George gained +a reputation for humour by giving French versions of all the old Joe~Millers +and ancient stories he could remember. + +It was his intention to make the Bar his profession\footnotemarkN, and in October~1869 +\footnotetextN{He was called in 1874 but did not practise.}% +we find him reading with Mr~Tatham, in 1870~and~1872 with the late +Mr~Montague Crackenthorpe (then Cookson). Again, in November~1871, he +was a pupil of Mr~W.~G. Harrison. The most valued result of his legal work +was the friendship of Mr~and~Mrs Crackenthorpe, which he retained throughout +his life. During these years we find the first indications of the circumstances +which forced him to give up a legal career---namely, his failing health and +\index{Darwin, Sir George, genealogy!ill health}% +his growing inclination towards science\footnotemarkN. Thus in the summer of~1869, when +\footnotetextN{As a boy he had energetically collected Lepidoptera during the years 1858--64, but the first + vague indications of a leaning towards physical science may perhaps be found in his joining the + Sicilian eclipse expedition, Dec.~1870--Jan.~1871. It appears from \textit{Nature}, Dec.~1, 1870, that + George was told off to make sketches of the Corona.}% +we were all at Caerdeon in the Barmouth valley, he writes that he ``fell ill''; +and again in the winter of~1871. His health deteriorated markedly during +1872~and~1873. In the former year he went to Malvern and to Homburg +without deriving any advantage. I have an impression that he did not +expect to survive these attacks; but I cannot say at what date he made this +forecast of an early death. In January~1873 he tried Cannes: and ``came +back very ill.'' It was in the spring of this year that he first consulted Dr +(afterwards Sir~Andrew) Clark, from whom he received the kindest care. +George suffered from digestive troubles, sickness and general discomfort and +weakness. Dr~Clark's care probably did what was possible to make life more +bearable, and as time went on his health gradually improved. In 1894 he +consulted the late Dr~Eccles, and by means of the rest-cure, then something +of a novelty, his weight increased from $9$~stone to $9$~stone $11$~pounds. I gain +the impression that this treatment produced a permanent improvement, +although his health remained a serious handicap throughout his life. + +Meanwhile he had determined on giving up the Bar, and settled, in +October~1873, when he was $28$~years old, at Trinity in Nevile's Court next +the Library~(G\;4). His diary continues to contain records of ill-health and +of various holidays in search of improvement. Thus in 1873 we read ``Very +bad during January. Went to Cannes and stayed till the end of April.'' Again +in~1874, ``February to July very ill.'' In spite of unwellness he began in 1872--3 +to write on various subjects. He sent to \textit{Macmillan's Magazine}\footnoteN + {\textit{Macmillan's Magazine}, 1872, vol.~\Vol{XXVI.} pp.~410--416.} +an entertaining +article, ``Development in Dress,'' where the various survivals in modern +\DPPageSep{017}{xv} +costume were recorded and discussed from the standpoint of evolution. In +1873 he wrote ``On beneficial restriction to liberty of marriage\footnotemarkN,'' a eugenic +\footnotetextN{\textit{Contemporary Review}, 1873, vol.~\Vol{XXII.} pp.~412--426.}% +article for which he was attacked with gross unfairness and bitterness by the +late St~George Mivart. He was defended by Huxley, and Charles Darwin +formally ceased all intercourse with Mivart. We find mention of a ``Globe +Paper for the British Association'' in~1873. And in the following year he +read a contribution on ``Probable Error'' to the Mathematical Society\footnoteN{Not published.}---on +which he writes in his diary, ``found it was old.'' Besides another paper in the +\textit{Messenger of Mathematics}, he reviewed ``Whitney on Language\footnotemarkN,'' and wrote +\footnotetextN{\textit{Contemporary Review}, 1874, vol.~\Vol{XXIV.} pp.~894--904.}% +a ``defence of Jevons'' which I have not been able to trace. In 1875 he +was at work on the ``flow of pitch,'' on an ``equipotential tracer,'' on slide +rules, and sent a paper on ``Cousin Marriages'' to the Statistical Society\footnotemarkN. It +\footnotetextN{\textit{Journal of the Statistical Society}, 1875, vol.~\Vol{XXXVIII.} pt~2, pp.~158--182, also pp.~183--184, + and pp.~344--348.}% +is not my province to deal with these papers; they are here of interest as +showing his activity of mind and his varied interests, features in character +which were notable throughout his life. + +The most interesting entry in his diary for 1875 is ``Paper on Equipotentials +\index{Darwin, Sir George, genealogy!association with Lord Kelvin}% +\index{Kelvin, associated with Sir George Darwin}% +much approved by Sir~W. Thomson.'' This is the first notice of an +association of primary importance in George's scientific career. Then came +his memoir ``On the influence of geological changes in the earth's axis of +rotation.'' Lord Kelvin was one of the referees appointed by the Council of +the Royal Society to report on this paper, which was published in the \textit{Philosophical +Transactions} in~1877. + +In his diary, November~1878, George records ``paper on tides ordered to +be printed.'' This refers to his work ``On the bodily tides of viscous and +semi-elastic spheroids,~etc.,'' published in the \textit{Phil.\ Trans.} in~1879. It was in +regard to this paper that his father wrote to George on October~29th, 1878\footnotemarkN: +\footnotetextN{Probably he heard informally at the end of October what was not formally determined till + November.}% +\index{Darwin, Charles, ix; letters of}% + +\begin{Letter}{}{My dear old George,} + I have been quite delighted with your letter and read it all with + eagerness. You were very good to write it. All of us are delighted, + for considering what a man Sir~William Thomson is, it is most grand + that you should have staggered him so quickly, and that he should + speak of your `discovery,~etc.'\ldots\ Hurrah + for the bowels of the earth and their viscosity and for the moon and + for the Heavenly bodies and for my son George (F.R.S. very + soon)\ldots\footnotemarkN. +\end{Letter} +\footnotetextN{\textit{Emma Darwin, A Century of Family Letters}, 1915, vol.~\Vol{II.} p.~233.}% + +The bond of pupil and master between George Darwin and Lord Kelvin, +originating in the years 1877--8, was to be a permanent one, and developed +\DPPageSep{018}{xvi} +not merely into scientific co-operation\DPnote{[** TN: Hyphenated in original]} but into a close friendship. Sir~Joseph +\index{Darwin, Sir George, genealogy!friendships}% +Larmor has recorded\footnoteN + {\textit{Nature}, Dec.~12, 1912.} +that George's ``tribute to Lord Kelvin, to whom he +dedicated volume~\Vol{I} of his Collected Papers\footnotemarkN\ldots gave lively pleasure to his +master and colleague.'' His words were: +\footnotetextN{It was in 1907 that the Syndics of the Cambridge University Press asked George to prepare + \index{Darwin, Sir George, genealogy!at Cambridge}% + a reprint of his scientific papers, which the present volume brings to an end. George was + deeply gratified at an honour that placed him in the same class as Lord Kelvin, Stokes, Cayley, + Adams, Clerk Maxwell, Lord Rayleigh and other men of distinction.}% +\begin{Quote} +Early in my scientific career it was my good fortune to be brought +into close personal relationship with Lord Kelvin. Many visits to Glasgow +and to Largs have brought me to look up to him as my master, and +I cannot find words to express how much I owe to his friendship and to +his inspiration. +\end{Quote} + +During these years there is evidence that he continued to enjoy the +friendship of Lord Rayleigh and of Mr~Balfour. We find in his diary +records of visits to Terling and to Whittingehame, or of luncheons at +Mr~Balfour's house in Carlton Gardens for which George's scientific committee +work in London gave frequent opportunity. In the same way we +find many records of visits to Francis Galton, with whom he was united alike +by kinship and affection. + +Few people indeed can have taken more pains to cultivate friendship +than did George. This trait was the product of his affectionate and eminently +sociable nature and of the energy and activity which were his chief +characteristics. In earlier life he travelled a good deal in search of health\footnotemarkN, +\footnotetextN{Thus in 1872 he was in Homburg, 1873~in Cannes, 1874~in Holland, Belgium, Switzerland + and Malta, 1876~in Italy and Sicily.}% +and in after years he attended numerous congresses as a representative +of scientific bodies. He thus had unusual opportunities of making the +acquaintance of men of other nationalities, and some of his warmest friendships +were with foreigners. In passing through Paris he rarely failed to visit +M.~and~Mme d'Estournelles and ``the d'Abbadies.'' It was in Algiers in 1878~and~1879 +that he cemented his friendship with the late J.~F.~MacLennan, +author of \textit{Primitive Marriage}; and in 1880 he was at Davos with the same +friends. In~1881 he went to Madeira, where he received much kindness from +the Blandy family---doubtless through the recommendation of Lady~Kelvin. + +\Section{}{Cambridge.} + +We have seen that George was elected a Fellow of Trinity in October~1868, +and that five years later (Oct.~1873) he began his second lease of +a Cambridge existence. There is at first little to record: he held at this +time no official position, and when his Fellowship expired he continued to +live in College busy with his research work and laying down the earlier tiers +\DPPageSep{019}{xvii} +of the monumental series of papers in the present volumes. This soon led to +his being proposed (in Nov.~1877) for the Royal Society, and elected in June~1879. +The principal event in this stage of his Cambridge life was his +election\footnoteN + {The voting at University elections is in theory strictly confidential, but in practice this is + unfortunately not always the case. George records in his diary the names of the five who voted + for him and of the four who supported another candidate. None of the electors are now living. + The election occurred in January, and in June he had the great pleasure and honour of being + re-elected to a Trinity Fellowship. His daughter, Madame Raverat, writes: ``Once, when I was + walking with my father on the road to Madingley village, he told me how he had walked there, + on the first Sunday he ever was at Cambridge, with two or three other freshmen; and how, when + they were about opposite the old chalk pit, one of them betted him~£20 that he (my father) + would never be a professor of Cambridge University: and said my father, with great indignation, + `He never paid me.'\,"} +in 1883 as Plumian Professor of Astronomy and Experimental +Philosophy. His predecessor in the Chair was Professor Challis, who had +held office since~1836, and is now chiefly remembered in connection with +Adams and the planet Neptune. The professorship is not necessarily connected +with the Observatory, and practical astronomy formed no part of +George's duties. His lectures being on advanced mathematics usually +attracted but few students; in the Long Vacation however, when he +habitually gave one of his courses, there was often a fairly large class. + +George's relations with his class have been sympathetically treated by +Professor E.~W.~Brown, than whom no one can speak with more authority, +since he was one of my brother's favourite pupils. + +In the late~'70's George began to be appointed to various University +Boards and Syndicates. Thus from 1878--82 he was on the Museums and +Lecture Rooms Syndicate. In 1879 he was placed on the Observatory +Syndicate, of which he became an official member in 1883 on his election +to the Plumian Professorship. In the same way he was on the Special Board +for Mathematics. He was on the Financial Board from~1900--1 to~1903--4 +and on the Council of the Senate in 1905--6 and~1908--9. But he never +became a professional syndic---one of those virtuous persons who spend their +lives in University affairs. In his obituary of George (\textit{Nature}, Dec.~12, 1912), +Sir~Joseph Larmor writes: +\index{Darwin, Sir George, genealogy!university work, described by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}% +\index{Larmor, Sir Joseph, Sir George Darwin's work on university committees}% +\index{University committees, Sir George Darwin on, by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}% +\begin{Quote} +In the affairs of the University of which he was an ornament, +Sir George Darwin made a substantial mark, though it cannot be said +that he possessed the patience in discussion that is sometimes a +necessary condition to taking a share in its administration. But his wide +acquaintance and friendships among the statesmen and men of affairs of +the time, dating often from undergraduate days, gave him openings for +usefulness on a wider plane. Thus, at a time when residents were +bewailing even more than usual the inadequacy of the resources of the +University for the great expansion which the scientific progress of the +age demanded, it was largely on his initiative that, by a departure from +all precedent, an unofficial body was constituted in 1899 under the name +\DPPageSep{020}{xviii} +of the Cambridge University Association, to promote the further endowment +of the University by interesting its graduates throughout the +Empire in its progress and its more pressing needs. This important +body, which was organised under the strong lead of the late Duke of +Devonshire, then Chancellor, comprises as active members most of the +public men who owe allegiance to Cambridge, and has already by its +interest and help powerfully stimulated the expansion of the University +into new fields of national work; though it has not yet achieved +financial support on anything like the scale to which American seats +of learning are accustomed. +\end{Quote} +The Master of Christ's writes: +\index{Darwin, Sir George, genealogy!university work, described by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}% +\index{Master of Christ's, Sir George Darwin's work on university committees}% +\index{Newall, Prof., Sir George Darwin's work on university committees}% +\index{University committees, Sir George Darwin on, by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}% +\begin{Letter}{\textit{May}~31\textit{st}, 1915.}{} + My impression is that George did not take very much interest in the + petty details which are so beloved by a certain type of University + authority. `Comma hunting' and such things were not to his taste, + and at Meetings he was often rather distrait: but when anything of + real importance came up he was of extraordinary use. He was + especially good at drafting letters, and over anything he thought + promoted the advancement of the University along the right lines he + would take endless trouble---writing and re-writing\DPnote{[** TN: Hyphenated in original]} reports and + letters till he got them to his taste. The sort of movements which + interested him most were those which connected Cambridge with the + outside world. He was especially interested in the Appointments + Board. A good many of us constantly sought his advice and nearly + always took it: but, as I say, I do not think he cared much about + the `parish pump,' and was usually worried at long Meetings. +\end{Letter} +Professor Newall has also been good enough to give me his impressions: +\begin{Quote} +His weight in the Committees on which I have had personal +experience of his influence seems to me to have depended in large +measure on his realising very clearly the distinction between the +importance of ends to be aimed at and the difficulty of harmonising +the personal characteristics of the men who might be involved in the +work needed to attain the ends. The ends he always took seriously; +the crotchets he often took humorously, to the great easement of many +situations that are liable to arise on a Committee. I can imagine that +to those who had corns his direct progress may at times have seemed +unsympathetic and hasty. He was ready to take much trouble in formulating +statements of business with great precision---a result doubtless +of his early legal experiences. I recall how he would say, `If a thing has +to be done, the minute should if possible make some individual responsible +for doing it.' He would ask, `Who is going to do the work? If a +man has to take the responsibility, we must do what we can to help him +and not hamper him by unnecessary restrictions and criticisms.' His +helpfulness came from his quickness in seizing the important point and +his readiness to take endless trouble in the important work of looking +into details before and after the meetings. The amount of work that he +did in response to the requirements of various Committees was very +great, and it was curious to realise in how many cases he seemed to +have diffidence as to the value of his contributions. +\end{Quote} +\DPPageSep{021}{xix} + +But on the whole the work which, in spite of ill-health, he was able to +carry out in addition to professional duties and research, was given to matters +unconnected with the University, but of a more general importance. To +these we shall return. + +In 1884 he became engaged to Miss Maud Du~Puy of Philadelphia. +\index{Darwin, Sir George, genealogy!marriage}% +She came of an old Huguenot stock, descending from Dr~John Du~Puy +who was born in France in~1679 and settled in New York in~1713. They +were married on July~22nd, 1884, and this event happily coloured the +remainder of George's life. As time went on and existence became fuller +and busier, she was able by her never-failing devotion to spare him much +arrangement and to shield him from fatigue and anxiety. In this way he +was helped and protected in the various semi-public functions in which he +took a principal part. Nor was her help valued only on these occasions, for +indeed the comfort and happiness of every day was in her charge. There is +a charming letter\footnoteN + {\textit{Emma Darwin, A Century of Family Letters}, Privately printed, 1904, vol.~\Vol{II.} p.~350.} +from George's mother, dated April~15th, 1884: +\begin{Quote} +Maud had to put on her wedding-dress in order to say at the +Custom-house in America that she had worn it, so we asked her to +come down and show it to us. She came down with great simplicity +and quietness\ldots only really pleased at its being admired and at looking +pretty herself, which was strikingly the case. She was a little shy at +coming in, and sent in Mrs~Jebb to ask George to come out and see it +first and bring her in. It was handsome and simple. I like seeing +George so frivolous, so deeply interested in which diamond trinket +should be my present, and in her new Paris morning dress, in which he +felt quite unfit to walk with her. +\end{Quote} + +Later, probably in June, George's mother wrote\footnoteN + {\textit{Emma Darwin, A Century of Family Letters}, 1912, vol.~\Vol{II.} p.~266.} +to Miss Du~Puy, ``Your +visit here was a great happiness to me, as something in you (I don't know +what) made me feel sure you would always be sweet and kind to George +when he is ill and uncomfortable.'' These simple and touching words may +be taken as a forecast of his happy married life. + +In March 1885 George acquired by purchase the house Newnham +\index{Darwin, Sir George, genealogy!house at Cambridge}% +Grange\footnotemarkN, which remained his home to the end of his life. It stands at the +\footnotetextN{At that time it was known simply as \textit{Newnham}, but as this is the name of the College and + was also in use for a growing region of houses, the Darwins christened it Newnham Grange. The + name Newnham is now officially applied to the region extending from Silver Street Bridge to the + Barton Road.}% +southern end of the Backs, within a few yards of the river where it bends +eastward in flowing from the upper to the lower of the two Newnham water-mills. +I remember forebodings as to dampness, but they proved wrong---even +the cellars being remarkably dry. The house is built of faded +yellowish bricks with old tiles on the roof, and has a pleasant home-like air. +\DPPageSep{022}{xx} +It was formerly the house of the Beales family\footnotemarkN, one of the old merchant +\footnotetextN{The following account of Newnham Grange is taken from C.~H. Cooper's \textit{Memorials of + Cambridge}, 1866, vol.~\Vol{III.} p.~262 (note):---``The site of the hermitage was leased by the Corporation + to Oliver Grene, 20~Sep., 31~Eliz.\ [1589]. It was in~1790 leased for a long term to + Patrick Beales, from whom it came to his brother S.~P. Beales, Esq., who erected thereon a + substantial mansion and mercantile premises now occupied by his son Patrick Beales, Esq., + alderman, who purchased the reversion from the Corporation in~1839.'' Silver Street was formerly + known as Little Bridges Street, and the bridges which gave it this name were in charge of a + hermit, hence the above reference to the hermitage.}% +stocks of Cambridge. This fact accounts for the great barn-like granaries +which occupied much of the plot near the high road. These buildings were +in part pulled down, thus making room for a lawn tennis court, while what +was not demolished made a gallery looking on the court as well as play-room +for the children. At the eastern end of the property a cottage and part of +the granaries were converted into a small house of an attractively individual +character, for which I think tenants have hitherto been easily found among +personal friends. It is at present inhabited by Lady~Corbett. One of the +most pleasant features of the Grange was the flower-garden and rockery +on the other side of the river, reached by a wooden bridge and called ``the +Little Island\footnotemarkN.'' The house is conveniently close to the town, yet has a most +\footnotetextN{This was to distinguish it from the ``Big Island,'' both being leased from the town. Later + George acquired in the same way the small oblong kitchen garden on the river bank, and bought + the freehold of the Lammas land on the opposite bank of the river.}% +pleasant outlook, to the north over the Backs while there is the river and the +Fen to the south. The children had a den or house in the branches of a +large copper beech tree, overhanging the river. They were allowed to use +the boat, which was known as the \textit{Griffin} from the family crest with which +it was adorned. None of them were drowned, though accidents were not +unknown; in one of these an eminent lady and well-known writer, who was +inveigled on to the river by the children, had to wade to shore near Silver +Street bridge owing to the boat running aground. + +The Darwins had five children, of whom one died an infant: of the others, +\index{Darwin, Sir George, genealogy!children}% +Charles Galton Darwin has inherited much of his father's mathematical +ability, and has been elected to a Mathematical Lectureship at Christ's +College. He is now in the railway service of the Army in France. The +younger son, William, has a commission in the 18th~Battalion of the Durham +Light Infantry. George's elder daughter is married to Monsieur Jacques +Raverat. Her skill as an artist has perhaps its hereditary root in her +father's draughtsmanship. The younger daughter Margaret lives with her +mother. + +George's relations with his family were most happy. His diary never +fails to record the dates on which the children came home, or the black days +which took them back to school. There are constantly recurring entries in +his diary of visits to the boys at Marlborough or Winchester. Or of the +\DPPageSep{023}{xxi} +journeys to arrange for the schooling of the girls in England or abroad. +The parents took pains that their children should have opportunities of +learning conversational French and German. + +George's characteristic energy showed itself not only in these ways but +also in devising bicycling expeditions and informal picnics, for the whole +family, to the Fleam Dyke, to Whittlesford, or other pleasant spots near home---and +these excursions he enjoyed as much as anyone of the party. As he +always wished to have his children with him, one or more generally accompanied +him and his wife when they attended congresses or other scientific +gatherings abroad. + +His house was the scene of many Christmas dinners, the first of which +I find any record being in~1886. These meetings were often made an +occasion for plays acted by the children; of these the most celebrated was +a Cambridge version of \textit{Romeo and Juliet}, in which the hero and heroine +were scions of the rival factions of Trinity and St~John's. + +\Section{}{Games and Pastimes.} +\index{Darwin, Sir George, genealogy!games and pastimes}% + +As an undergraduate George played tennis---not the modern out-door +game, but that regal pursuit which is sometimes known as the game of +kings and otherwise as the king of games. When George came up as an +undergraduate there were two tennis courts in Cambridge, one in the East +Road, the other being the ancient one that gave its name to Tennis Court +Road and was pulled down to make room for the new buildings of Pembroke. +In this way was destroyed the last of the College tennis courts of which we +read in Mr~Clark's \textit{History}. I think George must have had pleasure in the +obvious development of the tennis court from some primaeval court-yard in +which the \textit{pent-house} was the roof of a shed, and the \textit{grille} a real window +or half-door. To one brought up on evolution there is also a satisfaction +about the French terminology which survives in e.g.\ the \textit{Tambour} and +the \textit{Dedans}. George put much thought into acquiring a correct style of +play---for in tennis there is a religion of attitude corresponding to that which +painfully regulates the life of the golfer. He became a good tennis player as +an undergraduate, and was in the running for a place in the inter-University +match. The marker at the Pembroke court was Henry Harradine, whom we +all sincerely liked and respected, but he was not a good teacher, and it was +only when George came under Henry's sons, John and Jim Harradine, at the +Trinity and Clare courts, that his game began to improve. He continued to +play tennis for some years, and only gave it up after a blow from a tennis +ball in January~1895 had almost destroyed the sight of his left eye. + +In 1910 he took up archery, and zealously set himself to acquire the +correct mode of standing, the position of the head and hands,~etc. He kept +an archery diary in which each day's shooting is carefully analysed and the +\DPPageSep{024}{xxii} +results given in percentages. In 1911 he shot on 131~days: the last occasion +on which he took out his bow was September~13, 1912. + +I am indebted to Mr~H. Sherlock, who often shot with him at Cambridge, +for his impressions. He writes: ``I shot a good deal with your brother the +year before his death; he was very keen on the sport, methodical and painstaking, +and paid great attention to style, and as he had a good natural +`loose,' which is very difficult to acquire, there is little doubt (notwithstanding +that he came to Archery rather late in life) that had he lived he would have +been above the average of the men who shoot fairly regularly at the public +Meetings.'' After my brother's death, Mr~Sherlock was good enough to look +at George's archery note-book. ``I then saw,'' he writes, ``that he had +analysed them in a way which, so far as I am aware, had never been done +before.'' Mr~Sherlock has given examples of the method in a sympathetic +obituary published (p.~273) in \textit{The Archer's Register}\footnotemarkN. George's point was +\footnotetextN{\textit{The Archer's Register} for 1912--1913, by H.~Walrond. London, \textit{The Field} Office, 1913.}% +that the traditional method of scoring is not fair in regard to the areas of the +coloured rings of the target. Mr~Sherlock records in his \textit{Notice} that George +joined the Royal Toxophilite Society in~1912, and occasionally shot in the +Regent's Park. He won the Norton Cup and Medal (144~arrows at 120~yards) +in~1912. + +There was a billiard table at Down, and George learned to play fairly +well though he had no pretension to real proficiency. He used to play at +the Athenaeum, and in 1911 we find him playing there in the Billiard +Handicap, but a week later he records in his diary that he was ``knocked +out.'' + +\Section{}{Scientific Committees.} +\index{Committees, Sir George Darwin on}% +\index{Darwin, Sir George, genealogy!work on scientific committees}% + +George served for many years on the Solar Physics Committee and on +the Meteorological Council. With regard to the latter, Sir~Napier Shaw +has at my request supplied the following note:--- +\index{Meteorological Council, by Sir Napier Shaw}% +\index{Shaw, Sir Napier, Meteorological Council}% +\begin{Quote} +It was in February~1885 upon the retirement of Warren De~la~Rue +that your brother George, by appointment of the Royal Society, joined +the governing body of the Meteorological Office, at that time the +Meteorological Council. He remained a member until the end of the +Council in~1905 and thereafter, until his death, he was one of the two +nominees of the Royal Society upon the Meteorological Committee, the +new body which was appointed by the Treasury to take over the control +of the administration of the Office. + +It will be best to devote a few lines to recapitulating the salient +features of the history of the official meteorological organisation because, +otherwise, it will be difficult for anyone to appreciate the position in +which Darwin was placed. +\DPPageSep{025}{xxiii} + +In 1854 a department of the Board of Trade was constituted under +Admiral R.~FitzRoy to collect and discuss meteorological information +from ships, and in~1860, impressed by the loss of the `Royal Charter,' +FitzRoy began to collect meteorological observations by telegraph from +land stations and chart them. Looking at a synchronous chart and +conscious that he could gather from it a much better notion of coming +weather than anyone who had only his own visible sky and barometer +to rely upon, he formulated `forecasts' which were published in the +newspapers and `storm warnings' which were telegraphed to the ports. + +This mode of procedure, however tempting it might be to the +practical man with the map before him, was criticised as not complying +with the recognised canons of scientific research, and on FitzRoy's +untimely death in 1865 the Admiralty, the Board of Trade and the +Royal Society elaborated a scheme for an office for the study of weather +in due form under a Director and Committee, appointed by the Royal +Society, and they obtained a grant in aid of~£10,000 for this purpose. +In this transformation it was Galton, I believe, who took a leading part +and to him was probably due the initiation of the new method of study +which was to bring the daily experience, as represented by the map, +into relation with the continuous records of the meteorological elements +obtained at eight observatories of the Kew type, seven of which were +immediately set on foot, and Galton devoted an immense amount of +time and skill to the reproduction of the original curves so that the +whole sequence of phenomena at the seven observatories could be taken +in at a glance. Meanwhile the study of maps was continued and a good +deal of progress was made in our knowledge of the laws of weather. + +But in spite of the wealth of information the generalisations were +empirical and it was felt that something more than the careful examination +of records was required to bring the phenomena of weather within +the rule of mathematics and physics, so in 1876 the constitution of the +Office was changed and the direction of its work was placed in Commission +with an increased grant. The Commissioners, collectively known +as the Meteorological Council, were a remarkably distinguished body of +fellows of the Royal Society, and when Darwin took the place of +De~la~Rue, the members were men subsequently famous, as Sir~Richard +Strachey, Sir~William Wharton, Sir~George Stokes, Sir~Francis Galton, +Sir~George Darwin, with E.~J.~Stone, a former Astronomer Royal for +the Cape. + +It was understood that the attack had to be made by new methods +and was to be entrusted partly to members of the Council themselves, +with the staff of the Office behind them, and partly to others outside +who should undertake researches on special points. Sir~Andrew Noble, +Sir~William Abney, Dr~W.~J. Russell, Mr~W.~H. Dines, your brother +Horace and myself came into connection with the Council in this way. + +Two important lines of attack were opened up within the Council +itself. The first was an attempt, under the influence of Lord Kelvin, +to base an explanation of the sequence of weather upon harmonic +analysis. As the phenomena of tides at any port could be synthesized +by the combinations of waves of suitable period and amplitude, so the +sequence of weather could be analysed into constituent oscillations the +general relations of which would be recognisable although the original +\DPPageSep{026}{xxiv} +composite result was intractable on direct inspection. It was while this +enterprise was in progress that Darwin was appointed to the Council. +His experience with tides and tidal analysis was in a way his title +to admission. He and Stokes were the mathematicians of the Council +and were looked to for expert guidance in the undertaking. At first +the individual curves were submitted to analysis in a harmonic analyser +specially built for the purpose, the like of which Darwin had himself +used or was using for his work on tides; but afterwards it was decided +to work arithmetically with the numbers derived from the tabulation of +the curves; and the identity of the individual curves was merged in +`five-day means.' The features of the automatic records from which so +much was hoped in~1865, after twelve years of publication in facsimile, +were practically never seen outside the room in the Office in which they +were tabulated. + +It is difficult at this time to point to any general advances in +meteorology which can be attributed to the harmonic analyser or its +arithmetical equivalent as a process of discussion, though it still remains +a powerful method of analysis. It has, no doubt, helped towards the +recognition of the ubiquity and simultaneity of the twelve-hour term in +the diurnal change of pressure which has taken its place among fundamental +generalisations of meteorology and the curious double diurnal +change in the wind at any station belongs to the same category; but +neither appears to have much to do with the control of weather. +Probably the real explanation of the comparative fruitlessness of the +effort lies in the fact that its application was necessarily restricted to +the small area of the British Isles instead of being extended, in some +way or other, to the globe. + +It is not within my recollection that Darwin was particularly +enthusiastic about the application of harmonic analysis. When I was +appointed to the Council in~1897, the active pursuit of the enterprise +had ceased. Strachey who had taken an active part in the discussion +of the results and contributed a paper on them to the Philosophical +Transactions, was still hopeful of basing important conclusions upon the +seasonal peculiarities of the third component, but the interest of other +members of the Council was at best languid. + +The other line of attack was in connection with synoptic charts. For +the year from August~1892 to August~1893 there was an international +scheme for circumpolar observations in the Northern Hemisphere, and +in connection therewith the Council undertook the preparation of daily +synoptic charts of the Atlantic and adjacent land areas. A magnificent +series of charts was produced and published from which great results +were anticipated. But again the conclusions drawn from cursory inspection +were disappointing. At that time the suggestion that weather +travelled across the Atlantic in so orderly a manner that our weather +could be notified four or five days in advance from New York had a +considerable vogue and the facts disclosed by the charts put an end to +any hope of the practical development of that suggestion. Darwin was +very active in endeavouring to obtain the help of an expert in physics +for the discussion of the charts from a new point of view, but he was +unsuccessful. + +Observations at High Level Stations were also included in the +\DPPageSep{027}{xxv} +Council's programme. A station was maintained at Hawes Junction +for some years, and the Observatories on Ben Nevis received their +support. But when I joined the Council in 1897 there was a pervading +sense of discouragement. The forecasting had been restored as the result +of the empirical generalisations based on the work of the years 1867~to~1878, +but the study had no attractions for the powerful analytical minds +of the Council; and the work of the Office had settled down into the +assiduous compilation of observations from sea and land and the regular +issue of forecasts and warnings in the accustomed form. The only part +which I can find assigned to Darwin with regard to forecasting is an +endeavour to get the forecast worded so as not to suggest more assurance +than was felt. + +I do not think that Darwin addressed himself spontaneously to +meteorological problems, but he was always ready to help. He was +very regular in his attendance at Council and the Minutes show that +after Stokes retired all questions involving physical measurement or +mathematical reasoning were referred to him. There is a short and +very characteristic report from him on the work of the harmonic +analyser and a considerable number upon researches by Mr~Dines or +Sir~G.~Stokes on anemometers. It is hardly possible to exaggerate +his aptitude for work of that kind. He could take a real interest in +things that were not his own. He was full of sympathy and appreciation +for efforts of all kinds, especially those of young men, and at the same +time, using his wide experience, he was perfectly frank and fearless not +only in his judgment but also in the expression of it. He gave one the +impression of just protecting himself from boredom by habitual loyalty +and a finely tempered sense of duty. My earliest recollection of him on +the Council is the thrilling production of a new version of the Annual +Report of the Council which he had written because the original had +become more completely `scissors and paste' than he could endure. + +After the Office came into my charge in~1900, so long as he lived, +I never thought of taking any serious step without first consulting him +and he was always willing to help by his advice, by his personal influence +and by his special knowledge. For the first six years of the time +I held a college fellowship with the peculiar condition of four public +lectures in the University each year and no emolument. One year, +when I was rather overdone, Darwin took the course for me and devoted +the lectures to Dynamical Meteorology. I believe he got it up for the +occasion, for he professed the utmost diffidence about it, but the progress +which we have made in recent years in that subject dates from those +lectures and the correspondence which arose upon them. + +In Council it was the established practice to proceed by agreement +and not by voting; he had a wonderful way of bringing a discussion to +a head by courageously `voicing' the conclusion to which it led and +frankly expressing the general opinion without hurting anybody's +feelings. + +This letter has, I fear, run to a great length, but it is not easy +to give expression to the powerful influence which he exercised upon +all departments of official meteorology without making formal contributions +to meteorological literature. He gave me a note on a curious +point in the evaluation of the velocity equivalents of the Beaufort Scale +\DPPageSep{028}{xxvi} +which is published in the Office Memoirs No.~180, and that is all I have +to show in print, but he was in and behind everything that was done +and personally, I need hardly add, I owe to him much more than this or +any other letter can fully express. +\end{Quote} + +On May~6, 1904, he was elected President of the British Association---the +\index{British Association, South African Meeting, 1905}% +\index{South African Meeting of the British Association, 1905}% +South African meeting. + +On July~29, 1905, he embarked with his wife and his son Charles and +arrived on August~15 at the Cape, where he gave the first part of his +Presidential Address. Here he had the pleasure of finding as Governor +Sir~Walter Hely-Hutchinson, whom he had known as a Trinity undergraduate. +He was the guest of the late Sir~David Gill, who remained a close friend for +the rest of his life. George's diary gives his itinerary---which shows the +trying amount of travel that he went through. A sample may be quoted: +\begin{center} +\footnotesize +\begin{tabular}{cl} +August 19 & Embark, \\ +\Ditto 22 & Arrive at Durban, \\ +\Ditto 23 & Mount Edgecombe, \\ +\Ditto 24 & Pietermaritzburg, \\ +\Ditto 26 & Colenso, \\ +\Ditto 27 & Ladysmith, \\ +\Ditto 28 & Johannesburg. +\end{tabular} +\end{center} + +At Johannesburg he gave the second half of his Address. Then on by +Bloemfontein, Kimberley, Bulawayo, to the Victoria Falls, where a bridge had +to be opened. Then to Portuguese Africa on September~16,~17, where he +made speeches in French and English. Finally he arrived at Suez on +October~4 and got home October~18. + +It was generally agreed that his Presidentship was a conspicuous success. +The following appreciation is from the obituary notice in \textit{The Observatory}, +Jan.~1913, p.~58: +\begin{Quote} +The Association visited a dozen towns, and at each halt its President +addressed an audience partly new, and partly composed of people who +had been travelling with him for many weeks. At each place this +latter section heard with admiration a treatment of his subject wholly +fresh and exactly adapted to the locality. +\end{Quote} +Such duties are always trying and it should not be forgotten that tact was +necessary in a country which only two years before was still in the throes +of war. + +In the autumn he received the honour of being made a~K.C.B\@. The +distinction was doubly valued as being announced to him by his friend +Mr~Balfour, then Prime Minister. + +From 1899~to~1900 he was President of the Royal Astronomical Society. +One of his last Presidential acts was the presentation of the Society's Medal +to his friend M.~Poincaré. +\DPPageSep{029}{xxvii} + +He had the unusual distinction of serving twice as President of the +Cambridge Philosophical Society, once in 1890--92 and again 1911--12. + +In 1891 he gave the Bakerian Lecture\footnoteN + {See Prof.~Brown's Memoir, \Pageref{xlix}.} +of the Royal Society, his subject +being ``Tidal Prediction.'' This annual prælection dates from~1775 and the +list of lecturers is a distinguished roll of names. + +In 1897 he lectured at the Lowell Institute at Boston, and this was +\index{Tides, The@\textit{Tides, The}}% +the origin of his book on \textit{Tides}, published in the following year. Of this +Sir~Joseph Larmor says\footnoteN + {\textit{Nature}, 1912. See also Prof.~Brown's Memoir, \Pageref{l}.} +that ``it has taken rank with the semi-popular +writings of Helmholtz and Kelvin as a model of what is possible in the +exposition of a scientific subject.'' It has passed through three English +editions, and has been translated into many foreign languages. + +\Section{}{International Associations.} + +During the last ten or fifteen years of his life George was much occupied +\index{Geodetic Association, International}% +\index{Larmor, Sir Joseph, Sir George Darwin's work on university committees!International Geodetic Association}% +with various International bodies, e.g.~the International Geodetic Association, +the International Association of Academics, the International Congress of +Mathematicians and the Seismological Congress. + +With regard to the last named it was in consequence of George's report +to the Royal Society that the British Government joined the Congress. It +was however with the Geodetic Association that he was principally connected. + +Sir~Joseph Larmor (\textit{Nature}, December~12, 1912) gives the following +account of the origin of the Association: +\begin{Quote} +The earliest of topographic surveys, the model which other national +surveys adopted and improved upon, was the Ordnance Survey of the +United Kingdom. But the great trigonometrical survey of India, started +nearly a century ago, and steadily carried on since that time by officers +of the Royal Engineers, is still the most important contribution to the +science of the figure of the earth, though the vast geodetic operations in +the United States are now following it closely. The gravitational and +other complexities incident on surveying among the great mountain +masses of the Himalayas early demanded the highest mathematical +assistance. The problems originally attacked in India by Archdeacon +Pratt were afterwards virtually taken over by the Royal Society, and its +secretary, Sir~George Stokes, of Cambridge, became from 1864 onwards +the adviser and referee of the survey as regards its scientific enterprises. +On the retirement of Sir~George Stokes, this position fell very largely to +Sir~George Darwin, whose relations with the India Office on this and +other affairs remained close, and very highly appreciated, throughout +the rest of his life. + +The results of the Indian survey have been of the highest importance +for the general science of geodesy\ldots. It came to be felt that closer +cooperation between different countries was essential to practical +progress and to coordination of the work of overlapping surveys. +\end{Quote} +\DPPageSep{030}{xxviii} + +The further history of George's connection with the Association is told in +\index{Bakhuyzen, Dr Van d.\ Sande, Sir George Darwin's connection with the International Geodetic Association}% +\index{Geodetic Association, International}% +the words of its Secretary, Dr~van~d.\ Sande Bakhuyzen, to whom I am greatly +indebted. +\begin{Quote} +On the proposal of the Royal Society, the British Government, after +having consulted the Director of the Ordnance Survey, in~1898, resolved +upon the adhesion of Great Britain to the International Geodetic Association, +and appointed as its delegate, G.~H.~Darwin. By his former +researches and by his high scientific character, he, more than any other, +was entitled to this position, which would afford him an excellent +opportunity of furthering, by his recommendations, the study of theoretical +geodesy. + +The meeting at Stuttgart in 1898 was the first which he attended, +and at that and the following conferences, Paris~1900, Copenhagen~1903, +Budapest~1906, London-Cambridge~1909, he presented reports on the +geodetic work in the British Empire. To Sir~David Gill's report on the +geodetic work in South Africa, which he delivered at Budapest, Darwin +added an appendix in which he relates that the British South Africa +Company, which had met all the heavy expense of the part of the survey +along the 30th~meridian through Rhodesia, found it necessary to make +various economies, so that it was probably necessary to suspend the +survey for a time. This interruption would be most unfortunate for the +operations relating to the great triangulation from the Southern part of +Cape Colony to Egypt, but, happily, by the cooperation of different +authorities, all obstacles had been overcome and the necessary money +found, so that the triangulation could be continued. So much for +Sir~George Darwin's communication; it is correct but incomplete, as it +does not mention that it was principally by Darwin's exertions and by +his personal offer of financial help that the question was solved and the +continuation of this great enterprise secured. + +To the different researches which enter into the scope of the Geodetic +Association belong the researches on the tides, and it is natural that +Darwin should be chosen as general reporter on that subject; two +elaborate reports were presented by him at the conferences of Copenhagen +and London. + +In Copenhagen he was a member of the financial committee, and at +the request of this body he presented a report on the proposal to determine +gravity at sea, in which he strongly recommended charging Dr~Hecker +with that determination using the method of Prof.~Mohn (boiling +temperature of water and barometer readings). At the meeting of~1906 +an interesting report was read by him on a question raised by +the Geological Congress: the cooperation of the Geodetic Association +in geological researches by means of the anomalies in the intensity +of gravitation. + +By these reports and recommendations Darwin exercised a useful +influence on the activity of the Association, but his influence was to be +still increased. In 1907 the Vice-president of the Association, General +Zacharias, died, and the permanent committee, whose duty it was to +nominate his provisional successor, chose unanimously Sir~George +Darwin, and this choice was confirmed by the next General Conference +in London. +\DPPageSep{031}{xxix} + +We cannot relate in detail his valuable cooperation as a member of +the council in the various transactions of the Association, for instance on +the junction of the Russian and Indian triangulations through Pamir, +but we must gratefully remember his great service to the Association +when, at his invitation, the delegates met in 1909 for the 16th~General +Conference in London and Cambridge. +\index{Mathematicians, International Congress of, Cambridge, 1912}% + +With the utmost care he prepared everything to render the Conference +as interesting and agreeable as possible, and he fully succeeded. +Through his courtesy the foreign delegates had the opportunity of making +the personal acquaintance of several members of the Geodetic staff of +England and its colonies, and of other scientific men, who were invited +to take part in the conference; and when after four meetings in London +the delegates went to Cambridge to continue their work, they enjoyed +the most cordial hospitality from Sir~George and Lady~Darwin, who, +with her husband, procured them in Newnham Grange happy leisure +hours between their scientific labours. + +At this conference Darwin delivered various reports, and at the +discussion on Hecker's determination of the variation of the vertical by +the attraction of the moon and sun, he gave an interesting account of +the researches on the same subject made by him and his brother Horace +more than 20~years ago, which unfortunately failed from the bad conditions +of the places of observation. + +In 1912 Sir~George, though already over-fatigued by the preparations +for the mathematical congress in Cambridge, and the exertions entailed +by it, nevertheless prepared the different reports on the geodetic work +in the British Empire, but alas his illness prevented him from assisting +at the conference at Hamburg, where they were presented by other +British delegates. The conference thanked him and sent him its best +wishes, but at the end of the year the Association had to deplore the loss +of the man who in theoretical geodesy as well as in other branches of +mathematics and astronomy stood in the first rank, and who for his +noble character was respected and beloved by all his colleagues in the +International Geodetic Association. +\end{Quote} +Sir~Joseph Larmor writes\footnoteN + {\textit{Nature}, Dec.~12, 1912.}: +\index{Larmor, Sir Joseph, Sir George Darwin's work on university committees!International Congress of Mathematicians at Cambridge 1912}% +\index{Congress, International, of Mathematicians at Cambridge, 1912!note by Sir Joseph Larmor}% +\begin{Quote} +Sir~George Darwin's last public appearance was as president of the +fifth International Congress of Mathematicians, which met at Cambridge +on August~22--28, 1912. The time for England to receive the congress +having obviously arrived, a movement was initiated at Cambridge, with +the concurrence of Oxford mathematicians, to send an invitation to the +fourth congress held at Rome in~1908. The proposal was cordially +accepted, and Sir~George Darwin, as \textit{doyen} of the mathematical school +at Cambridge, became chairman of the organising committee, and was +subsequently elected by the congress to be their president. Though +obviously unwell during part of the meeting, he managed to discharge +the delicate duties of the chair with conspicuous success, and guided +with great \textit{verve} the deliberations of the final assembly of what turned +out to be a most successful meeting of that important body. +\end{Quote} +\DPPageSep{032}{xxx} + +\Section{}{Personal Characteristics.} +\index{Darwin, Sir George, genealogy!personal characteristics}% +\index{Darwin, Margaret, on Sir George Darwin's personal characteristics}% +\index{Raverat, Madame, on Sir George Darwin's personal characteristics}% + +His daughter, Madame Raverat, writes: +\begin{Quote} +I think most people might not realise that the sense of adventure +and romance was the most important thing in my father's life, except his +love of work. He thought about all life romantically and his own life +in particular; one could feel it in the quality of everything he said +about himself. Everything in the world was interesting and wonderful +to him and he had the power of making other people feel it. + +He had a passion for going everywhere and seeing everything; +learning every language, knowing the technicalities of every trade; and +all this emphatically \textit{not} from the scientific or collector's point of view, but +from a deep sense of the romance and interest of everything. It was +splendid to travel with him; he always learned as much as possible of +the language, and talked to everyone; we had to see simply everything +there was to be seen, and it was all interesting like an adventure. For +instance at Vienna I remember being taken to a most improper music hall; +and at Schönbrunn hearing from an old forester the whole secret history of +the old Emperor's son. My father would tell us the stories of the places +we went to with an incomparable conviction, and sense of the reality +and dramaticness of the events. It is absurd of course, but in that +respect he always seemed to me a little like Sir~Walter Scott\footnotemarkN. +\footnotetextN{Compare Mr~Chesterton's \textit{Twelve Types}, 1903, p.~190. He speaks of Scott's critic in the + \textit{Edinburgh Review}: ``The only thing to be said about that critic is that he had never been + a little boy. He foolishly imagined that Scott valued the plume and dagger of Marmion for + Marmion's sake. Not being himself romantic, he could not understand that Scott valued + the plume because it was a plume and the dagger because it was a dagger.''}% + +The books he used to read to us when we were quite small, +and which we adored, were Percy's \textit{Reliques} and the \textit{Prologue to the +Canterbury Tales}. He used often to read Shakespeare to himself, +I think generally the historical plays, Chaucer, \textit{Don Quixote} in Spanish, +and all kind of books like Joinville's \textit{Life of St~Louis} in the old French. + +I remember the story of the death of Gordon told so that we all +cried, I think; and Gladstone could hardly be mentioned in consequence. +All kinds of wars and battles interested him, and I think he liked archery +more because it was romantic than because it was a game. + +During his last illness his interest in the Balkan war never failed. +Three weeks before his death he was so ill that the doctor thought him +dying. Suddenly he rallied from the half-unconscious state in which he +had been lying for many hours and the first words he spoke on opening +his eyes were: ``Have they got to Constantinople yet?'' This was very +characteristic. I often wish he was alive now, because his understanding +and appreciation of the glory and tragedy of this war would +be like no one else's. +\end{Quote} +His daughter Margaret Darwin writes: +\begin{Quote} +He was absolutely unselfconscious and it never seemed to occur to +him to wonder what impression he was making on others. I think it +was this simplicity which made him so good with children. He seemed +to understand their point of view and to enjoy \textit{with} them in a way that +\DPPageSep{033}{xxxi} +is not common with grown-up people. I shall never forget how when +our dog had to be killed he seemed to feel the horror of it just as I did, +and how this sense of his really sharing my grief made him able to +comfort me as nobody else could. + +He took a transparent pleasure in the honours that came to him, +especially in his membership of foreign Academies, in which he and +Sir~David Gill had a friendly rivalry or ``race,'' as they called it. I think +this simplicity was one of his chief characteristics, though most important +of all was the great warmth and width of his affections. He +would take endless trouble about his friends, especially in going to see +them if they were lonely or ill; and he was absolutely faithful and +generous in his love. +\end{Quote} + +After his mother came to live in Cambridge, I believe he hardly ever +missed a day in going to see her even though he might only be able to stay +a few minutes. She lived at some distance off and he was often both busy +and tired. This constancy was very characteristic. It was shown once more +in his many visits to Jim Harradine, the marker at the tennis court, on what +proved to be his death-bed. + +His energy and his kindness of heart were shown in many cases of distress. +For instance, a guard on the Great Northern Railway was robbed of his savings +by an absconding solicitor, and George succeeded in collecting some~£300 +for him. In later years, when his friend the guard became bedridden, George +often went to see him. Another man whom he befriended was a one-legged +man at Balsham whom he happened to notice in bicycling past. He took the +trouble to see the village authorities and succeeded in sending the man to +London to be fitted with an artificial leg. + +In these and similar cases there was always the touch of personal +sympathy. For instance he pensioned the widow of his gardener, and he +often made the payment of her weekly allowance the excuse for a visit. + +In another sort of charity he was equally kind-hearted, viz.~in answering +the people who wrote foolish letters to him on scientific subjects---and here +as in many points he resembled his father. + +His sister, Mrs~Litchfield, has truly said\footnoteN + {\textit{Emma Darwin, A Century of Family Letters}, 1915, vol.~\Vol{II.} p.~146.} +of George that he inherited his +father's power of work and much of his ``cordiality and warmth of nature +with a characteristic power of helping others.'' He resembled his father in +another quality, that of modesty. His friend and pupil E.~W.~Brown writes: +\begin{Quote} +He was always modest about the importance of his researches. +He would often wonder whether the results were worth the labour they +had cost him and whether he would have been better employed in some +other way. +\end{Quote} + +His nephew Bernard, speaking of George's way of taking pains to be +friendly and forthcoming to anyone with whom he came in contact, says: +\DPPageSep{034}{xxxii} +\begin{Quote} +He was ready to take other people's pleasantness and politeness at +its apparent value and not to discount it. If they seemed glad to see him, +he believed that they \textit{were} glad. If he liked somebody, he believed +that the somebody liked him, and did not worry himself by wondering +whether they really did like him. +\end{Quote} + +Of his energy we have evidence in the \textit{amount} of work contained in +\index{Darwin, Sir George, genealogy!energy}% +these volumes. There was nothing dilatory about him, and here he again +resembled his father who had markedly the power of doing things at the +right moment, and thus avoiding waste of time and discomfort to others. +George had none of a characteristic which was defined in the case of Henry +Bradshaw, as ``always doing something else.'' After an interruption he could +instantly reabsorb himself in his work, so that his study was not kept as a +place sacred to peace and quiet. + +His wife is my authority for saying that although he got so much done, +it was not by working long hours. Moreover the days that he was away +from home made large gaps in his opportunities for steady application. His +diaries show in another way that his researches by no means took all his +time. He made a note of the books he read and these make a considerable +record. Although he read much good literature with honest enjoyment, he +had not a delicate or subtle literary judgment. Nor did he care for music. +He was interested in travels, history, and biography, and as he could remember +what he read or heard, his knowledge was wide in many directions. His +linguistic power was characteristic. He read many European languages. +I remember his translating a long Swedish paper for my father. And he +took pleasure in the Platt Deutsch stories of Fritz Reuter. + +The discomfort from which he suffered during the meeting at Cambridge +of the International Congress of Mathematicians in August~1912, was in fact +the beginning of his last illness. An exploratory operation showed that he +was suffering from malignant disease. Happily he was spared the pain that +gives its terror to this malady. His nature was, as we have seen, simple and +direct with a pleasant residue of the innocence and eagerness of childhood. +In the manner of his death these qualities were ennobled by an admirable +and most unselfish courage. As his vitality ebbed away his affection only +showed the stronger. He wished to live, and he felt that his power of work +and his enjoyment of life were as strong as ever, but his resignation to the +sudden end was complete and beautiful. He died on Dec.~7, 1912, and was +buried at Trumpington. +\DPPageSep{035}{xxxiii} + + +\Heading{Honours, Medals, Degrees, Societies, etc.} +\index{Darwin, Sir George, genealogy!honours}% + +\Subsection{Order. \upshape K.C.B. 1905.} + +\Subsection{Medals\footnotemarkN.} +\footnotetextN{Sir~George's medals are deposited in the Library of Trinity College, Cambridge.} + +1883. Telford Medal of the Institution of Civil Engineers. + +1884. Royal Medal\footnotemarkN. +\footnotetextN{Given by the Sovereign on the nomination of the Royal Society.} + +1892. Royal Astronomical Society's Medal. + +1911. Copley Medal of the Royal Society. + +1912. Royal Geographical Society's Medal. + +\Subsection{Offices.} + +Fellow of Trinity College, Cambridge, and Plumian Professor in the +University. + +Vice-President of the International Geodetic Association, Lowell Lecturer +at Boston U.S.~(1897). + +Member of the Meteorological and Solar Physics Committees. + +Past President of the Cambridge Philosophical Society\footnotemarkN, Royal Astronomical +\footnotetextN{Re-elected in 1912.} +Society, British Association. + +\Subsection{Doctorates, etc.\ of Universities.} + +Oxford, Dublin, Glasgow, Pennsylvania, Padua (Socio onorario), Göttingen, +Christiania, Cape of Good Hope, Moscow (honorary member). + +\Subsection{Foreign or Honorary Membership of Academies, etc.} + +Amsterdam (Netherlands Academy), Boston (American Academy), +Brussels (Royal Society), Calcutta (Math.\ Soc.), Dublin (Royal Irish +Academy), Edinburgh (Royal Society), Halle (K.~Leop.-Carol.\ Acad.), +Kharkov (Math.\ Soc.), Mexico (Soc.\ ``Antonio Alzate''), Moscow (Imperial +Society of the Friends of Science), New York, Padua, Philadelphia (Philosophical +Society), Rome (Lincei), Stockholm (Swedish Academy), Toronto +(Physical Society), Washington (National Academy), Wellington (New +Zealand Inst.). + +\Subsection{Correspondent of Academies, etc.\ at} + +Acireale (Zelanti), Berlin (Prussian Academy), Buda Pest (Hungarian +Academy), Frankfort (Senckenberg.\ Natur.\ Gesell.), Göttingen (Royal Society), +Paris, St~Petersburg, Turin, Istuto Veneto, Vienna\footnotemarkN. +\footnotetextN{The above list is principally taken from that compiled by Sir~George for the Year-Book of + the Royal Society,~1912, and may not be quite complete. + + It should be added that he especially valued the honour conferred on him in the publication + of his collected papers by the Syndics of the University Press.} +\DPPageSep{036}{xxxiv} + + +%[** TN: Changed the running heads; original splits the title] +\Chapter{The Scientific Work of Sir George Darwin} +\BY{Professor E. W. Brown} +\index{Brown, Prof.\ E. W., Sir George Darwin's Scientific Work}% +\index{Darwin, Sir George, genealogy!scientific work, by Prof.\ E. W. Brown}% +\index{Darwin, Sir George, genealogy!characteristics of his work}% + +The scientific work of Darwin possesses two characteristics which cannot +fail to strike the reader who glances over the titles of the eighty odd papers +which are gathered together in the four volumes which contain most of his +publications. The first of these characteristics is the homogeneous nature +of his investigations. After some early brief notes, on a variety of subjects, +he seems to have set himself definitely to the task of applying the tests of +mathematics to theories of cosmogony, and to have only departed from it +when pressed to undertake the solution of practical problems for which there +was an immediate need. His various papers on viscous spheroids concluding +with the effects of tidal friction, the series on rotating masses of fluids, even +those on periodic orbits, all have the idea, generally in the foreground, of +developing the consequences of old and new assumptions concerning the past +history of planetary and satellite systems. That he achieved so much, in +spite of indifferent health which did not permit long hours of work at his +desk, must have been largely due to this single aim. + +The second characteristic is the absence of investigations undertaken for +their mathematical interest alone; he was an applied mathematician in the +strict and older sense of the word. In the last few decades another school of +applied mathematicians, founded mainly by Poincaré, has arisen, but it differs +essentially from the older school. Its votaries have less interest in the +phenomena than in the mathematical processes which are used by the student +of the phenomena. They do not expect to examine or predict physical +events but rather to take up the special classes of functions, differential +equations or series which have been used by astronomers or physicists, to +examine their properties, the validity of the arguments and the limitations +which must be placed on the results. Occasionally theorems of great physical +importance will emerge, but from the primary point of view of the investigations +these are subsidiary results. Darwin belonged essentially to the school which +studies the phenomena by the most convenient mathematical methods. Strict +logic in the modern sense is not applied nor is it necessary, being replaced in +most cases by intuition which guides the investigator through the dangerous +places. That the new school has done great service to both pure and applied +mathematics can hardly be doubted, but the two points of view of the subject +\DPPageSep{037}{xxxv} +will but rarely be united in the same man if much progress in either direction +is to be made. Hence we do not find and do not expect to find in Darwin's +work developments from the newer point of view. + +At the same time, he never seems to have been affected by the problem-solving +habits which were prevalent in Cambridge during his undergraduate +days and for some time later. There was then a large number of mathematicians +brought up in the Cambridge school whose chief delight was the +discovery of a problem which admitted of a neat mathematical solution. +The chief leaders were, of course, never very seriously affected by this +attitude; they had larger objects in view, but the temptation to work out +a problem, even one of little physical importance, when it would yield to +known mathematical processes, was always present. Darwin kept his aim +fixed. If the problem would not yield to algebra he has recourse to +arithmetic; in either case he never seemed to hesitate to embark on the +most complicated computations if he saw a chance of attaining his end. +The papers on ellipsoidal harmonic analysis and periodic orbits are instructive +examples of the labour which he would undertake to obtain a knowledge of +physical phenomena. + +One cannot read any of his papers without also seeing another feature, +his preference for quantitative rather than qualitative results. If he saw +any possibility of obtaining a numerical estimate, even in his most speculative +work, he always made the necessary calculations. His conclusions +thus have sometimes an appearance of greater precision than is warranted +by the degree of accuracy of the data. But Darwin himself was never +misled by his numerical conclusions, and he is always careful to warn his +readers against laying too great a stress on the numbers he obtains. + +In devising processes to solve his problems, Darwin generally adopted +those which would lead in a straightforward manner to the end he had +in view. Few ``short cuts'' are to be found in his memoirs. He seems to +have felt that the longer processes often brought out details and points +of view which would otherwise have been concealed or neglected. This is +particularly evident in the papers on Periodic Orbits. In the absence of +general methods for the discovery and location of the curves, his arithmetic +showed classes of orbits which would have been difficult to find by analysis, +and it had a further advantage in indicating clearly the various changes +which the members of any class undergo when the parameter varies. Yet, +in spite of the large amount of numerical work which is involved in many +of his papers, he never seemed to have any special liking for either algebraic +or numerical computation; it was something which ``had to be done.'' Unlike +J.~C.~Adams and G.~W.~Hill, who would often carry their results to a large +number of places of decimals, Darwin would find out how high a degree of +accuracy was necessary and limit himself to it. +\DPPageSep{038}{xxxvi} + +The influence which Darwin exerted has been felt in many directions. +\index{Cosmogony, Sir George Darwin's influence on}% +\index{Darwin, Sir George, genealogy!his first papers}% +\index{Darwin, Sir George, genealogy!his influence on cosmogony}% +The exhibition of the necessity for quantitative and thorough analysis of the +problems of cosmogony and celestial mechanics has been perhaps one of his +chief contributions. It has extended far beyond the work of the pupils who +were directly inspired by him. While speculations and the framing of new +hypotheses must continue, but little weight is now attached to those which +are defended by general reasoning alone. Conviction fails, possibly because +it is recognised that the human mind cannot reason accurately in these +questions without the aids furnished by mathematical symbols, and in any +case language often fails to carry fully the argument of the writer as against +the exact implications of mathematics. If for no other reason, Darwin's work +marks an epoch in this respect. + +To the pupils who owed their first inspiration to him, he was a constant +\index{Darwin, Sir George, genealogy!his relationship with his pupils}% +\index{Pupils, Darwin's relationship with his}% +friend. First meeting them at his courses on some geophysical or astronomical +subject, he soon dropped the formality of the lecture-room, and they +found themselves before long going to see him continually in the study at +Newnham Grange. Who amongst those who knew him will fail to remember +the sight of him seated in an armchair with a writing board and papers +strewn about the table and floor, while through the window were seen +glimpses of the garden filled in summer time with flowers? While his +lectures in the class-room were always interesting and suggestive, the chief +incentive, at least to the writer who is proud to have been numbered amongst +his pupils and friends, was conveyed through his personality. To have spent +an hour or two with him, whether in discussion on ``shop'' or in general +conversation, was always a lasting inspiration. And the personal attachment +of his friends was strong; the gap caused by his death was felt to be far +more than a loss to scientific progress. Not only the solid achievements +contained in his published papers, but the spirit of his work and the example +of his life will live as an enduring memorial of him. + +\tb + +Darwin's first five papers, all published in~1875, are of some interest as +showing the mechanical turn of his mind and the desire, which he never lost, +for concrete illustrations of whatever problem might be interesting him. +A Peaucellier's cell is shown to be of use for changing a constant force into +one varying inversely as the square of the distance, and it is applied to the +description of equipotential lines. A method for describing graphically the +second elliptic integral and one for map projection on the face of a polyhedron +are also given. There are also a few other short papers of the same kind but +of no special importance, and Darwin says that he only included them in his +collected works for the sake of completeness. + +His first important contributions obviously arose through the study +of the works of his predecessors, and though of the nature of corrections to +\DPPageSep{039}{xxxvii} +previously accepted or erroneous ideas, they form definite additions to the +subject of cosmogony. The opening paragraph of the memoir ``On the +influence of geological changes in the earth's axis of rotation'' describes the +situation which prompted the work. ``The subject of the fixity or mobility +of the earth's axis of rotation in that body, and the possibility of variations +in the obliquity of the ecliptic, have from time to time attracted the notice +of mathematicians and geologists. The latter look anxiously for some grand +cause capable of producing such an enormous effect as the glacial period. +Impressed by the magnitude of the phenomenon, several geologists have +postulated a change of many degrees in the obliquity of the ecliptic and +a wide variability in the position of the poles on the earth; and this, again, +they have sought to refer back to the upheaval and subsidence of continents.'' +He therefore subjects the hypothesis to mathematical examination under +various assumptions which have either been put forward by geologists or +which he considers \textit{à ~priori} probable. The conclusion, now well known to +astronomers, but frequently forgotten by geologists even at the present time, +is against any extensive wanderings of the pole during geological times. +``Geologists and biologists,'' writes Professor Barrell\footnotemarkN, ``may array facts +\footnotetextN{\textit{Science}, Sept.~4, 1914, p.~333.}% +\index{Barrell, Prof., Cosmogony as related to Geology and Biology}% +\index{Cosmogony, Sir George Darwin's influence on!as related to Geology and Biology, by Prof.\ Barrell}% +which suggest such hypotheses, but the testing of their possibility is really +a problem of mathematics, as much as are the movements of precession, +and orbital perturbations. Notwithstanding this, a number of hypotheses +concerning polar migration have been ingeniously elaborated and widely +promulgated without their authors submitting them to these final tests, or +in most cases even perceiving that an accordance with the known laws of +mechanics was necessary\ldots. A reexamination of these assumptions in the +light of forty added years of geological progress suggests that the actual +changes have been much less and more likely to be limited to a fraction +of the maximum limits set by Darwin. His paper seems to have checked +further speculation upon this subject in England, but, apparently unaware +of its strictures, a number of continental geologists and biologists have +carried forward these ideas of polar wandering to the present day. The +hypotheses have grown, each creator selecting facts and building up from +his particular assortment a fanciful hypothesis of polar migration unrestrained +even by the devious paths worked out by others.'' The methods +used by Darwin are familiar to those who investigate problems connected +with the figure of the earth, but the whole paper is characteristic of his style +in the careful arrangement of the assumptions, the conclusions deduced +therefrom, the frequent reduction to numbers and the summary giving the +main results. + +It is otherwise interesting because it was the means of bringing Darwin +\index{Darwin, Sir George, genealogy!association with Lord Kelvin}% +\index{Kelvin, associated with Sir George Darwin}% +into close connection with Lord Kelvin, then Sir~William Thomson. The +\DPPageSep{040}{xxxviii} +latter was one of the referees appointed by the Royal Society to report on it, +and, as Darwin says, ``He seemed to find that on these occasions the quickest +way of coming to a decision was to talk over the subject with the author +himself---at least this was frequently so as regards myself.'' Through his +whole life Darwin, like many others, prized highly this association, and he +considered that his whole work on cosmogony ``may be regarded as the +scientific outcome of our conversation of the year~1877; but,'' he adds, ``for +me at least science in this case takes the second place.'' + +Darwin at this time was thirty-two years old. In the three years since +he started publication fourteen memoirs and short notes, besides two statistical +papers on marriage between first cousins, form the evidence of his +activity. He seems to have reached maturity in his mathematical power +and insight into the problems which he attacked without the apprenticeship +which is necessary for most investigators. Probably the comparatively late +age at which he began to show his capacity in print may have something to +do with this. Henceforth development is rather in the direction of the full +working out of his ideas than growth of his powers. It seems better therefore +to describe his further scientific work in the manner in which he arranged +it himself, by subject instead of in chronological order. And here we have +the great advantage of his own comments, made towards the end of his +life when he scarcely hoped to undertake any new large piece of work. +Frequent quotation will be made from these remarks which occur in the +prefaces to the volumes, in footnotes and in his occasional addresses. + +The following account of the Earth-Moon series of papers is taken bodily +\index{Earth-Moon theory of Darwin, described by Mr S. S. Hough}% +from the Notice in the \textit{Proceedings of the Royal Society}\footnoteN + {Vol.~\Vol{89\;A}, p.~i.} +by Mr~S.~S. Hough, +who was himself one of Darwin's pupils. + +``The conclusions arrived at in the paper referred to above were based on +the assumption that throughout geological history, apart from slow geological +changes, the Earth would rotate sensibly as if it were rigid. It is shown that +a departure from this hypothesis might possibly account for considerable +excursions of the axis of rotation within the Earth itself, though these would +be improbable, unless, indeed, geologists were prepared to abandon the view +`that where the continents now stand they have always stood'; but no such +effect is possible with respect to the direction of the Earth's axis in space. +Thus the present condition of obliquity of the Earth's equator could in no +way be accounted for as a result of geological change, and a further cause +had to be sought. Darwin foresaw a possibility of obtaining an explanation +in the frictional resistance to which the tidal oscillations of the mobile parts +of a planet must be subject. The investigation of this hypothesis gave rise +to a remarkable series of papers of far-reaching consequence in theories of +cosmogony and of the present constitution of the Earth. +\DPPageSep{041}{xxxix} + +``In the first of these papers, which is of preparatory character, `On the +Bodily Tides of Viscous and Semi-elastic Spheroids, and on the Ocean Tides +on a Yielding Nucleus' (\textit{Phil.\ Trans.}, 1879, vol.~170), he adapts the analysis +of Sir~William Thomson, relating to the tidal deformations of an elastic +sphere, to the case of a sphere composed of a viscous liquid or, more generally, +of a material which partakes of the character either of a solid or a fluid +according to the nature of the strain to which it is subjected. For momentary +deformations it is assumed to be elastic in character, but the elasticity is +considered as breaking down with continuation of the strain in such a manner +that under very slow variations of the deforming forces it will behave sensibly +as if it were a viscous liquid. The exact law assumed by Darwin was dictated +rather by mathematical exigencies than by any experimental justification, but +the evidence afforded by the flow of rocks under continuous stress indicates +that it represents, at least in a rough manner, the mechanical properties +which characterise the solid parts of the Earth. + +``The chief practical result of this paper is summed up by Darwin himself +by saying that it is strongly confirmatory of the view already maintained by +Kelvin that the existence of ocean tides, which would otherwise be largely +masked by the yielding of the ocean bed to tidal deformation, points to +a high effective rigidity of the Earth as a whole. Its value, however, +lies further in the mathematical expressions derived for the reduction in +amplitude and retardation in phase of the tides resulting from viscosity +which form the starting-point for the further investigations to which the +author proceeded. + +``The retardation in phase or `lag' of the tide due to the viscosity +implies that a spheroid as tidally distorted will no longer present a +symmetrical aspect as if no such cause were operative. The attractive forces +on the nearer and more distant parts will consequently form a non-equilibrating +system with resultant couples tending to modify the state of +rotation of the spheroid about its centre of gravity. The action of these +couples, though exceedingly small, will be cumulative with lapse of time, +and it is their cumulative effects over long intervals which form the subject +of the next paper, `On the Precession of a Viscous Spheroid and on the +Remote History of the Earth' (\textit{Phil.\ Trans.}, 1879, vol.~170, Part~II, pp.~447--530). +The case of a single disturbing body (the Moon) is first considered, +but it is shown that if there are two such bodies raising tidal disturbances +(the Sun and Moon) the conditions will be materially modified from the +superposed results of the two disturbances considered separately. Under +certain conditions of viscosity and obliquity the obliquity of the ecliptic +will increase, and under others it will diminish, but the analysis further +yields `some remarkable results as to the dynamical stability or instability +of the system\ldots for moderate degrees of viscosity, the position of zero +\DPPageSep{042}{xl} +obliquity is unstable, but there is a position of stability at a high obliquity. +For large viscosities the position of zero obliquity becomes stable, and +(except for a very close approximation to rigidity) there is an unstable +position at a larger obliquity, and again a stable one at a still larger one.' + +``The reactions of the tidal disturbing force on the motion of the Moon +are next considered, and a relation derived connecting that portion of the +apparent secular acceleration of the Moon's mean motion, which cannot be +otherwise accounted for by theory, with the heights and retardations of the +several bodily tides in the Earth. Various hypotheses are discussed, but with +the conclusion that insufficient evidence is available to form `any estimate +having any pretension to accuracy\ldots as to the present rate of change due to +tidal friction.' + +``But though the time scale involved must remain uncertain, the nature +of the physical changes that are taking place at the present time is practically +free from obscurity. These involve a gradual increase in the length +of the day, of the month, and of the obliquity of the ecliptic, with a gradual +recession of the Moon from the Earth. The most striking result is that +these changes can be traced backwards in time until a state is reached when +the Moon's centre would be at a distance of only about $6000$~miles from the +Earth's surface, while the day and month would be of equal duration, +estimated at $5$~hours $36$~minutes. The minimum time which can have +elapsed since this condition obtained is further estimated at about $54$~million +years. This leads to the inevitable conclusion that the Moon and Earth at +one time formed parts of a common mass and raises the question of how and +why the planet broke up. The most probable hypothesis appeared to be +that, in accordance with Laplace's nebular hypothesis, the planet, being +partly or wholly fluid, contracted, and thus rotated faster and faster, until the +ellipticity became so great that the equilibrium was unstable. + +``The tentative theory put forward by Darwin, however, differs from the +nebular hypothesis of Laplace in the suggestion that instability might set +in by the rupture of the body into two parts rather than by casting off a +ring of matter, somewhat analogous to the rings of Saturn, to be afterwards +consolidated into the form of a satellite. + +``The mathematical investigation of this hypothesis forms a subject to +which Darwin frequently reverted later, but for the time he devoted himself +to following up more minutely the motions which would ensue after the +supposed planet, which originally consisted of the existing Earth and Moon +in combination, had become detached into two separate masses. In the +final section of a paper `On the Secular Changes in the Elements of the +Orbit of a Satellite revolving about a Tidally Distorted Planet' (\textit{Phil.\ +Trans.}, 1880, vol.~171), Darwin summarises the results derived in his +different memoirs. Various factors ignored in the earlier investigations, +\DPPageSep{043}{xli} +such as the eccentricity and inclination of the lunar orbit, the distribution +of the heat generated by tidal friction and the effects of inertia, were duly +considered and a complete history traced of the evolution resulting from +tidal friction of a system originating as two detached masses nearly in +contact with one another and rotating nearly as though they were parts +of one rigid body. Starting with the numerical data suggested by the +Earth-Moon System, `it is only necessary to postulate a sufficient lapse of +time, and that there is not enough matter diffused through space to resist +materially the motions of the Moon and Earth,' when `a system would +necessarily be developed which would bear a strong resemblance to our own.' +`A theory, reposing on \textit{verae causae}, which brings into quantitative correlation +the lengths of the present day and month, the obliquity of the ecliptic, +and the inclination and eccentricity of the lunar orbit, must, I think, have +strong claims to acceptance.' + +``Confirmation of the theory is sought and found, in part at least, in the +case of other members of the Solar System which are found to represent +various stages in the process of evolution indicated by the analysis. + +``The application of the theory of tidal friction to the evolution of the +Solar System and of planetary sub-systems other than the Earth-Moon +System is, however, reconsidered later, `On the Tidal Friction of a Planet +attended by Several Satellites, and on the Evolution of the Solar System' +(\textit{Phil.\ Trans.}, 1882, vol.~172). The conclusions drawn in this paper are +that the Earth-Moon System forms a unique example within the Solar +System of its particular mode of evolution. While tidal friction may +perhaps be invoked to throw light on the distribution of the satellites +among the several planets, it is very improbable that it has figured as the +dominant cause of change of the other planetary systems or in the Solar +System itself.'' + +For some years after this series of papers Darwin was busy with practical +tidal problems but he returned later ``to the problems arising in connection +with the genesis of the Moon, in accordance with the indications previously +arrived at from the theory of tidal friction. It appeared to be of interest to +trace back the changes which would result in the figures of the Earth and +Moon, owing to their mutual attraction, as they approached one another. +The analysis is confined to the consideration of two bodies supposed constituted +of homogeneous liquid. At considerable distances the solution of the +problem thus presented is that of the equilibrium theory of the tides, but, +as the masses are brought nearer and nearer together, the approximations +available for the latter problem cease to be sufficient. Here, as elsewhere, +when the methods of analysis could no longer yield algebraic results, Darwin +boldly proceeds to replace his symbols by numerical quantities, and thereby +succeeds in tracing, with considerable approximation, the forms which such +\DPPageSep{044}{xlii} +figures would assume when the two masses are nearly in contact. He even +carries the investigation farther, to a stage when the two masses in part +overlap. The forms obtained in this case can only he regarded as satisfying +the analytical, and not the true physical conditions of the problem, as, of +course, two different portions of matter cannot occupy the same space. +They, however, suggest that, by a very slight modification of conditions, +a new form could be found, which would fulfil all the conditions, in which +the two detached masses are united into a single mass, whose shape has been +variously described as resembling that of an hour-glass, a dumb-bell, or a pear. +This confirms the suggestion previously made that the origin of the Moon was +to be sought in the rupture of the parent planet into two parts, but the theory +was destined to receive a still more striking confirmation from another source. + +``While Darwin was still at work on the subject, there appeared the great +\index{Poincaré, reference to, by Sir George Darwin!on equilibrium of fluid mass in rotation}% +\index{Equilibrium of a rotating fluid}% +\index{Rotating fluid, equilibrium of}% +memoir by M.~Poincaré, `Sur l'équilibre d'une masse fluide animée d'un +mouvement de rotation' (\textit{Acta Math.}, vol.~7). + +``The figures of equilibrium known as Maclaurin's spheroid and Jacobi's +\index{Jacobi's ellipsoid}% +\index{Maclaurin's spheroid}% +ellipsoid were already familiar to mathematicians, though the conditions of +stability, at least of the latter form, were not established. By means of +analysis of a masterly character, Poincaré succeeded in enunciating and +applying to this problem the principle of exchange of stabilities. This principle +may be briefly indicated as follows: Imagine a dynamical system such as +a rotating liquid planet to be undergoing evolutionary change such as would +result from a gradual condensation of its mass through cooling. Whatever +be the varying element to which the evolutionary changes may be referred, +it may be possible to define certain relatively simple modes of motion, the +features associated with which will, however, undergo continuous evolution. +If the existence of such modes has been established, M.~Poincaré shows that +the investigation of their persistence or `stability' may be made to depend +on the evaluation of certain related quantities which he defines as coefficients +of stability. The latter quantities will be subject to evolutionary +change, and it may happen that in the course of such change one or more +of them assumes a zero value. Poincaré shows that such an occurrence +indicates that the particular mode of motion under consideration coalesces +at this stage with one other mode which likewise has a vanishing coefficient +of stability. Either mode will, as a rule, be possible before the change, but +whereas one will be stable the other will be unstable. The same will be +true after the change, but there will be an interchange of stabilities, whereby +that which was previously stable will become unstable, and \textit{vice versâ}. +An illustration of this principle was found in the case of the spheroids of +Maclaurin and the ellipsoids of Jacobi. The former in the earlier stages of +evolution will represent a stable condition, but as the ellipticity of surface +increases a stage is reached where it ceases to be stable and becomes unstable. +\DPPageSep{045}{xliii} +At this stage it is found to coalesce with Jacobi's form which involves in its +further development an ellipsoid with three unequal axes. Poincaré shows +that the latter form possesses in its earlier stages the requisite elements of +stability, but that these in their turn disappear in the later developments. +In accordance with the principle of exchange of stabilities laid down by +him, the loss of stability will occur at a stage where there is coalescence +with another form of figure, to which the stability will be transferred, and +this form he shows at its origin resembles the pear which had already been +indicated by Darwin's investigation. The supposed pear-shaped figure was +thus arrived at by two entirely different methods of research, that of Poincaré +tracing the processes of evolution forwards and that of Darwin proceeding +backwards in time. + +``The chain of evidence was all but complete; it remained, however, to +consider whether the pear-shaped figure indicated by Poincaré, stable in its +earlier forms, could retain its stability throughout the sequence of changes +necessary to fill the gap between these forms and the forms found by Darwin. + +``In later years Darwin devoted much time to the consideration of this +\index{Ellipsoidal harmonics}% +\index{Harmonics, ellipsoidal}% +problem. Undeterred by the formidable analysis which had to be faced, he +proceeded to adapt the intricate theory of Ellipsoidal Harmonics to a form in +which it would admit of numerical application, and his paper `Ellipsoid +Harmonic Analysis' (\textit{Phil.\ Trans.},~A, 1901, vol.~197), apart from the application +for which it was designed, in itself forms a valuable contribution +to this particular branch of analysis. With the aid of these preliminary +investigations he succeeded in tracing with greater accuracy the form of the +pear-shaped figure as established by Poincaré, `On the Pear-shaped Figure of +\index{Pear-shaped figure of equilibrium}% +Equilibrium of a Rotating Mass of Liquid' (\textit{Phil.\ Trans.},~A, 1901, vol.~198), +and, as he considered, in establishing its stability, at least in its earlier forms. +Some doubt, however, is expressed as to the conclusiveness of the argument +employed, as simultaneous investigations by M.~Lia\-pou\-noff pointed to an +\index{Liapounoff's work on rotating liquids}% +opposite conclusion. Darwin again reverts to this point in a further paper +`On the Figure and Stability of a Liquid Satellite' (\textit{Phil.\ Trans.},~A, 1906, +vol.~206), in which is considered the stability of two isolated liquid masses in +the stage at which they are in close proximity, i.e.,~the condition which would +obtain, in the Earth-Moon System, shortly after the Moon had been severed +from the Earth. The ellipsoidal harmonic analysis previously developed is +then applied to the determination of the approximately ellipsoidal forms +which had been indicated by Roche. The conclusions arrived at seem to +\index{Roche's ellipsoid}% +point, though not conclusively, to instability at the stage of incipient rupture, +but in contradistinction to this are quoted the results obtained by Jeans, who +\index{Jeans, J. H., on rotating liquids}% +considered the analogous problems of the equilibrium and rotation of infinite +rotating cylinders of liquid. This problem is the two-dimensional analogue +of the problems considered by Darwin and Poincaré, but involves far greater +\DPPageSep{046}{xliv} +simplicity of the conditions. Jeans finds solutions of his problem strictly +analogous to the spheroids of Maclaurin, the ellipsoids of Jacobi, and the +pear of Poincaré, and is able to follow the development of the latter until the +neck joining the two parts has become quite thin. He is able to establish +conclusively that the pear is stable in its early stages, while there is no +evidence of any break in the stability up to the stage when it divides itself +into two parts.'' + +Darwin's own final comments on this work next find a place here. +He is writing the preface to the second volume of his Collected Works in~1908, +after which time nothing new on the subject came from his pen. +``The observations of Dr~Hecker,'' he says, ``and of others do not afford +\index{Hecker's observations on retardation of tidal oscillations in the solid earth}% +evidence of any considerable amount of retardation in the tidal oscillations +of the solid earth, for, within the limits of error of observation, the +phase of the oscillation appears to be the same as if the earth were purely +elastic. Then again modern researches in the lunar theory show that the +secular acceleration of the moon's mean motion is so nearly explained by +means of pure gravitation as to leave but a small residue to be referred +to the effects of tidal friction. We are thus driven to believe that at present +\index{Tidal friction as a true cause of change}% +tidal friction is producing its inevitable effects with extreme slowness. But +we need not therefore hold that the march of events was always so leisurely, +and if the earth was ever wholly or in large part molten, it cannot have been +the case. + +``In any case frictional resistance, whether it be much or little and +whether applicable to the solid planet or to the superincumbent ocean, is +a true cause of change\ldots. + +``For the astronomer who is interested in cosmogony the important point +is the degree of applicability of the theory as a whole to celestial evolution. +To me it seems that the theory has rather gained than lost in the esteem of +men of science during the last 25~years, and I observe that several writers +are disposed to accept it as an established acquisition to our knowledge of +cosmogony. + +``Undue weight has sometimes been laid on the exact numerical values +assigned for defining the primitive configurations of the earth and moon. +In so speculative a matter close accuracy is unattainable, for a different +theory of frictionally retarded tides would inevitably load to a slight difference +in the conclusion; moreover such a real cause as the secular increase +in the masses of the earth and moon through the accumulation of meteoric +dust, and possibly other causes, are left out of consideration. + +``The exact nature of the process by which the moon was detached from +the earth must remain even more speculative. I suggested that the fission +of the primitive planet may have been brought about by the synchronism of +the solar tide with the period of the fundamental free oscillation of the +\DPPageSep{047}{xlv} +planet, and the suggestion has received a degree of attention which I never +anticipated. It may be that we shall never attain to a higher degree of +certainty in these obscure questions than we now possess, but I would +maintain that we may now hold with confidence that the moon originated +by a process of fission from the primitive planet, that at first she revolved in +an orbit close to the present surface of the earth, and that tidal friction +has been the principal agent which transformed the system to its present +configuration. + +``The theory for a long time seemed to lie open to attack on the ground +\index{Kelvin, associated with Sir George Darwin!cooling of earth}% +that it made too great demands on time, and this has always appeared to +me the greatest difficulty in the way of its acceptance. If we were still +compelled to assent to the justice of Lord Kelvin's views as to the period +of time which has elapsed since the earth solidified, and as to the age of the +solar system, we should also have to admit the theory of evolution under +tidal influence as inapplicable to its full extent. Lord Kelvin's contributions +to cosmogony have been of the first order of importance, but his arguments +on these points no longer carry conviction with them. Lord Kelvin contended +that the actual distribution of land and sea proves that the planet +solidified at a time when the day had nearly its present length. If this +were true the effects of tidal friction relate to a period antecedent to the +solidification. But I have always felt convinced that the earth would adjust +its ellipticity to its existing speed of rotation with close approximation.'' + +After some remarks concerning the effects of the discovery of radio-activity +and the energy resident in the atom on estimates of geological time, +he continues, ``On the whole then it may be maintained that deficiency +of time does not, according to our present state of knowledge, form a bar to +the full acceptability of the theory of terrestrial evolution under the influence +of tidal friction. + +``It is very improbable that tidal friction has been the dominant cause +of change in any of the other planetary sub-systems or in the solar system +itself, yet it seems to throw light on the distribution of the satellites amongst +the several planets. It explains the identity of the rotation of the moon +with her orbital motion, as was long ago pointed out by Kant and Laplace, +and it tends to confirm the correctness of the observations according to which +Venus always presents the same face to the sun.'' + +Since this was written much information bearing on the point has been +gathered from the stellar universe. The curious curves of light-changes in +certain classes of spectroscopic binaries have been well explained on the +assumption that the two stars are close together and under strong tidal +distortion. Some of these, investigated on the same hypothesis, even seem +to be in actual contact. In chap.~\Vol{XX} of the third edition~(1910) of his book +on the Tides, Darwin gives a popular summary of this evidence which had +\DPPageSep{048}{xlvi} +in the interval been greatly extended by the discovery and application of +the hypothesis to many other similar systems. In discussing the question +Darwin sets forth a warning. He points out that most of the densities +which result from the application of the tidal theory are very small compared +with that of the sun, and he concludes that these stars are neither homogeneous +nor incompressible. Hence the figures calculated for homogeneous +liquid can only be taken to afford a general indication of the kind of figure +which we might expect to find in the stellar universe. + +Perhaps Darwin's greatest service to cosmogony was the successful effort +\index{Numerical work on cosmogony}% +which he made to put hypotheses to the test of actual calculation. Even +though the mathematical difficulties of the subject compel the placing of +many limitations which can scarcely exist in nature, yet the solution of even +these limited problems places the speculator on a height which he cannot +hope to attain by doubtful processes of general reasoning. If the time +devoted to the framing and setting forth of cosmogonic hypotheses by various +writers had been devoted to the accurate solution of some few problems, the +newspapers and popular scientific magazines might have been less interesting +to their readers, but we should have had more certain knowledge of our +universe. Darwin himself engaged but little in speculations which were +not based on observations or precise conclusions from definitely stated +assumptions, and then only as suggestions for further problems to be +undertaken by himself or others. And this view of progress he communicated +to his pupils, one of whom, Mr~J.~H. Jeans, as mentioned above, is +continuing with success to solve those gravitational problems on similar +lines. + +The nebular hypothesis of Kant and Laplace has long held the field as +\index{Kant, Nebular Hypothesis}% +\index{Laplace, Nebular Hypothesis}% +the most probable mode of development of our solar system from a nebula. +At the present time it is difficult to say what are its chief features. Much +criticism has been directed towards every part of it, one writer changing +a detail here, another there, and still giving to it the name of the best known +exponent. The only salient point which seems to be left is the main hypothesis +that the sun, planets and satellites were somehow formed during the +process of contraction of a widely diffused mass of matter to the system as +we now see it. Some writers, including Darwin himself, regard a gaseous +nebula contracting under gravitation as the essence of Laplace's hypotheses, +distinguishing this condition from that which originates in the accretion +of small masses. Others believe that both kinds of matter may be present. +After all it is only a question of a name, but it is necessary in a discussion to +know what the name means. + +Darwin's paper, ``The mechanical conditions of a swarm of meteorites,'' +\index{Mechanical condition of a swarm of meteorites}% +is an attempt to show that, with reasonable hypotheses, the nebula and the +small masses under contraction by collisions may have led to the same result. +\DPPageSep{049}{xlvii} +In his preface to volume~\Vol{IV} he says with respect to this paper: ``Cosmogonists +are of course compelled to begin their survey of the solar system at some +arbitrary stage of its history, and they do not, in general, seek to explain +how the solar nebula, whether gaseous or meteoritic, came to exist. My +investigation starts from the meteoritic point of view, and I assume the +meteorites to be moving indiscriminately in all directions. But the doubt +naturally arises as to whether at any stage a purely chaotic motion of the +individual meteorites could have existed, and whether the assumed initial +condition ought not rather to have been an aggregate of flocks of meteorites +moving about some central condensation in orbits which intersect one another +at all sorts of angles. If this were so the chaos would not be one consisting +of individual stones which generate a quasi-gas by their collisions, but it +would be a chaos of orbits. But it is not very easy to form an exact picture +of this supposed initial condition, and the problem thus seems to elude +mathematical treatment. Then again have I succeeded in showing that a +pair of meteorites in collision will be endowed with an effective elasticity? +If it is held that the chaotic motion and the effective elasticity are quite +imaginary, the theory collapses. It should however be remarked that an +infinite gradation is possible between a chaos of individuals and a chaos +of orbits, and it cannot be doubted that in most impacts the colliding stones +would glance from one another. It seems to me possible, therefore, that my +two fundamental assumptions may possess such a rough resemblance to truth +as to produce some degree of similitude between the life-histories of gaseous +and meteoritic nebulae. If this be so the Planetesimal Hypothesis of +Chamberlain and Moulton is nearer akin to the Nebular Hypothesis than +\index{Chamberlain and Moulton, Planetesimal Hypothesis}% +\index{Moulton, Chamberlain and, Planetesimal Hypothesis}% +\index{Planetesimal Hypothesis of Chamberlain and Moulton}% +the authors of the former seem disposed to admit. + +``Even if the whole of the theory could be condemned as futile, yet the +paper contains an independent solution of the problem of Lane and Ritter; +and besides the attempt to discuss the boundary of an atmosphere, where +the collisions have become of vanishing rarity, may still perhaps be worth +something.'' + +In writing concerning the planetesimal hypothesis, Darwin seems to have +forgotten that one of its central assumptions is the close approach of two +stars which by violent tidal action drew off matter in spiral curves which +became condensed into the attendants of each. This is, in fact, one of the +most debatable parts of the hypothesis, but one on which it is possible to +get evidence from the distribution of such systems in the stellar system. +Controversy on the main issue is likely to exist for many years to come. + +Quite early in his career Darwin was drawn into practical tidal problems +\index{Tidal problems, practical}% +by being appointed on a Committee of the British Association with Adams, +to coordinate and revise previous reports drawn up by Lord Kelvin. He +evidently felt that the whole subject of practical analysis of tidal observations +\DPPageSep{050}{xlviii} +needed to be set forth in full and made clear. His first report consequently +contains a development of the equilibrium theory of the Tides, and later, +after a careful analysis of each harmonic component, it proceeds to outline in +detail the methods which should be adopted to obtain the constants of each +component from theory or observation, as the case needed. Schedules and +forms of reduction are given with examples to illustrate their use. + +There are in reality two principal practical problems to be considered. +The one is the case of a port with much traffic, where it is possible to obtain +tide heights at frequent intervals and extending over a long period. While +the accuracy needed usually corresponds to the number of observations, it is +always assumed that the ordinary methods of harmonic analysis by which all +other terms but that considered are practically eliminated can be applied; +the corrections when this is not the case are investigated and applied. The +other problem is that of a port infrequently visited, so that we have only +a short series of observations from which to obtain the data for the computation +of future tides. The possible accuracy here is of course lower than in +the former case but may be quite sufficient when the traffic is light. In his +third report Darwin takes up this question. The main difficulty is the +separation of tides which have nearly the same period and which could not +be disentangled by harmonic analysis of observations extending over a very +few weeks. Theory must therefore be used, not only to obtain the periods, +but also to give some information about the amplitudes and phases if this +separation is to be effected. The magnitude of the tide-generating force is +used for the purpose. Theoretically this should give correct results, but it is +often vitiated by the form of the coast line and other circumstances depending +on the irregular shape of the water boundary. Darwin shows however that +fair prediction can generally be obtained; the amount of numerical work is +of course much smaller than in the analysis of a year's observations. This +report was expanded by Darwin into an article on the Tides for the \textit{Admiralty +Scientific Manual}. + +Still another problem is the arrangement of the analysis when times and +heights of high and low water alone are obtainable; in the previous papers +the observations were supposed to be hourly or obtained from an automatically +recording tide-gauge. The methods to be used in this case are of course +well known from the mathematical side: the chief problem is to reduce the +arithmetical work and to put the instructions into such a form that the +ordinary computer may use them mechanically. The problem was worked +out by Darwin in~1890, and forms the subject of a long paper in the +\textit{Proceedings of the Royal Society}. + +A little later he published the description of his now well known abacus, +\index{Abacus}% +designed to avoid the frequent rewriting\DPnote{[** TN: Not hyphenated in original]} of the numbers when the harmonic +analysis for many different periods is needed. Much care was taken to obtain +\DPPageSep{051}{xlix} +the right materials. The real objection to this, and indeed to nearly all the +methods devised for the purpose, is that the arrangement and care of the +mechanism takes much longer time than the actual addition of the numbers +after the arrangement has been made. In this description however there +are more important computing devices which reduce the time of computation +to something like one-fifth of that required by the previous methods. +The principal of these is the one in which it is shown how a single set +of summations of $9000$~hourly values can be made to give a good many +terms, by dividing the sums into proper groups and suitably treating +them. + +Another practical problem was solved in his Bakerian Lecture ``On Tidal +\index{Bakerian lecture}\Pagelabel{xlix}% +Prediction.'' In a previous paper, referred to above, Darwin had shown how +the tidal constants of a port might be obtained with comparatively little +expense from a short series of high and low water observations. These, +however, are of little value unless the port can furnish the funds necessary +to predict the future times and heights of the tides. Little frequented ports +can scarcely afford this, and therefore the problem of replacing such predictions +by some other method is necessary for a complete solution. ``The +object then,'' says Darwin, ``of the present paper, is to show how a general +tide-table, applicable for all time, may be given in such a form that anyone, +with an elementary knowledge of the \textit{Nautical Almanac}, may, in a few +minutes, compute two or three tides for the days on which they are required. +The tables will also be such that a special tide-table for any year may be +computed with comparatively little trouble.'' + +This, with the exception of a short paper dealing with the Tides in the +Antarctic as shown by observations made on the \textit{Discovery}, concludes Darwin's +published work on practical tidal problems. But he was constantly in correspondence +about the subject, and devoted a good deal of time to government +work and to those who wrote for information. + +In connection with these investigations it was natural that he should +\index{Rigidity of earth, from fortnightly tides}% +\index{Tide, fortnightly}% +turn aside at times to questions of more scientific interest. Of these the +fortnightly tide is important because by it some estimate may be reached as +to the earth's rigidity. The equilibrium theory while effective in giving the +periods only for the short-period tides is much more nearly true for those of +long period. Hence, by a comparison of theory and observation, it is possible +to see how much the earth yields to distortion produced by the moon's +attraction. Two papers deal with this question. In the first an attempt is +made to evaluate the corrections to the equilibrium theory caused by the +continents; this involves an approximate division of the land and sea +surfaces into blocks to which calculation may be applied. In the second +tidal observations from various parts of the earth are gathered together for +comparison with the theoretical values. As a result, Darwin obtains the +\DPPageSep{052}{l} +oft-quoted expression for the rigidity of the earth's mass, namely, that it is +effectively about that of steel. An attempt made by George and Horace +Darwin to measure the lunar disturbance of gravity by means of the +pendulum is in reality another approach to the solution of the same problem. +The attempt failed mainly on account of the local tremors which were produced +by traffic and other causes. Nevertheless the two reports contain +much that is still interesting, and their value is enhanced by a historical +account of previous attempts on the same lines. Darwin had the satisfaction +of knowing that this method was later successful in the hands of Dr~Hecker +\index{Hecker's observations on retardation of tidal oscillations in the solid earth}% +whose results confirmed his first estimate. Since his death the remarkable +experiment of Michelson\footnoteN + {\textit{Astrophysical Journal}, March,~1914.} +\index{Michelson's experiment on rigidity of earth}% +\index{Rigidity of earth, from fortnightly tides!Michelson's experiment}% +with a pipe partly filled with water has given +a precision to the determination of this constant which much exceeds that +of the older methods; he concludes that the rigidity and viscosity are at least +equal to and perhaps exceed those of steel. + +It is here proper to refer to Darwin's more popular expositions of the +\index{Tides, The@\textit{Tides, The}}% +\index{Tides, articles on}\Pagelabel{l}% +work of himself and others. He wrote several articles on Tides, notably for +the \textit{Encyclopaedia Britannica} and for the \textit{Encyclopaedie der Mathematischen +Wissenschaften}, but he will be best remembered in this connection for his +volume \textit{The Tides} which reached its third edition not long before his +death. The origin of it was a course of lectures in~1897 before the Lowell +Institute of Boston, Massachusetts. An attempt to explain the foundations +and general developments of tidal theory is its main theme. It naturally +leads on to the subject of tidal friction and the origin of the moon, and +therewith are discussed numerous questions of cosmogony. From the point +of view of the mathematician, it is not only clear and accurate but gives the +impression, in one way, of a \textit{tour de force}. Although Darwin rarely has to +ask the reader to accept his conclusions without some description of the +nature of the argument by which they are reached, there is not a single +algebraic symbol in the whole volume, except in one short footnote where, on +a minor detail, a little algebra is used. The achievement of this, together +with a clear exposition, was no light task, and there are few examples to be +found in the history of mathematics since the first and most remarkable of all, +Newton's translation of the effects of gravitation into geometrical reasoning. +\textit{The Tides} has been translated into German (two editions), Hungarian, +Italian and Spanish. + +In 1877 the two classical memoirs of G.~W.~Hill on the motion of the +\index{Hill, G. W., Lunar Theory}% +moon were published. The first of these, \textit{Researches in the Lunar Theory}, +contains so much of a pioneer character that in writing of any later work on +celestial mechanics it is impossible to dismiss it with a mere notice. One +portion is directly concerned with a possible mode of development of the +lunar theory and the completion of the first step in the process. The usual +\DPPageSep{053}{li} +method of procedure has been to consider the problem of three bodies as an +extension of the case of two bodies in which the motion of one round the +other is elliptic. Hill, following a suggestion of Euler which had been +worked out by the latter in some detail, starts to treat the problem as a +very special particular case of the problem of three bodies. One of them, +the earth, is of finite mass; the second, the sun, is of infinite mass and at +an infinite distance but is revolving round the former with a finite and +constant angular velocity. The third, the moon, is of infinitesimal mass, but +moves at a finite distance from the earth. Stated in this way, the problem +of the moon's motion appears to bear no resemblance to reality. It is, +however, nothing but a limiting case where certain constants, which are +small in the case of the actual motion, have zero values. The sun is +actually of very great mass compared with the earth, it is very distant as +compared with the distance of the moon, its orbit round the earth (or \textit{vice +versâ}) is nearly circular, and the moon's mass is small compared with that +of the earth. The differential equations which express the motion of +the moon under these limitations are fairly simple and admit of many +transformations. + +Hill simplifies the equations still further, first by supposing the moon +so started that it always remains in the same fixed plane with the earth +and the sun (its actual motion outside this plane is small). He then uses +moving rectangular axes one of which always points in the direction of the +sun. Even with all these limitations, the differential equations possess many +classes of solutions, for there will be four arbitrary constants in the most +general values of the coordinates which are to be derived in the form of a +doubly infinite series of harmonic terms. His final simplification is the +choice of one of these classes obtained by giving a zero value to one of +the arbitrary constants; in the moon's motion this constant is small. The +orbit thus obtained is of a simple character but it possesses one important +property; relative to the moving axes it is closed and the body following +it will always return to the same point of it (relative to the moving axis) +after the lapse of a definite interval. In other words, the relative motion +is periodic. + +Hill develops this solution literally and numerically for the case of our +satellite with high accuracy. This accuracy is useful because the form of +the orbit depends solely on the ratio of the mean rates of motion of the sun +and moon round the earth, and these rates, determined from centuries of +observation, are not affected by the various limitations imposed at the outset. +The curve does not differ much from a circle to the eye but it includes the +principal part of one of the chief differences of the motion from that in a +circle with uniform velocity, namely, the inequality long known as the +``variation''; hence the name since given to it, ``the Variational Orbit.'' Hill, +\DPPageSep{054}{lii} +however, saw that it was of more general interest than its particular application +to our satellite. He proceeds to determine its form for other values +of the mean rates of motion of the two bodies. This gives a family of +periodic orbits whose form gradually varies as the ratio is changed; the +greater the ratio, the more the curve differs from a circle. + +It is this idea of Hill's that has so profoundly changed the whole outlook +of celestial mechanics. Poincaré took it up as the basis of his celebrated +prize essay of~1887 on the problem of three bodies and afterwards expanded +his work into the three volumes; \textit{Les méthodes nouvelles de la Mécanique +Céleste}. His treatment throughout is highly theoretical. He shows that +\index{Poincaré, reference to, by Sir George Darwin!\textit{Les Méthodes Nouvelles de la Mécanique Céleste}}% +there must be many families of periodic orbits even for specialised problems +in the case of three bodies, certain general properties are found, and much +information concerning them which is fundamental for future investigation +is obtained. + +It is doubtful if Darwin had paid any special attention to Hill's work +on the moon for at least ten years after its appearance. All this time he +was busy with the origin of the moon and with tidal work. Adams had +published a brief \textit{résumé} of his own work on lines similar to those of Hill +immediately after the memoirs of the latter appeared, but nothing further +on the subject came from his pen. The medal of the Royal Astronomical +Society was awarded to Hill in~1888, and Dr~Glaisher's address on his work +\index{Glaisher, Dr J. W. L., address on presenting the gold medal of the R.A.S. to G. W. Hill}% +\index{Hill, G. W., Lunar Theory!awarded gold medal of R.A.S.}% +\index{Hill, G. W., Lunar Theory!lectures by Darwin on Lunar Theory}% +contains an illuminating analysis of the methods employed and the ideas +which are put forward. Probably both Darwin and Adams had a considerable +share in making the recommendation. Darwin often spoke of his +difficulties in assimilating the work of others off his own beat and possibly +this address started him thinking about the subject, for it was at his recommendation +in the summer of 1888 that the writer took up the study of Hill's +papers. ``They seem to be very good,'' he said, ``but scarcely anyone knows +much about them.'' + +He lectured on Hill's work for the first time in the Michaelmas Term +of~1893, and writes of his difficulties in following parts of them, more +particularly that on the Moon's Perigee which contains the development of +the infinite determinant. He concludes, ``I can't get on with my own work +until these lectures are over---but Hill's papers are splendid.'' One of his +pupils on this occasion was Dr~P.~H. Cowell, now Director of the Nautical +Almanac office. The first paper of the latter was a direct result of these +lectures and it was followed later by a valuable series of memoirs in which +the constants of the lunar orbit and the coefficients of many of the periodic +terms were obtained with great precision. Soon after these lectures Darwin +started his own investigations on the subject. But they took a different +line. The applications to the motion of the moon were provided for and +Poincaré had gone to the foundations. Darwin felt, however, that the work of +\DPPageSep{055}{liii} +the latter was far too abstract to satisfy those who, like himself, frequently +needed more concrete results, either for application or for their own mental +satisfaction. In discussing periodic orbits he set himself the task of tracing +numbers of them in order, as far as possible, to get a more exact knowledge +of the various families which Poincaré's work had shown must exist. Some +of Hill's original limitations are dropped. Instead of taking a sun of infinite +mass and at an infinite distance, he took a mass ten times that of the +planet and at a finite distance from that body. The orbit of each round +the other is circular and of uniform motion, the third body being still of +infinitesimal mass. Any periodic orbit which may exist is grist to his mill +whether it circulate, about one body or both or neither. + +Darwin saw little hope of getting any extensive results by solutions of +\index{Numerical work, great labour of}% +\index{Periodic orbits, Darwin begins papers on}% +\index{Periodic orbits, Darwin begins papers on!great numerical difficulties of}% +\index{Periodic orbits, Darwin begins papers on!stability of}% +the differential equations in harmonic series. It was obvious that the slowness +of convergence or the divergence would render the work far too doubtful. +He adopted therefore the tedious process of mechanical quadratures, starting +at an arbitrary position on the $x$-axis with an arbitrary speed in a direction +parallel to the $y$-axis. Tracing the orbit step-by-step, he again reaches the +$x$-axis. If the final velocity there is perpendicular to the axis, the orbit is +periodic. If not, he starts again with a different speed and traces another +orbit. The process is continued, each new attempt being judged by the +results of the previous orbits, until one is obtained which is periodic. The +amount of labour involved is very great since the actual discovery of a +periodic orbit generally involved the tracing of from three to five or even +more non-periodic paths. Concerning one of the orbits he traced for his last +paper on the subject, he writes: ``You may judge of the work when I tell +you that I determined $75$~positions and each averaged $\frac{3}{4}$~hr.\ (allowing for +correction of small mistakes---which sometimes is tedious). You will see +that it is far from periodic\ldots. I have now got six orbits of this kind.'' And all +this to try and find only one periodic orbit belonging to a class of whose +existence he was quite doubtful. + +Darwin's previous work on figures of equilibrium of rotating fluids made +the question of the stability of the motion in these orbits a prominent factor +in his mind. He considered it an essential part in their classification. To +determine this property it was necessary, after a periodic orbit had been +obtained, to find the effect of a small variation of the conditions. For this +purpose, Hill's second paper of~1877, on the Perigee of the Moon, is used. +After finding the variation orbit in his first paper. Hill makes a start +towards a complete solution of his limited differential equations by finding +an orbit, not periodic and differing slightly from the periodic orbit already +obtained. The new portion, the difference between the two, when expressed +as a sum of harmonic terms, contains an angle whose uniform rate of change,~$c$, +depends only on the constants of the periodic orbit. The principal +\DPPageSep{056}{liv} +portion of Hill's paper is devoted to the determination of~$c$ with great +precision. For this purpose, the infinite determinant is introduced and +expanded into infinite series, the principal parts of which are expressed by +a finite number of well known functions; the operations Hill devised to +achieve this have always called forth a tribute to his skill. Darwin uses +this constant~$c$ in a different way. If it is real, the orbit is stable, if +imaginary, unstable. In the latter case, it may be a pure imaginary or a +complex number; hence the necessity for the two kinds of unstability. + +In order to use Hill's method, Darwin is obliged to analyse a certain +function of the coordinates in the periodic orbit into a Fourier series, and to +obtain the desired accuracy a large number of terms must be included. +For the discovery of~$c$ from the infinite determinant, he adopts a mode of +expansion of his own better suited to the purpose in hand. But in any case +the calculation is laborious. In a later paper, he investigates the stability +by a different method because Hill's method fails when the orbit has +sharp flexures. + +For the classification into families, Darwin follows the changes according +\index{Hough, S. S., Darwin's work on Earth-Moon Theory, xxxviii; Periodic Orbits}% +\index{Periodic orbits, Darwin begins papers on!classification of, by Jacobi's integral}% +to variations in the constant of relative energy,~$C$. The differential equations +referred to the moving axes admit a Jacobian integral, the constant of +which is~$C$. One property of this integral Hill had already developed, +namely, that the curve obtained by making the kinetic energy zero is one +which the body cannot cross. Darwin draws the curves for different values +of~$C$ with care. He is able to show in several cases the origin of the +families he has found and much use is made of Poincaré's proposition, that +all such families originate in pairs, for following the changes. But even +his material is sometimes insufficient, especially where two quite different +pairs of families originate near the same point on the $x$-axis, and some later +corrections of the classification partly by himself and partly by Mr~S.~S. Hough +were necessary. In volume~\Vol{IV} of his collected works these corrections are +fully explained. + +The long first memoir was published in~1896. Nothing further on the +subject appeared from his hand until 1909 when a shorter paper containing +a number of new orbits was printed in the Monthly Notices of the Royal +Astronomical Society. Besides some additions and corrections to his older +families he considers orbits of ejection and retrograde orbits. During the +interval others had been at work on similar lines while Darwin with +increasing duties thrust upon him only found occasional opportunities to +keep his calculations going. A final paper which appears in the present +volume was the outcome of a request by the writer that a trial should be +made to find a member of a librating class of orbits for the mass ratio~$1:10$ +which had been shown to exist and had been traced for the mass ratio~$1:1048$. +The latter arose in an attempt to consider the orbits of the Trojan group of +\DPPageSep{057}{lv} +asteroids. He failed to find one but in the course of his work discovered +another class of great interest, which shows the satellite ultimately falling +into the planet. He concludes, ``My attention was first drawn to periodic +orbits by the desire to discover how a Laplacian ring could coalesce into +a planet. With this object in view I tried to discover how a large planet +could affect the mean motion of a small one moving in a circular orbit at +the same mean distance. After various failures the investigation drifted +towards the work of Hill and Poincaré, so that the original point of view +was quite lost and it is not even mentioned in my paper on `Periodic Orbits.' +It is of interest, to me at least, to find that the original aspect of the problem +has emerged again.'' It is of even greater interest to one of his pupils to +find that after more than twenty years of work on different lines in celestial +mechanics, Darwin's last paper should be on the same part of the subject to +which both had been drawn from quite different points of view. + +Thus Darwin's work on what appeared to be a problem in celestial +mechanics of a somewhat unpractical nature sprang after all from and +finally tended towards the question which had occupied his thoughts nearly +all his life, the genesis and evolution of the solar system. +\DPPageSep{058}{lvi} +%[Blank Page] +\DPPageSep{059}{1} +\index{Orbits, periodic|see{Periodic orbits}} + + +\Chapter{Inaugural Lecture} +\index{Inaugural lecture}% +\index{Cambridge School of Mathematics}% +\index{Lecture, inaugural}% +\index{Mathematical School at Cambridge}% + +\Heading{(Delivered at Cambridge, in 1883, on election to the +Plumian Professorship)} + +\First{I propose} to take advantage of the circumstance that this is the first of +the lectures which I am to give, to say a few words on the Mathematical +School of this University, and especially of the position of a professor in +regard to teaching at the present time. + +There are here a number of branches of scientific study to which there +are attached laboratories, directed by professors, or by men who occupy the +position and do the duties of professors, but do not receive their pay from, +nor full recognition by, the University. Of these branches of science I have +comparatively little to say. + +You are of course aware of the enormous impulse which has been given +to experimental science in Cambridge during the last ten years. It would +indeed have been strange if the presence of such men as now stand at the +head of those departments had not created important Schools of Science. +And yet when we consider the strange constitution of our University, it +may be wondered that they have been able to accomplish this. I suspect +that there may be a considerable number of men who go through their +University course, whose acquaintance with the scientific activity of the place +is limited by the knowledge that there is a large building erected for some +obscure purpose in the neighbourhood of the Corn Exchange. Is it possible +that any student of Berlin should be heard to exclaim, ``Helmholtz, who is +Helmholtz?'' And yet some years ago I happened to mention the name of +one of the greatest living mathematicians, a professor in this University, +in the presence of a first class man and fellow of his College, and he made +just such an exclamation. + +This general state of apathy to the very existence of science here has +now almost vanished, but I do not think I have exaggerated what it was +some years ago. Is not there a feeling of admiration called for for\DPnote{[** TN: Double word OK]} those, who +by their energy and ability have raised up all the activity which we now see? +\DPPageSep{060}{2} + +For example, Foster arrived here, a stranger to the University, without +University post or laboratory. I believe that during his first term Balfour +and one other formed his whole class. And yet holding only that position +of a College lecturer which he holds at this minute, he has come to make +Cambridge the first Physiological School of Great Britain, and the range of +buildings which the University has put at his disposal has already proved +too small for his requirements\footnotemarkN. His pupil Balfour had perhaps a less +\footnotetextN{Sir Michael Foster was elected the first Professor of Physiology a few weeks after the + delivery of this lecture.}% +uphill game to play, for the germs of the School of Natural Science were +already laid when he began his work as a teacher. But he did not merely +aid in the further developments of what he found, for he struck out in a +new line---that line of study which his own original work has gone, I +believe, a very long way to transform and even create. He did not live +to see the full development of the important school and laboratory which +he had founded. But thanks to his impulse it is now flourishing, and will +doubtless prosper under the able hands into which the direction has fallen. +His name ought surely to live amongst us for what he did; for those who +had the fortune to be his friends the remembrance of him cannot die, for +what he was. + +I should be going too far astray were I to continue to expatiate on the +work of Rayleigh, Stuart, and the others who are carrying on the development +of practical work in various branches within these buildings. It must +suffice to say that each school has had its difficulties, and that those difficulties +have been overcome by the zeal of those concerned in the management. + +But now let us turn to the case of the scientific professors who have no +laboratories to direct, and I speak now of the mathematical professors. In +comparison with the prosperity of which I have been speaking, I think +it is not too much to say that there is no vitality. I belong to this class of +professors, and I am far from flattering myself that I can do much to impart +life to the system. But if I shall not succeed I may perhaps be pardoned +if I comfort myself by the reflection, that it may not be entirely my own fault. + +The University has however just entered on a new phase; I have the +honour to be the first professor elected under the new Statutes now in force. +A new scheme for the examinations in Mathematics is in operation, and it +may be that such an opportunity will now be afforded as has hitherto been +wanting. We can but try to avail ourselves of the chance. + +To what causes are we to assign the fact that our most eminent +teachers of mathematics have hitherto been very frequently almost without +classes? It surely cannot be that Stokes, Adams and Cayley have \textit{nothing} +to say worth hearing by students of mathematics. Granting the possibility +\DPPageSep{061}{3} +that a distinguished man may lack the power of exposition, yet it is inadmissible +that they are \textit{all} deficient in that respect. No, the cause is not far +to seek, it lies in the Mathematical Tripos. How far it is desirable that the +system should be so changed, that it will be advisable for students in their +own interest to attend professorial lectures, I am not certain; but it can +scarcely be doubted that if there were no Tripos, the attendance at such +lectures would be larger. + +In hearing the remarks which I am about to make on the Mathematical +\index{Mathematical School at Cambridge!Tripos}% +\index{Tripos, Mathematical}% +Tripos, you must bear in mind that I have hitherto taken no part in mathematical +teaching of any kind, and therefore must necessarily be a bad judge +of the possibilities of mathematical training, and of its effects on most minds. +A year and a half ago I took part as Additional Examiner in the Mathematical +Tripos, and I must confess that I was a good deal discouraged by what +I saw. Now do not imagine that I flatter myself I was one jot better in all +these respects than others, when I went through the mill. I too felt the +pressure of time, and scribbled down all I could in my three hours, and +doubtless presented to my examiners some very pretty muddles. I can only +congratulate myself that the men I examined were not my competitors. + +In order to determine whether anything can be done to improve this +state of things, let us consider the merits and demerits of our Mathematical +School. One of the most prominent evils is that our system of examination +has a strong tendency to make men regard the subjects more as a series of +isolated propositions than as a whole; and much attention has to be paid to a +point, which is really important for the examination, viz.~where to begin and +where to leave off in answering a question. The \textit{coup d'{\oe}il} of the whole +subject is much impaired; but this is to some extent inherent in any system +of examination. This result is, however, principally due to our custom of +setting the examinees to reproduce certain portions of the books which they +have studied; that is to say this evil arises from the ``bookwork'' questions. +I have a strong feeling that such questions should be largely curtailed, and +that the examinees should by preference be asked for transformations and +modifications of the results obtained in the books. I suppose a certain amount +of bookwork must be retained in order to permit patient workers, who are +not favoured by any mathematical ability, to exhibit to the examiners that +they have done their best. But for men with any mathematical power +there can be no doubt that such questions as I suggest would give a far +more searching test, and their knowledge of the subject would not have +to be acquired in short patches. + +I should myself like to see an examination in which the examinees were +allowed to take in with them any books they required, so that they need not +load their memories with formulae, which no original worker thinks of trying +\DPPageSep{062}{4} +to remember. A first step in this direction has been taken by the introduction +of logarithm tables into the Senate House; and I fancy that a +terrible amount of incompetence was exhibited in the result. I may remark +by the way that the art of computation is utterly untaught here, and that +readiness with figures is very useful in ordinary life. I have done a good +deal of such work myself, but I had to learn it by practice and from a few +useful hints from others who had mastered it. + +It is to be regretted that questions should be set in examinations which +are in fact mere conjuring tricks with symbols, a kind of double acrostic; +another objectionable class of question is the so-called physical question which +has no relation to actual physics. This kind of question was parodied once +by reference to ``a very small elephant, whose weight may be neglected,~etc.'' +Examiners have often hard work to find good questions, and their difficulties +are evidenced by such problems as I refer to. I think, however, that of late +this kind of exercise is much less frequent than formerly. + +I am afraid the impression is produced in the minds of many, that if +a problem cannot be solved in a few hours, it cannot be solved at all. At any +rate there seems to be no adequate realisation of the process by which most +original work is done, when a man keeps a problem before him for weeks, +months, years and gnaws away from time to time when any new light may +strike him. + +I think some of our text books are to blame in this; they impress the +\index{Mathematical School at Cambridge!text-books}% +\index{Text-books, mathematical}% +student in the same way that a high road must appear to a horse with +blinkers. The road stretches before him all finished and macadamised, +having existed for all he knows from all eternity, and he sees nothing of +by-ways and foot-paths. Now it is the fact that scarcely any subject is so +way worn that there are not numerous unexplored by-paths, which may lead +across to undiscovered countries. I do not advocate that the student should +be led along and made to examine all the cul-de-sacs and blind alleys, as he +goes; he would never got on if he did so, but I do protest against that tone +which I notice in many text books that mathematics is a spontaneously +growing fruit of the tree of knowledge, and that all the fruits along \textit{that} +road have been gathered years ago. Rather let him see that the whole +grand work is the result of the labours of an army of men, each exploring +his little bit, and that there are acres of untouched ground, where he too may +gather fruit: true, if he begins on original work, he may think that he has +discovered something new and may very likely find that someone has been +before him; but at least he \textit{too} will have had the enormous pleasure of +discovery. + +There is another fault in the system of examinations, but I hardly know +whether it can be appreciably improved. It is this:---the system gives very +\DPPageSep{063}{5} +little training in the really important problem both of practical life and of +mathematics, viz.~the determination of the exact nature of the question +which is to be attacked, the making up of your mind as to what you will do. +Everyone who has done original work knows that at first the subject generally +presents itself as a chaos of possible problems, and careful analysis +is necessary before that chaos is disentangled. The process is exactly that +of a barrister with his brief. A pile of papers is set before him, and from +that pile he has to extract the precise question of law or fact on which +the whole turns. When he has mastered the story and the precise point, +he has generally done the more difficult part of his work. In most cases, +it is exactly the same in mathematical work; and when the question has +been pared down until its characteristics are those of a Tripos question, of +however portentous a size, the battle is half won. It only remains to the +investigator then to avail himself of all the ``morbid aptitude for the +manipulation of symbols'' which he may happen to possess. + +In examination, however, the whole of this preparatory part of the work +is done by the examiner, and every examiner must call to mind the weary +threshing of the air which he has gone through in trying ``to get a question'' +out of a general idea. Now the limitation of time in an examination makes +this evil to a large extent irremediable; but it seems to me that some good +may be done by requesting men to write essays on particular topics, +because in this case their minds are not guided by a pair of rails carefully +prepared by an examiner. + +In the report on the Tripos for~1882, I spoke of the slovenliness of style +which characterised most of the answers. It appears to me that this is really +much more than a mere question of untidiness and annoyance to examiners. +The training here seems to be that form and style are matters of no moment, +and answers are accordingly sent up in examination which are little more +than rough notes of solutions. But I insist that a mathematical writer +should attend to style as much as a literary man. + +Some of our Cambridge writers on mathematics seem never to have +recovered from the ill effects of their early training, even when they devote +the rest of their life to original work. I wish some of you would look at the +artistic mode of presentation practised by Gauss, and compare it with the +standard of excellence which passes muster here. Such a comparison will +not prove gratifying to our national pride. + +Where there is slovenliness of style it is, I think, almost certain that +there will be wanting that minute attention to form on which the successful, +or at least easy, marshalling of a complex analytical development depends. +The art of carrying out such work has to be learnt by trial and error by +the men trained in our school, and yet the inculcation of a few maxims +\DPPageSep{064}{6} +would generally be of great service to students, provided they are made to +attend to them in their work. The following maxims contain the pith of +the matter, although they might be amplified with advantage if I were to +detain you over this point for some time. + +1st. Choose the notation with great care, and where possible use a +standard notation. + +2nd. Break up the analysis into a series of subsections, each of which +may be attended to in detail. + +3rd. Never attempt too many transformations in one operation. + +4th. Write neatly and not quickly, so that in passing from step to step +there may be no mistakes of copying. + +A man who undertakes any piece of work, and does not attend to some +such rules as these, doubles his chances of mistake; even to short pieces +of work such as examination questions the same applies, and I have little +doubt that many a score of questions have been wrongly worked out from +want of attention to these points. + +It is true that great mathematicians have done their work in very +various styles, but we may be sure that those who worked untidily gave +themselves much unnecessary trouble. Within my own knowledge I may +say that Thomson [Lord Kelvin] works in a copy-book, which is produced at +Railway Stations and other conveniently quiet places for studious pursuits; +Maxwell worked in part on the backs of envelopes and loose sheets of paper +crumpled up in his pocket\footnotemarkN; Adams' manuscript is as much a model of +\footnotetextN{I think that he must have been only saved from error by his almost miraculous physical + insight, and by a knowledge of the time when work must be done neatly. But his \textit{Electricity} + was crowded with errata, which have now been weeded out one by one.}% +neatness in mathematical writing as Porson's of Greek writing. There is, of +course, no infallibility in good writing, but believe me that untidiness surely +has its reward in mistakes. I have spoken only on the evils of slovenliness +in its bearing on the men as mathematicians---I cannot doubt that as a +matter of general education it is deleterious. + +I have dwelt long on the demerits of our scheme, because there is hope +of amending some of them, but of the merits there is less to be said because +they are already present. The great merit of our plan seems to me to be +reaped only by the very ablest men in the year. It is that the student is +enabled to get a wide view over a great extent of mathematical country, +and if he has not assimilated all his knowledge thoroughly, yet he knows +that it is so, and he has a fair introduction to many subjects. This +advantage he would have lost had he become a pure specialist and original +investigator very early in his career. But this advantage is all a matter +of degree, and even the ablest man cannot cover an indefinitely long course +\DPPageSep{065}{7} +in his three years. Year by year new subjects were being added to the +curriculum, and the limit seemed to have been exceeded; whilst the +disastrous effects on the weaker brethren were becoming more prominent. +I cannot but think that the new plan, by which a man shall be induced to +become a partial specialist, gives us better prospects. + +Another advantage we gain by our strict competition is that a man must +be bright and quick; he must not sit mooning over his papers; he is quickly +brought to the test,---either he can or he cannot do a definite problem in +a finite time---if he cannot he is found out. Then if our scheme checks +original investigation, it at least spares us a good many of those pests of +science, the man who churns out page after page of~$x, y, z,$ and thinks he +has done something in producing a mass of froth. That sort of man is +quickly found out here, both for his own good and the good of the world +at large. Lastly this place has the advantage of having been the training +school of nearly all the English mathematicians of eminence, and of having +always attracted---as it continues to attract---whatever of mathematical +ability is to be found in the country. These are great merits, and in the +endeavour to remove blemishes, we must see that we do not destroy them. + +A discussion of the Mathematical Tripos naturally brings us face to face +with a much abused word, namely ``Cram.'' + +The word connotes bad teaching, and accordingly teaching with reference +to examinations has been supposed to be bad because it has been called +cram. The whole system of private tuition commonly called coaching has +been nick-named cram, and condemned accordingly. I can only say for +myself that I went to a private tutor whose name is familiar to everyone +in Cambridge, and found the most excellent and thorough teaching; far +be it from me to pretend that I shall prove his equal as a teacher. Whatever +fault is to be found, it is not with the teaching, but it lies in the +system. It is obviously necessary that when a vast number of new subjects +are to be mastered the most rigorous economy in the partition of the student's +time must be practised, and he is on no account to be allowed to spend +more than the requisite minimum on any one subject, even if it proves +attractive to him. The private tutor must clearly, under the old regime, +act as director of studies for his pupils strictly in accordance with examination +requirements; for place in the Tripos meant pounds, shillings, and +pence to the pupil. The system is now a good deal changed, and we may +hope that it will be possible henceforth to keep the examination less +incessantly before the student, who may thus become a student of a subject, +instead of a student for a Tripos. + +And now I think you must see the peculiar difficulties of a professor of +mathematics; his vice has been that he tried to teach a subject \textit{only}, and +\DPPageSep{066}{8} +private tutors felt, and felt justly, that they could not, in justice to their +pupils' prospects, conscientiously recommend the attendance at more than +a very small number of professorial lectures. But we are now at the beginning +of a new regime and it may be that now the professors have their +chance. But I think it depends much more on the examiners than on the +professors. If examiners can and will conduct the examinations in such +a manner that it shall ``pay'' better to master something thoroughly, than +to have a smattering of much, we shall see a change in the manner of +learning. Otherwise there will not be much change. I do not know how +it will turn out, but I do know that it is the duty of professors to take such +a chance if it exists. + +My purpose is to try my best to lecture in such a way as will impart an +interest to the subject itself and to help those who wish to learn, so that +they may reap advantage in examinations---provided the examinations are +conducted wisely. +\DPPageSep{067}{9} + + +\Chapter{Introduction to Dynamical Astronomy} +\index{Introduction to Dynamical Astronomy}% +\index{Dynamical Astronomy, introduction to}% + +\First{The} field of dynamical astronomy is a wide one and it is obvious that +it will be impossible to consider even in the most elementary manner +all branches of it; for it embraces all those effects in the heavens which may +be attributed to the effects of gravitation. In the most extended sense of +the term it may be held to include theories of gravitation itself. Whether +or not gravitation is an ultimate fact beyond which we shall never penetrate +is as yet unknown, but Newton, whose insight into physical causation was +almost preternatural, regarded it as certain that some further explanation +was ultimately attainable. At any rate from the time of Newton down to +to-day men have always been striving towards such explanation---it must be +admitted without much success. The earliest theory of the kind was that +of Lesage, promulgated some $170$~years ago. He conceived all space to be +filled with what he called ultramundane corpuscles, moving with very great +velocities in all directions. They were so minute and so sparsely distributed +that their mutual collisions were of extreme rarity, whilst they bombarded +the grosser molecules of ordinary matter. Each molecule formed a partial +shield to its neighbours, and this shielding action was held to furnish an +explanation of the mutual attraction according to the law of the inverse +square of the distance, and the product of the areas of the sections of the +two molecules. Unfortunately for this theory it is necessary to assume that +there is a loss of energy at each collision, and accordingly there must be +a perpetual creation of kinetic energy of the motion of the ultramundane +corpuscles at infinity. The theory is further complicated by the fact that +the energy lost by the corpuscle at each collision must have been communicated +to the molecule of matter, and this must occur at such a rate as to +vaporize all matter in a small fraction of a second. Lord Kelvin has, however, +pointed out that there is a way out of this fundamental difficulty, for +if at each collision the ultramundane corpuscle should suffer no loss of total +kinetic energy but only a transformation of energy of translation into energy +of internal vibration, the system becomes conservative of energy and the +eternal creation of energy becomes unnecessary. On the other hand, gravitation +will not be transmitted to infinity, but only to a limited distance. +\DPPageSep{068}{10} +I will not refer further to this conception save to say that I believe that no +man of science is disposed to accept it as affording the true road. + +It may be proved that if space were an absolute plenum of incompressible +fluid, and that if in that fluid there were points towards which the fluid +streams from all sides and disappears, those points would be urged towards +one another with a force varying inversely as the square of the distance +and directly as the product of the intensities of the two inward streams. +Such points are called sinks and the converse, namely points from whence +the fluid streams, are called sources. Now two sources also attract one +another according to the same law; on the other hand a source and a sink +repel one another. If we could conceive matter to be all sources or all sinks +we should have a mechanical theory of gravitation, but no one has as yet +suggested any means by which this can be realised. Bjerknes of Christiania +has, however, suggested a mechanical means whereby something of the kind +may be realised. Imagine an elastic ball immersed in water to swell and +contract rhythmically, then whilst it is contracting the motion of the surrounding +water is the same as that due to a sink at its centre, and whilst +it is expanding the motion is that due to a source. Hence two balls which +expand and contract in exactly the same phase will attract according to the +law of gravitation on taking the average over a period of oscillation. If, +however, the pulsations are in opposite phases the resulting force is one of +repulsion. If then all matter should resemble in some way the pulsating +balls we should have an explanation, but the absolute synchronism of the +pulsations throughout all space imports a condition which does not commend +itself to physicists. I may mention that Bjerknes has actually realised these +conclusions by experiment. Although it is somewhat outside our subject +I may say that if a ball of invariable volume should execute a small +rectilinear oscillation, its advancing half gives rise to a source and the +receding half to a sink, so that the result is what is called a doublet. Two +oscillating balls will then exercise on one another forces analogous to that +of magnetic particles, but the forces of magnetism are curiously inverted. +This quasi-magnetism of oscillating balls has also been treated experimentally +by Bjerknes. However curious and interesting these speculations +and experiments may be, I do not think they can afford a working hypothesis +of gravitation. + +A new theory of gravitation which appears to be one of extraordinary +\index{Gravitation, theory of}% +ingenuity has lately been suggested by a man of great power, viz.~Osborne +Reynolds, but I do not understand it sufficiently to do more than point +out the direction towards which he tends. He postulates a molecular ether. +I conceive that the molecules of ether are all in oscillation describing orbits +in the neighbourhood of a given place. If the region of each molecule be +replaced by a sphere those spheres may be packed in a hexagonal arrangement +\DPPageSep{069}{11} +completely filling all space. We may, however, come to places where the +symmetrical piling is interrupted, and Reynolds calls this a region of misfit. + +Then, according to this theory, matter consists of misfit, so that matter is +the deficiency of molecules of ether. Reynolds claims to show that whilst +the particular molecules which don't fit are continually changing the amount +of misfit is indestructible, and that two misfits attract one another. The +theory is also said to explain electricity. Notwithstanding that Reynolds +is not a good exponent of his own views, his great achievements in science +are such that the theory must demand the closest scrutiny. + +The newer theories of electricity with which the name of Prof.~J.~J. +Thomson is associated indicate the possibility that mass is merely an electrodynamic +phenomenon. This view will perhaps necessitate a revision of all +our accepted laws of dynamics. At any rate it will be singular if we shall +have to regard electrodynamics as the fundamental science, and subsequently +descend from it to the ordinary laws of motion. How much these notions +are in the air is shown by the fact that at a congress of astronomers, held in +1902 at Göttingen, the greater part of one day's discussion was devoted +to the astronomical results which would follow from the new theory of +electrons. + +I have perhaps said too much about the theories of gravitation, but it +should be of interest to you to learn how it teems with possibilities and how +great is the present obscurity. + +Another important subject which has an intimate relationship with +Dynamical Astronomy is that of abstract dynamics. This includes the +general principles involved in systems in motion under the action of conservative +forces and the laws which govern the stability of systems. Perhaps +the most important investigators in this field are Lagrange and Hamilton, +and in more recent times Lord Kelvin and Poincaré. + +Two leading divisions of dynamical astronomy are the planetary theory +\index{Lunar and planetary theories compared}% +\index{Planetary and lunar theories compared}% +and the theory of the motion of the moon and of other satellites. A first +approximation in all these cases is afforded by the case of simple elliptic +motion, and if we are to consider the case of comets we must include +parabolic and hyperbolic motion round a centre. Such a first approximation +is, however, insufficient for the prediction of the positions of any of the bodies +in our solar system for any great length of time, and it becomes necessary +to include the effects of the disturbing action of one or more other bodies. +The problem of disturbed revolution may be regarded as a single problem +in all its cases, but the defects of our analysis are such that in effect its +several branches become very distinct from one another. It is usual to +speak of the problem of disturbed revolution as the problem of three bodies, +for if it were possible to solve the case where there are three bodies we +\DPPageSep{070}{12} +should already have gone a long way towards the solution of that more +complex case where there are any number of bodies. + +Owing to the defects of our analysis it is at present only possible to +obtain accurate results of a general character by means of tedious expansions. +All the planets and all the satellites have their motions represented with +more or less accuracy by ellipses, but this first approximation ceases to be +satisfactory for satellites much more rapidly than is the case for planets. +The eccentricities of the ellipses and the inclinations of the orbits are in most +cases inconsiderable. It is assumed then that it is possible to effect the +requisite expansions in powers of the eccentricities and of suitable functions +of the inclinations. Further than this it is found necessary to expand in +powers of the ratios of the mean distances of the disturbed and disturbing +bodies from the centre. It is at this point that the first marked separation +of the lunar and planetary theories takes place. In the lunar theory the +distance of the sun (disturber) from the earth is very great compared with +that of the moon, and we naturally expand in this ratio in order to start +with as few terms as possible. In the planetary theory the ratio of the +distances of the disturbed and disturbing bodies---two planets---from the sun +may be a large fraction. For example, the mean distances of Venus and the +earth are approximately in the ratio~$7:10$, and in order to secure sufficient +accuracy a large number of terms is needed. In the case of the planetary +theory the expansion is delayed as long as possible. + +Again, in the lunar theory the mass of the disturbing body is very +great compared with that of the primary, a ratio on which it is evident that +the amount of perturbation greatly depends. On the other hand, in the +planetary theory the disturbing body has a very small mass compared with +that of the primary, the sun. From these facts we are led to expect that +large terms will be present in the expressions for the motion of the moon +due to the action of the sun, and that the later terms in the expansion will +rapidly decrease; and in the planetary theory we expect large numbers of +terms all of about equal magnitude and none of them very great. This +expectation is, however, largely modified by some further remarks to be made. + +You know that a dynamical system may have various modes of free +oscillation of various periods. If then a disturbing force with a period differing +but little from that of one of the modes of free oscillation acts on the +system for a long time it will generate an oscillation of large amplitude. + +A familiar instance of this is in the roll of a ship at sea. If the incidence +of the waves on the ship is such that the succession of impulses is very +nearly identical in period with the natural period of the ship, the roll becomes +large. In analysis this physical fact is associated with a division by a small +divisor on integration. +\DPPageSep{071}{13} +As an illustration of the simplest kind suppose that the equation of motion +of a system under no forces were +\[ +\frac{d^{2}x}{dt^{2}} + n^{2}x = 0. +\] +Then we know that the solution is +\[ +x = A \cos nt + B \sin nt, +\] +that is to say the free period is~$\dfrac{2 \pi}{n}$. Suppose then such a system be acted on +by a perturbing force $F\cos(n - \epsilon)t$, where $\epsilon$~is small; the equation of motion is +\[ +\frac{d^{2}x}{dt^{2}} + n^{2}x = F\cos(n - \epsilon)t, +\] +and the solution corresponding to such a disturbing force is +\[ +x = \frac{F}{-(n - \epsilon)^{2} + n^{2}} \cos(n - \epsilon)t + = \frac{F}{2n\epsilon - \epsilon^{2}} \cos(n - \epsilon)t. +\] +If $\epsilon$~is small the amplitude becomes great, and this arises, as has been said, by +a division by a small divisor. + +Now in both lunar and planetary theories the coefficients of the periodic +terms become frequently much greater than might have been expected +\textit{à ~priori}. In the lunar theory before this can happen in such a way as to +cause much trouble the coefficients have previously become so small that it +is not necessary to consider them. But suppose in the planetary theory $n, n'$ +are the mean motions of two planets round the primary. Then coefficients +will continually be having multipliers of the forms +\[ +\frac{n'}{in ± i'n'} \text{ and } \left(\frac{n'}{in ± i'n'}\right)^{2}, +\] +where $i, i'$ are small positive integers. In general the larger $i, i'$ the smaller is +the coefficient to begin with, but owing to the fact that the ratio~$n : n'$ may +very nearly approach that of two small integers a coefficient may become very +great; e.g.~$5$~Jovian years nearly equal $2$~of Saturn, while the ratio of +the mean distances is~$6 : 11$. The result is a large long inequality with a +period of $913$~years in the motions of those two planets. The periods of the +principal terms in the moon's motion are generally short, but some have +large coefficients, so that the deviation from elliptic motion is well marked. + +The general problem of three bodies is in its infancy, and as yet but little +is known as to the possibilities in the way of orbits and as to their stabilities. + +Another branch of our subject is afforded by the precession and nutation +of the earth, or any other planet, under the influence of the attractions of +disturbing bodies. This is the problem of disturbed rotation and it presents +a strong analogy with the problem of disturbed elliptic motion. When a top +\DPPageSep{072}{14} +spins with absolute steadiness we say that it is asleep. Now the earth in its +rotation may be asleep or it may not be so---there is nothing but observation +which is capable of deciding whether it is so or not. This is equally true +whether the rotation takes place under external perturbation or not. If the +earth is asleep its motion presents a perfect analogy with circular orbital +motion; if it wobbles the analogy is with elliptic motion. The analogy is +such that the magnitude of the wobble corresponds with the eccentricity of +orbit and the position of greatest departure with the longitude of pericentre. +Until the last $20$~years it has always been supposed that the earth is asleep +in its rotation, but the extreme accuracy of modern observation, when subjected +to the most searching analysis by Chandler and others, has shown +that there is actually a small wobble. This is such that the earth's axis of +rotation describes a small circle about the pole of figure. The theory of +precession indicated that this circle should be described in a period of +$305$~days, and all the earlier astronomers scrutinised the observations with +the view of detecting such an inequality. It was this preconception, apparently +well founded, which prevented the detection of the small inequality +in question. It was Chandler who first searched for an inequality of unknown +period and found a clearly marked one with a period of $428$~days. +He found also other smaller inequalities with a period of a year. This +wandering of the pole betrays itself most easily to the observer by changes +in the latitude of the place of observation. + +The leading period in the inequality of latitude is then one of $428$~days. +\index{Latitude, variation of}% +\index{Variation, the!of latitude}% +The theoretical period of $305$~days was, as I have said, apparently well +established, but after the actual period was found to be $428$~days Newcomb +pointed out that if the earth is not absolutely rigid, but slightly changes +its shape as the axis of rotation wanders, such a prolongation of period +would result. Thus these purely astronomical observations end by affording +a measure of the effective rigidity of the earth's mass. + +The theory of the earth's figure and the variation of gravity as we vary +\index{Earth's figure, theory of}% +our position on the surface or the law of variation of gravity as we descend +into mines are to be classified as branches of dynamical astronomy, although +in these cases the velocities happen to be zero. This theory is intimately +connected with that of precession, for it is from this that we conclude that +the free wobble of the perfectly rigid earth should have a period of $305$~days. +The ellipticity of the earth's figure also has an important influence on the +motion of the moon, and the determination of a certain inequality in the +moon's motion affords the means of finding the amount of ellipticity of the +earth's figure with perhaps as great an accuracy as by any other means. +Indeed in the case of Jupiter, Saturn, Mars, Uranus and Neptune the +ellipticity is most accurately determined in this way. The masses also of the +planets may be best determined by the periods of their satellites. +\DPPageSep{073}{15} + +The theory of Saturn's rings is another branch. The older and now +\index{Saturn's rings}% +obsolete views that the rings are solid or liquid gave the subject various +curious and difficult mathematical investigations. The modern view---now +well established---that they consist of an indefinite number of meteorites +which collide together from time to time presents a number of problems of +great difficulty. These were ably treated by Maxwell, and there does not +seem any immediate prospect of further extension in this direction. + +Then the theory of the tides is linked to astronomy through the fact that +it is the moon and sun which cause the tides, so that any inequality in their +motions is reflected in the ocean. + +On the fringe of our subject lies the whole theory of figures of equilibrium +of rotating liquids with the discussion of the stability of the various +possible forms and the theory of the equilibrium of gaseous planets. In this +field there is yet much to discover. + +This subject leads on immediately to theories of the origin of planetary +systems and to cosmogony. Tidal theory, on the hypothesis that the tides +are resisted by friction, leads to a whole series of investigations in speculative +astronomy whose applications to cosmogony are of great interest. + +Up to a recent date there was little evidence that gravitation held good +\index{Gravitation, theory of!universal}% +outside the solar system, but recent investigations, carried out largely by +means of the spectroscopic determinations of velocities of stars in the line of +sight, have shewn that there are many other systems, differing very widely +from our own, where the motions seem to be susceptible of perfect explanation +by the theory of gravitation. These new extensions of gravitation +outside our system are leading to many new problems of great difficulty +and we may hope in time to acquire wider views as to the possibilities of +motion in the heavens. + +This hurried sketch of our subject will show how vast it is, and I cannot +hope in these lectures to do more than touch on some of the leading topics. +\DPPageSep{074}{16} + + +\Chapter{Hill's Lunar Theory} +\index{Hill, G. W., Lunar Theory!lectures by Darwin on Lunar Theory}% +\index{Hill, G. W., Lunar Theory!characteristics of his Lunar Theory}% +\index{Lunar Theory, lecture on}% + +\Section{§ 1. }{Introduction\footnotemark.} + +\footnotetext{The references in this section are to Hill's ``Researches in the Lunar Theory'' first published + (1878) in the \textit{American Journal of Mathematics}, vol.~\Vol{I.} pp.~5--26, 129--147 and reprinted in + \textit{Collected Mathematical Works}, vol.~\Vol{I.} pp.~284--335. Hill's other paper connected with these + lectures is entitled ``On the Part of the Motion of the Lunar Perigee which is a function of the + Mean Motions of the Sun and Moon,'' published separately in 1877 by John Wilson and~Son, + Cambridge, Mass., and reprinted in \textit{Acta Mathematica}, vol.~\Vol{VIII.} pp.~1--36, 1886 and in \textit{Collected + Mathematical Works}, vol.~\Vol{I.} pp.~243--270.} + +\First{An} account of Hill's \textit{Lunar Theory} can best be prefaced by a few +quotations from Hill's original papers. These will indicate the peculiarities +which mark off his treatment from that of earlier writers and also, to some +extent, the reasons for the changes he introduced. Referring to the well-known +expressions which give, for undisturbed elliptic motion, the rectangular +coordinates as explicit functions of the time---expressions involving nothing +more complicated than Bessel's functions of integral order---Hill writes: + +``Here the law of series is manifest, and the approximation can easily be +carried as far as we wish. But the longitude and latitude, variables employed +by nearly all lunar theorists, are far from having such simple expressions; in +fact their coefficients cannot be finitely expressed in terms of Besselian +functions. And if this is true in the elliptic theory how much more likely is +a similar thing to be true when the complexity of the problem is increased +by the consideration of disturbing forces?\ldots\ There is also another advantage +in employing coordinates of the former kind (rectangular): the differential +equations are expressed in purely algebraic functions, while with the latter +(polar) circular functions immediately present themselves.'' + +In connection with the parameters to be used in the expansions Hill +argues thus: + +``Again as to parameters all those who have given literal developments, +Laplace setting the example, have used the parameter~$\m$, the ratio of the +sidereal month to the sidereal year. But a slight examination, even of the +results obtained, ought to convince anyone that this is a most unfortunate +selection in regard to convergence. Yet nothing seems to render the +parameter desirable, indeed the ratio of the synodic month to the sidereal +year would appear to be more naturally suggested as a parameter.'' +\DPPageSep{075}{17} + +When considering the order of the differential equations and the method +of integration, Hill wrote: + +``Again the method of integration by undetermined coefficients is most +likely to give us the nearest approach to the law of series; and in this +method it is as easy to integrate a differential equation of the second order +as one of the first, while the labour is increased by augmenting the number +of variables and equations. But Delaunay's method doubles the number of +variables in order that the differential equations may be all of the first order. +Hence in this disquisition I have preferred to use the equations expressed in +terms of the coordinates rather than those in terms of the elements; and, in +general, always to diminish the number of unknown quantities and equations +by augmenting the order of the latter. In this way the labour of making a +preliminary development of~$R$ in terms of the elliptic elements is avoided.'' + +We may therefore note the characteristics of Hill's method as follows: + +(1) Use of rectangular coordinates. + +(2) Expansion of series in powers of the ratio of the synodic month to +the sidereal year. + +(3) Use of differential equations of the second order which are solved by +assuming series of a definite type and equating coefficients. + +In these lectures we shall obtain only the first approximation to the +solution of Hill's differential equations. The method here followed is not +that given by Hill, although it is based on the same principles as his method. +Our work only involves simple algebra, and probably will be more easily +understood than Hill's. If followed in detail to further approximations, it +would prove rather tedious, but it leads to the results we require without too +much labour. If it is desired to follow out the method further, reference +should be made to Hill's own writings. + +\Section{§ 2. }{Differential Equations of Motion and Jacobi's Integral.} +\index{Differential Equations of Motion}% +\index{Equations of motion}% + +Let $E, M, \m'$ denote the masses or positions of the earth, moon, and sun, +and let $G$~be the centre of inertia of $E$~and~$M$. Let $x, y, z$ be the rectangular +coordinates of~$M$ with $E$~as origin, and let $x', y', z'$ be the coordinates +of~$\m'$ referred to parallel axes through~$G$. The coordinates of~$M$ relative to +the axes through~$G$ are clearly~$\dfrac{E}{E + M} x$, $\dfrac{E}{E + M} y$, $\dfrac{E}{E + M} z$; those of~$E$ are +$-\dfrac{E}{E + M} x$, $-\dfrac{E}{E + M} y$, $-\dfrac{E}{E + M} z$. The distances $EM, E\m', M\m'$\DPnote{** TN: Inconsistent overlines in original} are denoted +\DPPageSep{076}{18} +by $r, r_1, \Delta$ respectively. It is assumed that $G$~describes a Keplerian ellipse +round~$\m'$ so that $x', y', z'$ are known functions of the time. The accelerations +of~$M$ relative to~$E$ are shewn in the diagram. +\begin{figure}[hbt!] +\centering +\Input[0.75\textwidth]{p018} +\caption{Fig.~1.} +\end{figure} + +We have +\begin{gather*} +r^{2} = x^{2} + y^{2} + z^{2}, \\ +\begin{aligned} +r_{1}^{2} + &= \left(x' + \frac{Mx}{E + M}\right)^{2} + + \left(y' + \frac{My}{E + M}\right)^{2} + + \left(z' + \frac{Mz}{E + M}\right)^{2}, \\ +\Delta^{2} + &= \left(x' - \frac{Ex}{E + M}\right)^{2} + + \left(y' - \frac{Ey}{E + M}\right)^{2} + + \left(z' - \frac{Ez}{E + M}\right)^{2}. +\end{aligned} +\end{gather*} + +Hence +\begin{gather*} +\frac{\dd r}{\dd x} = \frac{x}{r}, \\ +\begin{aligned} +\frac{E + M}{M}\, \frac{\dd r_{1}}{\dd x} + &= \frac{x' + \dfrac{Mx}{E + M}}{r_{1}}, \\ +-\frac{E + M}{M}\, \frac{\dd \Delta}{\dd x} + &= \frac{x' - \dfrac{Ex}{E + M}}{\Delta}; +\end{aligned} +\end{gather*} +\begin{alignat*}{3} +\text{$\therefore$ the direction cosines of }& EM &&\text{ are }&& + \frac{\dd r}{\dd x},\ \frac{\dd r}{\dd y},\ \frac{\dd r}{\dd z},\\ +% +\Ditto[the ]\Ditto[direction ]\Ditto[cosines of ]& E\m' &&\text{ are }&& + \Neg\frac{E+M}{M}\left(\frac{\dd r_{1}}{\dd x},\ \frac{\dd r_{1}}{\dd y},\ \frac{\dd r_{1}}{\dd z}\right),\\ +\Ditto[the ]\Ditto[direction ]\Ditto[cosines of ]& M\m' &&\text{ are }&& + -\frac{E+M}{M}\left(\frac{\dd \Delta}{\dd x},\: \frac{\dd \Delta}{\dd y},\: \frac{\dd \Delta}{\dd z}\right). +\end{alignat*} + +If $X, Y, Z$ denote the components of acceleration of~$M$ relative to axes +through~$E$, +\DPPageSep{077}{19} +\[ +\left. +\begin{aligned} + X &= -\frac{E+M}{r^{2}}\, \frac{\partial r}{\partial x} + - \frac{\m'}{\Delta^{2}}\, \frac{E + M}{E} + \frac{\partial\Delta}{\partial x} - \frac{\m'}{r_{1}^{2}}\, \frac{E + M}{M}\, + \frac{\partial r_{1}}{\partial x}\\ + &= \frac{\partial F}{\partial x},\\ +& +\lintertext{where}{\phantom{= -\frac{E+M}{r^{2}} \frac{\partial r}{\partial x} + - \frac{\m'}{\Delta^{2}} \frac{E+M}{E} + \frac{\partial\Delta}{\partial x} - \frac{\m'}{r_{1}^{2}} \frac{E+M}{M} + \frac{\partial r_{1}}{\partial x}}} \\ +F &= \frac{E+M}{r} + \frac{\m'}{\Delta}\, \frac{E+M}{E} + + \frac{\m'}{r_{1}}\, \frac{E + M}{M}. \\ +&\lintertext{\indent Similarly,}{\phantom{= -\frac{E+M}{r^{2}} \frac{\partial r}{\partial x} + - \frac{\m'}{\Delta^{2}} \frac{E+M}{E} + \frac{\partial\Delta}{\partial x} - \frac{\m'}{r_{1}^{2}} \frac{E+M}{M} + \frac{\partial r_{1}}{\partial x}}} \\ +Y &= \frac{\partial F}{\partial y},\ +Z =\frac{\partial F}{\partial z}. +\end{aligned} +\right\} +\Tag{(1)} +\] + +Let $r'$~be the distance between $G$~and~$\m'$, and let $\theta$~be the angle~$\m'GM$; +then +\begin{align*} +r'^{2} &= x'^{2} + y'^{2} + z'^{2} \text{ and } + \cos\theta = \frac{xx' + yy' + zz'}{rr'}, \\ +r_{1}^{2} &= r'^{2} + \frac{2M}{E + M}\, rr' \cos\theta + \left(\frac{Mr}{E + M}\right)^{2}, \\ +\Delta^{2} &= r'^{2} - \frac{2E}{E + M}\, rr' \cos\theta + \left(\frac{Er}{E + M}\right)^{2}. +\end{align*} + +Since $r$~is very small compared with~$r'$, +\begin{gather*} +\begin{aligned} +\frac{1}{r_{1}} + &= \frac{1}{r'} \left\{1 - \frac{M}{E + M}\, \frac{r}{r'} \cos\theta + + \left(\frac{M}{E + M} · \frac{r}{r'} \right)^{2} + (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots\right\}, \\ +% +\frac{1}{\Delta} + &= \frac{1}{r'} \left\{1 + \frac{E}{E + M}\, \frac{r}{r'} \cos\theta + + \left(\frac{E}{E + M} · \frac{r}{r'} \right)^{2} + (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots\right\}. +\end{aligned} \\ +% +\therefore \frac{1}{E\Delta} + \frac{1}{Mr_{1}} + = \frac{E + M}{EM} · \frac{1}{r'} + + \frac{1}{E + M} · \frac{r^{2}}{r'^{3}} + (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots. +\end{gather*} + +Hence +\[ +F = \frac{E + M}{r} + \frac{\m'(E + M)^{2}}{EMr'} + + \frac{\m'r^{2}}{r'^{3}} (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots. +\] + +But the second term does not involve $x, y, z$, and may be dropped. +\[ +\therefore +F = \frac{E + M}{r} + + \frac{\m'r^{2}}{r'^{3}} (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2}), +\Tag{(2)} +\] +neglecting terms in~$\dfrac{r^{3}}{r'^{4}}$. + +We will now find an approximate expression for~$F$, paying attention to +the magnitude of the various terms in the actual earth-moon-sun system. +As a first rough approximation, $r'$~is a constant~$\a'$, and $G\m'$~rotates with +uniform angular velocity~$n'$. This neglects the effect on the sun of the earth +and moon not being collected at~$G$ (this effect is very small), and it neglects +the eccentricity of the solar orbit. In order that the coordinates of the sun +relative to the earth might be nearly constant, we introduce axes $x, y$ +\DPPageSep{078}{20} +rotating with angular velocity~$n'$ in the plane of the sun's orbit round the +earth; the $x$-axis being so chosen that it passes through the sun. When +required, a $z$-axis is taken perpendicular to the plane of~$x, y$. As before, let +$x, y, z$ be the coordinates of the moon; the sun's coordinates will be approximately +$\a', 0, 0$. In this approximation $r\cos\theta = x$ and +\[ +F = \frac{E + M}{r} + + \tfrac{3}{2}\, \frac{\m'}{\a'^{3}} x^{2} + - \tfrac{1}{2} \m' \frac{r^{2}}{\a'^{3}}. +\] + +This suggests the following general form for~$F$, instead of that given in +equation~\Eqref{(2)}: +\begin{align*} +F = \frac{E + M}{r} + &+ \tfrac{3}{2}\, \frac{\m'}{\a'^{3}} x^{2} + + \tfrac{3}{2} \m' \left( \frac{r^{2} \cos^{2}\theta}{r'^{3}} - \frac{x^{2}}{\a'^{3}} \right) \\ + &- \tfrac{1}{2}\, \frac{\m'}{\a'^{3}} (x^{2} + y^{2}) + - \tfrac{1}{2}\, \frac{\m'}{\a'^{3}} z^{2}\DPnote{** Why aren't previous terms combined?} + + \tfrac{1}{2} \m' r^{2} \left(\frac{1}{\a'^{3}} - \frac{1}{r'^{3}}\right). +\end{align*} + +For the sake of future developments, we now introduce a new notation. +Let $\nu$~be the moon's synodic mean motion and put $m = \dfrac{n'}{\nu} = \dfrac{n'}{n - n'}$\footnotemark. In the +\footnotetext{In the lunar theory $n'$~is supposed to be a known constant, while $n$ (or~$m$) is one of the + constants of integration the value of which is not yet determined and can only be determined + from the observations. So far $n$ (or~$m$) is quite arbitrary.}% +case of our moon, $m$~is approximately~$\frac{1}{12}$: this is a small quantity in +powers of which our expressions will be obtained. If we neglect $E$~and~$M$ +compared with~$\m'$, we have $\m' = n'^{2} \a'^{3}$, whence $\dfrac{\m'}{\a'^{3}} = n'^{2} = \nu^{2} m^{2}$. Let us also +write $E + M = \kappa \nu^{2}$, and then we get +\begin{align*}%[** TN: Re-broken] +F &+ \tfrac{1}{2} n'^{2} (x^{2} + y^{2}) \\ + &= \nu^{2} \biggl[\frac{\kappa}{r} + \tfrac{1}{2} m^{2} (3x^{2} - z^{2}) + + \tfrac{3}{2} m^{2} \left(\frac{\a'^{3}}{r'^{3}}\, r^{2} \cos^{2}\theta - x^{2}\right) + + \tfrac{1}{2} m^{2} r^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right)\biggr]. +\end{align*} + +For convenience we write +\Pagelabel{20} +\[ +\Omega + = \tfrac{3}{2} m^{2} \left(\frac{\a'^{3}}{r'^{3}} r^{2} \cos^{2}\theta - x^{2}\right) + + \tfrac{1}{2} m^{2} r^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right), +\] +and then +\[ +F + \tfrac{1}{2} n'^{2} (x^{2} + y^{2}) + = \nu^{2} \left[\frac{\kappa}{r} + \tfrac{1}{2} m^{2} (3x^{2} - z^{2}) + \Omega\right]. +\] + +The equations of motion for uniformly rotating axes\footnote + {See any standard treatise on Dynamics.} +are +\[ +\left. +\begin{alignedat}{3} +\frac{d^{2}x}{dt^{2}} &- 2n' \frac{dy}{dt} &&- n'^{2} x + &&= \frac{\dd F}{\dd x}\Add{,} \\ +\frac{d^{2}y}{dt^{2}} &- 2n' \frac{dx}{dt} &&- \DPtypo{n'}{n'^{2}} y + &&= \frac{\dd F}{\dd y}\Add{,} \\ +\frac{d^{2}z}{dt^{2}} & && + &&= \frac{\dd F}{\dd z}\Add{,} +\end{alignedat} +\right\} +\] +\DPPageSep{079}{21} +\index{Jacobi's ellipsoid!integral}% +which give +\begin{alignat*}{5} +&\frac{d^{2}x}{dt^{2}}-2n'\,\frac{dy}{dt} + &&=\frac{\dd}{\dd x}\left[F + \tfrac{1}{2}n'^{2}(x^{2} + y^{2})\right] + &&=\nu^{2}\biggl[-\frac{\kappa x}{r^{3}} &+{}&& 3m^{2}x &+ \frac{\dd \Omega}{\dd x}\biggr],\\ +% +&\frac{d^{2}y}{dt^{2}}+2n'\,\frac{dx}{dt} + &&=\frac{\dd}{\dd y}\left[F + \tfrac{1}{2}n'^{2}(x^{2} + y^{2})\right] + &&=\nu^{2}\biggl[-\frac{\kappa y}{r^{3}} &&&&+\frac{\dd \Omega}{\dd y}\biggr],\\ +% +&\frac{d^{2}z}{dt^{2}} + &&=\frac{\dd}{\dd z}\left[F + \tfrac{1}{2}n'^{2}(x^{2} + y^{2})\right] + &&=\nu^{2}\biggl[-\frac{\kappa z}{r^{3}} &-{}&& m^{2}z &+ \frac{\dd \Omega}{\dd z}\biggr]. +\end{alignat*} + +We might write $\tau = \nu t$ and on dividing the equations by~$\nu^2$ use $\tau$~henceforth +as equivalent to time; or we might choose a special unit of time such +that $\nu$~is unity. In either case our equations become +\[ +\left. +\begin{alignedat}{4} +\frac{d^{2}x}{d\tau^{2}} + & - 2m\frac{dy}{d\tau} + &&+ \frac{\kappa x}{r^{3}} + &&-& 3m^{2}x + =& \frac{\dd \Omega}{\dd x}\Add{,} \\ +% +\frac{d^{2}y}{d\tau^{2}} + & + 2m\frac{dx}{d\tau} + &&+ \frac{\kappa y}{r^{3}} && + &=& \frac{\dd \Omega}{\dd y}\Add{,} \\ +% +\frac{d^{2}z}{d\tau^{2}} & + &&+ \frac{\kappa z}{r^{3}} + &&+& m^{2}z + =& \frac{\dd \Omega}{\dd z}\Add{.} +\end{alignedat} +\right\} +\Tag{(3)} +\] + +If we multiply these equations respectively by $2\dfrac{dx}{d\tau}$, $2\dfrac{dy}{d\tau}$, $2\dfrac{dz}{d\tau}$ and add +them, we have +\begin{multline*}%[** TN: Slightly wide] +\frac{d}{d\tau}\Biggl\{ + \left(\frac{dx}{d\tau}\right)^{2} + + \left(\frac{dy}{d\tau}\right)^{2} + + \left(\frac{dz}{d\tau}\right)^{2}\Biggr\} + - 2\kappa \frac{d}{d\tau}\left(\frac{1}{r}\right) + - 3m^{2} \frac{d}{d\tau}(x^{2}) + + m^{2} \frac{d}{d\tau}(z^{2})\\ + =2\left(\frac{\dd \Omega}{\dd x}\,\frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\,\frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\,\frac{dz}{d\tau}\right). +\end{multline*} + +The whole of the left-hand side is a complete differential; the right-hand +side needs the addition of the term $2\dfrac{\dd \Omega}{\dd \tau}$. + +Let us put for brevity +\[ +V^{2} + = \left(\frac{dx}{d\tau}\right)^{2} + + \left(\frac{dy}{d\tau}\right)^{2} + + \left(\frac{dz}{d\tau}\right)^{2}. +\] + +Then +\[ +V^{2} = \frac{2\kappa}{r} + 3m^{2}x^{2} - m^{2}z^{2} + + 2\int_{0}^{\tau} \left[ + \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau} \right]d\tau + C. +\Tag{(4)} +\] + +If the earth moved round the sun with uniform angular velocity~$n'$, the +axis of~$x$ would always pass through the sun, and therefore we should have +\[ +x' = r' = \a',\quad +y' = z' = 0\Add{,} +\] +and +\[ +r\cos\theta = \frac{xx' + yy' + zz'}{r'} = x, +\] +\DPPageSep{080}{22} +giving +\[ +\frac{\a'^{3}}{r'^{3}} r^{2} \cos^{2}\theta - x^{2} = 0. +\] + +In this case $\Omega$~would vanish. It follows that $\Omega$~must involve as a factor +the eccentricity of the solar orbit. + +It is proposed as a first approximation to neglect that eccentricity, and +this being the case, our equations become +\[ +\left. +\begin{alignedat}{5} +\frac{d^{2}x}{d\tau^{2}} + &- 2m \frac{dy}{d\tau} &+ \frac{\kappa x}{r^{3}} &-& 3m^{2} x &= 0\Add{,} \\ +\frac{d^{2}y}{d\tau^{2}} + &+ 2m \frac{dx}{d\tau} &+ \frac{\kappa y}{r^{3}} && &= 0\Add{,} \\ +\frac{d^{2}z}{d\tau^{2}} + & &+ \frac{\kappa z}{r^{3}} &+& m^{2} z &= 0\Add{.} +\end{alignedat} +\right\} +\Tag{(5)} +\] + +Of these equations one integral is known, viz.\ Jacobi's integral, +\[ +V^{2} = 2\frac{\kappa}{r} + 3m^{2} x^{2} - m^{2} z^{2} + C. +\] + +\Section{§ 3. }{The Variational Curve.} +\index{Variational curve, defined}% + +In ordinary theories the position of a satellite is determined by the +departure from a simple ellipse---fixed or moving. The moving ellipse is +preferred to the fixed one, because it is found that the departures of the +actual body from the moving ellipse are almost of a periodic nature. But +the moving ellipse is not the solution of any of the equations of motion +occurring in the theory. Instead of referring the true orbit to an ellipse, +Hill introduced as the orbit of reference, or intermediate orbit, a curve +suggested by his differential equations, called the ``variational curve.'' + +We have already neglected the eccentricity of the solar orbit, and will +now go one step further and neglect the inclination of the lunar orbit to the +ecliptic, so that $z$~disappears. If the path of a body whose motion satisfies +\[ +\left. +\begin{alignedat}{2} +\frac{d^{2}x}{d\tau^{2}} - 2m \frac{dy}{d\tau} + &+ \left(\frac{\kappa}{r^{3}} - 3m^{2} \right) x &&= 0\\ +\frac{d^{2}y}{d\tau^{2}} + 2m \frac{dx}{d\tau} + &+ \frac{\kappa y}{r^{3}} &&= 0 +\end{alignedat} +\right\} +\Tag{(6)} +\] +intersects the $x$-axis at right angles, the circumstances of the motion before +and after intersection are identical, but in reverse order. Thus, if time +be counted from the intersection, $x = f(\tau^{2})$, $y = \tau f(\tau^{2})$; for if in the differential +equations the signs of $y$~and~$\tau$ are reversed, but $x$~left unchanged, +the equations are unchanged. + +A similar result holds if the path intersects~$y$ at right angles, for if +$x$~and~$\tau$ have signs changed, but $y$~is unaltered, the equations are unaltered. +\DPPageSep{081}{23} + +Now it is evident that the body may start from a given point on the +$x$-axis, and at right angles to it, with different velocities, and that within +certain limits it may reach the axis of~$y$ and cross it at correspondingly +different angles. If the right angle lie between some of these, we judge +from the principle of continuity that there is some intermediate velocity with +which the body would arrive at and cross the $y$-axis at right angles. + +If the body move from one axis to the other, crossing both at right +\index{Variational curve, defined!determined}% +angles, it is plain that the orbit is a closed curve symmetrical to both axes. +Thus is obtained a particular solution of the differential equations. This +solution is the ``variational curve.'' While the general integrals involve four +arbitrary constants, the variational curve has but two, which may be taken to +be the distance from the origin at the $x$~crossing and the time of crossing. + +For the sake of brevity, we may measure time from the instant of +crossing~$x$. + +Then since $x$~is an even function of~$\tau$ and $y$~an odd one, both of +period~$2\pi$, it must be possible to expand $x$~and~$y$ by Fourier Series---thus +\begin{alignat*}{4} +x &= A_{0} \cos \tau &&+ A_{1} \cos 3\tau &&+ A_{2} \cos 5\tau &&+ \ldots\ldots, \\ +y &= B_{0} \sin \tau &&+ B_{1} \sin 3\tau &&+ B_{2} \sin 5\tau &&+ \ldots\ldots. +\end{alignat*} + +When $\tau$~is a multiple of~$\pi$, $y = 0$; and when it is an odd multiple +of~$\dfrac{\pi}{2}$, $x = 0$: also in the first case $\dfrac{dx}{d\tau} = 0$ and in the second $\dfrac{dy}{d\tau} = 0$. Thus +these conditions give us the kind of curve we want. It will be noted that +there are no terms with even multiples of~$\tau$; such terms have to be omitted +if $x, \dfrac{dx}{d\tau}$ are to vanish at $\tau = \pi/2$,~etc.\DPnote{** Slant fraction} + +We do not propose to follow Hill throughout the arduous analysis by +which he determines the nature of this curve with the highest degree of +accuracy, but will obtain only the first rough approximation to its form---thereby +merely illustrating the principles involved. + +Accordingly we shall neglect all terms higher than those in~$3\tau$. It is +also convenient to change the constants into another form. Thus we write +\begin{align*} +A_{0} &= a_{0} + a_{-1},\quad A_{1} = a_{1}, \\ +B_{0} &= a_{0} - a_{-1},\quad B_{1} = a_{1}. +\end{align*} +We have one constant less than before, but it will be seen that this is +sufficient, for in fact $A_{1}$~and~$B_{1}$ only differ by terms of an order which we +are going to neglect. We assume $a_{1}, a_{-1}$ to be small quantities. + +Hence +\begin{align*} +x &= (a_{0} + a_{-1}) \cos\tau + a_{1} \cos 3\tau, \\ +y &= (a_{0} - a_{-1}) \sin\tau + a_{1} \sin 3\tau. +\end{align*} +\DPPageSep{082}{24} + +Since +\begin{alignat*}{4} +\cos 3\tau &= && 4\cos^{3}\tau - 3\cos\tau &&= &&\cos\tau(1 - 4\sin^{2}\tau), \\ +\sin 3\tau &= -&& 4\sin^{3}\tau + 3\sin\tau &&= -&&\sin\tau(1 - 4\cos^{2}\tau), +\end{alignat*} +we have +\[ +\left. +\begin{aligned} +x = a_{0} \cos\tau &\left[1 + \frac{a_{1} + a_{-1}}{a_{0}} + - \frac{4a_{1}}{a_{0}} \sin^{2}\tau\right]\Add{,} \\ +y = a_{0} \sin\tau &\left[1 - \frac{a_{1} + a_{-1}}{a_{0}} + + \frac{4a_{1}}{a_{0}} \cos^{2}\tau\right]\Add{.} +\end{aligned} +\right\} +\] + +Neglecting powers of $a_{1}, a_{-1}$ higher than the first, we deduce +\begin{align*} +r^{2} &= a_{0}^{2} \left[1 + 2\frac{a_{1} + a_{-1}}{a_{0}} \cos2\tau\right], +\Allowbreak +\frac{1}{r^{3}} + &= \frac{1}{a_{0}^{3}} \left[1 - 3\frac{a_{1} + a_{-1}}{a_{0}} \cos2\tau\right] \\ + &= \frac{1}{a_{0}^{3}} \left[1 - 3\frac{a_{1} + a_{-1}}{a_{0}} + 6\frac{a_{1} + a_{-1}}{a_{0}} \sin^{2}\tau\right] \\ + &= \frac{1}{a_{0}^{3}} \left[1 + 3\frac{a_{1} + a_{-1}}{a_{0}} - 6\frac{a_{1} + a_{-1}}{a_{0}} \cos^{2}\tau\right]; +\Allowbreak +\frac{\kappa x}{r^{3}} + &= \frac{\kappa}{a_{0}^{2}} \cos\tau + \left[1 - \frac{2a_{1} + 2a_{-1}}{a_{0}} + + \frac{2a_{1} + 6a_{-1}}{a_{0}} \sin^{2}\tau\right], \\ +\frac{\kappa y}{r^{3}} + &= \frac{\kappa}{a_{0}^{2}} \sin\tau + \left[1 + \frac{2a_{1} + 2a_{-1}}{a_{0}} + - \frac{2a_{1} + 6a_{-1}}{a_{0}} \cos^{2}\tau\right], +\Allowbreak +%[** TN: Added breaks at second equalities] +\frac{d^{2} x}{d\tau^{2}} + &= -\left[\left(a_{0} + a_{-1}\right) \cos\tau + 9a_{1} \cos3\tau\right] \\ + &= -\cos\tau \left[a_{0} + 9a_{1} + a_{-1} - 36a_{1} \sin^{2}\tau\right], +\Allowbreak +\frac{d^{2} y}{d\tau^{2}} + &= -\left[\left(a_{0} - a_{-1}\right) \sin\tau + 9a_{1} \sin3\tau\right] \\ + &= -\sin\tau \left[a_{0} - 9a_{1} + a_{-1} - 36a_{1} \cos^{2}\tau\right]. +\end{align*} + +With the required accuracy +\[ +-2m \frac{dy}{d\tau} = -2m a_{0}\cos\tau,\ + 2m \frac{dx}{d\tau} = -2m a_{0} \sin\tau, \text{ and } + 3m^{2} x = 3m^{2} a_{0} \cos\tau. +\] + +Substituting these results in the differential equations,~\Eqref{(6)}, we get +\begin{multline*} +a_{0}\cos\tau + \biggl[-1 - \frac{9a_{1} + a_{-1}}{a_{0}} + \frac{36a_{1}}{a_{0}}\sin^{2}\tau - 2m \\ + + \frac{\kappa}{a_{0}^{3}} \left(1 - \frac{2a_{1} + 2a_{-1}}{a_{0}} + + \frac{2a_{1} + 6a_{-1}}{a_{0}} \sin^{2}\tau\right) - 3m^{2}\biggr] = 0, +\end{multline*} +\begin{multline*} +a_{0}\sin\tau + \biggl[-1 + \frac{9a_{1} + a_{-1}}{a_{0}} - \frac{36a_{1}}{a_{0}}\cos^{2}\tau - 2m \\ + + \frac{\kappa}{a_{0}^{3}} \left(1 + \frac{2a_{1} + 2a_{-1}}{a_{0}} + - \frac{2a_{1} + 6a_{-1}}{a_{0}} \cos^{2}\tau\right)\biggr] = 0. +\end{multline*} +\DPPageSep{083}{25} + +Equating to zero the coefficients of $\cos\tau$, $\cos\tau \sin^{2}\tau$, $\sin\tau$, $\sin\tau \cos^{2}\tau$, +we get +\[ +\left. +\begin{gathered} +\begin{alignedat}{2} +&-1 - \frac{9a_{1} + a_{-1}}{a_{0}} - 2m + + \frac{\kappa}{a_{0}^{3}} \left(1 - \frac{2a_{1} + 2a_{-1}}{a_{0}}\right) + & -3m^{2} &= 0\Add{,} \\ +&-1 + \frac{9a_{1} + a_{-1}}{a_{0}} - 2m + + \frac{\kappa}{a_{0}^{3}} \left(1 + \frac{2a_{1} + 2a_{-1}}{a_{0}}\right) + &&= 0\Add{,} +\end{alignedat} +\\ +% +\frac{36a_{1}}{a_{0}} + + \frac{\kappa}{a_{0}^{2}} \left(\frac{2a_{1} + 6a_{-1}}{a_{0}}\right) = 0\Add{.} +\end{gathered} +\right\} +\Tag{(7)} +\] + +As there are only three equations for the determination of $\dfrac{\kappa}{a_{0}^{3}}$, $\dfrac{a_{1}}{a_{0}}$, $\dfrac{a_{-1}}{a_{0}}$ +our assumption that $A_{1} = B_{1} = a_{1}$ is justified to the order of small quantities +considered. + +Half the sum and difference of the first two give +\begin{gather*} +-1 - 2m - \tfrac{3}{2} m^{2} + \frac{\kappa}{a_{0}^{3}} = 0, \\ +\frac{9a_{1} + a_{-1}}{a_{0}} + \frac{2\kappa}{a_{0}^{3}}\, \frac{a_{1} + a_{-1}}{a_{0}} + + \tfrac{3}{2} m^{2} = 0. +\end{gather*} + +Therefore +\begin{align*} +&\frac{\kappa}{a_{0}^{3}} = 1 + 2m + \tfrac{3}{2} m^{2}, \\ +&\frac{11a_{1}}{a_{0}} + \frac{3a_{-1}}{a_{0}} = -\tfrac{3}{2}m^{2}, + \text{ to our order of accuracy, viz.~$m^{2}$}; \\ +\intertext{also} +&\frac{19a_{1}}{a_{0}} + \frac{3a_{-1}}{a_{0}} = 0, + \text{ from the third equation;} +\end{align*} +\begin{gather*} +\therefore \frac{8a_{1}}{a_{0}} = \tfrac{3}{2} m^{2}, \\ +\left. +\begin{aligned} +\frac{a_{1}}{a_{0}} + &= \tfrac{3}{16} m^{2},\quad \frac{a_{-1}}{a_{0}} + = -\tfrac{19}{16} m^{2}\Add{,} \\ +\frac{\kappa}{a_{0}^{3}} + &= 1 + 2m + \tfrac{3}{2} m^{2}\Add{.} +\end{aligned} +\right\} +\Tag{(8)} +\end{gather*} + +Hence +\begin{align*} +x &= a_{0}\left[(1 - \tfrac{19}{16} m^{2}) \cos\tau + + \tfrac{3}{16} m^{2} \cos 3\tau\right], \\ +y &= a_{0}\left[(1 + \tfrac{19}{16} m^{2}) \sin\tau + + \tfrac{3}{16} m^{2} \sin 3\tau\right], +\end{align*} +or perhaps more conveniently for future work +\[ +\left. +\begin{aligned} +x &= a_{0}\cos\tau + \left[1 - m^{2} - \tfrac{3}{4} m^{2} \sin^{2}\tau \right]\Add{,} \\ +y &= a_{0}\sin\tau + \left[1 + m^{2} + \tfrac{3}{4} m^{2} \cos^{2}\tau \right]\Add{.} +\end{aligned} +\right\} +\Tag{(9)} +\] + +It will be seen that those are the equations to an oval curve, the semi-axes +of which are $a_{0}(1 - m^{2})$, $a_{0}(1 + m^{2})$ along and perpendicular to the line +joining the earth and sun. If $r, \theta$~be the polar coordinates of a point on the +curve, +\begin{align*} +r^{2} &= a_{0}^{2}[1 - 2m^{2} \cos 2\tau], \\ +\intertext{giving} +r &= a_{0}[1 - m^{2} \cos 2\tau]. +\Tag{(10)} +\end{align*} +\DPPageSep{084}{26} +Also +\begin{gather*} +\begin{aligned} +\tan\theta &= \frac{y}{x} = \tan\tau \bigl[1 + 2m^{2} + \tfrac{3}{4} m^{2}\bigr] \\ + &= \bigl(1 + \tfrac{11}{4}\bigr) \tan\tau. +\end{aligned} \\ +\therefore \tan(\theta - \tau) + = \frac{\tan\tau}{1 + \tan^{2}\tau} · \tfrac{11}{4} m^{2} + = \tfrac{11}{8} \sin 2\tau, +\end{gather*} +giving +\[ +\theta = \tau + \tfrac{11}{8} m^{2} \sin 2\tau. +\Tag{(11)} +\] + +If $\a$~be the mean distance corresponding to a mean motion~$n$ in an +undisturbed orbit, Kepler's third law gives +\[ +n^{2}\a^{3} = E + M = \kappa \nu^{2}. +\Tag{(12)} +\] + +But +\[ +\frac{n}{\nu} = \frac{n - n' + n'}{n - n'} = 1 + m. +\] +Hence +\begin{gather*} +(1 + m)^{2} \a^{3} = \kappa = a_{0}^{3} (1 + 2m + \tfrac{3}{2} m^{2}), \\ +\frac{a_{0}^{3}}{\a^{3}} = \frac{1 + 2m + m^{2}}{1 + 2m + m^{2} + \tfrac{1}{2} m^{2}}, \\ +\intertext{and} +a_{0} = \a(1 - \tfrac{1}{6} m^{2}). +\Tag{(13)} +\end{gather*} + +This is a relation between $a_{0}$ and the undisturbed mean distance. + + +\Section{§ 4. }{Differential Equations \texorpdfstring{\protect\\}{} +for Small Displacements from the Variational Curve.} +\index{Small displacements from variational curve}% +\index{Variational curve, defined!small displacements from}% + +If the solar perturbations were to vanish, $m$~would be zero and we should +have $x = a_{0}\cos\tau$, $y = a_{0}\sin\tau$ so that the orbit would be a circle. We may +therefore consider the orbit already found as a circular orbit distorted by solar +influence. [We have indeed put $\Omega = 0$, but the terms neglected are small +and need not be considered at present.] As the circular orbit is only a +special solution of the problem of two bodies, we should not expect the +variational curve to give the actual motion of the moon. In fact it is known +that the moon moves rather in an ellipse of eccentricity~$\frac{1}{20}$ than in a circle or +variational curve. The latter therefore will only serve as an approximation +to the real orbit in the same way as a circle serves as an approximation to an +ellipse. An ellipse of small eccentricity can be obtained by ``free oscillations'' +about a circle, and what we proceed to do is to determine free oscillations +about the variational curve. We thus introduce two new arbitrary constants---determining +the amplitude and phase of the oscillations---and so get the +general solution of our differential equations~\Eqref{(6)}. The procedure is exactly +similar to that used in dynamics for the discussion of small oscillations about +a steady state, i.e.,~the moon is initially supposed to lie near the variational +curve, and its subsequent motion is determined relatively to this curve. At +first only first powers of the small quantities will be used---an approximation +\DPPageSep{085}{27} +which corresponds to the first powers of the eccentricity in the elliptic theory. +If required, further approximations can be made. + +Suppose then that $x, y$ are the coordinates of a point on the variational +curve which we have found to satisfy the differential equations of motion and +that $x + \delta x$, $y + \delta y$ are the coordinates of the moon in her actual orbit, then +since $x, y$~satisfy the equations it is clear that the equations to be satisfied +by~$\delta x, \delta y$ are +\[ +\left. +\begin{alignedat}{2} +&\frac{d^{2}}{d\tau^{2}}\, \delta x - 2m \frac{d}{d\tau}\, \delta y + + \kappa \delta \left(\frac{x}{r^{3}}\right) &- 3m^{2}\, \delta x &= 0\Add{,} \\ +% +&\frac{d^{2}}{d\tau^{2}}\, \delta y + 2m \frac{d}{d\tau}\, \delta x + + \kappa \delta \left(\frac{y}{r^{3}}\right) &&= 0\Add{.} +\end{alignedat} +\right\} +\Tag{(14)} +\] + +\begin{wrapfigure}[14]{r}{1.75in} + \centering + \Input[1.75in]{p027} + \caption{Fig.~2.} +\end{wrapfigure} +Now it is not convenient to proceed immediately +from these equations as you may see by +considering how you would proceed if the orbit of +reference were a simple undisturbed circle. The +obvious course is to replace~$\delta x, \delta y$ by normal +and tangential displacements~$\delta p, \delta s$. + +Suppose then that $\phi$~denotes the inclination +of the outward normal of the variational curve to +the $x$-axis. Then we have +\[ +\left. +\begin{aligned} +\delta x &= \delta p \cos\phi - \delta s \sin\phi\Add{,} \\ +\delta y &= \delta p \sin\phi + \delta s \cos\phi\Add{.} +\end{aligned} +\right\} +\Tag{(15)} +\] + +Multiply the first differential equation~\Eqref{(14)} by~$\cos\phi$ and the second by~$\sin\phi$ +and add; and again multiply the first by~$\sin\phi$ and the second by~$\cos\phi$ +and subtract. We have +\[ +\left. +\begin{aligned} +\cos\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}} + &+ \sin\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} + - 2m \left[\cos\phi\, \frac{d\, \delta y}{d\tau} + - \sin\phi\, \frac{d\, \delta x}{d\tau}\right] \\ +% + &+ \kappa \cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa \sin\phi\, \delta\left(\frac{y}{r^{3}}\right) + - 3m^{2}\cos\phi\, \delta x = 0, \\ +% +-\sin\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}} + &+ \cos\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} + + 2m \left[\sin\phi\, \frac{d\, \delta y}{d\tau} + + \cos\phi\, \frac{d\, \delta x}{d\tau}\right] \\ +% + &- \kappa \sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + + 3m^{2}\sin\phi\, \delta x = 0. +\end{aligned} +\right\} +\Tag{(16)} +\] + +Now we have from~\Eqref{(15)} +\[ +\delta p = \delta x \cos\phi + \delta y \sin\phi,\quad +\delta s = -\delta x \sin\phi + \delta y \cos\phi. +\] + +Therefore +\begin{align*} +\frac{d\, \delta p}{d\tau} + &= \Neg\cos\phi\, \frac{d\, \delta x}{d\tau} + + \sin\phi\, \frac{d\, \delta y}{d\tau} + + (-\delta x \sin\phi + \delta y \cos\phi)\, \frac{d\phi}{d\tau}, \\ + % +\frac{d\, \delta s}{d\tau} + &= -\sin\phi\, \frac{d\, \delta x}{d\tau} + + \cos\phi\, \frac{d\, \delta y}{d\tau} + - (\Neg\delta x \cos\phi + \delta y \sin\phi)\, \frac{d\phi}{d\tau}. +\end{align*} +\DPPageSep{086}{28} + +Hence the two expressions which occur in the second group of terms of~\Eqref{(16)} +are +\begin{align*} +\cos\phi\, \frac{d\, \delta y}{d\tau} - \sin\phi\, \frac{d\, \delta x}{d\tau} + &= \frac{d\, \delta s}{d\tau} + \delta p\, \frac{d\phi}{d\tau}, \\ +% +\sin\phi\, \frac{d\, \delta y}{d\tau} + \cos\phi\, \frac{d\, \delta x}{d\tau} + &= \frac{d\, \delta p}{d\tau} - \delta s\, \frac{d\phi}{d\tau}. +\end{align*} + +When we differentiate these again, we obtain the first group of terms in~\Eqref{(16)}. +Inverting the order of the equations we have +\begin{align*} +\cos\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}} + &+ \sin\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} \\ + &= \frac{d^{2}\, \delta p}{d\tau^{2}} + - \frac{d\, \delta s}{d\tau}\, \frac{d\phi}{d\tau} + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} + - \left(\cos\phi\, \frac{d\, \delta y}{d\tau} + - \sin\phi\, \frac{d\, \delta x}{d\tau}\right) \frac{d\phi}{d\tau} \\ + &= \frac{d^{2}\, \delta p}{d\tau^{2}} + - 2\frac{d\, \delta s}{d\tau}\, \frac{d\phi}{d\tau} + - \delta p\, \left(\frac{d\phi}{d\tau}\right)^{2} + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}}, +\Allowbreak +-\sin\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}} + &+ \cos\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} \\ + &= \frac{d^{2}\, \delta s}{d\tau^{2}} + + \frac{d\, \delta p}{d\tau}\, \frac{d\phi}{d\tau} + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} + + \left(\sin\phi\, \frac{d\, \delta y}{d\tau} + + \cos\phi\, \frac{d\, \delta x}{d\tau}\right) \frac{d\phi}{d\tau} \\ + &= \frac{d^{2}\, \delta s}{d\tau^{2}} + + 2\frac{d\, \delta p}{d\tau}\, \frac{d\phi}{d\tau} + - \delta s\, \left(\frac{d\phi}{d\tau}\right)^{2} + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}}. +\end{align*} + +Substituting in~\Eqref{(16)}, we have as our equations +\[ +\left. +\begin{aligned} +\frac{d^{2}\, \delta p}{d\tau^{2}} + &- \delta p \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m \frac{d\phi}{d\tau}\right] + - 2\frac{d\, \delta s}{d\tau} \left(\frac{d\phi}{d\tau} + m\right) + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} \\ +&\qquad + + \kappa\cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa\sin\phi\, \delta\left(\frac{y}{r^{3}}\right) + - 3m^{2}\cos\phi\, \delta x = 0\Add{,} \\ +% +\frac{d^{2}\, \delta s}{d\tau^{2}} + &- \delta s \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m \frac{d\phi}{d\tau}\right] + + 2\frac{d\, \delta p}{d\tau} \left(\frac{d\phi}{d\tau} + m\right) + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} \\ +&\qquad + - \kappa\sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa\cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + + 3m^{2}\sin\phi\, \delta x = 0\Add{.} +\end{aligned} +\right\} +\Tag{(17)} +\] + +Variation of the Jacobian integral +\[ +V^{2} + = \left(\frac{dx}{d\tau}\right)^{2} + + \left(\frac{dy}{d\tau}\right)^{2} + = \frac{2\kappa}{r} + 3m^{2}x^{2} + C +\] +gives +\[ +\frac{dx}{d\tau}\, \frac{d\, \delta x}{d\tau} + +\frac{dy}{d\tau}\, \frac{d\, \delta y}{d\tau} + = -\frac{\kappa}{r^{3}}\, \delta r + + 3m^{2}x\, \delta x.\footnotemark%[** TN: Moved mark after period] +\] +\footnotetext{We could introduce a term~$\delta C$, but the variation of the orbit which we are introducing + is one for which $C$~is unaltered.} + +Now +\[ +\frac{dx}{d\tau} = -V\sin\phi,\quad +\frac{dy}{d\tau} = V\cos\phi, +\] +\DPPageSep{087}{29} +and +\begin{alignat*}{4} + \frac{d\, \delta x}{d\tau} + &= \cos\phi\, \frac{d\, \delta p}{d\tau} + &&- \delta s \cos\phi\, \frac{d\phi}{d\tau} + &&- \sin\phi\, \frac{d\, \delta s}{d\tau} + &&- \sin\phi\, \delta p\, \frac{d\phi}{d\tau}, \\ +% + \frac{d\, \delta y}{d\tau} + &= \sin\phi\, \frac{d\, \delta p}{d\tau} + &&- \delta s \sin\phi\, \frac{d\phi}{d\tau} + &&+ \cos\phi \frac{d\, \delta s}{d\tau} + &&+ \cos\phi\, \delta p\, \frac{d\phi}{d\tau}. +\end{alignat*} + +Hence +\[ +\frac{dx}{d\tau}\, \frac{d\, \delta x}{d\tau} + +\frac{dy}{d\tau}\, \frac{d\, \delta y}{d\tau} + = V \left(\frac{d\, \delta s}{d\tau} + + \delta p\, \frac{d\phi}{d\tau}\right). +\] + +Also +\begin{align*} +-\frac{\kappa\, \delta r}{r^{2}} + &= -\frac{\kappa}{r^{3}}(x\, \delta x + y\, \delta y) \\ %[** TN: Added break] + &= -\frac{\kappa x}{r^{3}}(\delta p \cos\phi - \delta s \sin\phi) + -\frac{\kappa y}{r^{3}}(\delta p \sin\phi + \delta s \cos\phi) \\ + &= -\frac{\kappa}{r^{3}} \bigl[\delta p\, (x \cos\phi + y \sin\phi) + + \delta s\, (-x \sin\phi + y \cos\phi)\bigr]. +\end{align*} + +Thus, retaining the term $3m^{2} x\, \delta x$ in its original form, the varied Jacobian +integral becomes +\Pagelabel{29} +\begin{multline*} +V\left(\frac{d\, \delta s}{d\tau} + \delta p\, \frac{d\phi}{d\tau}\right) \\ + = -\frac{\kappa}{r^{3}} \bigl[\delta p\, (x \cos\phi + y \sin\phi) + + \delta s\, (-x \sin\phi + y \cos\phi)\bigr] + 3m^{2} x\, \delta x. +\Tag{(18)} +\end{multline*} + +Before we can solve the differential equations~\Eqref{(17)} for $\delta p, \delta s$ we require to +express all the other variables occurring in them, in terms of~$\tau$ by means of +the equations obtained in~\SecRef{3}. + + +\Section{§ 5. }{Transformation of the equations in \SecRef{4}.} + +We desire to transform the differential equations~\Eqref{(17)} so that the only +variables involved will be $\delta p, \delta s, \tau$. We shall then be in a position to solve +for $\delta p, \delta s$ in terms of~$\tau$. + +We have +\[ +r\, \delta r = x\, \delta x + y\, \delta y + = ( x \cos\phi + y \sin\phi)\, \delta p + + (-x \sin\phi + y \cos\phi)\, \delta s. +\] + +Hence +\begin{align*} +\cos\phi\, &\delta\left(\frac{x}{r^{3}}\right) + +\sin\phi\, \delta\left(\frac{y}{r^{3}}\right) \\ + &= \frac{1}{r^{3}} (\delta x \cos\phi + \delta y \sin\phi) + - \frac{3}{r^{5}} (x \cos\phi + y \sin\phi) r\, \delta r +\Allowbreak + &= \frac{\delta p}{r^{3}} - \frac{3}{r^{5}} + \biggl[(x^{2} \cos^{2} \phi + y^{2} \sin^{2} \phi + + 2xy \sin\phi \cos\phi)\, \delta p \\ + &\qquad \rlap{$\displaystyle + + (- x^{2} \sin\phi \cos\phi + + xy \cos^{2}\phi + - xy \sin^{2}\phi + + y^{2} \sin\phi \cos\phi)\, \delta s\biggr]$} +\Allowbreak + &= \frac{\delta p}{r^{3}} - \frac{3}{r^{5}} \biggl[ + \bigl\{\tfrac{1}{2}(x^{2} + y^{2}) + + \tfrac{1}{2}(x^{2} - y^{2}) \cos 2\phi + + xy \sin 2\phi\bigr\}\, \delta p \\ + &\qquad\qquad\qquad + + \bigl\{-\tfrac{1}{2}(x^{2} - y^{2}) \sin 2\phi + + xy \cos 2\phi\bigr\}\, \delta s \biggr] +\Allowbreak + &= \frac{\delta p}{r^{3}} \left[ + -\tfrac{1}{2} - \tfrac{3}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \cos 2\phi + - \frac{3xy}{r^{2}} \sin 2\phi + \right] \\ + &\qquad\qquad\qquad + - \frac{3\delta s}{r^{3}} \left[ + \frac{xy}{r^{2}} \cos 2\phi + - \tfrac{1}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \sin 2\phi + \right], +\Tag{(19)} +\Allowbreak +\DPPageSep{088}{30} +-\sin\phi\, &\delta\left(\frac{x}{r^{3}}\right) + + \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) \\ + &= \frac{1}{r^{3}} (-\delta x \sin\phi + \delta y \cos\phi) + - \frac{3}{r^{3}} (-x \sin\phi + y \cos\phi)r\, \delta r +\Allowbreak + &= \frac{\delta s}{r^{3}} - \frac{3}{r^{5}} \biggl[ + (-x^{2} \sin\phi \cos\phi + - xy \sin^{2}\phi + xy \cos^{2}\phi + + y^{2} \sin\phi \cos\phi)\, \rlap{$\delta p$} \\ + &\qquad\qquad\qquad + + (x^{2} \sin^{2}\phi + y^{2} \cos^{2}\phi + - 2xy \sin\phi \cos\phi)\, \delta s\biggr] +\Allowbreak + &= \frac{\delta s}{r^{3}} + - \frac{3}{r^{5}} \biggl[ + \bigl\{-\tfrac{1}{2}(x^{2} - y^{2}) \sin 2\phi + xy\cos 2\phi\bigr\}\, \delta p \\ + &\qquad\qquad\qquad + + \bigl\{\tfrac{1}{2}(x^{2} + y^{2}) - \tfrac{1}{2}(x^{2} - y^{2})\cos 2\phi + - xy\sin 2\phi\bigr\}\, \delta s \biggr] +\Allowbreak + &= -\frac{3\, \delta p}{r^{3}} \biggl[\frac{xy}{r^{2}}\cos 2\phi + - \tfrac{1}{2} \frac{x^{2} - y^{2}}{r^{2}} \sin 2\phi\biggr] \\ + &\qquad\qquad\qquad + + \frac{\delta s}{r^{3}} \biggl[ + -\tfrac{1}{2} + \tfrac{3}{2} \frac{x^{2} - y^{2}}{r^{2}} \cos 2\phi + + \frac{3xy}{r^{2}} \sin 2\phi \biggr]. +\Tag{(20)} +\end{align*} + +We shall consider the terms $3m^{2}\, \delta x \begin{array}{@{\,}c@{\,}}\cos\\ \sin\end{array} \phi$ later (\Pageref{33}). + +The next step is to substitute throughout the differential equations~\Eqref{(17)} +the values of~$x, y$ and~$\phi$ which correspond to the undisturbed orbit. For +simplicity in writing we drop the linear factor~$a_{0}$. It can be easily +introduced when required. + +We have already found, in~\Eqref{(9)}, +\begin{alignat*}{2} +x &= \cos\tau (1 - \tfrac{19}{16} m^{2}) + \tfrac{3}{16} m^{2}\cos 3\tau + &&= \cos\tau (1 - m^{2} - \tfrac{3}{4} m^{2}\sin^{2}\tau), \\ +x &= \sin\tau (1 + \tfrac{19}{16} m^{2}) + \tfrac{3}{16} m^{2}\sin 3\tau + &&= \sin\tau (1 + m^{2} + \tfrac{3}{4} m^{2}\cos^{2}\tau). +\end{alignat*} + +Then +\begin{align*} +\frac{dx}{d\tau} + &= -\sin\tau(1 - \tfrac{7}{4} m^{2} + \tfrac{9}{4} m^{2}\cos^{2} \tau) + = -\sin\tau(1 + \tfrac{1}{2} m^{2} - \tfrac{9}{4} m^{2}\sin^{2} \tau), \\ +% +\frac{dy}{d\tau} + &= \Neg\cos\tau(1 + \tfrac{7}{4} m^{2} - \tfrac{9}{4} m^{2}\sin^{2} \tau) + = \Neg\cos\tau(1 - \tfrac{1}{2} m^{2} + \tfrac{9}{4} m^{2}\cos^{2} \tau). +\end{align*} + +Whence +\begin{align*} +V^{2} + &= \left(\frac{dx}{d\tau}\right)^{2} + \left(\frac{dy}{d\tau}\right)^{2} \\ +%[** TN: Added break] + &= \sin^{2}\tau (1 + m^{2} - \tfrac{9}{2} m^{2}\sin^{2}\tau) + + \cos^{2}\tau (1 - m^{2} + \tfrac{9}{2} m^{2}\cos^{2}\tau) \\ +% + &= 1 - m^{2} \cos 2\tau + \tfrac{9}{2} m^{2}\cos 2\tau + = 1 + \tfrac{7}{2} m^{2}\cos 2\tau \\ +% + &= 1 + \tfrac{7}{2} m^{2} - 7 m^{2}\sin^{2}\tau + = 1 - \tfrac{7}{2} m^{2} + 7 m^{2}\cos^{2}\tau. +\end{align*} + +Therefore +\[ +\frac{1}{V} + = 1 + \tfrac{7}{4} m^{2} - \tfrac{7}{2} m^{2}\cos^{2}\tau + = 1 - \tfrac{7}{4} m^{2} + \tfrac{7}{2} m^{2}\sin^{2}\tau + = 1 - \tfrac{7}{4} m^{2} \cos 2\tau. +\] +\DPPageSep{089}{31} + +Now +\[ +\sin\phi = -\frac{1}{V}\, \frac{dx}{d\tau},\quad +\cos\phi = \frac{1}{V}\, \frac{dy}{d\tau}. +\] + +Therefore +\begin{align*} +\sin\phi + &= \sin\tau(1 + \tfrac{1}{2} m^{2} - \tfrac{9}{4} m^{2}\sin^{2}\tau + - \tfrac{7}{4} m^{2} + \tfrac{7}{2} m^{2}\sin^{2}\tau) \\ + &= \sin\tau(1 - \tfrac{5}{4} m^{2} + \tfrac{5}{4} m^{2}\sin^{2}\tau) + = \sin\tau(1 - \tfrac{5}{4} m^{2}\cos^{2}\tau), +\Allowbreak +\cos\phi + &= \cos\tau(1 - \tfrac{1}{2} m^{2} + \tfrac{9}{4} m^{2}\cos^{2}\tau + + \tfrac{7}{4} m^{2} - \tfrac{7}{2} m^{2}\cos^{2}\tau) \\ + &= \cos\tau(1 + \tfrac{5}{4} m^{2} - \tfrac{5}{4} m^{2}\cos^{2}\tau) + = \cos\tau(1 + \tfrac{5}{4} m^{2}\sin^{2}\tau); +\Allowbreak +\sin2\phi + &= \sin2\tau(1 - \tfrac{5}{4} m^{2}\cos2\tau), \\ +% +\cos2\phi + &= \cos2\tau + \tfrac{5}{4} m^{2}\sin^{2}2\tau); +\Allowbreak +\cos\phi\, \frac{d\phi}{d\tau} + &= \Neg\cos\tau(1 - \tfrac{5}{4} m^{2} + \tfrac{15}{4} m^{2} \sin^{2}\tau), \\ +% +\sin\phi\, \frac{d\phi}{d\tau} + &= -\sin\tau(1 + \tfrac{5}{4} m^{2} - \tfrac{15}{4} m^{2} \cos^{2}\tau). +\end{align*} + +Summing the squares of these, +\begin{align*} +\left(\frac{d\phi}{d\tau}\right)^{2} + &= \cos^{2}\tau(1 - \tfrac{5}{2} m^{2} + \tfrac{15}{2} m^{2} \sin^{2}\tau) + + \sin^{2}\tau(1 + \tfrac{5}{2} m^{2} - \tfrac{15}{2} m^{2} \cos^{2}\tau) \\ + &= 1 - \tfrac{5}{2} m^{2} \cos2\tau, +\end{align*} +and thence +\[ +\frac{d\phi}{d\tau} = 1 - \tfrac{5}{4} m^{2} \cos2\tau. +\Tag{(21)} +\] + +Differentiating again +\[ +\frac{d^{2}\phi}{d\tau^{2}} = \tfrac{5}{2} m^{2} \sin 2\tau. +\] + +We are now in a position to evaluate all the earlier terms in the +differential equations~\Eqref{(17)}. + +Thus +\[ +\left. +\begin{aligned}%[** TN: Re-broken] +\frac{d^{2}\, \delta p}{d\tau^{2}} + &- \delta p \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m\frac{d\phi}{d\tau}\right] + - 2\frac{d\, \delta s}{d\tau}\left(\frac{d\phi}{d\tau} + m\right) + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} \\ +% +&= \frac{d^{2}\, \delta p}{d^{2}} + \delta p \bigl[-1 + \tfrac{5}{2} m^{2}\cos 2\tau - 2m\bigr] \\ + &\qquad\qquad + - 2\frac{d\, \delta s}{d\tau}(1 + m - \tfrac{5}{4} m^{2}\cos2\tau) + - \tfrac{5}{2} m^{2}\sin2\tau\, \delta s\Add{,} \\ +% +\frac{d^{2}\, \delta s}{d\tau^{2}} + &- \delta s \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m\frac{d\phi}{d\tau}\right] + + 2\frac{d\, \delta p}{d\tau}\left(\frac{d\phi}{d\tau} + m\right) + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} \\ +% +&= \frac{d^{2}\, \delta s}{d^{2}} + \delta s \bigl[-1 + \tfrac{5}{2} m^{2}\cos 2\tau - 2m\bigr] \\ + &\qquad\qquad + + 2\frac{d\, \delta p}{d\tau}(1 + m - \tfrac{5}{4} m^{2}\cos2\tau) + + \tfrac{5}{2} m^{2}\sin2\tau\, \delta p\Add{.} +\end{aligned} +\right\} +\Tag{(22)} +\] +\DPPageSep{090}{32} + +We now have to evaluate the several terms involving $x$~and~$y$ in \Eqref{(18)},~\Eqref{(19)},~\Eqref{(20)}. + +\begin{align*} +x \cos\phi + y \sin\phi + &= \cos^{2}\tau (1 - m^{2} - \tfrac{3}{4} m^{2} \sin^{2}\tau + \tfrac{5}{4} m^{2} \sin^{2}\tau) \\ + &\,+ \sin^{2}\tau (1 + m^{2} + \tfrac{3}{4} m^{2} \cos^{2}\tau - \tfrac{5}{4} m^{2} \cos^{2}\tau) \\ + &= 1 - m^{2} \cos 2\tau, +\Allowbreak +% +-x \sin\phi + y \cos\phi + &= -\sin\tau \cos\tau (1 - m^{2} - \tfrac{3}{4} m^{2} \sin^{2}\tau - \tfrac{5}{4} m^{2} \cos^{2}\tau) \\ + &\quad+ \sin\tau \cos\tau (1 + m^{2} + \tfrac{3}{4} m^{2} \cos^{2}\tau + \tfrac{5}{4} m^{2} \sin^{2}\tau) \\ + &= 2m^{2} \sin 2\tau; +\Allowbreak +% +r^{2} = x^{2} + y^{2} &= 1 - 2m^{2} \cos 2\tau, +\Allowbreak +% +x^{2} - y^{2} &= \cos^{2}\tau(1 - 2m^{2} - \tfrac{3}{2} m^{2}\sin^{2}\tau) \\ + &\,- \sin^{2}\tau (1 + 2m^{2} + \tfrac{3}{2} m^{2}\cos^{2}\tau) \\ + &= \cos 2\tau - 2m^{2} - \tfrac{3}{4} m^{2}\sin^{2} 2\tau, +\Allowbreak +% +xy &= \tfrac{1}{2}\sin 2\tau(1 + \tfrac{3}{4} m^{2}\cos 2\tau); +\Allowbreak +% +(x^{2} - y^{2}) \cos 2\phi + &= \begin{aligned}[t] + \cos^{2}2\tau - 2m^{2} \cos 2\tau + &- \tfrac{3}{4} m^{2} \sin^{2}2\tau \cos 2\tau \\ + &+ \tfrac{5}{4} m^{2} \sin^{2}2\tau \cos 2\tau + \end{aligned} \\ + &= \cos 2\tau (\cos 2\tau - 2m^{2} + \tfrac{1}{2} m^{2} \sin^{2}2\tau), +\Allowbreak +% +(x^{2} - y^{2}) \sin 2\phi + &= \sin 2\tau (\cos 2\tau - 2m^{2} - \tfrac{3}{4} m^{2} \sin^{2}2\tau - \tfrac{5}{4} m^{2} \cos^{2}2\tau) \\ + &= \sin 2\tau (\cos 2\tau - \tfrac{11}{4} m^{2} - \tfrac{1}{2} m^{2} \cos^{2}2\tau); +\Allowbreak +% +xy \cos 2\phi + &= \tfrac{1}{2} \sin 2\tau (\cos 2\tau + \tfrac{5}{4} m^{2} \sin^{2}2\tau + \tfrac{3}{4} m^{2} \cos^{2}2\tau) \\ + &= \tfrac{1}{2} \sin 2\tau (\cos 2\tau + \tfrac{5}{4} m^{2} - \tfrac{1}{2} m^{2} \cos^{2}2\tau), \\ +% +xy \sin 2\phi + &= \tfrac{1}{2}\sin^{2}2\tau (1 - \tfrac{5}{4} m^{2}\cos 2\tau + \tfrac{3}{4} m^{2}\cos 2\tau) \\ + &= \tfrac{1}{2}\sin^{2}2\tau (1 - \tfrac{1}{2} m^{2}\cos 2\tau). +\end{align*} + +Therefore +\begin{gather*} +\begin{aligned} +&\tfrac{1}{2}(x^{2} - y^{2})\cos 2\phi + xy \sin 2\phi \\ +% + &= \tfrac{1}{2}\cos^{2}2\tau - m^{2}\cos 2\tau + \tfrac{1}{4} m^{2}\sin^{2}2\tau \cos 2\tau + + \tfrac{1}{2}\sin^{2}2\tau - \tfrac{1}{4} m^{2}\sin^{2}2\tau \cos 2\tau \\ +% + &= \tfrac{1}{2}(1 - 2m^{2}\cos 2\tau) = \tfrac{1}{2}r^{2}, +\end{aligned} \\ +% + \therefore + -\tfrac{1}{2} \mp \tfrac{3}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \cos 2\phi \mp \frac{3xy}{r^{2}}\sin 2\phi + = -\tfrac{1}{2} \mp \tfrac{3}{2} = -2 \text{ or } +1. +\end{gather*} + +These are the coefficients of~$\dfrac{\delta p}{r^{3}}$ in the expression~\Eqref{(19)} for +\[ +\cos\phi\, \delta \left(\frac{x}{r^{3}}\right) + +\sin\phi\, \delta \left(\frac{y}{r^{3}}\right), +\] +and of~$\dfrac{\delta s}{r^{3}}$ in the expression~\Eqref{(20)} for $-\sin\phi\, \delta\left(\dfrac{x}{r^{3}}\right) + \cos\phi\, \delta\left(\dfrac{y}{r^{3}}\right)$. +\DPPageSep{091}{33} + +Again +\begin{align*} +-\tfrac{1}{2}(x^{2} - y^{2}) \sin 2\phi &+ xy \cos 2\phi \\ + &= + \begin{alignedat}[t]{3} + -\tfrac{1}{2} \sin 2\tau + &(\cos 2\tau &&- \tfrac{11}{4} m^{2} &&- \tfrac{1}{2} m^{2} \cos^{2} 2\tau) \\ + +\tfrac{1}{2} \sin 2\tau + &(\cos 2\tau &&+ \tfrac{5}{4} m^{2} &&- \tfrac{1}{2} m^{2} \cos^{2} 2\tau) + \end{alignedat} \\ + &= 2m^{2} \sin 2\tau. +\end{align*} + +Then since to the order zero, $r^{3} = 1$, we have +\[ +3\left(\frac{xy}{r^{2}} \cos 2\phi - \tfrac{1}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \sin 2\phi\right) + = 6m^{2} \sin 2\tau. +\] + +This is the coefficient of~$-\dfrac{\delta s}{r^{3}}$ in $\cos\phi\, \delta\left(\dfrac{x}{r^{3}}\right) + \sin\phi\, \delta\left(\dfrac{y}{r^{3}}\right)$ and of~$-\dfrac{\delta p}{r^{3}}$ in +$-\sin\phi\, \delta\left(\dfrac{x}{r^{3}}\right) + \cos\phi\, \delta\left(\dfrac{y}{r^{3}}\right)$. + +Hence we have +\[ +\left. +\begin{aligned} +\cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + +\sin\phi\, \delta\left(\frac{y}{r^{3}}\right) + &= -2\frac{\delta p}{r^{3}} - \frac{6m^{2}}{r^{3}}\, \delta s \sin 2\tau \\ + &= -2\delta p\, (1 + 3m^{2} \cos 2\tau) + - 6m^{2}\, \delta s \sin 2\tau\Add{,} \\ +% +-\sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + &= -\frac{\delta p}{r^{3}} · 6m^{2} \sin 2\tau + \frac{\delta s}{r^{3}} \\ + &= -6m^{2}\, \delta p \sin 2\tau + \delta s\, (1 + 3m^{2} \cos 2\tau)\Add{.} +\end{aligned} +\right\} +\Tag{(23)} +\] + +These two expressions are to be multiplied by~$\kappa$ in the differential +equations~\Eqref{(17)}. + +{\stretchyspace +The other terms which occur in the differential equations are $-3m^{2}\cos\phi\, \delta x$ +and~$+3m^{2}\sin\phi\, \delta x$.\Pagelabel{33}} + +Since $m^{2}$~occurs in the coefficient we need only go to the order zero of +small quantities in $\cos\phi\, \delta x$ and~$\sin\phi\, \delta x$. + +Thus +\begin{align*}%[** TN: Added two breaks] +3m^{2}\, \delta x \cos\phi + &= 3m^{2} (\delta p \cos\tau - \delta s \sin\tau) \cos\tau \\ + &= \tfrac{3}{2} m^{2}\, \delta p\, (1 + \cos 2\tau) + - \tfrac{3}{2} m^{2}\, \delta s \sin 2\tau, \\ +% +3m^{2}\, \delta x \sin\phi + &= 3m^{2} (\delta p \cos\tau - \delta s \sin\tau) \sin\tau \\ + &= \tfrac{3}{2} m^{2}\, \delta p \sin 2\tau + - \tfrac{3}{2} m^{2}\, \delta s\, (1 - \cos 2\tau). +\end{align*} + +Now $\kappa = 1 + 2m + \frac{3}{2} m^{2}$, and hence +\begin{align*} + \kappa \cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + &+ \kappa \sin\phi\, \delta\left(\frac{y}{r^{3}}\right) - 3m^{2}\, \delta x \cos\phi \\ + &= -2\delta p\, (1 + 3m^{2} \cos 2\tau + 2m + \tfrac{3}{2} m^{2}) + - 6m^{2}\, \delta s \sin 2\tau \\ + &\quad -\tfrac{3}{2} m^{2}\, \delta p\, (1 + \cos 2\tau) + \tfrac{3}{2} m^{2}\, \delta s \sin 2\tau \\ + &= -2\delta p\, [1 + 2m + \tfrac{9}{4} m^{2} + \tfrac{15}{4} m^{2} \cos 2\tau] + - \tfrac{9}{2} m^{2}\, \delta s \sin 2\tau, +\Allowbreak +\DPPageSep{092}{34} +-\kappa \sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + &+ \kappa \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + 3m^{2}\, \delta x \sin\phi \\ +% + &= -6m^{2}\, \delta p \sin 2\tau + + \delta s\, (1 + 2m + \tfrac{3}{2} m^{2} + 3m^{2} \cos2\tau) \\ + &\quad + \tfrac{3}{2} m^{2}\, \delta p \sin 2\tau + - \delta s\, (\tfrac{3}{2} m^{2} - \tfrac{3}{2} m^{2} \cos2\tau) \\ +% + &= -\tfrac{9}{2} m^{2}\, \delta p \sin2\tau + + \delta s\, (1 + 2m + \tfrac{9}{2} m^{2} \cos2\tau). +\end{align*} + +Hence +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + - \delta p \left[\left(\frac{d\phi}{d\tau}\right)^{2} + + 2m\left(\frac{d\phi}{d\tau}\right)\right] + - 2 \frac{d\, \delta s}{d\tau} \left(\frac{d\phi}{d\tau} + m\right) + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} \\ +% + + \kappa \cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa \sin\phi\, \delta\left(\frac{y}{r^{3}}\right) + - 3m^{2} \cos\phi\, \delta x = 0 +\end{multline*} +becomes +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + - \delta p\, [1 + 2m - \tfrac{5}{2} m^{2} \cos 2\tau] + - 2 \frac{d\, \delta s}{d\tau}(1 + m - \tfrac{5}{4} m^{2} \cos 2\tau) + - \tfrac{5}{2} m^{2}\, \delta s \sin 2\tau \\ +% + - 2\delta p\, [1 + 2m + \tfrac{9}{4} m^{2} + \tfrac{15}{4} m^{2} \cos 2\tau] + - \tfrac{9}{2} m^{2}\, \delta s \sin 2\tau = 0 +\end{multline*} +or +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + - \delta p\, [3 + 6m + \tfrac{9}{2} m^{2} + 5m^{2} \cos 2\tau] + - 2 \frac{d\, \delta s}{d\tau} (1 + m - \tfrac{5}{4} m^{2} \cos 2\tau) \\ +% + - 7m^{2}\, \delta s \sin 2\tau = 0. +\Tag{(24)} +\end{multline*} + +This is the first of our equations transformed. + +Again the second equation is +\begin{multline*} +\frac{d^{2}\, \delta s}{d\tau^{2}} + - \delta s \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m \frac{d\phi}{d\tau}\right] + + 2 \frac{d\, \delta p}{d\tau} \left(\frac{d\phi}{d\tau} + m \right) + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} \\ +% + - \kappa \sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + + 3m^{2} \sin\phi\, \delta x = 0, +\end{multline*} +and it becomes +\begin{multline*} +\frac{d^{2}\, \delta s}{d\tau^{2}} + + \delta s\, (-1 - 2m + \tfrac{5}{2} m^{2} \cos 2\tau) + + 2 \frac{d\, \delta p}{d\tau}(1 + m - \tfrac{5}{4} m^{2} \cos 2\tau) + + \tfrac{5}{2} m^{2}\, \delta p \sin 2\tau \\ +% + - \tfrac{9}{2} m^{2}\, \delta p \sin 2\tau + + \delta s\, (1 + 2m + \tfrac{9}{2} m^{2} \cos 2\tau) = 0. +\end{multline*} + +Whence +\[ +\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos 2\tau + + 2 \frac{d\, \delta p}{d\tau} (1 + m -\tfrac{5}{4} m^{2} \cos 2\tau) + - 2m^{2}\, \delta p \sin 2\tau = 0. +\Tag{(25)} +\] + +This is the second of our equations transformed. + +The Jacobian integral gives +\begin{align*}%[** TN: Rebroken] +\frac{d\, \delta s}{d\tau} &+ \delta p\, \frac{d\phi}{d\tau} \\ + &= \frac{3m^{2} x\, \delta x}{V} + - \frac{\kappa}{V r^{3}} \bigl[\delta p\, (x \cos\phi + y \sin\phi) + \delta s\, (-x \sin\phi + y \cos\phi)\bigr] +\Allowbreak + &= 3m^{2} \cos\tau (\delta p \cos\tau - \delta s \sin\tau) \\ + &\qquad\qquad + - (1 + 2m + \tfrac{3}{2} m^{2} - \tfrac{7}{4} m^{2} \cos2\tau + + 3m^{2} \cos2\tau) \\ + &\qquad\qquad\qquad\qquad\Add{·} + \bigl[\delta p\, (1 - m^{2} \cos2\tau) + 2m^{2}\, \delta s \sin2\tau\bigr] +\Allowbreak +\DPPageSep{093}{35} + &= \frac{3m^{2}}{2}\, \delta p\, (1 + \cos 2\tau) + - \frac{3m^{2}}{2}\, \delta s \sin 2\tau \\ + &\qquad -\delta p\, (1 + 2m + \tfrac{3}{2} m^{2} + + \tfrac{5}{4} m^{2} \cos2\tau - m^{2} \cos2\tau) - 2m^{2}\, \delta s \sin2\tau +\Allowbreak + &= -\delta p\, (1 + 2m - \tfrac{5}{4} m^{2} \cos2\tau) + - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau. +\end{align*} + +Substituting for~$\dfrac{d\phi}{d\tau}$ its value from~\Eqref{(21)} +\begin{align*} +\frac{d\, \delta s}{d\tau} + &= -\delta p\, (1 + 2m - \tfrac{5}{4} m^{2} \cos2\tau) + - \delta p\, (1 - \tfrac{5}{4} m^{2} \cos2\tau) + - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau \\ +% + &= -\delta p\, (2 + 2m - \tfrac{5}{2} m^{2} \cos2\tau) + - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau \\ +% +\frac{2d\, \delta s}{d\tau} + &= -4\delta p\, (1 + m - \tfrac{5}{4} m^{2} \cos2\tau) + - 7m^{2}\, \delta s \sin2\tau \\ +\frac{2d\, \delta s}{d\tau} &(1 + m - \tfrac{5}{4} m^{2} \cos2\tau) + = -4\delta p\, (1 + 2m + m^{2} - \tfrac{5}{2} m^{2} \cos2\tau) + - 7m^{2}\, \delta s \sin2\tau \\ +% +\frac{2d\, \delta s}{d\tau} &(1 + m - \tfrac{5}{4} m^{2} \cos2\tau) + + 7m^{2}\, \delta s \sin2\tau + = -4\delta p\, (1 + 2m + m^{2} - \tfrac{5}{2} m^{2} \cos2\tau). +\Tag{(26)} +\end{align*} + +This expression occurs in~\Eqref{(24)}, and therefore can be used to eliminate +$\dfrac{d\, \delta s}{d\tau}$ from it. + +Substituting we get +\begin{gather*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + + \delta p\, \bigl[-3 - 6m - \tfrac{9}{2} m^{2} - 5m^{2} \cos2\tau + + 4 + 8m + 4m^{2} - 10 m^{2} \cos2\tau\bigr] = 0, +\Allowbreak +\left. +\begin{gathered} +\lintertext{i.e.} +{\frac{d^{2}\, \delta p}{d\tau^{2}} + \delta p\, \bigl[1 + 2m - \tfrac{1}{2} m^{2} - 15m^{2} \cos 2\tau\bigr] = 0.} \\ +\lintertext{And}{\frac{d\, \delta s}{d\tau} + = -2\delta p\, (1 + m - \tfrac{5}{4} m^{2} \cos2\tau) + - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau.} +\end{gathered} +\right\} +\Tag{(27)} +\end{gather*} + +If we differentiate the second of these equations, which it is to be +remembered was derived from Jacobi's integral and therefore involves our +second differential equation, we get +\Pagelabel{35} +\begin{align*}%[** TN: Rebroken] +\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos2\tau + &+ \tfrac{7}{2} m^{2} \sin 2\tau\, \frac{d\, \delta s}{d\tau} \\ + &+ 2 (1 + m - \tfrac{5}{4} m^{2} \cos 2\tau)\, \frac{d\, \delta p}{d\tau} + + 5 m^{2}\, \delta p \sin 2\tau = 0, +\end{align*} +and eliminating~$\dfrac{d\, \delta s}{d\tau}$ +\begin{align*} +\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos2\tau + &- 7m^{2}\, \delta p \sin 2\tau \\ + &+ 2 (1 + m - \tfrac{5}{4} m^{2} \cos2\tau)\, \frac{d\, \delta p}{d\tau} + + 5m^{2}\, \delta p \sin 2\tau = 0, +\end{align*} +\DPPageSep{094}{36} +or +\[ +\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos 2\tau + + 2(1 + m - \tfrac{5}{4} m^{2} \cos 2\tau)\, \frac{d\, \delta p}{d\tau} + - 2m^{2}\, \delta p \sin 2\tau = 0, +\] +and this is as might be expected our second differential equation which was +found above. Hence we only require to consider the equations~\Eqref{(27)}. + +\Section{§ 6. }{Integration of an important type of Differential Equation.} +\index{Differential Equation, Hill's}% +\index{Hill, G. W., Lunar Theory!Special Differential Equation}% + +The differential equation for~$\delta p$ belongs to a type of great importance +in mathematical physics. We may write the typical equation in the form +\[ +\frac{d^{2}x}{dt^{2}} + + (\Theta_{0} + 2\Theta_{1} \cos 2t + 2\Theta_{2} \cos 4t + \dots) x = 0, +\] +where $\Theta_{0}, \Theta_{1}, \Theta_{2}, \dots$ are constants depending on increasing powers of a small +quantity~$m$. It is required to find a solution such that $x$~remains small for +all values of~$t$. + +Let us attempt the apparently obvious process of solution by successive +approximations. + +Neglecting $\Theta_{1}, \Theta_{2}, \dots$, we get as a first approximation +\[ +x = A \cos(t \sqrt{\Theta_{0}} + \epsilon). +\] + +Using this value for~$x$ in the term multiplied by~$\Theta_{1}$, and neglecting $\Theta_{2}, +\Theta_{3}, \dots$, we get +\[ +\frac{d^{2}x}{dt^{2}} + + \Theta_{0} x + A\Theta_{1} \left\{ + \cos\bigl[t(\sqrt{\Theta_{0}} + 2) + \epsilon\bigr] + + \cos\bigl[t(\sqrt{\Theta_{0}} - 2) + \epsilon\bigr]\right\} = 0. +\] + +Solving this by the usual rules we get the second approximation +\begin{align*}%[** TN: Rebroken] +x = A\biggl\{\cos\left[t\sqrt{\Theta_{0}} + \epsilon\right] + &+ \frac{\Theta_{1} \cos\left[t(\sqrt{\Theta_{0}} + 2) + \epsilon\right]} + {4(\sqrt{\Theta_{0}} + 1)} \\ + &- \frac{\Theta_{1} \cos\left[t(\sqrt{\Theta_{0}} - 2) + \epsilon\right]} + {4(\sqrt{\Theta_{0}} - 1)} + \biggr\}. +\end{align*} + +Again using this we have the differential equation +\[ +\begin{split} +\frac{d^{2}x}{dt^{2}} + &+ \Theta_{0} x + A\Theta_{1}\left\{ + \cos\bigl[t(\sqrt{\Theta_{0}} + 2) + \epsilon\bigr] + + \cos\bigl[t(\sqrt{\Theta_{0}} - 2) - \epsilon\bigr] + \right\} \\ +% + &+ \frac{A\Theta_{1}^{2}}{4(\sqrt{\Theta_{0}} + 1)} \left\{ + \cos\bigl[t(\sqrt{\Theta_{0}} + 4) + \epsilon\bigr] + + \cos(t\sqrt{\Theta_{0}} + \epsilon) + \right\} \\ +% + &- \frac{A\Theta_{1}^{2}}{4(\sqrt{\Theta_{0}} - 1)} \left\{ + \cos(t\sqrt{\Theta_{0}} + \epsilon) + + \cos\bigl[t(\sqrt{\Theta_{0}} - 4) + \epsilon\bigr] + \right\} \\ +% + &+ A\Theta_{2} \left\{ + \cos\bigl[t(\sqrt{\Theta_{0}} + 4) + \epsilon\bigr] + + \cos\bigl[t(\sqrt{\Theta_{0}} - 4) + \epsilon\bigr] + \right\} = 0. +\end{split} +\] + +Now this equation involves terms of the form~$B \cos(t\sqrt{\Theta_{0}} + \epsilon)$; on +integration terms of the form~$Ct\sin(t\sqrt{\Theta_{0}} + \epsilon)$ will arise. But these terms +are not periodic and do not remain small when $t$~increases. $x$~will therefore +not remain small and the argument will fail. The assumption on which these +approximations have been made is that the period of the principal term of~$x$ +can be determined from $\Theta_{0}$~alone and is independent of~$\Theta_{1}, \Theta_{2}, \dots$. But the +\DPPageSep{095}{37} +appearance of secular terms leads us to revise this assumption and to take as +a first approximation +\[ +x = A \cos (ct \sqrt{\Theta_{0}} + \epsilon), +\] +where $c$~is nearly equal to~$1$ and will be determined, if possible, to prevent +secular terms arising. + +It will, however, be more convenient to write as a first approximation +\[ +x = A \cos (ct + \epsilon), +\] +where $c$~is nearly equal to~$\Surd{\Theta_{0}}$. + +Using this value of~$x$ in the term involving~$\Theta_{1}$, our equation becomes +\[ +\frac{d^{2}x}{dt^{2}} + + \Theta_{0} x + A\Theta_{1}\left\{ + \cos\bigl[(c + 2)t + \epsilon\bigr] + \cos\bigl[(c - 2)t + \epsilon\bigr] + \right\} = 0, +\] +and the second approximation is +\begin{align*} +x = A \cos (ct + \epsilon) + &+ \frac{A\Theta_{1}}{(c + 2)^{2} - \Theta_{0}} \cos\bigl[(c + 2)t + \epsilon\bigr] \\ + &+ \frac{A\Theta_{1}}{(c - 2)^{2} - \Theta_{0}} \cos\bigl[(c - 2)t + \epsilon\bigr].\footnotemark +\end{align*} +\footnotetext{This is not a solution of the previous equation, unless we actually put $c=\sqrt{\Theta_{0}}$ in the + first term.}% + +Proceeding to another approximation with this value of~$x$, we get +\[ +\begin{split} +\frac{d^{2}x}{dt^{2}} + &+ \Theta_{0}x + A\Theta_{1}\left\{ + \cos\bigl[(c + 2)t + \epsilon\bigr] + \cos\bigl[(c - 2)t + \epsilon\bigr] + \right\} \\ +% + &+ \frac{A\Theta_{1}^{2}}{(c + 2)^{2} - \Theta_{0}} \left\{ + \cos\bigl[(c + 4)t + \epsilon\bigr] + \cos(ct + \epsilon)\right\} \\ +% + &+ \frac{A\Theta_{1}^{2}}{(c - 2)^{2} - \Theta_{0}} \left\{ + \cos(ct + \epsilon) + \cos\bigl[(c - 4)t + \epsilon\bigr]\right\} \\ +% + &+ A\Theta_{2}\left\{ + \cos\bigl[(c + 4)t + \epsilon\bigr] + \cos\bigl[(c - 4)t + \epsilon\bigr] + \right\} =0. +\end{split} +\] + +We might now proceed to further approximations but just as a term in +$\cos (ct + \epsilon)$ generates in the solution terms in +\[ +\cos\bigl[(c ± 2)t + \epsilon\bigr]\quad \text{and}\quad +\cos\bigl[(c ± 4)t + \epsilon\bigr], +\] +terms in +\[ +\cos\bigl[(c ± 2)t + \epsilon\bigr]\quad \text{and}\quad +\cos\bigl[(c ± 4)t + \epsilon\bigr] +\] +will generate new terms in~$\cos(ct + \epsilon)$, i.e.~terms of exactly the same nature +as the term initially assumed. Hence to get our result it will be best to +begin by assuming a series containing all the terms which will arise. + +Various writers have found it convenient to introduce exponential instead +of trigonometric functions. Following their example we shall therefore write +the differential equation in the form +\[ +\frac{d^{2}x}{dt^{2}} + + x\sum_{-\infty}^{+\infty} \Theta_{i} e^{2it\Surd{-1}} = 0, +\Tag{(28)} +\] +where +\[ +\Theta_{-i} = \Theta_{i}, +\] +\DPPageSep{096}{38} +and the solution is assumed to be +\[ +x = \sum_{-\infty}^{+\infty} A_{j} e^{(c + 2j)t \sqrt{-1}}, +\] +where the ratios of all the coefficients~$A_{j}$, and~$c$, are to be determined by +equating coefficients of different powers of~$e^{t\sqrt{-1}}$. + +Substituting this expression for~$x$ in the differential equation, we get +\[ +-\sum_{-\infty}^{+\infty} (c + 2j)^{2} A_{j} e^{(c + 2j)t\sqrt{-1}} + + \sum_{-\infty}^{+\infty} A_{j} e^{(c + 2j)t \sqrt{-1}} + \sum_{-\infty}^{+\infty} \Theta_{i} e^{2i t\sqrt{-1}} = 0, +\] +and equating to zero the coefficient of~$e^{(c + 2j)t \sqrt{-1}}$, +\begin{multline*} +-(c + 2j)^{2}A_{j} + A_{j}\Theta_{0} + + A_{j-1}\Theta_{1} + A_{j-2}\Theta_{2} + A_{j-3}\Theta_{3} + \dots \\ + + A_{j+1}\Theta_{-1} + A_{j+2}\Theta_{-2} + A_{j+3}\Theta_{-3} + \dots = 0. +\end{multline*} + +Hence the succession of equations is +\index{Hill, G. W., Lunar Theory!infinite determinant}% +\index{Infinite determinant, Hill's}% +\iffalse +\begin{align*} +\dots &+ \bigl[\Theta_{0} - (c-4)^2\bigr]A_{-2} + \Theta_{-1}A_{-1} + \Theta_{-2}A_{0} + \Theta_{-3}A_{1} + \Theta_{-4}A_{2} + \dots = 0, \\ +\dots &+ \Theta_{1}A_{-2} + \bigl[\Theta_{0} - (c-2)^2\bigr]A_{-1} + \Theta_{-1}A_{0} + \Theta_{-2}A_{1} + \Theta_{-3}A_{2} + \dots = 0, \\ +\dots &+ \Theta_{2}A_{-2} + \Theta_{1}A_{-1} + (\Theta_{0} - c^2)A_{0} + \Theta_{-1}A_{1} + \Theta_{-2}A_{2} + \dots = 0, \\ +\dots &+ \Theta_3A_{-2} + \Theta_{2}A_{-1} + \Theta_{1}A_{0} + \bigl[\Theta_{0} - (c+2)^2\bigr]A_{1} + \Theta_{-1}A_{2} + \dots = 0, \\ +\dots &+ \Theta_4A_{-2} + \Theta_3A_{-1} + \Theta_{2}A_{0} + \Theta_{1}A_{1} + \bigl[\Theta_{0} - (c+4)^2\bigr]A_{2} + \dots = 0. +\end{align*} +\fi +{\small +\[ +\begin{array}{@{\,}*{17}{c@{\,}}} +\hdotsfor{17} \\ +\dots &+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c-4)^2\bigr]A_{-2}} &+& \Theta_{-1}A_{-1} &+& \Theta_{-2}A_{0} &+& \Theta_{-3}A_{1} &+& \Theta_{-4}A_{2} &+& \dots &=& 0, \\ +\dots &+& \Theta_{1}A_{-2}&+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c-2)^2\bigr]A_{-1}} &+& \Theta_{-1}A_{0} &+& \Theta_{-2}A_{1} &+& \Theta_{-3}A_{2} &+& \dots &=& 0, \\ +\dots &+& \Theta_{2}A_{-2}&+& \Theta_{1}A_{-1} &+& \multicolumn{3}{c}{(\Theta_{0} - c^2)A_{0}} &+& \Theta_{-1}A_{1} &+& \Theta_{-2}A_{2} &+& \dots &=& 0, \\ +\dots &+& \Theta_3A_{-2} &+& \Theta_{2}A_{-1} &+& \Theta_{1}A_{0} &+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c+2)^2\bigr]A_{1}} &+& \Theta_{-1}A_{2} &+& \dots &=& 0, \\ +\dots &+& \Theta_4A_{-2} &+& \Theta_3A_{-1} &+& \Theta_{2}A_{0} &+& \Theta_{1}A_{1} &+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c+4)^2\bigr]A_{2}} &+& \dots &=& 0. \\ +\hdotsfor{17} +\end{array} +\]} + +We clearly have an infinite determinantal equation for~$c$. + +If we take only three columns and rows, we get +\begin{multline*} +\bigl[\Theta_{0} - (c - 2)^{2}\bigr] \bigl[\Theta_{0} - c^{2}\bigr] \bigl[\Theta_{0} - (c + 2)^{2}\bigr] + - \Theta_{1}^{2} \bigl[\Theta_{0} - (c - 2)^{2}\bigr] - \Theta_{1}^{2} \bigl[\Theta_{0} - (c + 2)^{2}\bigr] \\ +% + - \Theta_{2}^{2}(\Theta_{0} - c^{2}) + 2\Theta_{1}^{2} \Theta_{2} = 0, \\ +% +\bigl[(\Theta_{0} - c^{2} - 4)^{2} - 16c^{2}\bigr] \bigl[\Theta_{0} - c^{2}\bigr] + - 2\Theta_{1}^{2}(\Theta_{0} - c^{2} - 4) + - \Theta_{2}^{2}(\Theta_{0} - c^{2}) + 2\Theta_{1}^{2} \Theta_{2} = 0. +\end{multline*} + +If we neglect $(\Theta_{0} - c^{2})^{3}$ which is certainly small +\begin{multline*} +\bigl[-8(\Theta_{0} - c^{2}) + 16 + 16(\Theta_{0} - c^{2}) - 16\Theta_{0}\bigr] \bigl[\Theta_{0} - c^{2}\bigr] \\ +% + \shoveright{ -(\Theta_{0} - c^{2}) \bigl[2\Theta_{1}^{2} + \Theta_{2}^{2}\bigr] + 8\Theta_{1}^{2} + 2\Theta_{1}^{2} \Theta_{2} = 0,} \\ +% + \shoveright{8(\Theta_{0} - c^{2})^{2} + (\Theta_{0} - c^{2})(16 - 16\Theta_{0} - 2\Theta_{1}^{2} - \Theta_{2}^{2}) + 8\Theta_{1}^{2} + 2\Theta_{1}^{2} \Theta_{2} = 0,} \\ +% +(\Theta_{0} - c^{2})^2 + 2(\Theta_{0} - c^{2})(1 - \Theta_{0} - \tfrac{1}{8}\Theta_{1}^{2} - \tfrac{1}{16}\Theta_{2}^{2}) + \Theta_{1}^{2} + \tfrac{1}{4}\Theta_{1}^{2} \Theta_{2} = 0. +\end{multline*} + +Since $\Theta_{1}^{2}, \Theta_{2}^{2}$ are small compared with~$1 - \Theta_{0}$, and $\Theta_{2}$~compared with~$1$, we +have as a rougher approximation +\[ +(c^{2} - \Theta_{0})^{2} + 2(\Theta_{0} - 1) (c^{2} - \Theta_{0}) = -\Theta_{1}^{2}, +\] +\DPPageSep{097}{39} +whence +\begin{gather*} +c^{2} - \Theta_{0} + = -(\Theta_{0} - 1) ± \sqrt{(\Theta_{0} - 1)^{2} - \Theta_{1}^{2}}, \\ +% +c^{2} = 1 ± \sqrt{(\Theta_{0} - 1)^{2} - \Theta_{1}^{2}}. +\end{gather*} + +Now $c^{2} = \Theta_{0}$ when $\Theta_{1} = 0$. Hence we take the positive sign and get +\[ +c = \sqrt{1 + \sqrt{(\Theta_{0} - 1)^{2} - \Theta_{1}^{2}}}, +\Tag{(29)} +\] +which is wonderfully nearly correct. + +For further discussion of the equation for~$c$, see Notes~1,~2, pp.~\Pgref{note:1},~\Pgref{note:2}. %[** TN: pp 53, 55 in original] + +\Section{§ 7. }{Integration of the Equation for $\delta p$.} + +We now return to the Lunar Theory and consider the solution of our +differential equation. Assume it to be +\[ +\delta p = A_{-1}\cos\bigl[(c - 2)\tau + \epsilon\bigr] + + A_{0}\cos(c\tau + \epsilon) + + A_{1}\cos\bigl[(c + 2)\tau + \epsilon\bigr]. +\] + +On substitution in~\Eqref{(27)} we get +\begin{align*} + A_{-1} \bigl[(1 + 2m - \tfrac{1}{2} m^{2} - 15 m^{2}\cos2\tau) &- (c - 2)^{2}\bigr]\cos\bigl[(c - 2)\tau + \epsilon\bigr] \\ +% ++ A_{0} \bigl[(1 + 2m - \tfrac{1}{2} m^{2} - 15 m^{2}\cos2\tau) &- c^{2}\bigr]\cos(c\tau + \epsilon) \\ +% ++ A_{1} \bigl[(1 + 2m - \tfrac{1}{2} m^{2} - 15 m^{2}\cos2\tau) &- (c + 2)^{2}\bigr]\cos\big[(c + 2)\tau + \epsilon\bigr] = 0. +\end{align*} + +Then we equate to zero the coefficients of the several cosines. + +1st~$\cos(c\tau + \epsilon)$ gives +\[ +-\tfrac{15}{2} m^{2}A_{-1} + + A_{0}(1 + 2m - \tfrac{1}{2} m^{2} - c^{2}) + - \tfrac{15}{2} m^{2}A_{1} = 0. +\] + +2nd~$\cos \bigl[(c - 2)\tau + \epsilon\bigr]$ gives +\[ +A_{-1} \bigl[1 + 2m - \tfrac{1}{2} m^{2} - (c - 2)^{2}\bigr] + - \tfrac{15}{2} m^{2}A_{0} = 0. +\] + +3rd~$\cos \bigl[(c + 2)\DPtypo{t}{\tau}\bigr] + \epsilon]$ gives +\[ +-\tfrac{15}{2} m^{2}A_{0} + A_{1}\bigl[1 + 2m - \tfrac{1}{2} m^{2} - (c + 2)^{2}] = 0. +\] + +If we neglect terms in~$m^{2}$ the first equation gives us $c^{2} = 1 + 2m$, and +\Pagelabel{39} +therefore $c = 1 + m$, $c - 2 = -(1 - m)$, $c + 2 = 3 + m$. + +The second and third equations then reduce to +\[ +4m A_{-1} = 0;\quad A_{1}(-8 - 4m) = 0. +\] + +From this it follows that $A_{-1}$~is at least of order~$m$ and $A_{1}$~at least of +order~$m^{2}$. + +Then since we are neglecting higher powers than~$m^{2}$, the first equation +reduces to +\[ +A_{0}(1 + 2m - \tfrac{1}{2} m^{2} - c^{2}) = 0, +\] +so that +\[ +c^{2} = 1 + 2m - \tfrac{1}{2} m^{2}\quad \text{or}\quad +c = 1 + m - \tfrac{3}{4} m^{2}. +\] + +Thus +\[ +(c - 2)^{2} = (1 - m + \tfrac{3}{4} m^{2})^{2} + = 1 - 2m + \tfrac{5}{2} m^{2}, +\] +and +\[ +1 + 2m - \tfrac{1}{2} m^{2} - (c - 2)^2 + = 4m - 3m^{2}. +\] +\DPPageSep{098}{40} + +Hence the second equation becomes +\[ +A_{-1}(4m - 3m^{2}) = \tfrac{15}{2} m^{2}A_{0}; +\] +and since $A_{-1}$~is of order~$m$, the term~$-3m^{2}A_{-1}$ is of order~$m^{3}$ and therefore +negligible. Hence +\[ +4m A_{-1} = \tfrac{15}{2} m^{2} A_{0} \quad \text{or}\quad +A_{-1} = \tfrac{15}{8} m A_{0}, +\] +and we cannot obtain $A_{-1}$~to an order higher than the first. + +The third equation is +\[ +-\tfrac{15}{2} m^{2} A_{0} + A_{1}[1 - 9] = 0, +\] +or +\[ +A_{1} = -\tfrac{15}{16} m^{2} A_{0}. +\] + +We have seen that $A_{-1}$~can only be obtained to the first order; so it is +useless to retain terms of a higher order in~$A_{1}$. Hence our solution is +\[ +A_{-1} = \tfrac{15}{8} m A_{0},\quad +A_{1} = 0. +\] + +Hence +\[ +\delta p = A_0 \left\{\cos(c\tau + \epsilon) + \tfrac{15}{8} m \cos\bigl[(c - 2)\tau + \epsilon\bigr]\right\}. +\Tag{(30)} +\] + +In order that the solution may agree with the more ordinary notation we +write $A_{0} = -a_{0}e$, and obtain +\[ +\left. +\begin{gathered} +\delta p = -a_{0}e \cos(c\tau + \epsilon) - \tfrac{15}{8} m a_{0} e \cos\bigl[(c - 2)\tau + \epsilon\bigr]\Add{,} \\ +\lintertext{where} +{c = 1 + m - \tfrac{3}{4} m^{2}\Add{.}} +\end{gathered} +\right\} +\Tag{(31)} +\] + +To the first order of small quantities the equation~\Eqref{(27)} for~$\delta s$ was +\begin{align*} +\frac{d\, \delta s}{d\tau} + &= -2(1 + m)\, \delta p \\ + &= 2(1 + m)a_{0}e \cos(c\tau + \epsilon) + + \tfrac{15}{4} m a_{0} e \cos\bigl[(c - 2)\tau + \epsilon\bigr]. +\end{align*} + +If we integrate and note that $c = 1 + m$ so that $c - 2 = -(1 - m)$, we have +\Pagelabel{40} +\[ +\delta s = 2a_{0} e \sin(c\tau + \epsilon) + - \tfrac{15}{4} m a_{0} e \sin\bigl[(c - 2)\tau + \epsilon\bigr]. +\Tag{(32)} +\] + +We take the constant of integration zero because $e = 0$ will then correspond +to no displacement along the variational curve. + +In order to understand the physical meaning of the results let us consider +the solution when~$m = 0$, i.e.~when the solar perturbation vanishes. + +Then +\[ +\delta p = -a_{0} e \cos (c\tau + \epsilon),\quad +\delta s = 2a_{0} e \sin (c\tau + \epsilon). +\] + +In the undisturbed orbit +\[ +x = a_{0} \cos\tau,\quad +y = a_{0} \sin\tau \quad \text{so that}\quad +\phi = \tau, +\] +and +\begin{gather*} +\begin{aligned} +\delta x &= \delta p \cos\phi - \delta s \sin\phi, \\ +\delta y &= \delta p \sin\phi + \delta s \cos\phi; +\end{aligned} \\ +\begin{aligned} +\delta x &= -a_{0} e \cos(c\tau + \epsilon)\cos\tau - 2a_{0} e \sin(c\tau + \epsilon)\sin\tau, \\ +\delta y &= -a_{0} e \cos(c\tau + \epsilon)\sin\tau + 2a_{0} e \sin(c\tau + \epsilon)\cos\tau. +\end{aligned} +\end{gather*} +\DPPageSep{099}{41} + +Therefore writing $X = x + \delta x$, $Y = y + \delta y$, $X = R \cos\Theta$, $Y = R \sin\Theta$, +\begin{alignat*}{3} +X &= a_{0}\bigl[\cos\tau &&- e \cos(c\tau + \epsilon)\cos\tau + &&- 2e \sin(c\tau + \epsilon)\sin\tau\bigr], \\ +% +Y &= a_{0}\bigl[\sin\tau &&- e \cos(c\tau + \epsilon)\sin\tau + &&+ 2e \sin(c\tau + \epsilon)\cos\tau\bigr]. +\end{alignat*} + +Therefore +\[ +R^{2} = a_{0}^{2} \bigl[1 - 2e \cos(c\tau + \epsilon)\bigr] +\] +or +\[ +R = a_{0} \bigl[1 - e \cos(c\tau + \epsilon)\bigr] + = \frac{a_{0}}{1 + e \cos(c\tau + \epsilon)}. +\Tag{(33)} +\] + +Again +\begin{alignat*}{2} +\cos\Theta &= \cos\tau &&- 2e \sin (c\tau + \epsilon)\sin\tau, \\ +\sin\Theta &= \sin\tau &&+ 2e \sin (c\tau + \epsilon)\cos\tau. +\end{alignat*} + +Hence +\[ +\sin(\Theta - \tau) = 2e \sin(c\tau + \epsilon), +\] +giving +\[ +\Theta = \tau + 2e \sin(c\tau + \epsilon). +\Tag{(34)} +\] + +It will be noted that the equations for $R, \Theta$ are of the same form as the +first approximation to the radius vector and true longitude in undisturbed +elliptic motion. When we neglect the solar perturbation by putting $m = 0$ +we see that $e$~is to be identified with the eccentricity and $c\tau + \epsilon$~with the +mean anomaly. + +\footnotemark~We can interpret~$c$ in terms of the symbols of the ordinary lunar theories. +%[** TN: Minor rewording coded using \DPtypo] +\footnotetext{\DPtypo{From here till the foot of this page} + {In the next three paragraphs} a slight knowledge of ordinary lunar theory is + supposed. The results given are not required for the further development of Hill's theory.}% +When no perturbations are considered the moon moves in an ellipse. The +\index{Apse, motion of}% +perturbations cause the moon to deviate from this simple path. If a fixed +ellipse is taken, these deviations increase with the time. It is found, +however, that if we consider the ellipse to be fixed in shape and size but with +the line of apses moving with uniform angular velocity, the actual motion of +the moon differs from this modified elliptic motion only by small periodic +quantities. If $n$~denote as before the mean sidereal motion of the moon and +$\dfrac{d\varpi}{dt}$~the mean motion of the line of apses, the argument entering into the +elliptic inequalities is~$\left(n - \dfrac{d\varpi}{dt}\right)t + \epsilon$. This must be the same as~$c\tau + \epsilon$, i.e.~as +$c(n - n')t + \epsilon$. + +Hence +\[ +n -\frac{d\varpi}{dt} = c(n - n'), +\] +giving +\begin{align*} +\frac{d\varpi}{n\, dt} + &= 1 - c \frac{n - n'}{n} \\ + &= 1 - \frac{c}{1 + m}\quad \text{since} \quad +m = \frac{n'}{ n - n'}. +\end{align*} + +A determination of~$c$ is therefore equivalent to a determination of the rate +of change of perigee; the value of~$c$ we have already obtained gives +\index{Perigee, motion of}% +\[ +\frac{d\varpi}{n\, dt} = \tfrac{3}{4} m^{2}. +\] +\DPPageSep{100}{42} + +Returning to our solution, and for simplicity again dropping the factor~$a_{0}$, +we have from \Eqref{(31)},~\Eqref{(32)} +\begin{align*} +\delta p &= -\tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr] - e \cos(c\tau + \epsilon), \\ +% +\delta s &= -\tfrac{15}{4} m e \sin\bigl[(c - 2)\tau + \epsilon\bigr] + 2e \sin(c\tau + \epsilon). +\end{align*} + +Also $\cos\phi = \cos\tau$, $\sin\phi = \sin\tau$ to the first order of small quantities, and +\[ +\delta x = \delta p \cos\phi - \delta s \sin\phi,\quad +\delta y = \delta p \sin\phi + \delta s \cos\phi. +\] +Therefore +\begin{multline*} +\delta x + = -\tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr]\cos\tau + - e \cos(c\tau + \epsilon) \cos\tau \\ +% + + \tfrac{15}{4} m e \sin\bigl[(c - 2)\tau + \epsilon\bigr]\sin\tau + - 2e \sin(c\tau + \epsilon) \sin\tau, +\end{multline*} +\begin{multline*} +\delta y + = -\tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr] \sin\tau + - e \cos(c\tau + \epsilon) \sin\tau \\ +% + - \tfrac{15}{4} m e \sin\bigl[(c - 2)\tau + \epsilon\bigr] \cos\tau + + 2e \sin(c\tau + \epsilon) \cos\tau. +\end{multline*} + +Now let $X = x + \delta x$, $Y = y + \delta y$ and we have by means of the values of $x, +y$ in the variational curve +\begin{align*} +X &= \cos\tau \bigl[1 - m^{2} + - \tfrac{3}{4} m^{2} \sin^{2}\tau + - \tfrac{15}{8} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - e \cos(c\tau + \epsilon)\bigr] \\ + &\qquad\qquad\qquad\qquad + + \sin\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr], \\ +% +Y &= \sin\tau \bigl[1 + m^{2} + + \tfrac{3}{4} m^{2} \cos^{2}\tau + - \tfrac{15}{8} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - e \cos(c\tau + \epsilon)\bigr] \\ + &\qquad\qquad\qquad\qquad + - \cos\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr]. +\end{align*} + +Writing $R^{2} = X^{2} + Y^{2}$, we obtain to the requisite degree of approximation +\begin{align*} +R^{2} &= \cos^{2}\tau \bigl[1 - 2m^{2} + - \tfrac{3}{2} m^{2} \sin^{2}\tau + - \tfrac{15}{4} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \cos(c\tau + \epsilon)\bigr] \\ +% + &+ \sin^{2}\tau \bigl[1 + 2m^{2} + + \tfrac{3}{2} m^{2} \cos^{2}\tau + - \tfrac{15}{4} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \cos(c\tau + \epsilon)\bigr] \\ +% + &+ \sin 2\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr] \\ +% + &- \sin 2\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr], \\ +% +R^{2} &= 1 - 2m^{2} \cos 2\tau + - \tfrac{15}{4} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \cos(c\tau + \epsilon). +\end{align*} + +Hence reintroducing the factor~$a_{0}$ which was omitted for the sake of brevity +\[ +R = a_{0}\bigl[1 - e \cos(c\tau + \epsilon) + - \tfrac{15}{8} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - m^{2} \cos 2\tau\bigr]. +\Tag{(35)} +\] + +This gives the radius vector; it remains to find the longitude. + +We multiply the expressions for $X, Y$ by~$1/R$,\DPnote{** Slant fraction} i.e.~by +\[ +1 + e \cos(c\tau + \epsilon) + + \tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr] + + m^{2} \cos 2\tau, +\] +and remembering that +\[ +m^{2} \cos 2\tau + = m^{2} - 2m^{2} \sin^{2}\tau + = 2m^{2} \cos^{2}\tau - m^{2}, +\] +we get +\begin{align*} +\cos\Theta + &= \cos\tau \bigl[1 - \tfrac{11}{4} m^{2} \sin^{2}\tau\bigr] + - \sin\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr], \\ +% +\sin\Theta + &= \sin\tau \bigl[1 + \tfrac{11}{4} m^{2} \cos^{2}\tau\bigr] + - \cos\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr]. +\end{align*} + +Whence +\[ +\sin(\Theta - \tau) + = \tfrac{11}{8} m^{2} \sin 2\tau + - \tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + + 2e \sin(c\tau + \epsilon), +\] +\DPPageSep{101}{43} +or to our degree of approximation +\[ +\Theta = \tau + \tfrac{11}{8} m^{2} \sin 2\tau + - \tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + + 2e \sin(c\tau + \epsilon). +\Tag{(36)} +\] + +We now transform these results into the ordinary notation. +\index{Equation, annual!of the centre}% +\index{Latitude of the moon}% + +\footnotemark~Let $l, v$ be the moon's mean and true longitudes, and $l'$~the sun's mean +\footnotetext{From here till the end of this paragraph is not a part of Hill's theory, it is merely a + comparison with ordinary lunar theories.}% +longitude. Then $\Theta$~being the moon's true longitude relatively to the moving +axes, we have +\[ +v = \Theta + l'. +\] + +Also +\begin{gather*} +\tau + l' = (n - n')t + n't =l, \\ +\therefore \tau = l - l'. +\end{gather*} + +We have seen that $c\tau + \epsilon$ is the moon's mean anomaly, or~$l - \varpi$, +\[ +\therefore (c - 2)\tau + \epsilon = l - \varpi - 2(l - l') = -(l + \varpi - 2l'). +\] + +Then substituting these values in the expressions for $R$~and~$\Theta$ and +adding~$l'$ to the latter we have on noting that $a_{0} = \a(1 - \frac{1}{6} m^{2})$ +\index{Evection}% +\[ +\left. +\begin{aligned} +R &= \a\bigl[1 - \tfrac{1}{6} m^{2} + - \UnderNote{e \cos(l - \varpi)}{equation of centre} + - \UnderNote{\tfrac{15}{8} m e \cos(l - 2l' + \varpi)}{evection} + - \UnderNote{m^{2} \cos 2(l - l')\bigr]}{variation}\Add{,} \\ +% +v &= l + \UnderNote{2e \sin (l - \varpi)}{equation of centre} + + \UnderNote{\tfrac{15}{4} m e \sin(l - 2l' + \varpi)}{evection} + + \UnderNote{\tfrac{11}{8} m^{2} \sin 2(l - l')}{variation}\Add{.} +\end{aligned} +\right\} +\Tag{(37)} +\] + +The names of the inequalities in radius vector and longitude are written +below, and the values of course agree with those found in ordinary lunar +theories. + +\Section{§ 8. }{Introduction of the Third Coordinate.} +\index{Third coordinate introduced}% +\index{Variation, the}% + +Still keeping $\Omega=0$, consider the differential equation for~$z$ in~\Eqref{(5)} +\[ +\frac{d^{2}z}{d\tau^{2}} + \frac{\kappa z}{r^{3}} + m^{2}z = 0. +\] + +From~\Eqref{(8)} +\[ +\frac{\kappa}{a_{0}^{3}} = 1 + 2m + \tfrac{3}{2} m^{2}, +\] +and from~\Eqref{(10)} +\[ +\frac{a_{0}^{3}}{r^{3}} = 1 + 3m^{2} \cos 2\tau. +\] + +The equation may therefore be written +\[ +\frac{d^{2}z}{d\tau^{2}} + z(1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau) = 0. +\] + +This is an equation of the type considered in~\SecRef{6} and therefore we +assume +\[ +z = B_{-1} \cos\bigl\{(g - 2)\tau + \zeta\bigr\} + + B_{0} \cos(g\tau + \zeta) + + B_{1} \cos\bigl\{(g + 2)\tau + \zeta\bigr\}. +\] +\DPPageSep{102}{44} + +On substitution we get +\begin{align*} +B_{-1} &\bigl[-(g - 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau \bigr] \cos\bigl[(g - 2)\tau + \zeta \bigr] \\ +% ++ B_{0} &\bigl[-g^{2} + 1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau \bigr] \cos(g\tau + \zeta) \\ +% ++ B_{1} &\bigl[-(g + 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau \bigr] \cos \bigl[(g + 2)\tau + \zeta \bigr] = 0. +\end{align*} + +The coefficients of $\cos(g\tau + \zeta)$, $\cos \bigl[(g - 2)\tau + \zeta\bigr]$, $\cos \bigl[(g + 2)\tau + \zeta\bigr]$ give +respectively +\[ +\left. +\begin{alignedat}{2} +&\tfrac{3}{2} m^{2} B_{-1} + B_{0} \bigl[-g^{2} + 1 + 2m + \tfrac{5}{2} m^{2}\bigr] + \tfrac{3}{2} m^{2} B_{1} &&= 0\Add{,} \\ +% +&B_{-1} \bigl[-(g - 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2} ] + \tfrac{3}{2} m^{2} B_{0} &&= 0\Add{,} \\ +% +&\tfrac{3}{2} m^{2} B_{0} + B_{1} \bigl[-(g + 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2}\bigr] &&= 0\Add{.} +\end{alignedat} +\right\} +\Tag{(38)} +\] + +As a first approximation drop the terms in~$m^{2}$. The first of these equations +then gives $g^{2} = 1 + 2m$. The third equation then shews that $\dfrac{B_{1}}{B_{0}}$~is of +order~$m^{2}$. But a factor~$m$ can be removed from the second equation shewing +that $\dfrac{B_{-1}}{B_{0}}$~is of order~$m$ and can only be determined to this order. Hence +$B_{1}$~can be dropped. [Cf.~pp.~\Pgref{39},~\Pgref{40}.] + +Considering terms in~$m^{2}$ we now get from the first equation +\[ +g^{2} = 1 + 2m + \tfrac{5}{2} m^{2}. +\] + +Therefore +\begin{gather*} +g = 1 + m + \tfrac{5}{4} m^{2} - \tfrac{1}{2} m^{2} + = 1 + m + \tfrac{3}{4} m^{2}, \\ +(g - 2)^{2} = (1 - m)^{2} = 1 - 2m, \text{ neglecting terms in~$m^{2}$}. +\end{gather*} + +The second equation then gives +\[ +B_{-1} = -\tfrac{3}{8} m B_{0}, +\] +and the solution is +\[ +z = B_{0} \bigl[\cos(g\tau + \zeta) - \tfrac{3}{8} m \cos\bigl\{(g - 2)\tau + \zeta\bigr\}\bigr]. +\Tag{(39)} +\] + +We shall now interpret this equation geometrically. To do so we neglect +the solar perturbation and we get +\[ +z = B_{0} \cos(g\tau + \zeta). +\Tag{(40)} +\] + +\begin{wrapfigure}{r}{1.5in} + \centering + \Input[1.5in]{p044} + \caption{Fig.~3.} +\end{wrapfigure} +Now consider the moon to move in a plane orbit inclined at angle~$i$ to +the ecliptic and let $\Omega$~be the longitude of the lunar +node, $l$~the longitude of the moon, $\beta$~the latitude. + +The right-angled spherical triangle gives +\[ +\tan\beta = \tan i \sin(l - \Omega) +\] +and therefore +\[ +z = r \tan\beta = r \tan i \sin (l - \Omega). +\] +\DPPageSep{103}{45} + +As we are only dealing with a first approximation we may put $r = a_{0}$ and +so we interpret +\begin{gather*} +B_{0} = a_{0} \tan i, \\ +g\tau + \zeta = l - \Omega -\tfrac{1}{2}\pi. +\end{gather*} + +\footnotemark~We can easily find the significance of~$g$, for differentiating this equation +\footnotetext{From here till end of paragraph is a comparison with ordinary lunar theories.}% +with respect to the time we get +\begin{gather*} +g(n - n') = n - \frac{d\Omega}{dt}, \\ +\begin{aligned} +\therefore \frac{d\Omega}{n\, dt} + &= 1 - \frac{g(n - n')}{n} \\ + &= 1 + \frac{g}{1 + m} \\ + &= -\tfrac{3}{4} m^{2} \text{ to our approximation.} +\end{aligned} +\end{gather*} +Thus we find that the node has a retrograde motion. + +We have +\begin{align*} +g\tau + \zeta + &= l - \Omega - \tfrac{1}{2}\pi, \\ +% +(g - 2)\tau + \zeta + &= l - \Omega - \tfrac{1}{2}\pi - 2(l - l') \\ +% + &= -(l - 2l' + \Omega) - \tfrac{1}{2}\pi. +\end{align*} + +If we write $s = \tan\beta$, $k = \tan i$, we find +\[ +s = k \sin(l - \Omega) + \tfrac{3}{8} m k \sin(l - 2l' + \Omega). +\Tag{(41)} +\] + +The last term in this equation is called the evection in latitude. +\index{Evection!in latitude}% + +\Section{§ 9. }{Results obtained.} + +We shall now shortly consider the progress we have made towards the +actual solution of the moon's motion. We obtained first of all a special +solution of the differential equations assuming the motion to be in the ecliptic +and neglecting certain terms in the force function denoted by~$\Omega$\footnotemark. This gave +\footnotetext{The $\Omega$~of \Pageref{20}, not that of the preceding paragraph.}% +us a disturbed circular orbit in the plane of the ecliptic. We have since +introduced the first approximation to two free oscillations about this motion, +the one corresponding to eccentricity of the orbit, the other to an inclination +of the orbit to the ecliptic. + +It is found to be convenient to refer the motion of the moon to the projection +on the ecliptic. We will denote by~$r_{1}$ the curtate radius vector, so +that $r_{1}^{2} = x^{2} + y^{2}$, $r^{2} = r_{1}^{2} + z^{2}$; the $x, y$~axes rotating as before with angular +velocity~$n'$ in the plane of the ecliptic. In determining the variational curve,~\SecRef{3}, +we put $\Omega = 0$, $r = r_{1}$. It will appear therefore that in finding the actual +motion of the moon we shall require to consider not only~$\Omega$ but new terms in~$z^{2}$. +In the next section we shall discuss the actual motion of the moon, making +use of the approximations we have already obtained. +\DPPageSep{104}{46} + +\Section{§ 10. }{General Equations of Motion and their solution.} +\index{Equations of motion}% + +We have +\[ +r_{1}^{2} = x^{2} + y^{2} \text{ and } +r^{2} = r_{1}^{2} + z^{2}. +\] + +Hence +\[ +\frac{1}{r^{3}} + = \frac{1}{r_{1}^{3}} \left(1 - \frac{3}{2}\, \frac{z^{2}}{r_{1}^{2}}\right); + \text{ and } +\frac{1}{r} + = \frac{1}{r_{1}} \left(1 - \frac{1}{2}\, \frac{z^{2}}{r_{1}^{2}}\right), +\] +to our order of accuracy. + +The original equations~\Eqref{(3)} may now be written +\[ +\left. +\begin{alignedat}{4} +\frac{d^{2}x}{d\tau^{2}} + &- 2m\, \frac{dy}{d\tau} &&+ \frac{\kappa x}{r_{1}^{3}} &&- 3m^{2}x + &&= \frac{\dd \Omega}{\dd x} + \frac{3}{2}\, \frac{\kappa z^{2}x}{r_{1}^{5}}\Add{,} \\ +% +\frac{d^{2}y}{d\tau^{2}} + &+ 2m\, \frac{dx}{d\tau} &&+ \frac{\kappa y}{r_{1}^{3}} && + &&= \frac{\dd \Omega}{\dd y} + \frac{3}{2}\, \frac{\kappa z^{2}y}{r_{1}^{5}}\Add{,} \\ +% +\frac{d^{2}z}{d\tau^{2}} + & &&+ \frac{\kappa z}{r_{1}^{3}} &&+ m^{2}z + &&= \frac{\dd \Omega}{\dd z} + \frac{3}{2}\, \frac{\kappa z^{2}}{r_{1}^{5}}\Add{.} +\end{alignedat} +\right\} +\Tag{(42)} +\] + +If we multiply by $2\dfrac{dx}{d\tau}$, $2\dfrac{dy}{d\tau}$, $2\dfrac{dz}{d\tau}$ and add, we find that the Jacobian +integral becomes +\[ +V^{2} = 2\frac{\kappa}{r_{1}} + m^{2}(3x^{2} - z^{2}) + - \frac{\kappa z^{2}}{r_{1}^{3}} + + 2\int_{0}^{\tau} \left( + \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dz}{d\tau} + \right) d\tau + C, +\Tag{(43)} +\] +where +\[ +V^{2} = V_{1}^{2} + \left(\frac{dz}{d\tau}\right)^{2} + = \left(\frac{dx}{d\tau}\right)^{2} + + \left(\frac{dy}{d\tau}\right)^{2} + + \left(\frac{dz}{d\tau}\right)^{2}. +\] + +Now +\[ +\Omega = \tfrac{3}{2} m^{2} \left(\frac{\a'^{3}}{r'^{3}}\, r^{2}\cos^{2} - x^{2}\right) + + \tfrac{1}{2} m^{2} r^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right), +\] +and +\[ +\cos\theta = \frac{xx' + yy' + zz'}{rr'} + = \frac{xx' + yy'}{rr'}, \text{ since $z' = 0$}. +\] + +Hence +\[ +\Omega = \tfrac{3}{2} m^{2} \left\{\frac{\a'^{3}}{r'^{3}}(xx' + yy')^{2} - x^{2}\right\} + + \tfrac{1}{2} m^{2} (x^{2} + y^{2}) \left(1 - \frac{\a'^{3}}{r'^{3}}\right) + + \tfrac{1}{2} m^{2} z^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right). +\] + +When we neglected $\Omega$~and~$z$, we found the solution +\begin{alignat*}{2} +x &= a_{0}\bigl[(1 - \tfrac{19}{16} m^{2})\cos\tau + &&+ \tfrac{3}{16} m^{2}\cos 3\tau\bigr], \\ +y &= a_{0}\bigl[(1 + \tfrac{19}{16} m^{2})\sin\tau + &&+ \tfrac{3}{16} m^{2}\sin 3\tau\bigr]. +\end{alignat*} + +We now require to determine the effect of the terms introduced on the +right, and for brevity we write +\[ +X = \frac{\dd \Omega}{\dd x} + \frac{3}{2}\, \frac{\kappa z^{2}x}{r_{1}^{5}},\quad +Y = \frac{\dd \Omega}{\dd y} + \frac{3}{2}\, \frac{\kappa z^{2}y}{r_{1}^{5}}. +\] + +When we refer to~\SecRef{4} and consider how the differential equations for~$\delta p, \delta s$ +were formed from those for~$\delta x, \delta y$, we see that the new terms~$X, Y$ on +the right-hand sides of the differential equations for~$\delta x, \delta y$ will lead to new +terms $X\cos\phi - Y\sin\phi$, $-X\sin\phi + Y\cos\phi$ on the right-hand sides of those +for~$\delta p, \delta s$. +\DPPageSep{105}{47} + +Hence taking the equations \Eqref{(24)}~and~\Eqref{(25)} for $\delta p$~and~$\delta s$ and introducing +these new terms, we find +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + + \delta p\, \bigl[-3 - 6m - \tfrac{9}{2}m^{2} - 5m^{2}\cos 2\tau\bigr] + - 2\frac{d\, \delta s}{d\tau}\, (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) \\ +\shoveright{-7m^{2}\, \delta s \sin 2\tau = X\cos\phi + Y\sin\phi,} \\ +% +\shoveleft{\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos 2\tau + + 2\frac{d\, \delta p}{d\tau}\, (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) + - 2m^{2}\, \delta p \sin 2\tau} \\ + = -X\sin\phi + Y\cos\phi. +\end{multline*} + +In this analysis we shall include all terms to the order~$m k^{2}$, where $k$~is the +small quantity in the expression for~$z$. Terms involving~$m^{2}z^{2}$ will therefore +be neglected. In the variation of the Jacobian integral the term~$\dfrac{dz}{d\tau}\, \dfrac{d\, \delta z}{d\tau}$ can +obviously be neglected. The variation of the Jacobian integral therefore +gives (cf.~pp.~\Pgref{29},~\Pgref{35}) +\begin{multline*} +\frac{d\, \delta s}{d\tau} + = -2\delta p\, (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) + - \tfrac{7}{2}m^{2}\, \delta s \sin 2\tau \\ +% + + \frac{1}{V_{1}} \biggl[\int_{0}^{\tau}\!\! + \left(\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau}\right) d\tau + + \tfrac{1}{2} \biggl\{\delta C - \frac{\kappa z^{2}}{r_{1}^{3}} + - \left(\frac{dz}{d\tau}\right)^{2}\biggr\} + \biggr], +\Tag{(44)} +\end{multline*} +where $\delta C$~will be chosen as is found most convenient. [In the previous work +we chose $\delta C = 0$.] + +By means of this equation we can eliminate~$\delta s$ from the differential +equation for~$\delta p$. For +\begin{align*} +2\frac{d\, \delta s}{d\tau}\, (1 &+ m - \tfrac{5}{4}m^{2}\cos 2\tau) + 7m^{2}\, \delta s \sin 2\tau \\ +% + &= -4\delta p\, (1 + 2m + m^{2} - \tfrac{5}{2}m^{2} \cos 2\tau) \\ +% + &+ \frac{2}{V_{1}} (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) + \biggl[\int_{0}^{\tau}\left( + \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau} \right) d\tau \\ +% + &+ \tfrac{1}{2} \left\{\delta C - \frac{\kappa z^{2}}{r_{1}^{3}} - \left(\frac{dz}{d\tau}\right)^{2}\right\} + \biggr], +\end{align*} +and therefore +\begin{align*} +\frac{d^{2}\delta p}{d\tau^{2}} + &+ \delta p\, (1 + 2m - \tfrac{1}{2}m^{2} - 15m^{2}\cos 2\tau) + = X\cos\phi + Y\sin\phi \\ +% + &+ \frac{2}{V_{1}} (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) + \biggl[\int_{0}^{\tau} \left( + \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau}\right) d\tau \\ +% + &+ \tfrac{1}{2}\left\{\delta C - \frac{\kappa z^{2}}{r_{1}^{3}} - \left(\frac{dz}{d\tau}\right)^{2}\right\} + \biggr]. +\Tag{(45)} +\end{align*} + +We first neglect~$\Omega$ and consider $X, Y$~as arising only from terms +in~$z^{2}$, i.e.\ +\begin{gather*} +X = \frac{3}{2}\, \frac{\kappa z^{2}x}{r_{1}^{5}},\quad +Y = \frac{3}{2}\, \frac{\kappa z^{2}y}{r_{1}^{5}}. \\ +% +\therefore X\cos\phi + Y\sin\phi + = \frac{3}{2}\, \frac{\kappa z^{2}}{r_{1}^{5}}(x\cos\phi + y\sin\phi). +\end{gather*} +\DPPageSep{106}{48} + +To the required order of accuracy. +\begin{gather*} +z = ka_{0} \cos(g\tau + \zeta),\quad \frac{\kappa}{a_{0}^{3}} = 1 + 2m, \\ +% +r_{1} = a_{0},\quad \phi = \tau,\quad x = a_{0}\cos\tau,\quad y = a_{0}\sin\tau. \\ +% +\therefore X \cos\phi + Y \sin\phi + = \tfrac{3}{4}(1 + 2m)k^{2}a_{0} \bigl[1 + \cos 2(g\tau + \zeta)\bigr]. +\end{gather*} + +Also to order~$m$ +\begin{align*} +\frac{\kappa z^{2}}{r_{1}^{3}} + \left(\frac{dz}{d\tau}\right)^{2} + &= (1 + 2m) k^{2}a_{0}^{2} \cos^{2}(g\tau + \zeta) + + g^{2}k^{2}a_{0}^{2} \sin^{2}(g\tau + \zeta) \\ +% + &= (1 + 2m) k^{2}a_{0}^{2}, +\end{align*} +since $g^{2} = 1 + 2m$. + +The equation for~$\delta p$ becomes therefore, as far as regards the new terms +now introduced, +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + \delta p\, (1 + 2m) + = \tfrac{3}{4}(1 + 2m) k^{2}a_{0} \bigl[1 + \cos 2(g\tau + \zeta)\bigr] \\ + + \frac{(1 + m)}{a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr]. +\end{multline*} + +Hence +\[ +\delta p - \tfrac{3}{4} k^{2}a_{0} + - \frac{(1 - m)}{a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr] + = \tfrac{3}{4}\frac{1 + 2m}{1 + 2m + 4g^{2}} k^{2}a_{0} \cos 2(g\tau + \zeta), \footnotemark +\] +\footnotetext{It is of course only the special integral we require. The general integral when the right-hand + side is zero has already been dealt with,~\SecRef{7}.}% +but +\begin{gather*} +g^{2} = 1 + 2m, \text{ and therefore } +1 + 2m - 4g^{2} = -3(1 + 2m), \\ +% +\therefore \delta p = \tfrac{3}{4} k^{2}a_{0} + + \frac{(1 - m)}{a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr] + - \tfrac{1}{4} k^{2}a_{0} \cos 2(g\tau + \zeta). +\end{gather*} + +Again the varied Jacobian integral is +\begin{align*} +\frac{d\, \delta s}{d\tau} + &= -2(1 + m)\, \delta p + + \frac{1}{2a_{0}} \bigl[\delta C - (1 - 2m) k^{2}a_{0}^{2}\bigr] \\ +% + &= -\tfrac{3}{2}(1 + m) k^{2}a_{0} + - \frac{3}{2a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr] + + \tfrac{1}{2}(1 + m) k^{2}a_{0} \cos 2(g\tau + \zeta). +\end{align*} + +In order that $\delta s$~may not increase with the time we choose~$\delta C$ so that the +constant term is zero, +\begin{align*} +\therefore \delta C &= m k^{2}a_{0}, +\intertext{and} +\frac{d\, \delta s}{d\tau} + &= \tfrac{1}{2}(1 - m) k^{2}a_{0} \cos 2(g\tau + \zeta), \\ +% +\intertext{giving} +\delta s &= \tfrac{1}{4} k^{2}a_{0} \sin 2(g\tau + \zeta), +\Tag{(46)} +\end{align*} +as there is no need to introduce a new constant\footnotemark. Using the value of~$\delta C$ just +\footnotetext{Cf.\ same point in connection with equation~\Eqref{(32)}.}% +found we get +\[ +\delta p = -\tfrac{1}{4} k^{2}a_{0} + - \tfrac{1}{4} k^{2}a_{0} \cos 2(g\tau + \zeta). +\Tag{(47)} +\] + +Having obtained $\delta p$~and~$\delta s$, we now require~$\delta x, \delta y$. These are +\begin{align*} +\delta x &= \delta p \cos\phi - \delta s \sin\phi, \\ +\delta y &= \delta p \sin\phi + \delta s \cos\phi. +\end{align*} +\DPPageSep{107}{49} + +In this case with sufficient accuracy $\phi = \tau$, +\begin{alignat*}{3} +\delta x + &= - \tfrac{1}{4} a_{0}k^{2} \cos\tau + &&- \tfrac{1}{4} a_{0}k^{2} \cos\tau \cos 2(g\tau + \zeta) + &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \sin 2(g\tau + \zeta), \\ +% +\delta y + &= - \tfrac{1}{4} a_{0}k^{2} \sin\tau + &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \cos 2(g\tau + \zeta) + &&+ \tfrac{1}{4} a_{0}k^{2} \cos\tau \sin 2(g\tau + \zeta). +\end{alignat*} + +Dropping the recent use of~$X, Y$ in connection with the forces and using +as before $X = x + \delta x$, $Y = y + \delta y$ we have +\begin{alignat*}{3} +X &= a_{0}\cos\tau(1 - \tfrac{1}{4}k^{2}) + &&- \tfrac{1}{4} a_{0}k^{2} \cos\tau \cos 2(g\tau + \zeta) + &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \sin 2(g\tau + \zeta), \\ +% +Y &= a_{0}\sin\tau(1 - \tfrac{1}{4}k^{2}) + &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \cos 2(g\tau + \zeta) + &&+ \tfrac{1}{4} a_{0}k^{2} \cos\tau \sin 2(g\tau + \zeta), \\ +% +R^{2} &= \rlap{$X^{2} + Y^{2} + = a_{0}^{2}(1 - \tfrac{1}{2}k^{2}) + - \tfrac{1}{2} a_{0}^{2}k^{2} \cos 2(g\tau + \zeta)$,}&&&& \\ +% +R &= \rlap{$a_{0}\bigl[1 - \tfrac{1}{4}k^{2} + - \tfrac{1}{4}k^{2} \cos 2(g\tau + \zeta)\bigr]$.}&&&& +\Tag{(48)} +\end{alignat*} + +We thus get corrected result in radius vector as projected on to the ecliptic. + +Again +\begin{alignat*}{2} +\cos\Theta &= \frac{X}{R} + &&= \cos\tau - \tfrac{1}{4} k^{2} \sin\tau \sin 2(g\tau + \zeta), \\ +% +\sin\Theta &= \frac{Y}{R} + &&= \sin\tau + \tfrac{1}{4} k^{2} \cos\tau \sin 2(g\tau + \zeta), \\ +% +\Theta - \tau + &= \rlap{$\sin(\Theta - \tau) = \tfrac{1}{4} k^{2} \sin 2(g\tau + \zeta)$.}&& +\Tag{(49)} +\end{alignat*} + +Hence we have as a term in the moon's longitude $\frac{1}{4}k^{2}\sin 2(g\tau + \zeta)$. Terms +\index{Reduction, the}% +of this type are called the reduction; they result from referring the moon's +orbit to the ecliptic. + +We have now only to consider the terms depending on~$\Omega$. We have seen +that $\Omega$~vanishes when the solar eccentricity,~$e'$, is put equal to zero. We shall +only develop~$\Omega$ as far as first power of~$e'$. + +The radius vector~$r'$, and the true longitude~$v'$, of the sun are given to the +required approximation by +\begin{align*} +r' &= \a' \bigl\{1 - e'\cos(n't - \varpi')\bigr\}, \\ +v' &= n't + 2e'\sin(n't - \varpi'). +\end{align*} + +Hence +\begin{alignat*}{2} +x' &= r'\cos(v' - n't) = r' &&= \a' \bigl\{1 - e'\cos(n't - \varpi')\bigr\}, \\ +y' &= r'\sin(v' - n't) &&= 2\a'e' \sin(n't - \varpi'). +\end{alignat*} +And +\begin{gather*} +n't = m\tau; \\ +\begin{aligned} +\therefore \frac{xx' + yy'}{\a'} + &= x - e'x \cos(m\tau - \varpi) + 2e'y \sin(m\tau - \varpi), \\ +\left(\frac{xx' + yy'}{\a'}\right)^{2} + &= x^{2} - 2e'x^{2} \cos(m\tau - \varpi) + 4e'xy \sin(m\tau - \varpi), \\ +\frac{\a'^{5}}{r'^{5}} + &= 1 + 5e' \cos(m\tau - \varpi), +\end{aligned} +\Allowbreak +\DPPageSep{108}{50} +\frac{3m^{2}}{2} \left\{\frac{\a'^{3}}{r'^{5}} (xx' + yy')^{2} - x^{2}\right\} + = \frac{9m^{2}}{2} e' x^{2} \cos(m\tau - \varpi') + + 6m^{2} e'xy \sin(m\tau - \varpi'), \\ +% +\tfrac{1}{2} m^{2} (x^{2} + y^{2} + z^{2}) \left(1 - \frac{\a'^{3}}{r'^{3}}\right) + = -\tfrac{3}{2} m^{2} (x^{2} + y^{2} + z^{2}) e' \cos(m\tau - \varpi'), \\ +\Omega + = m^{2} e' \bigl[3x^{2} \cos(m\tau - \varpi') + + 6xy \sin(m\tau - \varpi') - \tfrac{3}{2} y^{2} \cos(m\tau - \varpi') \bigr], +\end{gather*} +for we neglect~$m^{2}z^{2}$ when multiplied by~$e'$, +\begin{align*} +\frac{\dd \Omega}{\dd x} + &= 6m^{2}e' \bigl[x \cos(m\tau - \varpi') + y \sin(m\tau - \varpi')\bigr], \\ +% +\frac{\dd \Omega}{\dd y} + &= 6m^{2}e' \bigl[x \sin(m\tau - \varpi') - \tfrac{1}{2} y \cos(m\tau - \varpi')\bigr]. +\end{align*} + +It is sufficiently accurate for us to take +\begin{align*} +x &= a_{0} \cos \tau,\quad +y = a_{0} \sin \tau, \\ +\phi &= \tau; +\end{align*} +\begin{multline*} +\therefore +\frac{\dd \Omega}{\dd x} \cos\phi + +\frac{\dd \Omega}{\dd y} \sin\phi + = 6m^{2} e' a_{0} \bigl[\cos^{2}\tau \cos(m\tau - \varpi') + + \cos\tau \sin\tau \sin(m\tau - \varpi') \\ +% +\shoveright{+ \cos\tau \sin\tau \sin(m\tau - \varpi') + - \tfrac{1}{2} \sin^{2}\tau \cos(m\tau - \varpi') \bigr]} \\ +% +\shoveleft{= 3m^{2} e'a_{0} \bigl[\cos(m\tau - \varpi') + + \cos 2\tau \cos(m\tau - \varpi') + 2\sin 2\tau \sin(m\tau - \varpi') \bigr]} \\ +% +\shoveright{- \tfrac{1}{2} \cos(m\tau - \varpi') + \tfrac{1}{2} \cos2\tau \cos(m\tau - \varpi')} \\ +% +\shoveleft{= 3m^{2} e' a_{0} \bigl[\tfrac{1}{2} \cos(m\tau - \varpi') + + \tfrac{3}{4} \cos \bigl\{(2 + m)\tau - \varpi' \bigr\} + + \tfrac{3}{4} \cos \bigl\{(2 - m)\tau + \varpi' \bigr\}} \\ +% +\shoveright{+ \cos \bigl\{(2 - m)\tau + \varpi' \bigr\} + - \cos \bigl\{(2 + m)\tau - \varpi' \bigr\} \bigr]} \\ +% +\shoveleft{= \tfrac{3}{2} m^{2} e'a_{0} \bigl[\cos(m\tau - \varpi') + - \tfrac{1}{2} \cos \bigl\{(2 + m)\tau - \varpi' \bigr\} + + \tfrac{7}{2} \cos \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr].} +\end{multline*} + +Again +\begin{multline*} +\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + +\frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + = 6m^{2} e'a_{0} \bigl[-\sin\tau \cos\tau \cos(m\tau - \varpi') + - \sin^{2} \tau \sin(m\tau - \varpi') \\ +% +\shoveright{+ \cos^{2} \tau \sin(m\tau - \varpi') + - \tfrac{1}{2} \sin\tau \cos\tau \cos(m\tau - \varpi') \bigr]} \\ +% +\shoveleft{= 3m^{2} e'a_{0} \bigl[-\tfrac{3}{2} \sin 2\tau \cos(m\tau - \varpi') + + 2 \cos 2\tau \sin(m\tau - \varpi') \bigr]} \\ +% +\shoveleft{= \tfrac{3}{2} m^{2} e'a_{0} \bigl[-\tfrac{3}{2} \sin \bigl\{(2 + m)\tau - \varpi' \bigr\} + - \tfrac{3}{2} \sin \bigl\{(2 - m)\tau + \varpi' \bigr\}} \\ +% +\shoveright{+ 2\sin \bigl\{(2 + m)\tau - \varpi' \bigr\} + - 2\sin \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr]} \\ +% +\shoveleft{= \tfrac{3}{2} m^{2} e'a_{0} \bigl[\tfrac{1}{2} \sin \bigl\{(2 + m)\tau - \varpi' \bigr\} + - \tfrac{7}{2} \sin \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr],} \\ +% +\shoveleft{2 \int \left(\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}\right) d\tau + = -\tfrac{3}{2} m^{2} e'a_{0} \bigl[\tfrac{1}{2} \cos \bigl\{(2 + m)\tau - \varpi' \bigr\}} \\ +% +\shoveright{- \tfrac{7}{2} \cos \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr];} \\ +% +\shoveleft{\therefore + \frac{\dd \Omega}{\dd x} \cos\phi + + \frac{\dd \Omega}{\dd y} \sin\phi + + 2 \int \left(\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}\right) d\tau + = \tfrac{3}{2} m^{2} e'a_{0} \bigl[\cos(m\tau - \varpi')} \\ +% + - \cos \bigl\{(2 + m)\tau - \varpi' \bigr\} + + 7\cos \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr]. +\end{multline*} +\DPPageSep{109}{51} + +Hence to the order required +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + (1 + 2m)\, \delta p = \tfrac{3}{2} m^{2} e'a_{0} +\bigl[ +\cos(m\tau - \varpi') - \cos \left\{(2 + m) \tau - \varpi'\right\} \\ + + 7 \cos \left\{(2 - m)\tau + \varpi'\right\}\bigr], +\end{multline*} +\[ +\begin{aligned} +\delta p &= \tfrac{3}{2} m^{2} e'a_{0} + \left[\frac{\cos(m\tau - \varpi')}{-m^{2} + 1 + 2m} + - \frac{ \cos\left\{(2 + m)\tau - \varpi'\right\}}{-(4 + 4m) + 1 + 2m} + + \frac{7\cos\left\{(2 - m)\tau + \varpi'\right\}}{-(4 - 4m) + 1 + 2m}\right] \\ +% + &= \tfrac{3}{2} m^{2} e'a_{0} + \left[\cos(m\tau - \varpi') + + \tfrac{1}{3} \cos \left\{(2 + m)\tau - \varpi'\right\} + - \tfrac{7}{3} \cos \left\{(2 - m)\tau + \varpi'\right\}\right]\Add{,} +\end{aligned} +\Tag{(50)} +\] +{\setlength{\abovedisplayskip}{0pt}% +\setlength{\belowdisplayskip}{0pt}% +\begin{multline*} +\frac{d\, \delta s}{d\tau} + = -2\delta p\, (1 + m) + + \frac{1}{V}\int \left(\frac{d\Omega}{dx}\, \frac{dx}{d\tau} + +\frac{d\Omega}{dy}\, \frac{dy}{d\tau}\right) d\tau \\ +% +\shoveleft{= -3m^{2} e'a_{0} \left[\cos(m\tau - \varpi') + + \tfrac{1}{3}\cos\left\{(2 + m)\tau - \varpi'\right\} + - \tfrac{7}{3}\cos\left\{(2 - m)\tau + \varpi'\right\}\right]} \\ +% +\shoveright{- \tfrac{3}{4} m^{2} e'\left[\tfrac{1}{2} \cos\left\{(2 + m)\tau - \varpi'\right\} + - \tfrac{7}{2} \cos \left\{(2 - m)\tau + \varpi'\right\}\right]} \\ +% +\shoveleft{= -3m^{2}e'a_0 \bigl[\cos(m\tau - \varpi') + + \tfrac{11}{24} \cos\left\{(2 + m)\tau - \varpi'\right\}} \\ +% +\shoveright{-\tfrac{77}{24} \cos\left\{(2 - m)\tau + \varpi'\right\}\bigr];} \\ +\end{multline*} +\begin{multline*} +\therefore \delta s = - 3m e'a_{0} \sin(m\tau - \varpi') + - 3m^{2} e'a_{0} \bigl[\tfrac{11}{48} \sin \left\{(2 + m) \tau - \varpi'\right\} \\ + - \tfrac{77}{48} \sin\left\{(2 - m)\tau + \varpi'\right\}\bigr]\Add{.} +\Tag{(51)} +\end{multline*}} + +Hence to order~$m e'$, to which order only our result is correct, +\[ +\delta p = 0, \quad +\delta s = -3m e'a_{0} \sin (m\tau - \varpi'). +\] + +And following our usual method for obtaining new terms in radius vector +and longitude +\begin{align*} +\delta x &= \delta p \cos \phi - \delta s \sin \phi, \quad +\delta y = \delta p \sin \phi + \delta s \cos \phi, \\ +\delta x &= +%[** TN: Hack to align second equation with previous second equation] + \settowidth{\TmpLen}{$\delta p \cos \phi - \delta s \sin \phi$,\quad} + \makebox[\TmpLen][l]{$- \delta s \sin \tau$,}\, +\delta y = \delta s \cos \tau, \\ +X &= a_{0} \left[\cos \tau + 3m e' \sin \tau \sin (m\tau - \varpi')\right], \\ +Y &= a_{0} \left[\sin \tau - 3m e' \cos \tau \sin (m\tau - \varpi')\right], \\ +R^{2} &= a_{0}^{2} \left[1 + 3m e' \sin 2\tau \sin (m\tau - \varpi') + - 3m e' \sin 2\tau \sin (m\tau - \varpi')\right] = a_{0}^{2}, \\ +\Tag{(52)} +\end{align*} +and to the order required there is no term in radius vector +\begin{align*} +\cos \Theta &= \cos \tau + 3m e' \sin \tau \sin (m\tau - \varpi'),\\ +\sin \Theta &= \sin \tau - 3m e' \cos \tau \sin (m\tau - \varpi'),\\ +\sin (\Theta - \tau) &= - 3m e' \sin (m\tau - \varpi'),\\ +\Theta &= \tau - 3m e' \sin(m\tau - \varpi'). +\Tag{(53)} +\end{align*} + +The new term in the longitude is~$-3m e' \sin (l' - \varpi')$. This term is called +the annual equation. +\index{Annual Equation}% +\index{Equation, annual}% +\DPPageSep{110}{52} + +\Section{§ 11. }{Compilation of Results.} + +Let $v$~be the longitude, $s$~the tangent of the latitude (or to our order +simply the latitude). When we collect our results we find +\begin{align*} +v &= \settowidth{\TmpLen}{longitude}% + \UnderNote{\makebox[\TmpLen][c]{$l$}}{% + \parbox[c]{\TmpLen}{\centering(mean\\ longitude\\ ${}= nt + \epsilon$)}} + + \UnderNote{2e \sin (l - \varpi)}{% + \settowidth{\TmpLen}{equation to}% + \parbox[c]{\TmpLen}{\centering equation to\\ the centre}} + + \UnderNote{\tfrac{15}{4} m e \sin(l - 2l' + \varpi)}{evection} + + \UnderNote{\tfrac{11}{8} m^2 \sin2(l - l')}{variation} \\ +% + &\qquad\qquad\qquad + \UnderNote{-\tfrac{1}{4} k^{2} \sin 2(l - \Omega)}{reduction} + - \UnderNote{3m e' \sin(l' - \varpi')}{annual equation}, \\ +% +s &= k \sin(l - \Omega) + + \UnderNote{\tfrac{3}{8} m k \sin(l - 2l' + \Omega)}{evection in latitude}. +\end{align*} + +For~$R$, the projection of the radius vector on the ecliptic, we get +\begin{multline*} +R = \a\bigl[1 - \tfrac{1}{6} m^{2} - \tfrac{1}{4} k^{2} + - \UnderNote{e \cos(l - \varpi)}{% + \settowidth{\TmpLen}{equation to the}% + \parbox[c]{\TmpLen}{\centering equation to the\\ centre}} + - \UnderNote{\tfrac{15}{8} m e \cos(l - 2l' - \varpi)}{evection} + - \UnderNote{m^{2} \cos 2(l - l')}{variation} \\ +% + + \UnderNote{\tfrac{1}{4} k^{2} \cos 2(l - \Omega)}{reduction}\bigr]. +\Tag{(54)} +\end{multline*} + +To get the actual radius vector we require to multiply by~$\sec\beta$, i.e.~by +\[ +1 + \tfrac{1}{2} k^{2} \sin^{2}(l - \Omega) \text{ or } +1 + \tfrac{1}{4} k^{2} - \tfrac{1}{4} k^{2} \cos 2(l - \Omega). +\] + +This amounts to removing the terms $-\frac{1}{4}k^{2} + \frac{1}{4}k^{2}\cos2(l - \Omega)$. The radius +vector then is +\[ +\a \bigl[1 - \tfrac{1}{6} m^{2} - e \cos(l - \varpi) + - \tfrac{15}{8} m e \cos(l - 2l' + \varpi) - m^{2} \cos2(l - l')\bigr]. +\] + +This is independent of~$k$, but $k$~will enter into product terms of higher +order than we have considered. The perturbations are excluded by putting +$m = 0$ and the value of the radius vector is then independent of~$k$ as it +should be. The quantity of practical importance is not the radius vector but +its reciprocal. To our degree of approximation it is +\[ +\frac{1}{\a}\bigl[1 + \tfrac{1}{6} m^{2} + e \cos(l - \varpi) + + \tfrac{15}{8} m e \cos(l - 2l' + \varpi) + m^{2}\cos2(l - l')\bigr]. +\] + +It may be noted in conclusion that the terms involving only~$e$ in the +coefficient, and designated the equation to the centre, are not perturbations +but the ordinary elliptic inequalities. There are terms in~$e^{2}$ but these have +not been included in our work. +\DPPageSep{111}{53} + +\Note{1.}{On the Infinite Determinant of \SecRef{5}.} +\index{Hill, G. W., Lunar Theory!infinite determinant}% +\index{Infinite determinant, Hill's}% + +We assume (as has been justified by Poincaré) that we may treat the +infinite determinant as though it were a finite one. + +For every row corresponding to~$+i$ there is another corresponding to~$-i$, +and there is one for~$i =0$. + +If we write~$-c$ for~$c$ the determinant is simply turned upside down. +Hence the roots occur in pairs and if $c_{0}$~is a root $-c_{0}$~is also a root. + +If for $c$ we write~$c ± 2j$, where $j$~is an integer, we simply shift the centre +of the determinant. + +Hence if $c_{0}$~is a root, $± c_{0} ± 2j$~are also roots. + +But these are the roots of $\cos \pi c = \cos \pi c_{0}$. + +Therefore the determinant must be equal to +\[ +k(\cos \pi c - \cos \pi c_{0}). +\] + +If all the roots have been enumerated, $k$~is independent of~$c$. + +Now the number of roots cannot be affected by the values assigned to +the~$\Theta$'s. Let us put $\Theta_{1} = \Theta_{2} = \Theta_{3} = \dots = 0$. + +The determinant then becomes equal to the product of the diagonal terms +and the equation is +\[ +\dots \bigl[\Theta_{0} - (c - 2)^{2}\bigr] + \bigl[\Theta_{0} - c^{2}\bigr] + \bigl[\Theta_{0} - (c + 2)^{2}\bigr] \dots = 0. +\] + +$c_{0} = ±\Surd{\Theta_{0}}$ is one pair of roots, and all the others are given by~$c_{0} ± 2i$. + +Hence there are no more roots and $k$~is independent of~$c$. + +The determinant which we have obtained is inconvenient because the +diagonal elements increase as we pass away from the centre while the non-diagonal +elements are of the same order of magnitude for all the rows. But +the roots of the determinant are not affected if the rows are multiplied by +numerical constants and we can therefore introduce such numerical multipliers +as we may find convenient. + +The following considerations indicate what multipliers may prove useful. +If we take a finite determinant from the centre of the infinite one it can be +completely expanded by the ordinary processes. Each of the terms in the +expansion will only involve~$c$ through elements from the principal diagonal +and the term obtained by multiplying all the elements of this diagonal will +contain the highest power of~$c$. When the determinant has $(2i + 1)$ rows +and columns, the highest power of~$c$ will be~$-c^{2(2i + 1)}$. We wish to associate +the infinite determinant with~$\cos \pi c$. Now +\[ +\cos \pi c + = \left(1 - \frac{4c^{2}}{1}\right) + \left(1 - \frac{4c^{2}}{9}\right) + \left(1 - \frac{4c^{2}}{25}\right) \dots. +\] +\DPPageSep{112}{54} + +The first $2i + 1$~terms of this product may be written +\[ +\left(1 - \frac{2c}{4i + 1}\right) +\left(1 - \frac{2c}{4i - 1}\right) \dots +\left(1 + \frac{2c}{4i - 1}\right) +\left(1 + \frac{2c}{4i + 1}\right), +\] +and the highest power of~$c$ in this product is +\[ +\frac{4c^{2}}{(4i)^{2} - 1} · \frac{4c^{2}}{\bigl\{4(i - 1)\bigr\}^{2} - 1} \dots \frac{4c^{2}}{(4i)^{2} - 1}. +\] + +Hence we multiply the $i$th~row below or above the central row by~$\dfrac{-4}{(4i)^{2} - 1}$. +The $i$th~diagonal term below the central term will now be~$\dfrac{4\bigl[(2i + c)^{2} - \Theta_{0}\bigr]}{(4i)^{2} - 1}$ +and will be denoted by~$\{i\}$. It clearly tends to unity as $i$~tends to infinity by +positive or negative values. The $i$th~row below the central row will now +read +\[ +\dots +\frac{-4\Theta_{2}}{(4i)^{2} - 1},\quad +\frac{-4\Theta_{1}}{(4i)^{2} - 1},\quad \{i\},\quad +\frac{-4\Theta_{1}}{(4i)^{2} - 1},\quad +\frac{-4\Theta_{2}}{(4i)^{2} - 1},\dots. +\] + +The new determinant which we will denote by~$\nabla (c)$ has the same roots +as the original one and so we may write +\[ +\nabla (c) = k' \{\cos \pi c - \cos \pi c_{0}\}, +\] +where $k'$~is a new numerical constant. But it is easy to see that~$k' = 1$. +This was the object of introducing the multipliers and that it is true is easily +proved by taking the case of $\Theta_{1} = \Theta_{2} = \dots = 0$ and $\Theta_{0} = \frac{1}{4}$, in which case the +determinant reduces to~$\cos \pi c$. We thus have the equation +\[ +\nabla (c) = \cos \pi c - \cos \pi c_{0}, +\] +which can be considered as an identity in~$c$. + +Putting $c = 0$ we get +\[ +\nabla (0) = 1 - \cos \pi c_{0}. +\] + +$\nabla (0)$~depends only on the~$\Theta$'s; written so as to shew the principal elements +it is +\[ +\left\lvert +\begin{array}{@{}c *{5}{r} c@{}} +\multicolumn{7}{c}{\dotfill} \\ +\dots & \tfrac{4}{63}(16-\Theta_{0}),& -\tfrac{4}{63}\Theta_{1},& -\tfrac{4}{63}\Theta_{2},& -\tfrac{4}{63}\Theta_{3},& -\tfrac{4}{63}\Theta_{4},& \dots \\ +\dots & -\tfrac{4}{15}\Theta_{1},& \tfrac{4}{15}(4-\Theta_{0}),& -\tfrac{4}{15}\Theta_{1},& -\tfrac{4}{15}\Theta_{2},& -\tfrac{4}{15}\Theta_{3},& \dots \\ +\dots & 4\Theta_{2},& 4\Theta_{1},& 4\Theta_{0},& 4\Theta_{1},& 4\Theta_{2},& \dots \\ +\dots & -\tfrac{4}{15}\Theta_{3},& -\tfrac{4}{15}\Theta_{2},& -\tfrac{4}{15}\Theta_{1},& \tfrac{4}{15}(4-\Theta_{0}),& -\tfrac{4}{15}\Theta_{1},& \dots \\ +\dots & -\tfrac{4}{63}\Theta_{4},& -\tfrac{4}{63}\Theta_{3},& -\tfrac{4}{63}\Theta_{2},& -\tfrac{4}{63}\Theta_{1},& \tfrac{4}{63}(16-\Theta_{0}),& \dots \\ +\multicolumn{7}{c}{\dotfill} +\end{array} +\right\rvert +\] + +{\stretchyspace +If $\Theta_{1}, \Theta_{2}$,~etc.\ vanish, the solution of the differential equation is $\cos(\Surd{\Theta_{0}} + \epsilon)$ +or~$c = \Surd{\Theta_{0}}$. But in this case the determinant has only diagonal terms and +the product of the diagonal terms of~$\nabla (0)$ is~$1 - \cos \pi \Surd{\Theta_{0}}$ or~$2 \sin^{2} \frac{1}{2}\pi\Surd{\Theta_{0}}$.} +\DPPageSep{113}{55} + +Hence we may divide each row by its diagonal member and put +$2 \sin^{2} \frac{1}{2} \pi \Surd{\Theta_{0}}$ outside. + +If therefore +{\small +\begin{align*} +\Delta(0) &= \left\lvert +\begin{array}{@{}c *{5}{>{\ }c@{,\ }} c} +\multicolumn{7}{c}{\dotfill} \\ +\dots & 1 & -\dfrac{\Theta_{1}}{16-\Theta_{0}}& -\dfrac{\Theta_{2}}{16-\Theta_{0}}& -\dfrac{\Theta_{3}}{16-\Theta_{0}}& -\dfrac{\Theta_{4}}{16-\Theta_{0}} & \dots \\ +\dots & -\dfrac{\Theta_{1}}{4-\Theta_{0}} & 1 & -\dfrac{\Theta_{1}}{4-\Theta_{0}} & -\dfrac{\Theta_{2}}{4-\Theta_{0}} & -\dfrac{\Theta_{3}}{4-\Theta_{0}} & \dots \\ +\dots & \dfrac{\Theta_{2}}{\Theta_{0}} & \dfrac{\Theta_{1}}{\Theta_{0}} & 1 & \dfrac{\Theta_{1}}{\Theta_{0}} & \dfrac{\Theta_{2}}{\Theta_{0}} & \dots \\ +\dots & -\dfrac{\Theta_{3}}{4-\Theta_{0}} & -\dfrac{\Theta_{2}}{4-\Theta_{0}} & -\dfrac{\Theta_{1}}{4-\Theta_{0}} & 1 & -\dfrac{\Theta_{1}}{4-\Theta_{0}} & \dots \\ +\multicolumn{7}{c}{\dotfill} +\end{array} +\right\rvert +\\ +\nabla(0) &= 2 \sin^{2} \tfrac{1}{2} \pi\Surd{\Theta_{0}} \Delta(0). +\end{align*}} + +Now since +\[ +\cos \pi c_{0} = 1 - \nabla (0) + = 1 - 2 \sin^{2} \frac{1}{2} \pi \Surd{\Theta_{0}} \Delta(0), +\] +we have +\Pagelabel{55} +\[ +\frac{\sin^{2} \frac{1}{2} \pi c_{0}}{\sin^{2} \frac{1}{2} \pi \Surd{\Theta_{0}}} + = \Delta(0), +\] +an equation to be solved for~$c_{0}$ (or~$c$). + +Clearly for stability $\Delta(0)$~must be positive and $\Delta(0) < \cosec^2 \frac{1}{2} \pi \Surd{\Theta_{0}}$. +Hill gives other transformations. + +\Note{2\footnotemark.}{On the periodicity of the integrals of the equation +\footnotetext{This treatment of the subject was pointed out to Sir~George Darwin by Mr~S.~S. Hough.} +\begin{gather*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + \Theta\, \delta p = 0, \\ +\lintertext{where} +{\Theta = \Theta_{0} + + \Theta_{1} \cos 2\tau + + \Theta_{2} \cos 4\tau + \dots.} +\end{gather*}} +\index{Differential Equation, Hill's!periodicity of integrals of}% +\index{Hill, G. W., Lunar Theory!periodicity of integrals of}% +\index{Periodicity of integrals of Hill's Differential Equation}% + +Since the equation remains unchanged when $\tau$ becomes~$\tau + \pi$, it follows +that if $\delta p = F(\tau)$ is a solution $F(\tau + \pi)$ is also a solution. + +Let $\phi(\tau)$~be a solution subject to the conditions that when +\[ +\tau=0,\quad +\delta p = 1,\quad +\frac{d\, \delta p}{d\tau} = 0; \text{ i.e.\ } \phi(0) = 1,\quad +\phi'(0) = 0. +\] + +Let $\psi(\tau)$~be a second solution subject to the conditions that when +\[ +\tau=0,\quad +\delta p = 0,\quad +\frac{d\, \delta p}{d\tau} = 1; \text{ i.e.\ } \psi(0) = 0,\quad +\psi'(0) = 1. +\] +\DPPageSep{114}{56} + +It is clear that $\phi(\tau)$ is an even function of~$\tau$, and $\psi(\tau)$~an odd one, so +that +\begin{alignat*}{2} +\phi (-\tau) &= \Neg\phi(\tau),&\qquad \psi(-\tau)&= -\psi(\tau),\\ +\phi'(-\tau) &= -\phi(\tau),&\qquad \psi'(-\tau)&= \Neg\psi(\tau). +\end{alignat*} +Then the general solution of the equation is +\[ +\delta p = F(\tau) = A\phi(\tau) + B\psi(\tau), +\] +where $A$~and~$B$ are two arbitrary constants. + +Since $\phi(\tau + \pi)$, $\psi(\tau + \pi)$ are also solutions of the equation, it follows +that +\[ +\left. +\begin{aligned} +\phi(\tau + \pi) &= \alpha\phi(\tau) + \beta \psi(\tau)\Add{,} \\ +\psi(\tau + \pi) &= \gamma\phi(\tau) + \delta\psi(\tau)\Add{,} +\end{aligned} +\right\} +\Tag{(55)} +\] +where $\alpha, \beta, \gamma, \delta$ are definite constants. + +If possible let $A : B$ be so chosen that +\[ +F(\tau + \pi) = \nu F(\tau), +\] +where $\nu$~is a numerical constant. + +When we substitute for~$F$ its values in terms of $\phi$~and~$\psi$, we obtain +\[ +A\phi(\tau + \pi) + B\psi(\tau + \pi) = \nu\bigl[A\phi(\tau) + B\psi(\tau)\bigr]. +\] + +Further, substituting for $\phi(\tau + \pi)$, $\psi(\tau + \pi)$ their values, we have +\[ +A\bigl[\alpha\phi(\tau) + \beta \psi(\tau)\bigr] + +B\bigl[\gamma\phi(\tau) + \delta\psi(\tau)\bigr] + = \nu\bigl[A\phi(\tau) + B\psi(\tau)\bigr], +\] +whence +\[ +\bigl[A(\alpha - \nu) + B\gamma\bigr] \phi(\tau) + + \bigl[A\beta + B(\delta - \nu)\bigr] \psi(\tau) = 0. +\] + +Since this is satisfied for all values of~$\tau$, +\begin{align*} +A(\alpha - \nu) + B\gamma &= 0,\\ +A\beta + B(\delta - \nu) &= 0,\\ +\therefore(\alpha - \nu)(\delta - \nu) - \beta\gamma &= 0,\\ +\text{i.e.}\quad +\nu^{2} - (\alpha + \delta)\nu + \alpha\delta - \beta\gamma &= 0, +\end{align*} +an equation for $\nu$ in terms of the constants $\alpha, \beta, \gamma, \delta$. This equation can be +simplified. + +Since +\[ +\frac{d^{2}\phi}{d\tau^{2}} + \Theta\phi = 0,\qquad +\frac{d^{2}\psi}{d\tau^{2}} + \Theta\psi = 0, +\] +we have +\[ +\phi \frac{d^{2}\psi}{d\tau^{2}} - \psi \frac{d^{2}\phi}{d\tau^{2}} = 0. +\] +On integration of which +\[ +\phi\psi' - \psi\phi' = \text{const.} +\] +But +\[ +\phi(0) = 1,\quad +\psi'(0) = 1,\quad +\psi(0) = 0,\quad +\phi'(0) = 0. +\] + +Therefore the constant is unity; and +\[ +\phi(\tau)\psi'(\tau) - \psi(\tau)\phi'(\tau) = 1. +\Tag{(56)} +\] +\DPPageSep{115}{57} +But putting $\tau = 0$ in the equations~\Eqref{(55)}, and in the equations obtained by +differentiating them, +\begin{alignat*}{3} +\phi(\pi) &= \alpha\,\phi\,(0) &&+ \beta\,\psi(0) &&= \alpha,\\ +\psi(\pi) &= \gamma\,\phi\,(0) &&+ \delta\,\psi\,(0) &&= \gamma,\\ +\phi'(\pi) &= \alpha\phi'(0) &&+ \beta\psi'(0) &&= \beta,\\ +\psi'(\pi) &= \gamma\phi'(0) &&+ \delta\,\psi'(0) &&= \delta. +\end{alignat*} + +Therefore by~\Eqref{(56)}, +\[ +\alpha\delta - \beta\gamma = 1. +\] +Accordingly our equation for~$\nu$ is +\[ +\nu^{2} - (\alpha + \delta)\nu + 1 = 0 +\] +or +\[ +\tfrac{1}{2} \left(\nu + \frac{1}{\nu)}\right) = \tfrac{1}{2} (\alpha + \delta). +\] + +If now we put $\tau = -\frac{1}{2}\pi$ in~\Eqref{(55)} and the equations obtained by +differentiating them, +\begin{align*} +&\begin{alignedat}{4} +\phi(\tfrac{1}{2}\pi) + &= \alpha\phi(-\tfrac{1}{2}\pi) &&+ \beta\psi(-\tfrac{1}{2}\pi) + &&= \Neg\alpha\phi(\tfrac{1}{2}\pi) &&- \beta\psi(\tfrac{1}{2}\pi), \\ +% +\psi(\tfrac{1}{2}\pi) + &= \gamma\phi(-\tfrac{1}{2}\pi) &&+ \delta\psi(-\tfrac{1}{2}\pi) + &&= \Neg\gamma\phi(\tfrac{1}{2}\pi) &&- \delta\psi(\tfrac{1}{2}\pi), \\ +% +\phi'(\tfrac{1}{2}\pi) + &= \alpha\phi'(-\tfrac{1}{2}\pi) &&+ \beta\psi'(-\tfrac{1}{2}\pi) + &&= -\alpha\phi'(\tfrac{1}{2}\pi) &&+ \beta\psi'(\tfrac{1}{2}\pi), \\ +% +\psi'(\tfrac{1}{2}\pi) + &= \gamma\phi'(-\tfrac{1}{2}\pi) &&+ \delta\psi'(-\tfrac{1}{2}\pi) + &&= -\gamma\phi'(\tfrac{1}{2}\pi) &&+ \delta\psi'(\tfrac{1}{2}\pi),\\ +\end{alignedat} +\Allowbreak +&\frac{\phi(\tfrac{1}{2}\pi)}{\psi(\tfrac{1}{2}\pi)} + = \frac{\beta}{\alpha - 1} + = \frac{\delta + 1}{\gamma},\quad +\frac{\psi'(\tfrac{1}{2}\pi)}{\phi'(\tfrac{1}{2}\pi)} + = \frac{\alpha + 1}{\beta} + = \frac{\gamma}{\delta - 1}, \\ +% +&\frac{\phi(\tfrac{1}{2}\pi) \psi'(\tfrac{1}{2}\pi)} + {\psi(\tfrac{1}{2}\pi) \phi'(\tfrac{1}{2}\pi)} + = \frac{\alpha + 1}{\alpha - 1} = \frac{\delta + 1 }{\delta - 1}. +\end{align*} + +But since $\phi(\frac{1}{2}\pi)\psi'(\frac{1}{2}\pi) - \phi'(\frac{1}{2}\pi)\psi(\frac{1}{2}\pi) = 1$ we have +\[ +\alpha = \delta = \tfrac{1}{2}(\alpha + \delta) + = \phi (\tfrac{1}{2}\pi) \psi'(\tfrac{1}{2}\pi) + + \phi'(\tfrac{1}{2}\pi) \psi (\tfrac{1}{2}\pi). +\] + +Hence the equation for~$\nu$ may be written in five different forms, viz.\ +\begin{align*} +\tfrac{1}{2}\left(\nu + \frac{1}{\nu}\right) + &= \phi(\pi) = \psi'(\pi) + = \phi (\tfrac{1}{2}\pi)\psi'(\tfrac{1}{2}\pi) + + \phi'(\tfrac{1}{2}\pi)\psi (\tfrac{1}{2}\pi) \\ + &= 1 + 2\phi'(\tfrac{1}{2}\pi)\psi(\tfrac{1}{2}\pi) + = 2\phi(\tfrac{1}{2}\pi) \psi'(\tfrac{1}{2}\pi) - 1. +\Tag{(57)} +\end{align*} + +It remains to determine the meaning of~$\nu$ in terms of the~$c$ introduced in +the solution by means of the infinite determinant. + +The former solution was +\[ +\delta p = \sum_{-\infty}^{+\infty} + \bigl\{A_{j} \cos(c + 2j)\tau + B_{j} \sin(c + 2j)\tau\bigr\}, +\] +where +\[ +A_{j} : B_{j} \text{ as } -\cos\epsilon : \sin\epsilon. +\] +In the solution $\phi(\tau)$ we have $\phi(0) = 1$, $\phi'(0) = 0$, and $\phi(\tau)$~is an even +function of~$\tau$. Hence to get~$\phi(\tau)$ from~$\delta p$ we require to put $\sum A_{j} = 1$, and +$B_{j} = 0$ for all values of~$j$. +\DPPageSep{116}{58} + +This gives +\begin{align*} +\phi(\pi) &= \sum \bigl\{A_{j} \cos(c + 2j)\pi\bigr\} \\ + &=\cos\pi c \sum A_{j} = \cos\pi c. +\end{align*} +Similarly we may shew that $\psi'(\pi) = \cos\pi c$. + +It follows from equations~\Eqref{(57)} that +\begin{align*} +\cos\pi c &= \phi(\pi) = \psi'(\pi),\\ +\cos^{2} \tfrac{1}{2}\pi c + &= \phi(\tfrac{1}{2}\pi)\psi'(\tfrac{1}{2}\pi);\quad +\sin^{2} \tfrac{1}{2}\pi c + = -\phi'(\tfrac{1}{2}\pi)\psi(\tfrac{1}{2}\pi). +\end{align*} + +We found on \Pageref{55} that $\sin^{2} \frac{1}{2}\pi c = \sin^{2} \frac{1}{2}\pi \sqrt{\Theta_{0}} · \Delta(0)$, where $\Delta(0)$~is a +certain determinant. Hence the last solution being of this form, we have +the value of the determinant~$\Delta(0)$ in terms of $\phi$~and~$\psi$, viz.\ +\[ +\Delta(0) = - \frac{\phi'(\frac{1}{2}\pi)\psi(\frac{1}{2}\pi)} + {\sin^{2} \frac{1}{2}\pi \sqrt{\Theta_{0}}}. +\] + +From this new way of looking at the matter it appears that the value of~$c$ +may be found by means of the two special solutions $\phi$~and~$\psi$. +\DPPageSep{117}{59} + + +\Chapter{On Librating Planets and on a New Family +of Periodic Orbits} +\SetRunningHeads{On Librating Planets}{and on a New Family of Periodic Orbits} + +\Section{§ 1. }{Librating Planets.} +\index{Brown, Prof.\ E. W., Sir George Darwin's Scientific Work!new family of periodic orbits}% +\index{Librating planets}% +\index{Periodic orbits, Darwin begins papers on!new family of}% + +\First{In} Professor Ernest Brown's interesting paper on ``A New Family of +Periodic Orbits'' (\textit{M.N.}, \textit{R.A.S.}, vol.~\Vol{LXXI.}, 1911, p.~438) he shews how to +obtain the orbit of a planet which makes large oscillations about the vertex +of the Lagrangian equilateral triangle. In discussing this paper I shall +depart slightly from his notation, and use that of my own paper on ``Periodic +Orbits'' (\textit{Scientific Papers}, vol.~\Vol{IV.}, or \textit{Acta Math.}, vol.~\Vol{LI.}). ``Jove,''~J, of +mass~$1$, revolves at distance~$1$ about the ``Sun,''~S, of mass~$\nu$, and the orbital +angular velocity is~$n$, where~$n^{2} = \nu + 1$. + +{\stretchyspace +The axes of reference revolve with SJ~as axis of~$x$, and the heliocentric +and jovicentric rectangular coordinates of the third body are $x, y$ and +$x - 1, y$ respectively. The heliocentric and jovicentric polar co-ordinates\DPnote{[** TN: Hyphenated in original]} are +respectively $r, \theta$ and $\rho, \psi$. The potential function for relative energy is~$\Omega$.} + +The equations of motion and Jacobian integral, from which Brown +proceeds, are +\[ +\left. +\begin{gathered} +\begin{aligned} +\frac{d^{2}r}{dt^{2}} + - r \frac{d\theta}{dt} \left(\frac{d\theta}{dt} + 2n\right) + &= \frac{\dd \Omega}{\dd r}\Add{,} \\ +% +\frac{d}{dt} \left[r^{2} \left(\frac{d\theta}{dt} + n\right)\right] + &= \frac{\dd \Omega}{\dd \theta}\Add{,} \\ +% +\left(\frac{dr}{dt}\right)^{2} + + \left(r \frac{d\theta}{dt}\right)^{2} &= 2\Omega - C\Add{,} +\end{aligned} \\ +\lintertext{where}{2\Omega + = \nu\left(r^{2} + \frac{2}{r}\right) + \left(\rho^{2} + \frac{2}{\rho}\right)\Add{,}} +\end{gathered} +\right\} +\Tag{(1)} +\] + +The following are rigorous transformations derived from those equations, +virtually given by Brown in approximate forms in equation~(13), and at the +foot of p.~443:--- +\DPPageSep{118}{60} +\begin{align*} +\left(\frac{d\theta}{dt} + n\right)^{2} + &= A + \frac{1}{r}\, \frac{d^{2}r}{dt^{2}}, +\Tag{(2)} +\Allowbreak +% +\frac{dr}{dt} \left(L + 3 \frac{d^{2}r}{dt^{2}}\right) + &= B + D \left(\frac{d\theta}{dt} + n\right) - r \frac{d^{3}r}{dt^{3}}, +\Tag{(3)} +\Allowbreak +% +\frac{d^{2}r}{dt^{2}} \left(L + 3 \frac{d^{2}r}{dt^{2}}\right) + &= E \left(\frac{dr}{dt}\right)^{2} + + F \frac{dr}{dt}\, \frac{d\theta}{dt} + + G \left(\frac{d\theta}{dt}\right)^{2} + + H \frac{dr}{dt} + J \frac{d\theta}{dt} + K \\ + &\qquad\qquad\qquad\qquad + - 4 \frac{dr}{dt}\, \frac{d^{3}r}{dt^{3}} - r \frac{d^{4}r}{dt^{4}}, +\Tag{(4)} +\end{align*} +where +\begin{align*} +A &= n^{2} - \frac{\dd \Omega}{r\, \dd r} + = \frac{\nu}{r^{3}} + 1 + - \frac{1}{r} \left(\rho - \frac{1}{\rho^{2}}\right)\cos(\theta-\psi), +\Allowbreak +% +B &= -nr \frac{\dd^{2}\Omega}{\dd r\, \dd \theta} + = -n \sin\psi \left[\left(\rho - \frac{1}{\rho^{2}}\right) + \frac{3r}{\rho^{3}} \cos(\theta - \psi)\right], +\Allowbreak +% +D &= r \frac{\dd^{2}\Omega}{\dd r\, \dd \theta} + 2 \frac{\dd \Omega}{\dd \theta} + = 3 \sin\psi \left[\left(\rho - \frac{1}{\rho^{2}}\right) + \frac{r}{\rho^{3}} \cos(\theta - \psi)\right], +\Allowbreak +% +%[** TN: Added break] +L &= 4n^2r - r \frac{\dd^{2} \Omega}{\dd r^{2}} - 3 \frac{\dd \Omega}{\dd r} \\ + &= \frac{\nu}{r^{2}} + 3r + \frac{r}{\rho^{3}} + - 3\left(\rho - \frac{1}{\rho^{2}}\right) \cos(\theta - \psi) + - \frac{3r}{\rho^{3}} \cos^{2}(\theta - \psi), +\Allowbreak +% +E &= r \frac{\dd^{3} \Omega}{\dd r^{3}} + 4 \frac{\dd^{2} \Omega}{\dd r^{2}} - 4n^{2} \\ + &= \frac{2\nu}{r^{3}} + \frac{4}{\rho^{3}}\bigl[3 \cos^{2}(\theta - \psi) - 1\bigr] % \\ +% ++ \frac{3r}{\rho^{4}} \cos(\theta - \psi) \bigl[3 - 5\cos^{2}(\theta - \psi) \bigr], \\ +% +F &= 2r \frac{\dd^{3} \Omega}{\dd r^{2}\, \dd \theta} + + 4\frac{\dd^{2} \Omega}{\dd r\, \dd \theta} - 4 \frac{\dd \Omega}{r\, \dd \theta} + = \frac{6}{\rho^{4}} \sin\psi \bigl[5r \sin^{2}(\theta - \psi) - 4\cos\theta\bigr], \\ +% +G &= r \frac{\dd^{3} \Omega}{\dd r\, \dd \theta^{2}} + 2\frac{\dd^{2} \Omega}{\dd \theta^{2}} + = \frac{3r}{\rho} \left[\left(\rho - \frac{1}{\rho^{2}}\right)\cos\theta + - \frac{r}{\rho^{3}} \sin\psi(5 \sin^{2}(\theta - \psi) - 1)\right], +\Allowbreak +% +H &= -\frac{4n}{r}\, \frac{\dd \Omega}{\dd \theta} + = 4n\left(\rho - \frac{1}{\rho^{2}}\right) \sin(\theta - \psi), \\ +% +J &= 2n \frac{\dd^{2} \Omega}{\dd \theta^{2}} + = \frac{2nr}{\rho} \left[\left(\rho - \frac{1}{\rho^{2}}\right) \cos\theta + - \frac{3}{\rho^{2}} \sin\psi \sin(\theta - \psi)\right], \\ +% +K &= \frac{\dd \Omega}{r^{2}\, \dd \theta} \left(r\frac{\dd^{2} \Omega}{\dd r\, \dd \theta} + 2\frac{\dd \Omega}{\dd \theta}\right) + = \frac{3}{r} \left(\rho - \frac{1}{\rho^{2}}\right) + \sin\theta \sin\psi \left(1 + \frac{1}{\rho^{4}}\cos\psi\right). +\end{align*} +A great diversity of forms might be given to these functions, but the foregoing +seemed to be as convenient for computations as I could devise. + +It is known that when $\nu$~is less than~$24.9599$\footnote + {``Periodic Orbits,'' \textit{Scientific Papers}, vol.~\Vol{IV.}, p.~73.} +the vertex of the equilateral +triangle is an unstable solution of the problem, and if the body is +displaced from the vertex it will move away in a spiral orbit. Hence for +small values of~$\nu$ there are no small closed periodic orbits of the kind +considered by Brown. But certain considerations led him to conjecture that +\DPPageSep{119}{61} +there might still exist large oscillations of this kind. The verification of +such a conjecture would be interesting, and in my attempt to test his idea +I took $\nu$~equal to~$10$. This value was chosen because the results will thus +form a contribution towards that survey of periodic orbits which I have made +in previous papers for $\nu$~equal to~$10$. + +Brown's system of approximation, which he justifies for large values of~$\nu$, +may be described, as far as it is material for my present object, as follows:--- + +We begin the operation at any given point~$r, \theta$, such that $\rho$~is greater +than unity. + +Then in \Eqref{(2)}~and~\Eqref{(3)} $\dfrac{d^{2}r}{dt^{2}}$ and $\dfrac{d^{3}r}{dt^{3}}$ are neglected, and we thence find +$\dfrac{dr}{dt}$,~$\dfrac{d\theta}{dt}$. + +By means of these values of the first differentials, and neglecting $\dfrac{d^{3}r}{dt^{3}}$ +and $\dfrac{d^{4}r}{dt^{4}}$ in~\Eqref{(4)}, we find~$\dfrac{d^{2}r}{dt^{2}}$ from~\Eqref{(4)}. + +Returning to \Eqref{(2)}~and~\Eqref{(3)} and using this value of~$\dfrac{d^{2}r}{dt^{2}}$, we re-determine the +first differentials, and repeat the process until the final values of $\dfrac{dr}{dt}$ and $\dfrac{d\theta}{dt}$ +remain unchanged. We thus obtain the velocity at this point~$r, \theta$ on the +supposition that $\dfrac{d^{3}r}{dt^{3}}$, $\dfrac{d^{4}r}{dt^{4}}$ are negligible, and on substitution in the last of~\Eqref{(1)} +we obtain the value of~$C$ corresponding to the orbit which passes through the +chosen point. + +Brown then shews how the remainder of the orbit may be traced with all +desirable accuracy in the case where $\nu$~is large. It does not concern me to +follow him here, since his process could scarcely be applicable for small values +of~$\nu$. But if his scheme should still lead to the required result, the remainder +of the orbit might be traced by quadratures, and this is the plan which +I have adopted. If the orbit as so determined proves to be clearly non-periodic, +it seems safe to conclude that no widely librating planets can exist +for small values of~$\nu$. + +I had already become fairly confident from a number of trials, which will +be referred to hereafter, that such orbits do not exist; but it seemed worth +while to make one more attempt by Brown's procedure, and the result appears +to be of sufficient interest to be worthy of record. + +For certain reasons I chose as my starting-point +\begin{alignat*}{2} +x_{0} &= -.36200,\quad& y_{0} &= .93441, \\ +\intertext{which give} +r_{0} &= 1.00205,& \rho_0 &= 1.65173. +\end{alignat*} +\DPPageSep{120}{62} +The successive approximations to~$C$ were found to be +\[ +33.6977,\quad 33.7285,\quad 33.7237,\quad 33.7246,\quad 33.7243. +\] +I therefore took the last value as that of~$C$, and found also that the direction +of motion was given by $\phi_{0} = 2°\,21'$. These values of $x_{0}, y_{0}, \phi_{0}$, and~$C$ then +furnish the values from which to begin the quadratures. + +\FigRef[Fig.]{1} shews the result, the starting-point being at~B. The curve was +traced backwards to~A and onwards to~C, and the computed positions are +shewn by dots connected into a sweeping curve by dashes. +\begin{figure}[hbt!] + \centering + \Input{p062} + \caption{Fig.~1. Results derived from Professor Brown's Method.} + \Figlabel{1} +\end{figure} + +From~A back to perijove and from~C on to~J the orbit was computed as +undisturbed by the Sun\footnotemark. Within the limits of accuracy adopted the body +\footnotetext{When the body has been traced to the neighbourhood of~J, let it be required to determine + its future position on the supposition that the solar perturbation is negligible. Since the axes + of reference are rotating, the solution needs care, and it may save the reader some trouble if I set + down how it may be done conveniently. + + Let the coordinates, direction of motion, and velocity, at the moment $t = 0$ when solar + perturbation is to be neglected, be given by $x_{0}, y_{0}$ (or $r_{0}, \theta_{0}$, and $\rho_{0}, \psi_{0}$), $\phi_{0}, V_{0}$; and generally + let the suffix~$0$ to any symbol denote its value at this epoch. Then the mean distance~$\a$, mean + motion~$\mu$, and eccentricity~$e$ are found from + \begin{gather*} + \frac{1}{\a} + = \frac{2}{\rho_{0}} + - \bigl[V_{0}^{2} + 2\pi \rho_{0} V_{0} \cos(\phi_{0} + - \psi_{0}) + n^{2} \rho_{0}^{2}\bigr],\quad + \mu^{2} \a^{3} = 1, \\ + % + \a (1 - e^{2}) + = \bigl[V_{0} \rho_{0} \cos(\phi_{0} - \psi_{0}) + n \rho_0^{2}\bigr]^{2}. + \end{gather*} + Let $t = \tau$ be the time of passage of perijove, so that when $\tau$~is positive perijove is later than the + epoch $t = 0$. + + At any time~$t$ let $\rho, v, E$ be radius vector, true and eccentric anomalies; then + \begin{align*} + \rho &= \a(1 - e \cos E), \\ + \rho^{\frac{1}{2}} \cos \tfrac{1}{2} v + &= \a^{\frac{1}{2}}(1 - e)^{\frac{1}{2}}\cos \tfrac{1}{2} E, \\ +% + \rho^{\frac{1}{2}} \sin \tfrac{1}{2} v + &= \a^{\frac{1}{2}}(1 + e)^{\frac{1}{2}}\sin \tfrac{1}{2} E, \\ +% + \mu(t - \tau) &= E - e \sin E, \\ + \psi &= \psi_{0}- v_{0} + v - nt. + \end{align*} + + On putting $t = 0$, $E_{0}$~and~$\tau$ may be computed from these formulae, and it must be noted that + when $\tau$~is positive $E_{0}$~and~$v_{0}$ are to be taken as negative. + + The position of the body as it passes perijove is clearly given by + \[ + x - 1 = \a(1 - e)\cos(\psi_{0} - v_{0} - n\tau),\quad + y = \a(1 - e)\sin(\psi_{0} - v_{0} - n\tau). + \] + Any other position is to be found by assuming a value for~$E$, computing $\rho, v, t, \psi$, and using the + formulae + \[ + x - 1 = \rho \cos\psi,\quad y = \rho \sin\psi. + \] + + In order to find $V$~and~$\phi$ we require the formulae + \[ + \frac{1}{\rho}\, \frac{d\rho}{dt} = \frac{\a e\sin E}{\rho} · \frac{\mu \a}{\rho};\quad + \frac{dv}{dt} = \frac{\bigl[\a(1 - e^{2})\bigr]^{\frac{1}{2}}}{\rho} · \frac{\a^{\frac{1}{2}}}{\rho} · \frac{\mu \a}{\rho}, \\ + \] + and + \begin{align*} + V\sin \phi + &= -\frac{(x - 1)}{\rho}\, \frac{d\rho}{dt} + + y\left(\frac{dv}{dt} - n\right), \\ + % + V\cos \phi &= \Neg\frac{y}{\rho}\, \frac{d\rho}{dt} + + (x - 1) \left(\frac{dv}{dt} - n\right). + \end{align*} + + The value of~$V$ as computed from these should be compared with that derived from + \[ + V^{2} = \nu\left(r^{2} + \frac{2}{r}\right) + + \left(\rho^{2} + \frac{2}{\rho}\right) - C, + \] + and if the two agree pretty closely, the assumption as to the insignificance of solar perturbation + is justified. + + If the orbit is retrograde about~J, care has to be taken to use the signs correctly, for $v$~and~$E$ + will be measured in a retrograde direction, whereas $\psi$~will be measured in the positive direction. + + A similar investigation is applicable, \textit{mutatis mutandis}, when the body passes very close to~S\@.}% +collides with~J\@. +\DPPageSep{121}{63} + +Since the curve comes down on to the negative side of the line of syzygy~SJ +it differs much from Brown's orbits, and it is clear that it is not periodic. +Thus his method fails, and there is good reason to believe that his conjecture +is unfounded. + +After this work had been done Professor Brown pointed out to me in +a letter that if his process be translated into rectangular coordinates, the +approximate expressions for $dx/dt$~and~$dy/dt$\DPnote{** slant fractions} will have as a divisor the +function +\[ +Q = \left(4n^{2} - \frac{\dd^{2} \Omega}{\dd x^{2}}\right) + \left(4n^{2} - \frac{\dd^{2} \Omega}{\dd y^{2}}\right) + - \left(\frac{\dd^{2} \Omega}{\dd x\, \dd y}\right)^{2}. +\] +The method will then fail if~$Q$ vanishes or is small. +\DPPageSep{122}{64} + +I find that if we write $\Gamma = \dfrac{\nu}{r^{3}} + \dfrac{1}{\rho^{3}}$, the divisor may be written in the +form +\[ +Q = (3n^{2} + \Gamma)(3n^{2} - 2\Gamma) + \frac{\rho \nu}{r^{5}\rho^{5}} \sin\theta \sin\psi. +\] + +Now, Mr~T.~H. Brown, Professor Brown's pupil, has traced one portion of +the curve $Q = 0$, corresponding to $\nu = 10$, and he finds that it passes rather +near to the orbit I have traced. This confirms the failure of the method +which I had concluded otherwise. + +\Section{§ 2. }{Variation of an Orbit.} +\index{Orbit, variation of an}% +\index{Variation, the!of an orbit}% + +A great difficulty in determining the orbits of librating planets by +quadratures arises from the fact that these orbits do not cut the line of +syzygies at right angles, and therefore the direction of motion is quite indeterminate +at every point. I endeavoured to meet this difficulty by a method +of variation which is certainly feasible, but, unfortunately, very laborious. +In my earlier attempts I had drawn certain orbits, and I attempted to utilise +the work by the method which will now be described. + +The stability of a periodic orbit is determined by varying the orbit. The +form of the differential equation which the variation must satisfy does not +depend on the fact that the orbit is periodic, and thus the investigation in +§§~8,~9 of my paper on ``Periodic Orbits'' remains equally true when the +varied orbit is not periodic. + +Suppose, then, that the body is displaced from a given point of a non-periodic +orbit through small distances $\delta q\, V^{-\frac{1}{2}}$ along the outward normal and +$\delta s$~along the positive tangent, then we must have +\begin{gather*} +\frac{d^{2}\, \delta q}{ds^{2}} + \Psi\, \delta q = 0, \\ +% +\frac{d}{ds}\left(\frac{\delta s}{V}\right) + = -\frac{2\delta q}{V^{\frac{3}{2}}} \left(\frac{1}{R} + \frac{n}{V}\right), +\end{gather*} +where +\[ +\Psi = \frac{5}{2} \left(\frac{1}{R} + \frac{n}{V}\right)^{2} + - \frac{3}{2V^{2}} \left[\frac{\nu}{r^{3}}\cos^{2}(\phi - \theta) + + \frac{1}{\rho^{3}}\cos^{2}(\phi - \psi)\right] + + \frac{3}{4} \left(\frac{dV}{V\, ds}\right)^{2}, +\] +and +\[ +\frac{dV}{V\, ds} + = \frac{\nu}{V^{2}} \left(\frac{1}{r^{2}} - r\right)\sin(\phi - \theta) + + \frac{1}{V^{2}} \left(\frac{1}{\rho^{2}} - \rho\right) \sin(\phi - \psi). +\] +Also +\[ +\delta \phi = -\frac{1}{V^{\frac{1}{2}}} + \left[\frac{d\, \delta q}{ds} + - \tfrac{1}{2}\, \delta q \left(\frac{dV}{V\, ds}\right)\right] + + \frac{\delta s}{R}. +\] +\DPPageSep{123}{65} + +Since it is supposed that the coordinates, direction of motion, and radius +of curvature~$R$ have been found at a number of points equally distributed +along the orbit, it is clear that $\Psi$~may be computed for each of those +points. + +At the point chosen as the starting-point the variation may be of two +kinds:--- +\begin{alignat*}{2} +(1)\quad \delta q_0 &= \a, \qquad + \frac{d\delta q_{0}}{ds} &&= 0, \text{ where $\a$ is a constant}, \\ +% +(2)\quad \delta q_0 &=0, \qquad + \frac{d\delta q_{0}}{ds} &&= b, \text{ where $b$ is a constant}. +\end{alignat*} +Each of these will give rise to an independent solution, and if in either of +them $\a$~or~$b$ is multiplied by any factor, that factor will multiply all the +succeeding results. It follows, therefore, that we need not concern ourselves +with the exact numerical values of $\a$~or~$b$, but the two solutions will give us +all the variations possible. In the first solution we start parallel with the +original curve at the chosen point on either side of it, and at any arbitrarily +chosen small distance. In the second we start from the chosen point, but at +any arbitrary small inclination on either side of the original tangent. + +The solution of the equations for $\delta q$~and~$\delta s$ have to be carried out step by +step along the curve, and it may be worth while to indicate how the work +may be arranged. + +The length of arc from point to point of the unvaried orbit may be +denoted by~$\Delta s$, and we may take four successive values of~$\Psi$, say $\Psi_{n-1}, +\Psi_{n}, \Psi_{n+1}, \Psi_{n+2}$, as affording a sufficient representation of the march +of the function~$\Psi$ throughout the arc~$\Delta s$ between the points indicated by +$n$~to~$n+1$. + +If the differential equation for~$\delta q$ be multiplied by~$(\Delta s)^{2}$, and if we +introduce a new independent variable~$z$ such that~$dz = ds/\Delta s$,\DPnote{** slant fractions} and write +$X = \Psi(\Delta s)^{2}$, the equation becomes +\[ +\frac{d^{2}\, \delta q}{dz^{2}} = -X\, \delta q, +\] +and $z$~increases by unity as the arc increases by~$\Delta s$. + +Suppose that the integration has been carried as far as the point~$n$, and +that $\delta q_{0}, d\, \delta q_{0}/dz$ are the values at that point; then it is required to find $\delta q_{1}, +d\, \delta q_{1}/dz$ at the point~$n + 1$. + +If the four adjacent values of~$X$ are $X_{-1}, X_{0}, X_{1}, X_{2}$, and if +\[ +\delta_{1} = X_{1} - X_{0},\quad +\delta_{2} = \tfrac{1}{2} \bigl[(X_{2} - 2X_{1} + X_{0}) + (X_{1} - 2X_{0} + X_{-1})\bigr], +\] +Bessel's formula for the function~$X$ is +\[ +X = X_{0} + (\delta_{1} - \tfrac{1}{2}\delta_{2})z + + \tfrac{1}{2}\delta_{2}z^{2}\DPtypo{}{.} +\] +\DPPageSep{124}{66} +We now assume that throughout the arc $n$~to~$n + 1$, +\[ +\delta q = \delta q_{0} + \frac{d\, \delta q_{0}}{dz} z + + Q_{2} z^{2} + Q_{3} z^{3} + Q_{4} z^{4}, +\] +where $Q_{2}, Q_{3}, Q_{4}$ have to be determined so as to satisfy the differential +equation. + +On forming the product~$X\, \delta q$, integrating, and equating coefficients, we +find $Q_{2} = -\frac{1}{2} X_{0}\, \delta q_{0}$, and the values of~$Q_{3}, Q_{4}$ are easily found. In carrying out +this work I neglect all terms of the second order except~$X_{0}^{2}$. + +\pagebreak[1] +The result may be arranged as follows:---\pagebreak[0] \\ +Let +\begin{align*} +A &= 1 - \tfrac{1}{2} X_{0} - \tfrac{1}{6}\delta_{1} + + \tfrac{1}{24} (\delta_{2} + X_{0}^{2}), \\ +% +B &= 1 - \tfrac{1}{6} X_{0} - \tfrac{1}{12} \delta_{1} + \tfrac{1}{24} \delta_{2}, \\ +% +C &= X_{0} + \tfrac{1}{2} \delta_{1} + \tfrac{1}{12} \delta_{2} - \tfrac{1}{6} (\delta_{2} + X_{0}^{2}), \\ +% +D &= 1 - \tfrac{1}{2} X_{0} - \tfrac{1}{8} \delta_{1} + \tfrac{1}{6} \delta_{2}; +\end{align*} +then, on putting $z =1$, we find +\begin{align*} +\delta q_{1} &= \Neg A\, \delta q_{0} + B \frac{d\, \delta q_{0}}{dz}, \\ +\frac{d\, \delta q_{1}}{dz} &= -C\, \delta q_{0} + D \frac{d\, \delta q_{0}}{dz}. +\end{align*} + +When the~$\Psi$'s have been computed, the~$X$'s and $A, B, C, D$ are easily +found at each point of the unvaried orbit. We then begin the two solutions +from the chosen starting-point, and thus trace $\delta q$~and~$d\, \delta q/dz$ from point to +point both backwards and forwards. The necessary change of procedure when +$\Delta s$~changes in magnitude is obvious. + +The procedure is tedious although easy, but the work is enormously +increased when we pass on further to obtain an intelligible result from the +integration. When $\delta q$~and~$d\, \delta q/dz$ have been found at each point, a further +integration has to be made to determine~$\delta s$, and this has, of course, to be done +for each of the solutions. Next, we have to find the normal displacement~$\delta p$ +(equal to~$\delta q\, V^{-\frac{1}{2}}$), and, finally, $\delta p, \delta s$~have to be converted into rectangular +displacements~$\delta x, \delta y$. + +The whole process is certainly very laborious; but when the result is +attained it does furnish a great deal of information as to the character of the +orbits adjacent to the orbit chosen for variation. I only carried the work +through in one case, because I had gained enough information by this single +instance. However, it does not seem worth while to record the numerical +results in that case. + +In the variation which has been described, $C$~is maintained unchanged, +\DPPageSep{125}{67} +but it is also possible to vary~$C$. If $C$~becomes $C + \delta C$, it will be found that +the equations assume the form +\begin{align*} +\frac{d^{2}\, \delta q}{ds^{2}} + \Psi\, \delta q + + \frac{\delta C}{V^{\frac{3}{2}}} \left(\frac{1}{R} + \frac{n}{V}\right) &= 0, \\ +% +\frac{d}{ds}\left(\frac{\delta s}{V}\right) + + \frac{2\delta q}{V^{\frac{3}{2}}} \left(\frac{1}{R} + \frac{n}{V}\right) + + \frac{\delta C}{2V^{2}} &= 0. +\end{align*} + +But this kind of variation cannot be used with much advantage, for +although it is possible to evaluate $\delta q$~and~$\delta s$ for specific initial values of~$\delta C, +\delta q, d\, \delta q/ds$ at a specific initial point, only one single varied orbit is so deducible. +In the previous case we may assign any arbitrary values, either positive or +negative, to the constants denoted by $\a$~and~$b$, and thus find a group of varied +orbits. + +\Section{§ 3. }{A New Family of Periodic Orbits.} +\index{Periodic orbits, Darwin begins papers on!new family of}% + +In attempting to discover an example of an orbit of the kind suspected +by Brown, I traced a number of orbits. Amongst these was that one which +was varied as explained in~\SecRef{2}, although when the variation was effected I did +not suspect it to be in reality periodic in a new way. It was clear that it +could not be one of Brown's orbits, and I therefore put the work aside and +made a fresh attempt, as explained in~\SecRef{1}. Finally, for my own satisfaction, +I completed the circuit of this discarded orbit, and found to my surprise that +it belonged to a new and unsuspected class of periodic. The orbit in question +is that marked~$33.5$ in \FigRef{3}, where only the half of it is drawn which lies on +the positive side of~SJ\@. + +It will be convenient to use certain terms to indicate the various parts +of the orbits under discussion, and these will now be explained. Periodic +orbits have in reality neither beginning nor end; but, as it will be convenient +to follow them in the direction traversed from an orthogonal crossing of the +line of syzygies, I shall describe the first crossing as the ``beginning'' and the +second orthogonal crossing of~SJ as the ``end.'' I shall call the large curve +surrounding the apex of the Lagrangian equilateral triangle the ``loop,'' and +this is always described in the clockwise or negative direction. The portions +of the orbit near~J will be called the ``circuit,'' or the ``half-'' or ``quarter-circuit,'' +as the case may be. The ``half-circuits'' about~J are described +counter-clockwise or positively, but where there is a complete ``circuit'' it is +clockwise or negative. For example, in \FigRef{3} the orbit~$33.5$ ``begins'' with +a positive quarter-circuit, passes on to a negative ``loop,'' and ``ends'' in a +positive quarter-circuit. Since the initial and final quarter-circuits both cut~SJ +at right angles, the orbit is periodic, and would be completed by a similar +curve on the negative side of~SJ\@. In the completed orbit positively described +\DPPageSep{126}{68} +half-circuits are interposed between negative loops described alternately on +the positive and negative sides of~SJ\@. + +%[** TN: Moved up two paragraphs to accommodate pagination] +\begin{figure}[hbt!] + \centering + \Input{p068} + \caption{Fig.~2. Orbits computed for the Case of $C = 33.25$.} + \Figlabel{2} +\end{figure} +Having found this orbit almost by accident, it was desirable to find other +orbits of this kind; but the work was too heavy to obtain as many as is +desirable. There seems at present no way of proceeding except by conjecture, +and bad luck attended the attempts to draw the curve when $C$~is~$33.25$. The +various curves are shewn in \FigRef{2}, from which this orbit may be constructed +with substantial accuracy. + +In \FigRef{2} the firm line of the external loop was computed backwards, +starting at right angles to~SJ from $x = .95$, $y=0$, the point to which $480°$~is +attached. After the completion of the loop, the curve failed to come down +close to~J as was hoped, but came to the points marked $10°$~and~$0°$. The +``beginnings'' of two positively described quarter-circuits about~J are shewn +as dotted lines, and an orbit of ejection, also dotted, is carried somewhat +further. Then there is an orbit, shewn in firm line, ``beginning'' with a +negative half-circuit about~J, and when this orbit had been traced half-way +through its loop it appeared that the body was drawing too near to the curve +of zero velocity, from which it would rebound, as one may say. This orbit is +continued in a sense by a detached portion starting from a horizontal tangent +at $x = .2$, $y = 1.3$. It became clear ultimately that the horizontal tangent +ought to have been chosen with a somewhat larger value for~$y$. From these +\DPPageSep{127}{69} +attempts it may be concluded that the periodic orbit must resemble the +broken line marked as conjectural, and as such it is transferred to \FigRef{3} and +shewn there as a dotted curve. I shall return hereafter to the explanation +of the degrees written along these curves. + +Much better fortune attended the construction of the orbit~$33.75$ shewn +in \FigRef{3}, for, although the final perijove does not fall quite on the line of +syzygies, yet the true periodic orbit can differ but little from that shewn. +It will be noticed that in this case the orbit ``ends'' with a negative half-circuit, +and it is thus clear that if we were to watch the march of these +\begin{figure}[hbt!] + \centering + \Input{p069} + \caption{Fig.~3. Three Periodic Orbits.} + \Figlabel{3} +\end{figure} +orbits as $C$~falls from~$33.75$ to~$33.5$ we should see the negative half-circuit +shrink, pass through the ejectional stage, and emerge as a positive quarter-circuit +when $C$~is~$33.5$. + +The three orbits shewn in \FigRef{3} are the only members of this family that +I have traced. It will be noticed that they do not exhibit that regular +progress from member to member which might have been expected from the +fact that the values of~$C$ are equidistant from one another. It might be +suspected that they are really members of different families presenting similar +characteristics, but I do not think this furnishes the explanation. +\DPPageSep{128}{70} + +In describing the loop throughout most of its course the body moves +roughly parallel to the curve of zero velocity. For the values of~$C$ involved +here that curve is half of the broken horse-shoe described in my paper on +``Periodic Orbits'' (\textit{Scientific Papers}, vol.~\Vol{IV.}, p.~11, or \textit{Acta Math.}, vol.~\Vol{XXI.} +(1897)). Now, for $\nu = 10$ the horse-shoe breaks when $C$~has fallen to~$34.91$, +and below that value each half of the broken horse-shoe, which delimits the +forbidden space, shrinks. Now, since the orbits follow the contour of the +horse-shoe, it might be supposed that the orbits would also shrink as $C$~falls +in magnitude. On the other hand, as $C$~falls from~$33.5$ to~$33.25$, our figures +shew that the loop undoubtedly increases in size. This latter consideration +would lead us to conjecture that the loop for~$33.75$ should be smaller than +that for~$33.5$. Thus, looking at the matter from one point of view, we should +expect the orbits to shrink, and from another to swell as $C$~falls in value. +It thus becomes intelligible that neither conjecture can be wholly correct, +and we may thus find an explanation of the interlacing of the orbits as shewn +in my \FigRef{3}. + +It is certain from general considerations that families of orbits must +originate in pairs, and we must therefore examine the origin of these orbits, +and consider the fate of the other member of the pair. + +It may be that for values of~$C$ greater than~$33.75$ the initial positive +quarter-circuit about~J is replaced by a negative half-circuit; but it is +unnecessary for the present discussion to determine whether this is so or not, +and it will suffice to assume that when $C$~is greater than~$33.75$ the ``beginning'' +is as shewn in my figure. The ``end'' of~$33.75$ is a clearly marked negative +half-circuit, and this shews that the family originates from a coalescent pair of +orbits ``ending'' in such a negative half-circuit, with identical final orthogonal +crossing of~SJ in which the body passes from the negative to the positive +side of~SJ\@. + +This coalescence must occur for some critical value of~$C$ between $34.91$ +and~$33.75$, and it is clear that as $C$~falls below that critical value one +of the ``final'' orthogonal intersections will move towards~S and the other +towards~J. + +In that one of the pair for which the intersection moves towards~S the +negative circuit increases in size; in the other in which it moves towards~J +the circuit diminishes in size, and these are clearly the orbits which have +been traced. We next see that the negative circuit vanishes, the orbit +becomes ejectional, and the motion about~J both at ``beginning'' and ``end'' +has become positive. + +It may be suspected that when $C$~falls below~$33.25$ the half-circuits +round~J increase in magnitude, and that the orbit tends to assume the +form of a sort of asymmetrical double figure-of-8, something like the figure +\DPPageSep{129}{71} +which Lord Kelvin drew as an illustration of his graphical method of curve-tracing\footnotemark. +\footnotetext{\textit{Popular Lectures}, vol.~\Vol{I.}, 2nd~ed., p.~31; \textit{Phil.\ Mag.}, vol.~\Vol{XXXIV.}, 1892, p.~443.}% + +In the neighbourhood of Jove the motion of the body is rapid, but the +loops are described very slowly. The number of degrees written along the +curves in \FigRef{2} represent the angles turned through by Jove about the Sun +since the moment corresponding to the position marked~$0°$. Thus the firm +line which lies externally throughout most of the loop terminates with~$480°$. +Since this orbit cuts~SJ orthogonally, it may be continued symmetrically on +the negative side of~SJ, and therefore while the body moves from the point~$0°$ +to a symmetrical one on the negative side Jove has turned through~$960°$ round +the Sun, that is to say, through $2\frac{2}{3}$~revolutions. + +Again, in the case of the orbit beginning with a negative half-circuit, +shewn as a firm line, Jove has revolved through~$280°$ by the time the point +so marked is reached. We may regard this as continued in a sense by the +detached portion of an orbit marked with~$0°, 113°, 203°$; and since $280° + 203°$ +is equal to~$483°$, we again see that the period of the periodic orbit must be +about~$960°$, or perhaps a little more. + +In the cases of the other orbits more precise values may be assigned. For +$C = 33.5$, the angle~$nT$ (where $T$~is the period) is~$1115°$ or $3.1$~revolutions of +Jove; and for $C = 33.75$, $nT$~is~$1235°$ or $3.4$~revolutions. + +It did not seem practicable to investigate the stability of these orbits, but +we may suspect them to be unstable. + +The numerical values for drawing the orbits $C = 33.5$ and~$33.75$ are given +in an appendix, but those for the various orbits from which the conjectural +orbit $C = 33.25$ is constructed are omitted. I estimate that it is as laborious +to trace one of these orbits as to determine fully half a dozen of the simpler +orbits shewn in my earlier paper. + +Although the present contribution to our knowledge is very imperfect, +yet it may be hoped that it will furnish the mathematician with an +intimation worth having as to the orbits towards which his researches must +lead him. + +The librating planets were first recognised as small oscillations about the +triangular positions of Lagrange, and they have now received a very remarkable +extension at the hands of Professor Brown. It appears to me that the +family of orbits here investigated possesses an interesting relationship to +these librating planets, for there must be orbits describing double, triple, +and multiple loops in the intervals between successive half-circuits about +Jove. Now, a body which describes its loop an infinite number of times, +\DPPageSep{130}{72} +before it ceases to circulate round the triangular point, is in fact a librating +planet. It may be conjectured that when the Sun's mass~$\nu$ is yet smaller +than~$10$, no such orbit as those traced is possible. When $\nu$~has increased +to~$10$, probably only a single loop is possible; for a larger value a double loop +may be described, and then successively more frequently described multiple +loops will be reached. When $\nu$~has reached~$24.9599$ a loop described an +infinite number of times must have become possible, since this is the smallest +value of~$\nu$ which permits oscillation about the triangular point. If this idea +is correct, and if $\mathrm{N}$~denotes the number expressing the multiplicity of the +loop, then as $\nu$~increases $d\mathrm{N}/d\nu$~must tend to infinity; and I do not see why +this should not be the case. + +These orbits throw some light on cosmogony, for we see how small planets +with the same mean motion as Jove in the course of their vicissitudes tend +to pass close to Jove, ultimately to be absorbed into its mass. We thus see +something of the machinery whereby a large planet generates for itself a clear +space in which to circulate about the Sun. + +My attention was first drawn to periodic orbits by the desire to discover +how a Laplacian ring could coalesce into a planet. With that object in view +I tried to discover how a large planet would affect the motion of a small one +moving in a circular orbit at the same mean distance. After various failures +the investigation drifted towards the work of Hill and Poincaré, so that the +original point of view was quite lost and it is not even mentioned in my paper +on ``Periodic Orbits.'' It is of interest, to me at least, to find that the original +aspect of the problem has emerged again. + +\Appendix{Numerical results of Quadratures.} + +\Heading{$C = 33.5$.} + +\noindent\begin{minipage}{\textwidth} +\centering\footnotesize +\settowidth{\TmpLen}{Time from perijove up to $s=-2.1$ is given by $nt=9°\, 25'$.}% +\parbox{\TmpLen}{Perijove $x_0=1.0171$, $y_0=-.0034$, taken as zero. \\ +Time from perijove up to $s=-2.1$ is given by $nt=9°\, 25'$.} +\end{minipage} +\begin{longtable}{.{1,3}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} ,{6,2}!{\extracolsep{\fill}} .{2,3}}\\ +\ColHead{s} & \ColHead{x} & \ColHead{y} & \ColHead{\Z\Z\Z\Z\phi} & \ColHead{2n/V}\\ + & & & \ColHead{\AngleHeading} & \\ +\endhead +-2.1 &+\Z.8282&+\Z.0980& {}+66, 10 & 2.408 \\ + 2.0 & .7409 & .1467 & 55, 53 & 2.829 \\ + 1.9 & .6625 & .2084 & 48, 36 & 2.876 \\ + 1.8 & .5894 & .2766 & 46,\Z3 & 2.768 \\ + 1.7 & .5171 & .3457 & 46, 55 & 2.655 \\ + 1.6 & .4425 & .4124 & 49, 46 & 2.584 \\ + 1.5 & .3641 & .4744 & 53, 39 & 2.568 \\ +-1.4 &+\Z.2814&+\Z.5306& {}+57, 56 & 2.613 \\ +\DPPageSep{131}{73} +-1.3 &+\Z.1948&+\Z.5805& {}+62,\Z8 & 2.728 \\ + 1.2 & .1049 & .6243 & 65, 51 & 2.930 \\ + 1.1 &+\Z.0126& .6628 & 68, 38 & 3.251 \\ + 1.0 &-\Z.0810& .6979 & 69, 46 & 3.760 \\ + .9 & .1747 & .7330 & 68,\Z7 & 4.598 \\ + .85 & .2207 & .7526 & 65, 13 & 5.240 \\ + .8 & .2653 & .7754 & 60,\Z1 & 6.133 \\ + .75 & .3068 & .8035 & 50, 51 & 7.377 \\ + .725& .3252 & .8203 & 44,\Z2 & 8.139 \\ + .7 & .3412 & .8395 & 35, 17 & 8.944 \\ + .675& .3537 & .8611 & 24, 33 & 9.664 \\ + .65 & .3617 & .8848 & 12, 27 & 10.129 \\ + .625& .3644 & .9096 & {}+\Z0, 13 & 10.224 \\ + .6 & .3620 & .9344 & {}-10, 56 & 10.009 \\ + .575& .3552 & .9584 & 20, 31 & 9.655 \\ + .55 & .3448 & .9811 & 28, 30 & 9.205 \\ + .5 & .3161 & 1.0220 & 40, 48 & 8.448 \\ + .45 & .2806 & 1.0571 & 49, 38 & 7.872 \\ + .4 & .2405 & 1.0869 & 56, 51 & 7.460 \\ + .3 & .1518 & 1.1326 & 68,\Z4 & 6.961 \\ + .2 &-\Z.0565& 1.1626 & 76, 47 & 6.730 \\ +-\Z.1 &+\Z.0421& 1.1791 & 83, 58 & 6.647 \\ + .0 & .1419 & 1.1842 & {}-90,\Z0 & 6.633 \\ ++\Z.05& .1919 & 1.1830 & 180°+87, 21 & 6.630 \\ + .1 & .2418 & 1.1797 & 84, 54 & 6.626 \\ + .15 & .2915 & 1.1742 & 82, 38 & 6.609 \\ + .2 & .3410 & 1.1669 & 80, 31 & 6.572 \\ + .3 & .4389 & 1.1470 & 76, 31 & 6.432 \\ + .4 & .5353 & 1.1203 & 72, 33 & 6.201 \\ + .5 & .6295 & 1.0869 & 68, 16 & 5.912 \\ + .6 & .7208 & 1.0461 & 63, 29 & 5.605 \\ + .7 & .8081 & .9974 & 58,\Z8 & 5.313 \\ + .8 & .8902 & .9404 & 52, 12 & 5.055 \\ + .9 & .9656 & .8748 & 45, 39 & 4.842 \\ + 1.0 & 1.0326 & .8006 & 38, 22 & 4.671 \\ + 1.1 & 1.0889 & .7181 & 30, 11 & 4.540 \\ + 1.2 & 1.1321 & .6280 & 20, 46 & 4.435 \\ + 1.3 & 1.1585 & .5318 & \Z9, 38 & 4.326 \\ + 1.35 & 1.1642 & .4821 &180°+\Z3, 16 & 4.250 \\ + 1.4 & 1.1641 & .4322 &180°-\Z3, 40 & 4.141 \\ + 1.45 & 1.1577 & .3826 & 11,\Z5 & 3.983 \\ + 1.5 & 1.1448 & .3343 & 18, 44 & 3.758 \\ + 1.55 & 1.1257 & .2881 & 26,\Z8 & 3.460 \\ + 1.6 & 1.1011 & .2446 & 32, 39 & 3.100 \\ + 1.65 & 1.0723 & .2038 & 37, 33 & 2.701 \\ + 1.7 & 1.0408 & .1650 & 40,\Z4 & 2.291 \\ ++1.75 &+1.0087 &+\Z.1267& 180°-39, 12 & 1.893 \\ +\end{longtable} +\noindent\begin{minipage}{\textwidth} +\centering\footnotesize +\settowidth{\TmpLen}{Time from $s = 1.75$ to perijove given by $nt = 5°58'$.}% +\parbox{\TmpLen}{Time from $s = 1.75$ to perijove given by $nt = 5°58'$. \\ +Coordinates of perijove $x = .9501$, $y = -.0029$.} +\end{minipage} +\DPPageSep{132}{74} + +The following additional positions were calculated backwards from a perijove at +$x = .95$, $y = 0$, $\phi = 180°$. + +\[ +\begin{array}{.{1,4} c<{\qquad} .{1,4} c<{\qquad} ,{6,2}} +\ColHead{x} && \ColHead{y} && \ColHead{\Z\Z\Z\Z\phi} \\ + && && \ColHead{\AngleHeading} \\ ++\Z.9500 && +.0000 && 180°+\Z0, \Z0 \\ + .9512 && .0531 && 180°- 22, 30 \\ + .9647 && .0797 && 30, 52 \\ + .9756 && .0966 && 34, 48 \\ + .9874 && .1127 && 37, 37 \\ + 1.0128 && .1436 && 40, 37 \\ + 1.0390 && .1738 && 40, 56 \\ + 1.0649 && .2043 && 39, 12 \\ + 1.0893 && .2360 && 35, 51 \\ + 1.1114 && .2693 && 31, 16 \\ + 1.1463 && .3412 && 20, 10 \\ ++ 1.1661 && +.4186 && 180°-\Z8, 40 \\ +\end{array} +\] + +This supplementary orbit becomes indistinguishable in a figure of moderate size from +the preceding orbit, which is therefore accepted as being periodic. The period is given by +$nT = 1115°.4 = 3.1$ revolutions of Jove. + +\Heading{$C = 33.75$.} + +This orbit was computed from a conjectural starting-point which seemed likely to lead +to the desired result; the computation was finally carried backwards from the starting-point. +The coordinates of perijove were found to be $x_{0} = 1.0106$, $y_{0} = .0006$, which may be +taken as virtually on the line of syzygies. The motion from perijove is direct. + +\begin{longtable}{.{1,3}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} ,{6,2}!{\extracolsep{\fill}} .{2,3}}\\ +\ColHead{s} & \ColHead{x} & \ColHead{y} & \ColHead{\Z\Z\Z\Z\phi} & \ColHead{2n/V}\\ + & & & \ColHead{\AngleHeading} & \\ +\endhead +\ColHead{\text{perijove}} + &+1.0106 &+\Z.0006& \Z0, \Z0& \ColHead{\text{very nearly}} \\ +-\Z.3 & .9652 & .0403 & 66, 38 & 1.140 \\ +-\Z.3 & .9184 & .0578 & 71, \Z6& 1.635 \\ +-\Z.2 & .8713 & .0744 & 69, 27 & 2.075 \\ +-\Z.2 & .8251 & .0936 & 65, \Z3& 2.447 \\ +-\Z.1 & .7391 & .1444 & 54, 15 & 2.882 \\ + 0.0 & .6625 & .2084 & 47, \Z0& 2.946 \\ + .1 & .5911 & .2785 & 44, 44 & 2.850 \\ + .2 & .5202 & .3490 & 46, \Z0& 2.749 \\ + .3 & .4465 & .4165 & 49, 13 & 2.686 \\ + .4 & .3685 & .4791 & 53, 29 & 2.675 \\ + .5 & .2858 & .5352 & 58, 10 & 2.723 \\ + .6 & .1987 & .5844 & 62, 52 & 2.838 \\ + .7 & .1081 & .6265 & 67, 13 & 3.036 \\ + .8 &+\Z.0147& .6622 & 70, 49 & 3.348 \\ + .9 &-\Z.0805& .6929 & 73, 11 & 3.834 \\ + 1.0 & .1764 & .7213 & 73, 25 & 4.631 \\ + 1.1 & .2713 & .7525 & 69, 17 & 6.090 \\ + 1.15 & .3173 & .7721 & 63, 50 & 7.333 \\ + 1.2 & .3601 & .7977 & 53, 25 & 9.236 \\ + 1.225 & .3791 & .8140 & 45, \Z6& 10.360 \\ + 1.25 &-\Z.3951&+\Z.8332& 33, 54 & 11.840 \\ +\DPPageSep{133}{75} + 1.275 &-\Z.4064&+\Z.8553& {}+19, 53 & 12.955 \\ + 1.3 & .4118 & .8796 & {}+\Z4, 42& 13.412 \\ + 1.325 & .4108 & .9046 & {}-\Z9, 14& 13.174 \\ + 1.35 & .4043 & .9287 & 20, 35 & 12.599 \\ + 1.375 & .3936 & .9513 & 29, 25 & 11.945 \\ + 1.4 & .3800 & .9723 & 36, 21 & 11.364 \\ + 1.45 & .3466 & 1.0096 & 46, 23 & 10.471 \\ + 1.5 & .3082 & 1.0416 & 53, 25 & 9.849 \\ + 1.6 & .2227 & 1.0940 & 62, 21 & 9.034 \\ + 1.7 & .1317 & 1.1356 & 67, 59 & 8.347 \\ + 1.8 &-\Z.0377& 1.1696 & 72, \Z2& 7.618 \\ + 2.0 &+\Z.1563& 1.2184 & 79, 17 & 6.140 \\ + 2.2 & .3547 & 1.2407 & {}-88, 13 & 4.966 \\ + 2.4 & .5541 & 1.2300 & 180°+81, 54 & 4.182 \\ + 2.6 & .7487 & 1.1845 & 71, 49 & 3.665 \\ + 2.8 & .9322 & 1.1057 & 61, 40 & 3.305 \\ + 3.0 & 1.0989 & .9956 & 51, 24 & 3.052 \\ + 3.2 & 1.2429 & .8573 & 40, 54 & 2.873 \\ + 3.4 & 1.3588 & .6946 & 29, 55 & 2.751 \\ + 3.6 & 1.4402 & .5123 & 18, \Z1& 2.682 \\ + 3.8 & 1.4797 & .3168 & 180°+\Z4, 28& 2.670 \\ + 4.0 & 1.4674 & .1181 & 180°-12, 14 & 2.733 \\ + 4.1 & 1.4377 &+\Z.0227& 23, 43 & 2.806 \\ + 4.2 & 1.3894 &-\Z.0646& 35, 38 & 2.910 \\ + 4.3 & 1.3208 & .1366 & 52, 23 & 3.027 \\ + 4.35 & 1.2787 & .1635 & 62, 47 & 3.068 \\ + 4.4 & 1.2322 & .1817 & 74, 47 & 3.063 \\ + 4.45 & 1.1829 & .1892 & 180°-88, 15 & 2.983 \\ + 4.5 & 1.1332 & .1845 & {}+77, 25 & 2.780 \\ + 4.55 & 1.0863 & .1676 & 63, \Z8& 2.477 \\ + 4.6 & 1.0448 & .1399 & 49, 32 & 2.101 \\ + 4.65 & 1.0108 & .1034 & 36, 18 & 1.683 \\ + 4.7 & .9867 &-\Z.0598& 21, \Z1& 1.234 \\ +\ColHead{\text{perijove}} + & +\Z.990&+\Z.011 & \llap{\text{about }} 49, & \\ +\end{longtable} + +The orbit is not vigorously periodic, but an extremely small change at the beginning +would make it so. The period is given by $nT = 1234°.6 = 3.43$ revolutions of Jove. + +\normalsize +\DPPageSep{134}{76} + + +\Chapter{Address} +\index{Address to the International Congress of Mathematicians in Cambridge, 1912}% +\index{Cambridge School of Mathematics}% +\index{Congress, International, of Mathematicians at Cambridge, 1912}% +\index{Mathematical School at Cambridge}% +\index{Mathematicians, International Congress of, Cambridge, 1912}% + +\Heading{(Delivered before the International Congress of Mathematicians +at Cambridge in 1912)} + +\First{Four} years ago at our Conference at Rome the Cambridge Philosophical +Society did itself the honour of inviting the International Congress of +Mathematicians to hold its next meeting at Cambridge. And now I, as +President of the Society, have the pleasure of making you welcome here. +I shall leave it to the Vice-Chancellor, who will speak after me, to express +the feeling of the University as a whole on this occasion, and I shall +confine myself to my proper duty as the representative of our Scientific +Society. + +The Science of Mathematics is now so wide and is already so much +\index{Specialisation in Mathematics}% +specialised that it may be doubted whether there exists to-day any man +fully competent to understand mathematical research in all its many diverse +branches. I, at least, feel how profoundly ill-equipped I am to represent +our Society as regards all that vast field of knowledge which we classify as +pure mathematics. I must tell you frankly that when I gaze on some of the +papers written by men in this room I feel myself much in the same position +as if they were written in Sanskrit. + +But if there is any place in the world in which so one-sided a President +of the body which has the honour to bid you welcome is not wholly out of +place it is perhaps Cambridge. It is true that there have been in the past +at Cambridge great pure mathematicians such as Cayley and Sylvester, but +we surely may claim without undue boasting that our University has played +a conspicuous part in the advance of applied mathematics. Newton was +a glory to all mankind, yet we Cambridge men are proud that fate ordained +that he should have been Lucasian Professor here. But as regards the part +played by Cambridge I refer rather to the men of the last hundred years, +such as Airy, Adams, Maxwell, Stokes, Kelvin, and other lesser lights, who +have marked out the lines of research in applied mathematics as studied in +this University. Then too there are others such as our Chancellor, Lord +Rayleigh, who are happily still with us. +\DPPageSep{135}{77} + +Up to a few weeks ago there was one man who alone of all mathematicians +\index{Poincaré, reference to, by Sir George Darwin}% +might have occupied the place which I hold without misgivings as to his +fitness; I mean Henri Poincaré. It was at Rome just four years ago that +the first dark shadow fell on us of that illness which has now terminated so +fatally. You all remember the dismay which fell on us when the word passed +from man to man ``Poincaré is ill.'' We had hoped that we might again +have heard from his mouth some such luminous address as that which he +gave at Rome; but it was not to be, and the loss of France in his death +affects the whole world. + +It was in 1900 that, as president of the Royal Astronomical Society, +I had the privilege of handing to Poincaré the medal of the Society, and +I then attempted to give an appreciation of his work on the theory of the +tides, on figures of equilibrium of rotating fluid and on the problem of the +three bodies. Again in the preface to the third volume of my collected +papers I ventured to describe him as my patron Saint as regards the papers +contained in that volume. It brings vividly home to me how great a man +he was when I reflect that to one incompetent to appreciate fully one half of +his work yet he appears as a star of the first magnitude. + +It affords an interesting study to attempt to analyze the difference in the +\index{Galton, Sir Francis!analysis of difference in texture of different minds}% +textures of the minds of pure and applied mathematicians. I think that +I shall not be doing wrong to the reputation of the psychologists of half +a century ago when I say that they thought that when they had successfully +analyzed the way in which their own minds work they had solved the problem +before them. But it was Sir~Francis Galton who shewed that such a view is +erroneous. He pointed out that for many men visual images form the most +potent apparatus of thought, but that for others this is not the case. Such +visual images are often quaint and illogical, being probably often founded on +infantile impressions, but they form the wheels of the clockwork\DPnote{[** TN: Not hyphenated in original]} of many +minds. The pure geometrician must be a man who is endowed with great +powers of visualisation, and this view is confirmed by my recollection of the +difficulty of attaining to clear conceptions of the geometry of space until +practice in the art of visualisation had enabled one to picture clearly the +relationship of lines and surfaces to one another. The pure analyst probably +relies far less on visual images, or at least his pictures are not of a geometrical +character. I suspect that the mathematician will drift naturally to one branch +or another of our science according to the texture of his mind and the nature +of the mechanism by which he works. + +I wish Galton, who died but recently, could have been here to collect +from the great mathematicians now assembled an introspective account +of the way in which their minds work. One would like to know whether +students of the theory of groups picture to themselves little groups of dots; +or are they sheep grazing in a field? Do those who work at the theory +\DPPageSep{136}{78} +of numbers associate colour, or good or bad characters with the lower +ordinal numbers, and what are the shapes of the curves in which the +successive numbers are arranged? What I have just said will appear pure +nonsense to some in this room, others will be recalling what they see, and +perhaps some will now for the first time be conscious of their own visual +images. + +The minds of pure and applied mathematicians probably also tend to +differ from one another in the sense of aesthetic beauty. Poincaré has well +remarked in his \textit{Science et Méthode} (p.~57): +\index{Poincaré, reference to, by Sir George Darwin!\textit{Science et Méthode}, quoted}% + +``On peut s'étonner de voir invoquer la sensibilité apropos de démon\-stra\-tions +mathématiques qui, semble-t-il, ne peuvent intéresser que l'intelligence. +Ce serait oublier le sentiment de la beauté mathématique, de +l'harmonie des nombres et des formes, de l'élégance géometrique. C'est un +vrai sentiment esthétique que tous les vrais mathématiciens connaissent. +Et c'est bien là de la sensibilité.'' + +And again he writes: + +``Les combinaisons utiles, ce sont précisément les plus belles, je veux dire +celles qui peuvent le mieux charmer cette sensibilité spéciale que tous les +mathématiciens connaissent, mais que les profanes ignorent au point qu'ils +sont souvent tentés d'en sourire.'' + +Of course there is every gradation from one class of mind to the other, +and in some the aesthetic sense is dominant and in others subordinate. + +In this connection I would remark on the extraordinary psychological +interest of Poincaré's account, in the chapter from which I have already +quoted, of the manner in which he proceeded in attacking a mathematical +problem. He describes the unconscious working of the mind, so that his +conclusions appeared to his conscious self as revelations from another world. +I suspect that we have all been aware of something of the same sort, and +like Poincaré have also found that the revelations were not always to be +trusted. + +Both the pure and the applied mathematician are in search of truth, but +the former seeks truth in itself and the latter truths about the universe in +which we live. To some men abstract truth has the greater charm, to others +the interest in our universe is dominant. In both fields there is room for +indefinite advance; but while in pure mathematics every new discovery +is a gain, in applied mathematics it is not always easy to find the direction +in which progress can be made, because the selection of the conditions +essential to the problem presents a preliminary task, and afterwards there +arise the purely mathematical difficulties. Thus it appears to me at least, +that it is easier to find a field for advantageous research in pure than in +\DPPageSep{137}{79} +applied mathematics. Of course if we regard an investigation in applied +mathematics as an exercise in analysis, the correct selection of the essential +conditions is immaterial; but if the choice has been wrong the results lose +almost all their interest. I may illustrate what I mean by reference to +\index{Kelvin, associated with Sir George Darwin!cooling of earth}% +Lord Kelvin's celebrated investigation as to the cooling of the earth. He +was not and could not be aware of the radio-activity of the materials of which +the earth is formed, and I think it is now generally acknowledged that the +conclusions which he deduced as to the age of the earth cannot be maintained; +yet the mathematical investigation remains intact. + +The appropriate formulation of the problem to be solved is one of the +\index{Darwin, Sir George, genealogy!on his own work}% +greatest difficulties which beset the applied mathematician, and when he +has attained to a true insight but too often there remains the fact that +his problem is beyond the reach of mathematical solution. To the layman +the problem of the three bodies seems so simple that he is surprised to learn +that it cannot be solved completely, and yet we know what prodigies of +mathematical skill have been bestowed on it. My own work on the subject +cannot be said to involve any such skill at all, unless indeed you describe as +skill the procedure of a housebreaker who blows in a safe-door with dynamite +instead of picking the lock. It is thus by brute force that this tantalising +problem has been compelled to give up some few of its secrets, and great as +has been the labour involved I think it has been worth while. Perhaps this +work too has done something to encourage others such as Störmer\footnote + {\textit{Videnskabs Selskab}, Christiania, 1904.} +to similar +tasks as in the computation of the orbits of electrons in the neighbourhood +of the earth, thus affording an explanation of some of the phenomena of the +aurora borealis. To put at their lowest the claims of this clumsy method, +which may almost excite the derision of the pure mathematician, it +has served to throw light on the celebrated generalisations of Hill and +Poincaré. + +I appeal then for mercy to the applied mathematician and would ask +you to consider in a kindly spirit the difficulties under which he labours. +If our methods are often wanting in elegance and do but little to satisfy that +aesthetic sense of which I spoke before, yet they are honest attempts to +unravel the secrets of the universe in which we live. + +We are met here to consider mathematical science in all its branches. +Specialisation has become a necessity of modern work and the intercourse +which will take place between us in the course of this week will serve to +promote some measure of comprehension of the work which is being carried +on in other fields than our own. The papers and lectures which you will +hear will serve towards this end, but perhaps the personal conversations +outside the regular meetings may prove even more useful. +\DPPageSep{138}{80} +\backmatter +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} + +\Pagelabel{indexpage} + +\printindex + +\iffalse +%INDEX TO VOLUME V + +%A + +Abacus xlviii + +Address to the International Congress of Mathematicians in Cambridge, 1912#Address 76 + +Annual Equation 51 + +Apse, motion of 41 + +%B + +Bakerian lecture xlix + +Bakhuyzen, Dr Van d.\ Sande, Sir George Darwin's connection with the International Geodetic Association xxviii + +Barrell, Prof., Cosmogony as related to Geology and Biology xxxvii + +British Association, South African Meeting, 1905#British xxvi + +Brown, Prof.\ E. W., Sir George Darwin's Scientific Work xxxiv + new family of periodic orbits 59 + +%C + +Cambridge School of Mathematics 1, 76 + +Chamberlain and Moulton, Planetesimal Hypothesis xlvii + +Committees, Sir George Darwin on xxii + +Congress, International, of Mathematicians at Cambridge, 1912#Congress 76 + note by Sir Joseph Larmor xxix + +Cosmogony, Sir George Darwin's influence on xxxvi + as related to Geology and Biology, by Prof.\ Barrell xxxvii + +%D + +Darwin, Charles, ix; letters of xiii, xv + +Darwin, Sir Francis, Memoir of Sir George Darwin by ix + +Darwin, Sir George, genealogy ix + boyhood x + interested in heraldry xi + education xi + at Cambridge xii, xvi + friendships xiii, xvi + ill health xiv + marriage xix + children xx + house at Cambridge xix + games and pastimes xxi + personal characteristics xxx + energy xxxii + honours xxxiii + university work, described by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall xvii, xviii + work on scientific committees xxii + association with Lord Kelvin xv, xxxvii + scientific work, by Prof.\ E. W. Brown xxxiv + his first papers xxxvi + characteristics of his work xxxiv + his influence on cosmogony xxxvi + his relationship with his pupils xxxvi + on his own work 79 + +Darwin, Margaret, on Sir George Darwin's personal characteristics xxx + +Differential Equation, Hill's 36 + periodicity of integrals of 55 + +Differential Equations of Motion 17 + +Dynamical Astronomy, introduction to 9 + +%E + +Earth-Moon theory of Darwin, described by Mr S. S. Hough xxxviii + +Earth's figure, theory of 14 + +Ellipsoidal harmonics xliii + +Equation, annual 51 + of the centre 43 + +Equations of motion 17, 46 + +Equilibrium of a rotating fluid xlii + +Evection 43 + in latitude 45 + +%G + +Galton, Sir Francis ix + analysis of difference in texture of different minds 77 + +Geodetic Association, International xxvii, xxviii + +Glaisher, Dr J. W. L., address on presenting the gold medal of the R.A.S. to G. W. Hill lii + +Gravitation, theory of 9 + universal 15 + +%H + +Harmonics, ellipsoidal xliii + +Hecker's observations on retardation of tidal oscillations in the solid earth xliv, l + +Hill, G. W., Lunar Theory l + awarded gold medal of R.A.S. lii + lectures by Darwin on Lunar Theory lii, 16 + characteristics of his Lunar Theory 16 + Special Differential Equation 36 + periodicity of integrals of 55 + infinite determinant 38, 53 + +Hough, S. S., Darwin's work on Earth-Moon Theory, xxxviii; Periodic Orbits liv + +%I + +Inaugural lecture 1 + +Infinite determinant, Hill's 38, 53 + +Introduction to Dynamical Astronomy 9 +%\DPPageSep{139}{81} + +Jacobi's ellipsoid xlii + integral 21 + +Jeans, J. H., on rotating liquids xliii + +%K + +Kant, Nebular Hypothesis xlvi + +Kelvin, associated with Sir George Darwin xv, xxxvii + cooling of earth xlv, 79 + +%L + +Laplace, Nebular Hypothesis xlvi + +Larmor, Sir Joseph, Sir George Darwin's work on university committees xvii + International Geodetic Association xxvii + International Congress of Mathematicians at Cambridge 1912#Cambridge xxix + +Latitude of the moon 43 + +Latitude, variation of 14 + +Lecture, inaugural 1 + +Liapounoff's work on rotating liquids xliii + +Librating planets 59 + +Lunar and planetary theories compared 11 + +Lunar Theory, lecture on 16 + +%M + +Maclaurin's spheroid xlii + +Master of Christ's, Sir George Darwin's work on university committees xviii + +Mathematical School at Cambridge 1, 76 + text-books 4 + Tripos 3 + +Mathematicians, International Congress of, Cambridge, 1912#Cambridge xxix, 76 + +Mechanical condition of a swarm of meteorites xlvi + +Meteorological Council, by Sir Napier Shaw xxii + +Michelson's experiment on rigidity of earth l + +Moulton, Chamberlain and, Planetesimal Hypothesis xlvii + +%N + +Newall, Prof., Sir George Darwin's work on university committees xviii + +Numerical work on cosmogony xlvi + +Numerical work, great labour of liii + +%O + +Orbit, variation of an 64 + +Orbits, periodic, |see{Periodic} + +%P + +Pear-shaped figure of equilibrium xliii + +Perigee, motion of 41 + +Periodic orbits, Darwin begins papers on liii + great numerical difficulties of liii + stability of liii + classification of, by Jacobi's integral liv + new family of 59, 67 + +Periodicity of integrals of Hill's Differential Equation 55 + +Planetary and lunar theories compared 11 + +Planetesimal Hypothesis of Chamberlain and Moulton xlvii + +Poincaré, reference to, by Sir George Darwin 77 + on equilibrium of fluid mass in rotation xlii + \textit{Les Méthodes Nouvelles de la Mécanique Céleste} lii + \textit{Science et Méthode}, quoted 78 + +Pupils, Darwin's relationship with his xxxvi + +%R + +Raverat, Madame, on Sir George Darwin's personal characteristics xxx + +Reduction, the 49 + +Rigidity of earth, from fortnightly tides xlix + Michelson's experiment l + +Roche's ellipsoid xliii + +Rotating fluid, equilibrium of xlii + +%S + +Saturn's rings 15 + +Shaw, Sir Napier, Meteorological Council xxii + +Small displacements from variational curve 26 + +South African Meeting of the British Association, 1905#British xxvi + +Specialisation in Mathematics 76 + +%T + +Text-books, mathematical 4 + +Third coordinate introduced 43 + +Tidal friction as a true cause of change xliv + +Tidal problems, practical xlvii + +Tide, fortnightly xlix + +\textit{Tides, The} xxvii, l + +Tides, articles on l + +Tripos, Mathematical 3 + +%U + +University committees, Sir George Darwin on, by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall xvii, xviii + +%V + +Variation, the 43 + of an orbit 64 + of latitude 14 + +Variational curve, defined 22 + determined 23 + small displacements from 26 +\fi +\DPPageSep{140}{82} +\newpage +\null\vfill +\begin{center} +\scriptsize +\textgoth{Cambridge}: \\[4pt] +PRINTED BY JOHN CLAY, M.A. \\[4pt] +AT THE UNIVERSITY PRESS +\end{center} +\vfill +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\cleardoublepage +\phantomsection +\pdfbookmark[0]{PG License}{Project Gutenberg License} +\SetRunningHeads{Licensing}{Licensing} +\pagenumbering{Roman} + +\begin{PGtext} +End of the Project Gutenberg EBook of Scientific Papers by Sir George Howard +Darwin, by George Darwin + +*** END OF THIS PROJECT GUTENBERG EBOOK SCIENTIFIC PAPERS *** + +***** This file should be named 35588-pdf.pdf or 35588-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/5/5/8/35588/ + +Produced by Andrew D. 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EBook of Scientific Papers by Sir George Howard % +% Darwin, by George Darwin % +% % +% This eBook is for the use of anyone anywhere at no cost and with % +% almost no restrictions whatsoever. You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: Scientific Papers by Sir George Howard Darwin % +% Volume V. Supplementary Volume % +% % +% Author: George Darwin % +% % +% Commentator: Francis Darwin % +% E. W. Brown % +% % +% Editor: F. J. M. Stratton % +% J. Jackson % +% % +% Release Date: March 16, 2011 [EBook #35588] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK SCIENTIFIC PAPERS *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{35588} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. 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+\newcommand{\stretchyspace}{\spaceskip 0.375em plus 0.375em minus 0.25em} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} + +\pagestyle{empty} +\pagenumbering{Alph} + +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +The Project Gutenberg EBook of Scientific Papers by Sir George Howard +Darwin, by George Darwin + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Scientific Papers by Sir George Howard Darwin + Volume V. Supplementary Volume + +Author: George Darwin + +Commentator: Francis Darwin + E. W. Brown + +Editor: F. J. M. Stratton + J. Jackson + +Release Date: March 16, 2011 [EBook #35588] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK SCIENTIFIC PAPERS *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Laura Wisewell, Chuck Greif +and the Online Distributed Proofreading Team at +http://www.pgdp.net (The original copy of this book was +generously made available for scanning by the Department +of Mathematics at the University of Glasgow.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\phantomsection +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +\newpage +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\DPPageSep{001}{Unnumbered page} +\begin{center} +\null\vfill +\LARGE\textbf{SCIENTIFIC PAPERS} +\vfill +\end{center} +\newpage +\DPPageSep{002}{Unnumbered page} +\begin{center} +\null\vfill +\scriptsize\setlength{\TmpLen}{2pt}% +CAMBRIDGE UNIVERSITY PRESS \\[\TmpLen] +C. F. CLAY, \textsc{Manager} \\[\TmpLen] +\textgoth{London}: FETTER LANE, E.C. \\[\TmpLen] +\textgoth{Edinburgh}: 100 PRINCES STREET \\[\TmpLen] +\includegraphics[width=1in]{./images/cups.png} \\[\TmpLen] +\textgoth{New York}: G. P. PUTNAM'S SONS \\[\TmpLen] +\textgoth{Bombay, Calcutta and Madras}: MACMILLAN AND CO., \textsc{Ltd.} \\[\TmpLen] +\textgoth{Toronto}: J. M. DENT AND SONS, \textsc{Ltd.} \\[\TmpLen] +\textgoth{Tokyo}: THE MARUZEN-KABUSHIKI-KAISHA +\vfill +\textit{All rights reserved} +\end{center} +\frontmatter +\pagenumbering{roman} +\DPPageSep{003}{i} +%[Blank Page] +\DPPageSep{004}{ii} +\null\vfill +\begin{figure}[p!] + \centering + \Pagelabel{frontis} + \ifthenelse{\boolean{ForPrinting}}{% + \includegraphics[width=\textwidth]{./images/frontis.jpg} + }{% + \includegraphics[width=0.875\textwidth]{./images/frontis.jpg} + } +\iffalse +[Hand-written note: From a water-colour drawing +by his daughter +Mrs Jacques Raverat +G. H. Darwin] +\fi +\end{figure} +\vfill +\clearpage +\DPPageSep{005}{iii} +\begin{center} +\setlength{\TmpLen}{12pt}% +\textbf{\Huge SCIENTIFIC PAPERS} +\vfil +\footnotesize% +BY \\[\TmpLen] +{\normalsize SIR GEORGE HOWARD DARWIN} \\ +{\scriptsize K.C.B., F.R.S. \\ +FELLOW OF TRINITY COLLEGE \\ +PLUMIAN PROFESSOR IN THE UNIVERSITY OF CAMBRIDGE} +\vfil +VOLUME V \\ +SUPPLEMENTARY VOLUME \\[\TmpLen] + +{\scriptsize CONTAINING} \\ + +BIOGRAPHICAL MEMOIRS BY SIR FRANCIS DARWIN \\[2pt] +AND PROFESSOR E. W. BROWN, \\[2pt] +LECTURES ON HILL'S LUNAR THEORY, \textsc{etc.} +\vfil +EDITED BY \\ +F. J. M. STRATTON, M.A., \textsc{and} J. JACKSON, M.A., \textsc{B.Sc.} +\vfil\vfil +\normalsize +Cambridge: \\ +at the University Press \\ +1916 +\end{center} +\newpage +\DPPageSep{006}{iv} +\begin{center} +\null\vfill\scriptsize +\textgoth{Cambridge}: \\ +PRINTED BY JOHN CLAY, M.A. \\ +AT THE UNIVERSITY PRESS +\vfill +\end{center} +\newpage +\DPPageSep{007}{v} + + +\Chapter{Preface} + +\First{Before} his death Sir~George Darwin expressed the view that his +lectures on Hill's Lunar Theory should be published. He made no +claim to any originality in them, but he believed that a simple presentation +of Hill's method, in which the analysis was cut short while the fundamental +principles of the method were shewn, might be acceptable to students of +astronomy. In this belief we heartily agree. The lectures might also +with advantage engage the attention of other students of mathematics +who have not the time to enter into a completely elaborated lunar theory. +They explain the essential peculiarities of Hill's work and the method of +approximation used by him in the discussion of an actual problem of +nature of great interest. It is hoped that sufficient detail has been given +to reveal completely the underlying principles, and at the same time not +be too tedious for verification by the reader. + +During the later years of his life Sir~George Darwin collected his +principal works into four volumes. It has been considered desirable to +publish these lectures together with a few miscellaneous articles in a fifth +volume of his works. Only one series of lectures is here given, although +he lectured on a great variety of subjects connected with Dynamics, Cosmogony, +Geodesy, Tides, Theories of Gravitation,~etc. The substance of +many of these is to be found in his scientific papers published in the four +earlier volumes. The way in which in his lectures he attacked problems +of great complexity by means of simple analytical methods is well illustrated +in the series chosen for publication. + +Two addresses are included in this volume. The one gives a view of +the mathematical school at Cambridge about~1880, the other deals with +the mathematical outlook of~1912. +\DPPageSep{008}{vi} + +The previous volumes contain all the scientific papers by Sir~George +Darwin published before~1910 which he wished to see reproduced. They +do not include a large number of scientific reports on geodesy, the tides and +other subjects which had involved a great deal of labour. Although the +reports were of great value for the advancement and encouragement of +science, he did not think it desirable to reprint them. We have not +ventured to depart from his own considered decision; the collected lists +at the beginning of these volumes give the necessary references for such +papers as have been omitted. We are indebted to the Royal Astronomical +Society for permission to complete Sir~George Darwin's work on Periodic +Orbits by reproducing his last published paper. + +The opportunity has been taken of securing biographical memoirs of +Darwin from two different points of view. His brother, Sir~Francis Darwin, +writes of his life apart from his scientific work, while Professor E.~W.~Brown, +of Yale University, writes of Darwin the astronomer, mathematician and +teacher. + +\footnotesize +\settowidth{\TmpLen}{F. J. M. S.\quad}% +\null\hfill\parbox{\TmpLen}{F. J. M. S.\\ J. J.} + +\scriptsize +\textsc{Greenwich,} \\ +\indent\indent6 \textit{December} 1915. + +\normalsize +\newpage +\DPPageSep{009}{vii} +%[** TN: Table of Contents] + + +\Chapter{Contents} +\enlargethispage{36pt} +\ToCFrontis{Portrait of Sir George Darwin}%{Frontispiece} + +\ToCPAGE + +\ToCChap{Memoir of Sir George Darwin by his brother Sir Francis Darwin} +{chapter:3}%{ix} + +\ToCChap{The Scientific Work of Sir George Darwin by Professor E. W. +Brown}{chapter:4}%{xxxiv} + +\ToCChap{Inaugural lecture (Delivered at Cambridge, in 1883, on Election to +the Plumian Professorship)}{chapter:5}%{1} + +\ToCChap{Introduction to Dynamical Astronomy}{chapter:6}%{9} + +\ToCChap{Lectures on Hill's Lunar Theory}{chapter:7}%{16} + +\ToCSec{§ 1.}{Introduction}{1}%{16} + +\ToCSec{§ 2.}{Differential Equations of Motion and Jacobi's Integral} +{2}%{17} + +\ToCSec{§ 3.}{The Variational Curve}{3}%{22} + +\ToCSec{§ 4.}{Differential Equations for Small Displacements from the +Variational Curve}{4}%{26} + +\ToCSec{§ 5.}{Transformation of the Equations in § 4}{5}%{29} + +\ToCSec{§ 6.}{Integration of an important type of Differential Equation} +{6}%{36} + +\ToCSec{§ 7.}{Integration of the Equation for~$\delta p$}{7}%{39} + +\ToCSec{§ 8.}{Introduction of the Third Coordinate}{8}%{43} + +\ToCSec{§ 9.}{Results obtained}{9}%{45} + +\ToCSec{§ 10.}{General Equations of Motion and their solution} +{10}%{46} + +\ToCSec{§ 11.}{Compilation of Results}{11}%{52} + +\ToCNote{Note 1.}{On the Infinite Determinant of § 5}{note:1}%{53} + +\ToCNote{Note 2.}{On the periodicity of the integrals of the equation +\[ +\frac{d^{2}\, \delta p}{d\tau^{2}} + \Theta\, \delta p = 0, +\] +where $\Theta = \Theta_{0} + \Theta_{1} \cos 2\tau + + \Theta_{2} \cos 4\tau + \dots$.}{note:2}%{55} + +\ToCChap{On Librating Planets and on a New Family of Periodic Orbits} +{chapter:8}%{59} + +\ToCMisc{[\textit{Monthly Notices of the Royal Astronomical Society}, Vol.~72 (1912), pp.~642--658.]} + +\ToCChap{Address to the International Congress of Mathematicians at +Cambridge in 1912}{chapter:9}%{76} + +\ToCChap{Index}{indexpage}%{80} +\DPPageSep{010}{viii} +% [Blank Page] +\DPPageSep{011}{ix} + +\cleardoublepage +\phantomsection +\pdfbookmark[-1]{Main Matter}{Main Matter} + + +\Chapter{Memoir of Sir George Darwin} +\BY{His Brother Sir Francis Darwin} +\SetRunningHeads{Memoir of Sir George Darwin}{By Sir Francis Darwin} +\index{Darwin, Sir Francis, Memoir of Sir George Darwin by}% +\index{Darwin, Sir George, genealogy}% +\index{Galton, Sir Francis}% + +George Howard, the fifth\footnoteN + {The third of those who survived childhood.} +child of Charles and Emma Darwin, was +born at Down July~9th, 1845. Why he was christened\footnoteN + {At Maer, the Staffordshire home of his mother.} +George, I cannot +say. It was one of the facts on which we founded a theory that our parents +lost their presence of mind at the font and gave us names for which there +was neither the excuse of tradition nor of preference on their own part. +His second name, however, commemorates his great-grandmother, Mary +Howard, the first wife of Erasmus Darwin. It seems possible that George's +ill-health and that of his father were inherited from the Howards. This at +any rate was Francis Galton's view, who held that his own excellent health +was a heritage from Erasmus Darwin's second wife. George's second name, +Howard, has a certain appropriateness in his case for he was the genealogist +and herald of our family, and it is through Mary Howard that the +Darwins can, by an excessively devious route, claim descent from certain +eminent people, e.g.~John of~Gaunt. This is shown in the pedigrees which +George wrote out, and in the elaborate genealogical tree published in Professor +Pearson's \textit{Life of Francis Galton}. George's parents had moved to +Down in September~1842, and he was born to those quiet surroundings of +which Charles Darwin wrote ``My life goes on like clock-work\DPnote{[** TN: Hyphenated in original]} and I am +fixed on the spot where I shall end it.\footnotemarkN'' It would have been difficult to +\footnotetextN{\textit{Life and Letters of Charles Darwin}, vol.~\Vol{I.} p.~318.}% +find a more retired place so near London. In 1842 a coach drive of some +twenty miles was the only means of access to Down; and even now that +railways have crept closer to it, it is singularly out of the world, with little +to suggest the neighbourhood of London, unless it be the dull haze of smoke +that sometimes clouds the sky. In 1842 such a village, communicating with +the main lines of traffic only by stony tortuous lanes, may well have been +enabled to retain something of its primitive character. Nor is it hard to +believe in the smugglers and their strings of pack-horses making their way +up from the lawless old villages of the Weald, of which the memory then +still lingered. +\DPPageSep{012}{x} + +George retained throughout life his deep love for Down. For the lawn +\index{Darwin, Sir George, genealogy!boyhood}% +with its bright strip of flowers; and for the row of big lime trees that +bordered it. For the two yew trees between which we children had our +swing, and for many another characteristic which had become as dear and +as familiar to him as a human face. He retained his youthful love of +the ``Sand-walk,'' a little wood far enough from the house to have for us +a romantic character of its own. It was here that our father took his daily +exercise, and it has ever been haunted for us by the sound of his heavy +walking stick striking the ground as he walked. + +George loved the country round Down,---and all its dry chalky valleys +of ploughed land with ``shaws,'' i.e.~broad straggling hedges on their +crests, bordered by strips of flowery turf. The country is traversed by +many foot-paths, these George knew well and used skilfully in our walks, +in which he was generally the leader. His love for the house and the +neighbourhood was I think entangled with his deepest feelings. In later +years, his children came with their parents to Down, and they vividly +remember his excited happiness, and how he enjoyed showing them his +ancient haunts. + +In this retired region we lived, as children, a singularly quiet life +practically without friends and dependent on our brothers and sisters for +companionship. George's earliest recollection was of drumming with his +spoon and fork on the nursery table because dinner was late, while a +barrel-organ played outside. Other memories were less personal, for instance +the firing of guns when Sebastopol was supposed to have been taken. His +diary of~1852 shows a characteristic interest in current events and in the +picturesqueness of Natural History: +\begin{Quote} +\centering +The Duke is dead. Dodos are out of the world. +\end{Quote} +He perhaps carried rather far the good habit of re-reading one's\DPnote{[** TN: [sic]]} favourite +authors. He told his children that for a year or so he read through every +day the story of Jack the Giant Killer, in a little chap-book with coloured +pictures. He early showed signs of the energy which marked his character +in later life. I am glad to remember that I became his companion and +willing slave. There was much playing at soldiers, and I have a clear +remembrance of our marching with toy guns and knapsacks across the +field to the Sand-walk. There we made our bivouac with gingerbread, +and milk, warmed (and generally smoked) over a ``touch-wood'' fire. I was +a private while George was a sergeant, and it was part of my duty to stand +sentry at the far end of the kitchen-garden until released by a bugle-call +from the lawn. I have a vague remembrance of presenting my fixed bayonet +at my father to ward off a kiss which seemed to me inconsistent with my +military duties. Our imaginary names and heights were written up on the +wall of the cloak-room. George, with romantic exactitude, made a small +\DPPageSep{013}{xi} +foot rule of such a size that he could conscientiously record his height as +$6$~feet and mine as slightly less, in accordance with my age and station. + +Under my father's instruction George made spears with loaded heads +which he hurled with remarkable skill by means of an Australian throwing +stick. I used to skulk behind the big lime trees on the lawn in the character +of victim, and I still remember the look of the spears flying through the air +with a certain venomous waggle. Indoors, too, we threw at each other lead-weighted +javelins which we received on beautiful shields made by the village +carpenter and decorated with coats of arms. + +Heraldry was a serious pursuit of his for many years, and the London +\index{Darwin, Sir George, genealogy!interested in heraldry}% +Library copies of Guillim and Edmonson\footnoteN + {Guillim, John, \textit{A display of heraldry}, 6th~ed., folio~1724. Edmonson,~J., \textit{A complete body + of heraldry}, folio~1780.} +were generally at Down. He +retained a love of the science through life, and his copy of Percy's \textit{Reliques} +is decorated with coats of arms admirably drawn and painted. In later life +he showed a power of neat and accurate draughtsmanship, and some of the +illustrations in his father's books, e.g.~in \textit{Climbing Plants}, are by his hand. + +His early education was given by governesses: but the boys of the family +\index{Darwin, Sir George, genealogy!education}% +used to ride twice or thrice a week to be instructed in Latin by Mr~Reed, the +Rector of Hayes---the kindest of teachers. For myself, I chiefly remember +the cake we used to have at 11~o'clock and the occasional diversion of looking +at the pictures in the great Dutch bible. George must have impressed his +parents with his solidity and self-reliance, since he was more than once +allowed to undertake alone the $20$~mile ride to the house of a relative at +Hartfield in Sussex. For a boy of ten to bait his pony and order his +luncheon at the Edenbridge inn was probably more alarming than the +rest of the adventure. There is indeed a touch of David Copperfield in +his recollections, as preserved in family tradition. ``The waiter always said, +`What will you have for lunch, Sir?' to which he replied. `What is there?' +and the waiter said, `Eggs and bacon'; and, though he hated bacon more +than anything else in the world, he felt obliged to have it.'' + +On August~16th, 1856, George was sent to school. Our elder brother, +William, was at Rugby, and his parents felt his long absences from home +such an evil that they fixed on the Clapham Grammar School for their +younger sons. Besides its nearness to Down, Clapham had the merit of +giving more mathematics and science than could them be found in public +schools. It was kept by the Rev.~Charles Pritchard\footnotemarkN, a man of strong +\footnotetextN{Afterwards Savilian Professor of Astronomy at Oxford. Born~1808, died~1893.}% +character and with a gift for teaching mathematics by which George undoubtedly +profited. In (I think) 1861 Pritchard left Clapham and was +succeeded by the Rev.~Alfred Wrigley, a man of kindly mood but without +the force or vigour of Pritchard. As a mathematical instructor I imagine +\DPPageSep{014}{xii} +Wrigley was a good drill-master rather than an inspiring teacher. Under +him the place degenerated to some extent; it no longer sent so many boys +to the Universities, and became more like a ``crammer's'' and less like a public +school. My own recollections of George at Clapham are coloured by an abiding +gratitude for his kindly protection of me as a shrinking and very unhappy +``new boy'' in~1860. + +George records in his diary that in 1863 he tried in vain for a Minor +\index{Darwin, Sir George, genealogy!at Cambridge}% +Scholarship at St~John's College, Cambridge, and again failed to get one at +Trinity in~1864, though he became a Foundation Scholar in~1866. These +facts suggested to me that his capacity as a mathematician was the result of +slow growth. I accordingly applied to Lord Moulton, who was kind enough +to give me his impressions: +\begin{Quote} +My memories of your brother during his undergraduate career +correspond closely to your suggestion that his mathematical power +developed somewhat slowly and late. Throughout most if not the +whole of his undergraduate years he was in the same class as myself +and Christie, the ex-Astronomer Royal, at Routh's\footnotemarkN. We all recognised +\footnotetextN{The late Mr~Routh was the most celebrated Mathematical ``Coach'' of his +day.}% +him as one who was certain of being high in the Tripos, but he did not +display any of that colossal power of work and taking infinite trouble +that characterised him afterwards. On the contrary, he treated his +work rather jauntily. At that time his health was excellent and he +took his studies lightly so that they did not interfere with his enjoyment +of other things\footnotemarkN. I remember that as the time of the examination +\footnotetextN{Compare Charles Darwin's words: ``George has not slaved himself, which makes his + success the more satisfactory.'' (\textit{More Letters of C.~Darwin}, vol.~\Vol{II.} p.~287)}% +came near I used to tell him that he was unfairly handicapped in being +in such robust health and such excellent spirits. + +Even when he had taken his degree I do not think he realised his +innate mathematical power\ldots. It has been a standing wonder to me that +he developed the patience for making the laborious numerical calculations +on which so much of his most original work was necessarily +based. He certainly showed no tendency in that direction during his +undergraduate years. Indeed he told me more than once in later life +that he detested Arithmetic and that these calculations were as tedious +and painful to him as they would have been to any other man, but that +he realised that they must be done and that it was impossible to train +anyone else to do them. +\end{Quote} + +As a Freshman he ``kept'' (i.e.~lived) in~A\;6, the staircase at the N.W. +corner of the New Court, afterwards moving to~F\;3 in the Old Court, +pleasant rooms entered by a spiral staircase on the right of the Great Gate. +Below him, in the ground floor room, now used as the College offices, lived +Mr~Colvill, who remained a faithful but rarely seen friend as long as George +lived. + +Lord Moulton, who, as we have seen, was a fellow pupil of George's at +Routh's, was held even as a Freshman to be an assured Senior Wrangler, +\DPPageSep{015}{xiii} +a prophecy that he easily made good. The second place was held by George, +and was a much more glorious position than he had dared to hope for. In +those days the examiners read out the list in the Senate House, at an early +hour, 8~a.m.\ I think. George remained in bed and sent me to bring the +news. I remember charging out through the crowd the moment the magnificent +``Darwin of Trinity'' had followed the expected ``Moulton of St~John's.'' +I have a general impression of a cheerful crowd sitting on George's bed and +literally almost smothering him with congratulations. He received the +following characteristic letter from his father\footnotemarkN: +\footnotetextN{\textit{Emma Darwin, A Century of Family Letters}, vol.~\Vol{II.} p.~186.}% +\index{Darwin, Charles, ix; letters of}% +\begin{Letter} + {\textsc{Down}, \textit{Jan.}~24\textit{th} [1868].} + {My dear old fellow,} + + I am so pleased. I congratulate you with all my heart and soul. I + always said from your early days that such energy, perseverance and + talent as yours would be sure to succeed: but I never expected such + brilliant success as this. Again and again I congratulate you. But + you have made my hand tremble so I can hardly write. The telegram + came here at eleven. We have written to W.~and the boys. + + God bless you, my dear old fellow---may your life so continue. + + \Signature{Your affectionate Father,}{Ch.~Darwin.} +\end{Letter} + +In those days the Tripos examination was held in the winter, and the +successful candidates got their degrees early in the Lent Term; George +records in his diary that he took his~B.A. on January~25th, 1868: also +that he won the second of the two Smith's Prizes,---the first being the +natural heritage of the Senior Wrangler. There is little to record in this +year. He had a pleasant time in the summer coaching Clement Bunbury, +the nephew of Sir~Charles, at his beautiful place Barton Hall in Suffolk. +In the autumn he was elected a Fellow of Trinity, as he records, ``with +Galabin, young Niven, Clifford, [Sir~Frederick] Pollock, and [Sir~Sidney] +Colvin.'' W.~K.~Clifford was the well-known brilliant mathematician who +died comparatively early. + +Chief among his Cambridge friends were the brothers Arthur, Gerald +\index{Darwin, Sir George, genealogy!friendships}% +and Frank Balfour. The last-named was killed, aged~31, in a climbing +accident in~1882 on the Aiguille Blanche near Courmayeur. He was +remarkable both for his scientific work and for his striking and most lovable +personality. George's affection for him never faded. Madame Raverat remembers +her father (not long before his death) saving with emotion, ``I dreamed +Frank Balfour was alive.'' I imagine that tennis was the means of bringing +George into contact with Mr~Arthur Balfour. What began in this chance +way grew into an enduring friendship, and George's diary shows how much +kindness and hospitality he received from Mr~Balfour. George had also the +\DPPageSep{016}{xiv} +advantage of knowing Lord Rayleigh at Cambridge, and retained his friendship +through his life. + +In the spring of~1869 he was in Paris for two months working at French. +His teacher used to make him write original compositions, and George gained +a reputation for humour by giving French versions of all the old Joe~Millers +and ancient stories he could remember. + +It was his intention to make the Bar his profession\footnotemarkN, and in October~1869 +\footnotetextN{He was called in 1874 but did not practise.}% +we find him reading with Mr~Tatham, in 1870~and~1872 with the late +Mr~Montague Crackenthorpe (then Cookson). Again, in November~1871, he +was a pupil of Mr~W.~G. Harrison. The most valued result of his legal work +was the friendship of Mr~and~Mrs Crackenthorpe, which he retained throughout +his life. During these years we find the first indications of the circumstances +which forced him to give up a legal career---namely, his failing health and +\index{Darwin, Sir George, genealogy!ill health}% +his growing inclination towards science\footnotemarkN. Thus in the summer of~1869, when +\footnotetextN{As a boy he had energetically collected Lepidoptera during the years 1858--64, but the first + vague indications of a leaning towards physical science may perhaps be found in his joining the + Sicilian eclipse expedition, Dec.~1870--Jan.~1871. It appears from \textit{Nature}, Dec.~1, 1870, that + George was told off to make sketches of the Corona.}% +we were all at Caerdeon in the Barmouth valley, he writes that he ``fell ill''; +and again in the winter of~1871. His health deteriorated markedly during +1872~and~1873. In the former year he went to Malvern and to Homburg +without deriving any advantage. I have an impression that he did not +expect to survive these attacks; but I cannot say at what date he made this +forecast of an early death. In January~1873 he tried Cannes: and ``came +back very ill.'' It was in the spring of this year that he first consulted Dr +(afterwards Sir~Andrew) Clark, from whom he received the kindest care. +George suffered from digestive troubles, sickness and general discomfort and +weakness. Dr~Clark's care probably did what was possible to make life more +bearable, and as time went on his health gradually improved. In 1894 he +consulted the late Dr~Eccles, and by means of the rest-cure, then something +of a novelty, his weight increased from $9$~stone to $9$~stone $11$~pounds. I gain +the impression that this treatment produced a permanent improvement, +although his health remained a serious handicap throughout his life. + +Meanwhile he had determined on giving up the Bar, and settled, in +October~1873, when he was $28$~years old, at Trinity in Nevile's Court next +the Library~(G\;4). His diary continues to contain records of ill-health and +of various holidays in search of improvement. Thus in 1873 we read ``Very +bad during January. Went to Cannes and stayed till the end of April.'' Again +in~1874, ``February to July very ill.'' In spite of unwellness he began in 1872--3 +to write on various subjects. He sent to \textit{Macmillan's Magazine}\footnoteN + {\textit{Macmillan's Magazine}, 1872, vol.~\Vol{XXVI.} pp.~410--416.} +an entertaining +article, ``Development in Dress,'' where the various survivals in modern +\DPPageSep{017}{xv} +costume were recorded and discussed from the standpoint of evolution. In +1873 he wrote ``On beneficial restriction to liberty of marriage\footnotemarkN,'' a eugenic +\footnotetextN{\textit{Contemporary Review}, 1873, vol.~\Vol{XXII.} pp.~412--426.}% +article for which he was attacked with gross unfairness and bitterness by the +late St~George Mivart. He was defended by Huxley, and Charles Darwin +formally ceased all intercourse with Mivart. We find mention of a ``Globe +Paper for the British Association'' in~1873. And in the following year he +read a contribution on ``Probable Error'' to the Mathematical Society\footnoteN{Not published.}---on +which he writes in his diary, ``found it was old.'' Besides another paper in the +\textit{Messenger of Mathematics}, he reviewed ``Whitney on Language\footnotemarkN,'' and wrote +\footnotetextN{\textit{Contemporary Review}, 1874, vol.~\Vol{XXIV.} pp.~894--904.}% +a ``defence of Jevons'' which I have not been able to trace. In 1875 he +was at work on the ``flow of pitch,'' on an ``equipotential tracer,'' on slide +rules, and sent a paper on ``Cousin Marriages'' to the Statistical Society\footnotemarkN. It +\footnotetextN{\textit{Journal of the Statistical Society}, 1875, vol.~\Vol{XXXVIII.} pt~2, pp.~158--182, also pp.~183--184, + and pp.~344--348.}% +is not my province to deal with these papers; they are here of interest as +showing his activity of mind and his varied interests, features in character +which were notable throughout his life. + +The most interesting entry in his diary for 1875 is ``Paper on Equipotentials +\index{Darwin, Sir George, genealogy!association with Lord Kelvin}% +\index{Kelvin, associated with Sir George Darwin}% +much approved by Sir~W. Thomson.'' This is the first notice of an +association of primary importance in George's scientific career. Then came +his memoir ``On the influence of geological changes in the earth's axis of +rotation.'' Lord Kelvin was one of the referees appointed by the Council of +the Royal Society to report on this paper, which was published in the \textit{Philosophical +Transactions} in~1877. + +In his diary, November~1878, George records ``paper on tides ordered to +be printed.'' This refers to his work ``On the bodily tides of viscous and +semi-elastic spheroids,~etc.,'' published in the \textit{Phil.\ Trans.} in~1879. It was in +regard to this paper that his father wrote to George on October~29th, 1878\footnotemarkN: +\footnotetextN{Probably he heard informally at the end of October what was not formally determined till + November.}% +\index{Darwin, Charles, ix; letters of}% + +\begin{Letter}{}{My dear old George,} + I have been quite delighted with your letter and read it all with + eagerness. You were very good to write it. All of us are delighted, + for considering what a man Sir~William Thomson is, it is most grand + that you should have staggered him so quickly, and that he should + speak of your `discovery,~etc.'\ldots\ Hurrah + for the bowels of the earth and their viscosity and for the moon and + for the Heavenly bodies and for my son George (F.R.S. very + soon)\ldots\footnotemarkN. +\end{Letter} +\footnotetextN{\textit{Emma Darwin, A Century of Family Letters}, 1915, vol.~\Vol{II.} p.~233.}% + +The bond of pupil and master between George Darwin and Lord Kelvin, +originating in the years 1877--8, was to be a permanent one, and developed +\DPPageSep{018}{xvi} +not merely into scientific co-operation\DPnote{[** TN: Hyphenated in original]} but into a close friendship. Sir~Joseph +\index{Darwin, Sir George, genealogy!friendships}% +Larmor has recorded\footnoteN + {\textit{Nature}, Dec.~12, 1912.} +that George's ``tribute to Lord Kelvin, to whom he +dedicated volume~\Vol{I} of his Collected Papers\footnotemarkN\ldots gave lively pleasure to his +master and colleague.'' His words were: +\footnotetextN{It was in 1907 that the Syndics of the Cambridge University Press asked George to prepare + \index{Darwin, Sir George, genealogy!at Cambridge}% + a reprint of his scientific papers, which the present volume brings to an end. George was + deeply gratified at an honour that placed him in the same class as Lord Kelvin, Stokes, Cayley, + Adams, Clerk Maxwell, Lord Rayleigh and other men of distinction.}% +\begin{Quote} +Early in my scientific career it was my good fortune to be brought +into close personal relationship with Lord Kelvin. Many visits to Glasgow +and to Largs have brought me to look up to him as my master, and +I cannot find words to express how much I owe to his friendship and to +his inspiration. +\end{Quote} + +During these years there is evidence that he continued to enjoy the +friendship of Lord Rayleigh and of Mr~Balfour. We find in his diary +records of visits to Terling and to Whittingehame, or of luncheons at +Mr~Balfour's house in Carlton Gardens for which George's scientific committee +work in London gave frequent opportunity. In the same way we +find many records of visits to Francis Galton, with whom he was united alike +by kinship and affection. + +Few people indeed can have taken more pains to cultivate friendship +than did George. This trait was the product of his affectionate and eminently +sociable nature and of the energy and activity which were his chief +characteristics. In earlier life he travelled a good deal in search of health\footnotemarkN, +\footnotetextN{Thus in 1872 he was in Homburg, 1873~in Cannes, 1874~in Holland, Belgium, Switzerland + and Malta, 1876~in Italy and Sicily.}% +and in after years he attended numerous congresses as a representative +of scientific bodies. He thus had unusual opportunities of making the +acquaintance of men of other nationalities, and some of his warmest friendships +were with foreigners. In passing through Paris he rarely failed to visit +M.~and~Mme d'Estournelles and ``the d'Abbadies.'' It was in Algiers in 1878~and~1879 +that he cemented his friendship with the late J.~F.~MacLennan, +author of \textit{Primitive Marriage}; and in 1880 he was at Davos with the same +friends. In~1881 he went to Madeira, where he received much kindness from +the Blandy family---doubtless through the recommendation of Lady~Kelvin. + +\Section{}{Cambridge.} + +We have seen that George was elected a Fellow of Trinity in October~1868, +and that five years later (Oct.~1873) he began his second lease of +a Cambridge existence. There is at first little to record: he held at this +time no official position, and when his Fellowship expired he continued to +live in College busy with his research work and laying down the earlier tiers +\DPPageSep{019}{xvii} +of the monumental series of papers in the present volumes. This soon led to +his being proposed (in Nov.~1877) for the Royal Society, and elected in June~1879. +The principal event in this stage of his Cambridge life was his +election\footnoteN + {The voting at University elections is in theory strictly confidential, but in practice this is + unfortunately not always the case. George records in his diary the names of the five who voted + for him and of the four who supported another candidate. None of the electors are now living. + The election occurred in January, and in June he had the great pleasure and honour of being + re-elected to a Trinity Fellowship. His daughter, Madame Raverat, writes: ``Once, when I was + walking with my father on the road to Madingley village, he told me how he had walked there, + on the first Sunday he ever was at Cambridge, with two or three other freshmen; and how, when + they were about opposite the old chalk pit, one of them betted him~£20 that he (my father) + would never be a professor of Cambridge University: and said my father, with great indignation, + `He never paid me.'\,"} +in 1883 as Plumian Professor of Astronomy and Experimental +Philosophy. His predecessor in the Chair was Professor Challis, who had +held office since~1836, and is now chiefly remembered in connection with +Adams and the planet Neptune. The professorship is not necessarily connected +with the Observatory, and practical astronomy formed no part of +George's duties. His lectures being on advanced mathematics usually +attracted but few students; in the Long Vacation however, when he +habitually gave one of his courses, there was often a fairly large class. + +George's relations with his class have been sympathetically treated by +Professor E.~W.~Brown, than whom no one can speak with more authority, +since he was one of my brother's favourite pupils. + +In the late~'70's George began to be appointed to various University +Boards and Syndicates. Thus from 1878--82 he was on the Museums and +Lecture Rooms Syndicate. In 1879 he was placed on the Observatory +Syndicate, of which he became an official member in 1883 on his election +to the Plumian Professorship. In the same way he was on the Special Board +for Mathematics. He was on the Financial Board from~1900--1 to~1903--4 +and on the Council of the Senate in 1905--6 and~1908--9. But he never +became a professional syndic---one of those virtuous persons who spend their +lives in University affairs. In his obituary of George (\textit{Nature}, Dec.~12, 1912), +Sir~Joseph Larmor writes: +\index{Darwin, Sir George, genealogy!university work, described by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}% +\index{Larmor, Sir Joseph, Sir George Darwin's work on university committees}% +\index{University committees, Sir George Darwin on, by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}% +\begin{Quote} +In the affairs of the University of which he was an ornament, +Sir George Darwin made a substantial mark, though it cannot be said +that he possessed the patience in discussion that is sometimes a +necessary condition to taking a share in its administration. But his wide +acquaintance and friendships among the statesmen and men of affairs of +the time, dating often from undergraduate days, gave him openings for +usefulness on a wider plane. Thus, at a time when residents were +bewailing even more than usual the inadequacy of the resources of the +University for the great expansion which the scientific progress of the +age demanded, it was largely on his initiative that, by a departure from +all precedent, an unofficial body was constituted in 1899 under the name +\DPPageSep{020}{xviii} +of the Cambridge University Association, to promote the further endowment +of the University by interesting its graduates throughout the +Empire in its progress and its more pressing needs. This important +body, which was organised under the strong lead of the late Duke of +Devonshire, then Chancellor, comprises as active members most of the +public men who owe allegiance to Cambridge, and has already by its +interest and help powerfully stimulated the expansion of the University +into new fields of national work; though it has not yet achieved +financial support on anything like the scale to which American seats +of learning are accustomed. +\end{Quote} +The Master of Christ's writes: +\index{Darwin, Sir George, genealogy!university work, described by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}% +\index{Master of Christ's, Sir George Darwin's work on university committees}% +\index{Newall, Prof., Sir George Darwin's work on university committees}% +\index{University committees, Sir George Darwin on, by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall}% +\begin{Letter}{\textit{May}~31\textit{st}, 1915.}{} + My impression is that George did not take very much interest in the + petty details which are so beloved by a certain type of University + authority. `Comma hunting' and such things were not to his taste, + and at Meetings he was often rather distrait: but when anything of + real importance came up he was of extraordinary use. He was + especially good at drafting letters, and over anything he thought + promoted the advancement of the University along the right lines he + would take endless trouble---writing and re-writing\DPnote{[** TN: Hyphenated in original]} reports and + letters till he got them to his taste. The sort of movements which + interested him most were those which connected Cambridge with the + outside world. He was especially interested in the Appointments + Board. A good many of us constantly sought his advice and nearly + always took it: but, as I say, I do not think he cared much about + the `parish pump,' and was usually worried at long Meetings. +\end{Letter} +Professor Newall has also been good enough to give me his impressions: +\begin{Quote} +His weight in the Committees on which I have had personal +experience of his influence seems to me to have depended in large +measure on his realising very clearly the distinction between the +importance of ends to be aimed at and the difficulty of harmonising +the personal characteristics of the men who might be involved in the +work needed to attain the ends. The ends he always took seriously; +the crotchets he often took humorously, to the great easement of many +situations that are liable to arise on a Committee. I can imagine that +to those who had corns his direct progress may at times have seemed +unsympathetic and hasty. He was ready to take much trouble in formulating +statements of business with great precision---a result doubtless +of his early legal experiences. I recall how he would say, `If a thing has +to be done, the minute should if possible make some individual responsible +for doing it.' He would ask, `Who is going to do the work? If a +man has to take the responsibility, we must do what we can to help him +and not hamper him by unnecessary restrictions and criticisms.' His +helpfulness came from his quickness in seizing the important point and +his readiness to take endless trouble in the important work of looking +into details before and after the meetings. The amount of work that he +did in response to the requirements of various Committees was very +great, and it was curious to realise in how many cases he seemed to +have diffidence as to the value of his contributions. +\end{Quote} +\DPPageSep{021}{xix} + +But on the whole the work which, in spite of ill-health, he was able to +carry out in addition to professional duties and research, was given to matters +unconnected with the University, but of a more general importance. To +these we shall return. + +In 1884 he became engaged to Miss Maud Du~Puy of Philadelphia. +\index{Darwin, Sir George, genealogy!marriage}% +She came of an old Huguenot stock, descending from Dr~John Du~Puy +who was born in France in~1679 and settled in New York in~1713. They +were married on July~22nd, 1884, and this event happily coloured the +remainder of George's life. As time went on and existence became fuller +and busier, she was able by her never-failing devotion to spare him much +arrangement and to shield him from fatigue and anxiety. In this way he +was helped and protected in the various semi-public functions in which he +took a principal part. Nor was her help valued only on these occasions, for +indeed the comfort and happiness of every day was in her charge. There is +a charming letter\footnoteN + {\textit{Emma Darwin, A Century of Family Letters}, Privately printed, 1904, vol.~\Vol{II.} p.~350.} +from George's mother, dated April~15th, 1884: +\begin{Quote} +Maud had to put on her wedding-dress in order to say at the +Custom-house in America that she had worn it, so we asked her to +come down and show it to us. She came down with great simplicity +and quietness\ldots only really pleased at its being admired and at looking +pretty herself, which was strikingly the case. She was a little shy at +coming in, and sent in Mrs~Jebb to ask George to come out and see it +first and bring her in. It was handsome and simple. I like seeing +George so frivolous, so deeply interested in which diamond trinket +should be my present, and in her new Paris morning dress, in which he +felt quite unfit to walk with her. +\end{Quote} + +Later, probably in June, George's mother wrote\footnoteN + {\textit{Emma Darwin, A Century of Family Letters}, 1912, vol.~\Vol{II.} p.~266.} +to Miss Du~Puy, ``Your +visit here was a great happiness to me, as something in you (I don't know +what) made me feel sure you would always be sweet and kind to George +when he is ill and uncomfortable.'' These simple and touching words may +be taken as a forecast of his happy married life. + +In March 1885 George acquired by purchase the house Newnham +\index{Darwin, Sir George, genealogy!house at Cambridge}% +Grange\footnotemarkN, which remained his home to the end of his life. It stands at the +\footnotetextN{At that time it was known simply as \textit{Newnham}, but as this is the name of the College and + was also in use for a growing region of houses, the Darwins christened it Newnham Grange. The + name Newnham is now officially applied to the region extending from Silver Street Bridge to the + Barton Road.}% +southern end of the Backs, within a few yards of the river where it bends +eastward in flowing from the upper to the lower of the two Newnham water-mills. +I remember forebodings as to dampness, but they proved wrong---even +the cellars being remarkably dry. The house is built of faded +yellowish bricks with old tiles on the roof, and has a pleasant home-like air. +\DPPageSep{022}{xx} +It was formerly the house of the Beales family\footnotemarkN, one of the old merchant +\footnotetextN{The following account of Newnham Grange is taken from C.~H. Cooper's \textit{Memorials of + Cambridge}, 1866, vol.~\Vol{III.} p.~262 (note):---``The site of the hermitage was leased by the Corporation + to Oliver Grene, 20~Sep., 31~Eliz.\ [1589]. It was in~1790 leased for a long term to + Patrick Beales, from whom it came to his brother S.~P. Beales, Esq., who erected thereon a + substantial mansion and mercantile premises now occupied by his son Patrick Beales, Esq., + alderman, who purchased the reversion from the Corporation in~1839.'' Silver Street was formerly + known as Little Bridges Street, and the bridges which gave it this name were in charge of a + hermit, hence the above reference to the hermitage.}% +stocks of Cambridge. This fact accounts for the great barn-like granaries +which occupied much of the plot near the high road. These buildings were +in part pulled down, thus making room for a lawn tennis court, while what +was not demolished made a gallery looking on the court as well as play-room +for the children. At the eastern end of the property a cottage and part of +the granaries were converted into a small house of an attractively individual +character, for which I think tenants have hitherto been easily found among +personal friends. It is at present inhabited by Lady~Corbett. One of the +most pleasant features of the Grange was the flower-garden and rockery +on the other side of the river, reached by a wooden bridge and called ``the +Little Island\footnotemarkN.'' The house is conveniently close to the town, yet has a most +\footnotetextN{This was to distinguish it from the ``Big Island,'' both being leased from the town. Later + George acquired in the same way the small oblong kitchen garden on the river bank, and bought + the freehold of the Lammas land on the opposite bank of the river.}% +pleasant outlook, to the north over the Backs while there is the river and the +Fen to the south. The children had a den or house in the branches of a +large copper beech tree, overhanging the river. They were allowed to use +the boat, which was known as the \textit{Griffin} from the family crest with which +it was adorned. None of them were drowned, though accidents were not +unknown; in one of these an eminent lady and well-known writer, who was +inveigled on to the river by the children, had to wade to shore near Silver +Street bridge owing to the boat running aground. + +The Darwins had five children, of whom one died an infant: of the others, +\index{Darwin, Sir George, genealogy!children}% +Charles Galton Darwin has inherited much of his father's mathematical +ability, and has been elected to a Mathematical Lectureship at Christ's +College. He is now in the railway service of the Army in France. The +younger son, William, has a commission in the 18th~Battalion of the Durham +Light Infantry. George's elder daughter is married to Monsieur Jacques +Raverat. Her skill as an artist has perhaps its hereditary root in her +father's draughtsmanship. The younger daughter Margaret lives with her +mother. + +George's relations with his family were most happy. His diary never +fails to record the dates on which the children came home, or the black days +which took them back to school. There are constantly recurring entries in +his diary of visits to the boys at Marlborough or Winchester. Or of the +\DPPageSep{023}{xxi} +journeys to arrange for the schooling of the girls in England or abroad. +The parents took pains that their children should have opportunities of +learning conversational French and German. + +George's characteristic energy showed itself not only in these ways but +also in devising bicycling expeditions and informal picnics, for the whole +family, to the Fleam Dyke, to Whittlesford, or other pleasant spots near home---and +these excursions he enjoyed as much as anyone of the party. As he +always wished to have his children with him, one or more generally accompanied +him and his wife when they attended congresses or other scientific +gatherings abroad. + +His house was the scene of many Christmas dinners, the first of which +I find any record being in~1886. These meetings were often made an +occasion for plays acted by the children; of these the most celebrated was +a Cambridge version of \textit{Romeo and Juliet}, in which the hero and heroine +were scions of the rival factions of Trinity and St~John's. + +\Section{}{Games and Pastimes.} +\index{Darwin, Sir George, genealogy!games and pastimes}% + +As an undergraduate George played tennis---not the modern out-door +game, but that regal pursuit which is sometimes known as the game of +kings and otherwise as the king of games. When George came up as an +undergraduate there were two tennis courts in Cambridge, one in the East +Road, the other being the ancient one that gave its name to Tennis Court +Road and was pulled down to make room for the new buildings of Pembroke. +In this way was destroyed the last of the College tennis courts of which we +read in Mr~Clark's \textit{History}. I think George must have had pleasure in the +obvious development of the tennis court from some primaeval court-yard in +which the \textit{pent-house} was the roof of a shed, and the \textit{grille} a real window +or half-door. To one brought up on evolution there is also a satisfaction +about the French terminology which survives in e.g.\ the \textit{Tambour} and +the \textit{Dedans}. George put much thought into acquiring a correct style of +play---for in tennis there is a religion of attitude corresponding to that which +painfully regulates the life of the golfer. He became a good tennis player as +an undergraduate, and was in the running for a place in the inter-University +match. The marker at the Pembroke court was Henry Harradine, whom we +all sincerely liked and respected, but he was not a good teacher, and it was +only when George came under Henry's sons, John and Jim Harradine, at the +Trinity and Clare courts, that his game began to improve. He continued to +play tennis for some years, and only gave it up after a blow from a tennis +ball in January~1895 had almost destroyed the sight of his left eye. + +In 1910 he took up archery, and zealously set himself to acquire the +correct mode of standing, the position of the head and hands,~etc. He kept +an archery diary in which each day's shooting is carefully analysed and the +\DPPageSep{024}{xxii} +results given in percentages. In 1911 he shot on 131~days: the last occasion +on which he took out his bow was September~13, 1912. + +I am indebted to Mr~H. Sherlock, who often shot with him at Cambridge, +for his impressions. He writes: ``I shot a good deal with your brother the +year before his death; he was very keen on the sport, methodical and painstaking, +and paid great attention to style, and as he had a good natural +`loose,' which is very difficult to acquire, there is little doubt (notwithstanding +that he came to Archery rather late in life) that had he lived he would have +been above the average of the men who shoot fairly regularly at the public +Meetings.'' After my brother's death, Mr~Sherlock was good enough to look +at George's archery note-book. ``I then saw,'' he writes, ``that he had +analysed them in a way which, so far as I am aware, had never been done +before.'' Mr~Sherlock has given examples of the method in a sympathetic +obituary published (p.~273) in \textit{The Archer's Register}\footnotemarkN. George's point was +\footnotetextN{\textit{The Archer's Register} for 1912--1913, by H.~Walrond. London, \textit{The Field} Office, 1913.}% +that the traditional method of scoring is not fair in regard to the areas of the +coloured rings of the target. Mr~Sherlock records in his \textit{Notice} that George +joined the Royal Toxophilite Society in~1912, and occasionally shot in the +Regent's Park. He won the Norton Cup and Medal (144~arrows at 120~yards) +in~1912. + +There was a billiard table at Down, and George learned to play fairly +well though he had no pretension to real proficiency. He used to play at +the Athenaeum, and in 1911 we find him playing there in the Billiard +Handicap, but a week later he records in his diary that he was ``knocked +out.'' + +\Section{}{Scientific Committees.} +\index{Committees, Sir George Darwin on}% +\index{Darwin, Sir George, genealogy!work on scientific committees}% + +George served for many years on the Solar Physics Committee and on +the Meteorological Council. With regard to the latter, Sir~Napier Shaw +has at my request supplied the following note:--- +\index{Meteorological Council, by Sir Napier Shaw}% +\index{Shaw, Sir Napier, Meteorological Council}% +\begin{Quote} +It was in February~1885 upon the retirement of Warren De~la~Rue +that your brother George, by appointment of the Royal Society, joined +the governing body of the Meteorological Office, at that time the +Meteorological Council. He remained a member until the end of the +Council in~1905 and thereafter, until his death, he was one of the two +nominees of the Royal Society upon the Meteorological Committee, the +new body which was appointed by the Treasury to take over the control +of the administration of the Office. + +It will be best to devote a few lines to recapitulating the salient +features of the history of the official meteorological organisation because, +otherwise, it will be difficult for anyone to appreciate the position in +which Darwin was placed. +\DPPageSep{025}{xxiii} + +In 1854 a department of the Board of Trade was constituted under +Admiral R.~FitzRoy to collect and discuss meteorological information +from ships, and in~1860, impressed by the loss of the `Royal Charter,' +FitzRoy began to collect meteorological observations by telegraph from +land stations and chart them. Looking at a synchronous chart and +conscious that he could gather from it a much better notion of coming +weather than anyone who had only his own visible sky and barometer +to rely upon, he formulated `forecasts' which were published in the +newspapers and `storm warnings' which were telegraphed to the ports. + +This mode of procedure, however tempting it might be to the +practical man with the map before him, was criticised as not complying +with the recognised canons of scientific research, and on FitzRoy's +untimely death in 1865 the Admiralty, the Board of Trade and the +Royal Society elaborated a scheme for an office for the study of weather +in due form under a Director and Committee, appointed by the Royal +Society, and they obtained a grant in aid of~£10,000 for this purpose. +In this transformation it was Galton, I believe, who took a leading part +and to him was probably due the initiation of the new method of study +which was to bring the daily experience, as represented by the map, +into relation with the continuous records of the meteorological elements +obtained at eight observatories of the Kew type, seven of which were +immediately set on foot, and Galton devoted an immense amount of +time and skill to the reproduction of the original curves so that the +whole sequence of phenomena at the seven observatories could be taken +in at a glance. Meanwhile the study of maps was continued and a good +deal of progress was made in our knowledge of the laws of weather. + +But in spite of the wealth of information the generalisations were +empirical and it was felt that something more than the careful examination +of records was required to bring the phenomena of weather within +the rule of mathematics and physics, so in 1876 the constitution of the +Office was changed and the direction of its work was placed in Commission +with an increased grant. The Commissioners, collectively known +as the Meteorological Council, were a remarkably distinguished body of +fellows of the Royal Society, and when Darwin took the place of +De~la~Rue, the members were men subsequently famous, as Sir~Richard +Strachey, Sir~William Wharton, Sir~George Stokes, Sir~Francis Galton, +Sir~George Darwin, with E.~J.~Stone, a former Astronomer Royal for +the Cape. + +It was understood that the attack had to be made by new methods +and was to be entrusted partly to members of the Council themselves, +with the staff of the Office behind them, and partly to others outside +who should undertake researches on special points. Sir~Andrew Noble, +Sir~William Abney, Dr~W.~J. Russell, Mr~W.~H. Dines, your brother +Horace and myself came into connection with the Council in this way. + +Two important lines of attack were opened up within the Council +itself. The first was an attempt, under the influence of Lord Kelvin, +to base an explanation of the sequence of weather upon harmonic +analysis. As the phenomena of tides at any port could be synthesized +by the combinations of waves of suitable period and amplitude, so the +sequence of weather could be analysed into constituent oscillations the +general relations of which would be recognisable although the original +\DPPageSep{026}{xxiv} +composite result was intractable on direct inspection. It was while this +enterprise was in progress that Darwin was appointed to the Council. +His experience with tides and tidal analysis was in a way his title +to admission. He and Stokes were the mathematicians of the Council +and were looked to for expert guidance in the undertaking. At first +the individual curves were submitted to analysis in a harmonic analyser +specially built for the purpose, the like of which Darwin had himself +used or was using for his work on tides; but afterwards it was decided +to work arithmetically with the numbers derived from the tabulation of +the curves; and the identity of the individual curves was merged in +`five-day means.' The features of the automatic records from which so +much was hoped in~1865, after twelve years of publication in facsimile, +were practically never seen outside the room in the Office in which they +were tabulated. + +It is difficult at this time to point to any general advances in +meteorology which can be attributed to the harmonic analyser or its +arithmetical equivalent as a process of discussion, though it still remains +a powerful method of analysis. It has, no doubt, helped towards the +recognition of the ubiquity and simultaneity of the twelve-hour term in +the diurnal change of pressure which has taken its place among fundamental +generalisations of meteorology and the curious double diurnal +change in the wind at any station belongs to the same category; but +neither appears to have much to do with the control of weather. +Probably the real explanation of the comparative fruitlessness of the +effort lies in the fact that its application was necessarily restricted to +the small area of the British Isles instead of being extended, in some +way or other, to the globe. + +It is not within my recollection that Darwin was particularly +enthusiastic about the application of harmonic analysis. When I was +appointed to the Council in~1897, the active pursuit of the enterprise +had ceased. Strachey who had taken an active part in the discussion +of the results and contributed a paper on them to the Philosophical +Transactions, was still hopeful of basing important conclusions upon the +seasonal peculiarities of the third component, but the interest of other +members of the Council was at best languid. + +The other line of attack was in connection with synoptic charts. For +the year from August~1892 to August~1893 there was an international +scheme for circumpolar observations in the Northern Hemisphere, and +in connection therewith the Council undertook the preparation of daily +synoptic charts of the Atlantic and adjacent land areas. A magnificent +series of charts was produced and published from which great results +were anticipated. But again the conclusions drawn from cursory inspection +were disappointing. At that time the suggestion that weather +travelled across the Atlantic in so orderly a manner that our weather +could be notified four or five days in advance from New York had a +considerable vogue and the facts disclosed by the charts put an end to +any hope of the practical development of that suggestion. Darwin was +very active in endeavouring to obtain the help of an expert in physics +for the discussion of the charts from a new point of view, but he was +unsuccessful. + +Observations at High Level Stations were also included in the +\DPPageSep{027}{xxv} +Council's programme. A station was maintained at Hawes Junction +for some years, and the Observatories on Ben Nevis received their +support. But when I joined the Council in 1897 there was a pervading +sense of discouragement. The forecasting had been restored as the result +of the empirical generalisations based on the work of the years 1867~to~1878, +but the study had no attractions for the powerful analytical minds +of the Council; and the work of the Office had settled down into the +assiduous compilation of observations from sea and land and the regular +issue of forecasts and warnings in the accustomed form. The only part +which I can find assigned to Darwin with regard to forecasting is an +endeavour to get the forecast worded so as not to suggest more assurance +than was felt. + +I do not think that Darwin addressed himself spontaneously to +meteorological problems, but he was always ready to help. He was +very regular in his attendance at Council and the Minutes show that +after Stokes retired all questions involving physical measurement or +mathematical reasoning were referred to him. There is a short and +very characteristic report from him on the work of the harmonic +analyser and a considerable number upon researches by Mr~Dines or +Sir~G.~Stokes on anemometers. It is hardly possible to exaggerate +his aptitude for work of that kind. He could take a real interest in +things that were not his own. He was full of sympathy and appreciation +for efforts of all kinds, especially those of young men, and at the same +time, using his wide experience, he was perfectly frank and fearless not +only in his judgment but also in the expression of it. He gave one the +impression of just protecting himself from boredom by habitual loyalty +and a finely tempered sense of duty. My earliest recollection of him on +the Council is the thrilling production of a new version of the Annual +Report of the Council which he had written because the original had +become more completely `scissors and paste' than he could endure. + +After the Office came into my charge in~1900, so long as he lived, +I never thought of taking any serious step without first consulting him +and he was always willing to help by his advice, by his personal influence +and by his special knowledge. For the first six years of the time +I held a college fellowship with the peculiar condition of four public +lectures in the University each year and no emolument. One year, +when I was rather overdone, Darwin took the course for me and devoted +the lectures to Dynamical Meteorology. I believe he got it up for the +occasion, for he professed the utmost diffidence about it, but the progress +which we have made in recent years in that subject dates from those +lectures and the correspondence which arose upon them. + +In Council it was the established practice to proceed by agreement +and not by voting; he had a wonderful way of bringing a discussion to +a head by courageously `voicing' the conclusion to which it led and +frankly expressing the general opinion without hurting anybody's +feelings. + +This letter has, I fear, run to a great length, but it is not easy +to give expression to the powerful influence which he exercised upon +all departments of official meteorology without making formal contributions +to meteorological literature. He gave me a note on a curious +point in the evaluation of the velocity equivalents of the Beaufort Scale +\DPPageSep{028}{xxvi} +which is published in the Office Memoirs No.~180, and that is all I have +to show in print, but he was in and behind everything that was done +and personally, I need hardly add, I owe to him much more than this or +any other letter can fully express. +\end{Quote} + +On May~6, 1904, he was elected President of the British Association---the +\index{British Association, South African Meeting, 1905}% +\index{South African Meeting of the British Association, 1905}% +South African meeting. + +On July~29, 1905, he embarked with his wife and his son Charles and +arrived on August~15 at the Cape, where he gave the first part of his +Presidential Address. Here he had the pleasure of finding as Governor +Sir~Walter Hely-Hutchinson, whom he had known as a Trinity undergraduate. +He was the guest of the late Sir~David Gill, who remained a close friend for +the rest of his life. George's diary gives his itinerary---which shows the +trying amount of travel that he went through. A sample may be quoted: +\begin{center} +\footnotesize +\begin{tabular}{cl} +August 19 & Embark, \\ +\Ditto 22 & Arrive at Durban, \\ +\Ditto 23 & Mount Edgecombe, \\ +\Ditto 24 & Pietermaritzburg, \\ +\Ditto 26 & Colenso, \\ +\Ditto 27 & Ladysmith, \\ +\Ditto 28 & Johannesburg. +\end{tabular} +\end{center} + +At Johannesburg he gave the second half of his Address. Then on by +Bloemfontein, Kimberley, Bulawayo, to the Victoria Falls, where a bridge had +to be opened. Then to Portuguese Africa on September~16,~17, where he +made speeches in French and English. Finally he arrived at Suez on +October~4 and got home October~18. + +It was generally agreed that his Presidentship was a conspicuous success. +The following appreciation is from the obituary notice in \textit{The Observatory}, +Jan.~1913, p.~58: +\begin{Quote} +The Association visited a dozen towns, and at each halt its President +addressed an audience partly new, and partly composed of people who +had been travelling with him for many weeks. At each place this +latter section heard with admiration a treatment of his subject wholly +fresh and exactly adapted to the locality. +\end{Quote} +Such duties are always trying and it should not be forgotten that tact was +necessary in a country which only two years before was still in the throes +of war. + +In the autumn he received the honour of being made a~K.C.B\@. The +distinction was doubly valued as being announced to him by his friend +Mr~Balfour, then Prime Minister. + +From 1899~to~1900 he was President of the Royal Astronomical Society. +One of his last Presidential acts was the presentation of the Society's Medal +to his friend M.~Poincaré. +\DPPageSep{029}{xxvii} + +He had the unusual distinction of serving twice as President of the +Cambridge Philosophical Society, once in 1890--92 and again 1911--12. + +In 1891 he gave the Bakerian Lecture\footnoteN + {See Prof.~Brown's Memoir, \Pageref{xlix}.} +of the Royal Society, his subject +being ``Tidal Prediction.'' This annual prælection dates from~1775 and the +list of lecturers is a distinguished roll of names. + +In 1897 he lectured at the Lowell Institute at Boston, and this was +\index{Tides, The@\textit{Tides, The}}% +the origin of his book on \textit{Tides}, published in the following year. Of this +Sir~Joseph Larmor says\footnoteN + {\textit{Nature}, 1912. See also Prof.~Brown's Memoir, \Pageref{l}.} +that ``it has taken rank with the semi-popular +writings of Helmholtz and Kelvin as a model of what is possible in the +exposition of a scientific subject.'' It has passed through three English +editions, and has been translated into many foreign languages. + +\Section{}{International Associations.} + +During the last ten or fifteen years of his life George was much occupied +\index{Geodetic Association, International}% +\index{Larmor, Sir Joseph, Sir George Darwin's work on university committees!International Geodetic Association}% +with various International bodies, e.g.~the International Geodetic Association, +the International Association of Academics, the International Congress of +Mathematicians and the Seismological Congress. + +With regard to the last named it was in consequence of George's report +to the Royal Society that the British Government joined the Congress. It +was however with the Geodetic Association that he was principally connected. + +Sir~Joseph Larmor (\textit{Nature}, December~12, 1912) gives the following +account of the origin of the Association: +\begin{Quote} +The earliest of topographic surveys, the model which other national +surveys adopted and improved upon, was the Ordnance Survey of the +United Kingdom. But the great trigonometrical survey of India, started +nearly a century ago, and steadily carried on since that time by officers +of the Royal Engineers, is still the most important contribution to the +science of the figure of the earth, though the vast geodetic operations in +the United States are now following it closely. The gravitational and +other complexities incident on surveying among the great mountain +masses of the Himalayas early demanded the highest mathematical +assistance. The problems originally attacked in India by Archdeacon +Pratt were afterwards virtually taken over by the Royal Society, and its +secretary, Sir~George Stokes, of Cambridge, became from 1864 onwards +the adviser and referee of the survey as regards its scientific enterprises. +On the retirement of Sir~George Stokes, this position fell very largely to +Sir~George Darwin, whose relations with the India Office on this and +other affairs remained close, and very highly appreciated, throughout +the rest of his life. + +The results of the Indian survey have been of the highest importance +for the general science of geodesy\ldots. It came to be felt that closer +cooperation between different countries was essential to practical +progress and to coordination of the work of overlapping surveys. +\end{Quote} +\DPPageSep{030}{xxviii} + +The further history of George's connection with the Association is told in +\index{Bakhuyzen, Dr Van d.\ Sande, Sir George Darwin's connection with the International Geodetic Association}% +\index{Geodetic Association, International}% +the words of its Secretary, Dr~van~d.\ Sande Bakhuyzen, to whom I am greatly +indebted. +\begin{Quote} +On the proposal of the Royal Society, the British Government, after +having consulted the Director of the Ordnance Survey, in~1898, resolved +upon the adhesion of Great Britain to the International Geodetic Association, +and appointed as its delegate, G.~H.~Darwin. By his former +researches and by his high scientific character, he, more than any other, +was entitled to this position, which would afford him an excellent +opportunity of furthering, by his recommendations, the study of theoretical +geodesy. + +The meeting at Stuttgart in 1898 was the first which he attended, +and at that and the following conferences, Paris~1900, Copenhagen~1903, +Budapest~1906, London-Cambridge~1909, he presented reports on the +geodetic work in the British Empire. To Sir~David Gill's report on the +geodetic work in South Africa, which he delivered at Budapest, Darwin +added an appendix in which he relates that the British South Africa +Company, which had met all the heavy expense of the part of the survey +along the 30th~meridian through Rhodesia, found it necessary to make +various economies, so that it was probably necessary to suspend the +survey for a time. This interruption would be most unfortunate for the +operations relating to the great triangulation from the Southern part of +Cape Colony to Egypt, but, happily, by the cooperation of different +authorities, all obstacles had been overcome and the necessary money +found, so that the triangulation could be continued. So much for +Sir~George Darwin's communication; it is correct but incomplete, as it +does not mention that it was principally by Darwin's exertions and by +his personal offer of financial help that the question was solved and the +continuation of this great enterprise secured. + +To the different researches which enter into the scope of the Geodetic +Association belong the researches on the tides, and it is natural that +Darwin should be chosen as general reporter on that subject; two +elaborate reports were presented by him at the conferences of Copenhagen +and London. + +In Copenhagen he was a member of the financial committee, and at +the request of this body he presented a report on the proposal to determine +gravity at sea, in which he strongly recommended charging Dr~Hecker +with that determination using the method of Prof.~Mohn (boiling +temperature of water and barometer readings). At the meeting of~1906 +an interesting report was read by him on a question raised by +the Geological Congress: the cooperation of the Geodetic Association +in geological researches by means of the anomalies in the intensity +of gravitation. + +By these reports and recommendations Darwin exercised a useful +influence on the activity of the Association, but his influence was to be +still increased. In 1907 the Vice-president of the Association, General +Zacharias, died, and the permanent committee, whose duty it was to +nominate his provisional successor, chose unanimously Sir~George +Darwin, and this choice was confirmed by the next General Conference +in London. +\DPPageSep{031}{xxix} + +We cannot relate in detail his valuable cooperation as a member of +the council in the various transactions of the Association, for instance on +the junction of the Russian and Indian triangulations through Pamir, +but we must gratefully remember his great service to the Association +when, at his invitation, the delegates met in 1909 for the 16th~General +Conference in London and Cambridge. +\index{Mathematicians, International Congress of, Cambridge, 1912}% + +With the utmost care he prepared everything to render the Conference +as interesting and agreeable as possible, and he fully succeeded. +Through his courtesy the foreign delegates had the opportunity of making +the personal acquaintance of several members of the Geodetic staff of +England and its colonies, and of other scientific men, who were invited +to take part in the conference; and when after four meetings in London +the delegates went to Cambridge to continue their work, they enjoyed +the most cordial hospitality from Sir~George and Lady~Darwin, who, +with her husband, procured them in Newnham Grange happy leisure +hours between their scientific labours. + +At this conference Darwin delivered various reports, and at the +discussion on Hecker's determination of the variation of the vertical by +the attraction of the moon and sun, he gave an interesting account of +the researches on the same subject made by him and his brother Horace +more than 20~years ago, which unfortunately failed from the bad conditions +of the places of observation. + +In 1912 Sir~George, though already over-fatigued by the preparations +for the mathematical congress in Cambridge, and the exertions entailed +by it, nevertheless prepared the different reports on the geodetic work +in the British Empire, but alas his illness prevented him from assisting +at the conference at Hamburg, where they were presented by other +British delegates. The conference thanked him and sent him its best +wishes, but at the end of the year the Association had to deplore the loss +of the man who in theoretical geodesy as well as in other branches of +mathematics and astronomy stood in the first rank, and who for his +noble character was respected and beloved by all his colleagues in the +International Geodetic Association. +\end{Quote} +Sir~Joseph Larmor writes\footnoteN + {\textit{Nature}, Dec.~12, 1912.}: +\index{Larmor, Sir Joseph, Sir George Darwin's work on university committees!International Congress of Mathematicians at Cambridge 1912}% +\index{Congress, International, of Mathematicians at Cambridge, 1912!note by Sir Joseph Larmor}% +\begin{Quote} +Sir~George Darwin's last public appearance was as president of the +fifth International Congress of Mathematicians, which met at Cambridge +on August~22--28, 1912. The time for England to receive the congress +having obviously arrived, a movement was initiated at Cambridge, with +the concurrence of Oxford mathematicians, to send an invitation to the +fourth congress held at Rome in~1908. The proposal was cordially +accepted, and Sir~George Darwin, as \textit{doyen} of the mathematical school +at Cambridge, became chairman of the organising committee, and was +subsequently elected by the congress to be their president. Though +obviously unwell during part of the meeting, he managed to discharge +the delicate duties of the chair with conspicuous success, and guided +with great \textit{verve} the deliberations of the final assembly of what turned +out to be a most successful meeting of that important body. +\end{Quote} +\DPPageSep{032}{xxx} + +\Section{}{Personal Characteristics.} +\index{Darwin, Sir George, genealogy!personal characteristics}% +\index{Darwin, Margaret, on Sir George Darwin's personal characteristics}% +\index{Raverat, Madame, on Sir George Darwin's personal characteristics}% + +His daughter, Madame Raverat, writes: +\begin{Quote} +I think most people might not realise that the sense of adventure +and romance was the most important thing in my father's life, except his +love of work. He thought about all life romantically and his own life +in particular; one could feel it in the quality of everything he said +about himself. Everything in the world was interesting and wonderful +to him and he had the power of making other people feel it. + +He had a passion for going everywhere and seeing everything; +learning every language, knowing the technicalities of every trade; and +all this emphatically \textit{not} from the scientific or collector's point of view, but +from a deep sense of the romance and interest of everything. It was +splendid to travel with him; he always learned as much as possible of +the language, and talked to everyone; we had to see simply everything +there was to be seen, and it was all interesting like an adventure. For +instance at Vienna I remember being taken to a most improper music hall; +and at Schönbrunn hearing from an old forester the whole secret history of +the old Emperor's son. My father would tell us the stories of the places +we went to with an incomparable conviction, and sense of the reality +and dramaticness of the events. It is absurd of course, but in that +respect he always seemed to me a little like Sir~Walter Scott\footnotemarkN. +\footnotetextN{Compare Mr~Chesterton's \textit{Twelve Types}, 1903, p.~190. He speaks of Scott's critic in the + \textit{Edinburgh Review}: ``The only thing to be said about that critic is that he had never been + a little boy. He foolishly imagined that Scott valued the plume and dagger of Marmion for + Marmion's sake. Not being himself romantic, he could not understand that Scott valued + the plume because it was a plume and the dagger because it was a dagger.''}% + +The books he used to read to us when we were quite small, +and which we adored, were Percy's \textit{Reliques} and the \textit{Prologue to the +Canterbury Tales}. He used often to read Shakespeare to himself, +I think generally the historical plays, Chaucer, \textit{Don Quixote} in Spanish, +and all kind of books like Joinville's \textit{Life of St~Louis} in the old French. + +I remember the story of the death of Gordon told so that we all +cried, I think; and Gladstone could hardly be mentioned in consequence. +All kinds of wars and battles interested him, and I think he liked archery +more because it was romantic than because it was a game. + +During his last illness his interest in the Balkan war never failed. +Three weeks before his death he was so ill that the doctor thought him +dying. Suddenly he rallied from the half-unconscious state in which he +had been lying for many hours and the first words he spoke on opening +his eyes were: ``Have they got to Constantinople yet?'' This was very +characteristic. I often wish he was alive now, because his understanding +and appreciation of the glory and tragedy of this war would +be like no one else's. +\end{Quote} +His daughter Margaret Darwin writes: +\begin{Quote} +He was absolutely unselfconscious and it never seemed to occur to +him to wonder what impression he was making on others. I think it +was this simplicity which made him so good with children. He seemed +to understand their point of view and to enjoy \textit{with} them in a way that +\DPPageSep{033}{xxxi} +is not common with grown-up people. I shall never forget how when +our dog had to be killed he seemed to feel the horror of it just as I did, +and how this sense of his really sharing my grief made him able to +comfort me as nobody else could. + +He took a transparent pleasure in the honours that came to him, +especially in his membership of foreign Academies, in which he and +Sir~David Gill had a friendly rivalry or ``race,'' as they called it. I think +this simplicity was one of his chief characteristics, though most important +of all was the great warmth and width of his affections. He +would take endless trouble about his friends, especially in going to see +them if they were lonely or ill; and he was absolutely faithful and +generous in his love. +\end{Quote} + +After his mother came to live in Cambridge, I believe he hardly ever +missed a day in going to see her even though he might only be able to stay +a few minutes. She lived at some distance off and he was often both busy +and tired. This constancy was very characteristic. It was shown once more +in his many visits to Jim Harradine, the marker at the tennis court, on what +proved to be his death-bed. + +His energy and his kindness of heart were shown in many cases of distress. +For instance, a guard on the Great Northern Railway was robbed of his savings +by an absconding solicitor, and George succeeded in collecting some~£300 +for him. In later years, when his friend the guard became bedridden, George +often went to see him. Another man whom he befriended was a one-legged +man at Balsham whom he happened to notice in bicycling past. He took the +trouble to see the village authorities and succeeded in sending the man to +London to be fitted with an artificial leg. + +In these and similar cases there was always the touch of personal +sympathy. For instance he pensioned the widow of his gardener, and he +often made the payment of her weekly allowance the excuse for a visit. + +In another sort of charity he was equally kind-hearted, viz.~in answering +the people who wrote foolish letters to him on scientific subjects---and here +as in many points he resembled his father. + +His sister, Mrs~Litchfield, has truly said\footnoteN + {\textit{Emma Darwin, A Century of Family Letters}, 1915, vol.~\Vol{II.} p.~146.} +of George that he inherited his +father's power of work and much of his ``cordiality and warmth of nature +with a characteristic power of helping others.'' He resembled his father in +another quality, that of modesty. His friend and pupil E.~W.~Brown writes: +\begin{Quote} +He was always modest about the importance of his researches. +He would often wonder whether the results were worth the labour they +had cost him and whether he would have been better employed in some +other way. +\end{Quote} + +His nephew Bernard, speaking of George's way of taking pains to be +friendly and forthcoming to anyone with whom he came in contact, says: +\DPPageSep{034}{xxxii} +\begin{Quote} +He was ready to take other people's pleasantness and politeness at +its apparent value and not to discount it. If they seemed glad to see him, +he believed that they \textit{were} glad. If he liked somebody, he believed +that the somebody liked him, and did not worry himself by wondering +whether they really did like him. +\end{Quote} + +Of his energy we have evidence in the \textit{amount} of work contained in +\index{Darwin, Sir George, genealogy!energy}% +these volumes. There was nothing dilatory about him, and here he again +resembled his father who had markedly the power of doing things at the +right moment, and thus avoiding waste of time and discomfort to others. +George had none of a characteristic which was defined in the case of Henry +Bradshaw, as ``always doing something else.'' After an interruption he could +instantly reabsorb himself in his work, so that his study was not kept as a +place sacred to peace and quiet. + +His wife is my authority for saying that although he got so much done, +it was not by working long hours. Moreover the days that he was away +from home made large gaps in his opportunities for steady application. His +diaries show in another way that his researches by no means took all his +time. He made a note of the books he read and these make a considerable +record. Although he read much good literature with honest enjoyment, he +had not a delicate or subtle literary judgment. Nor did he care for music. +He was interested in travels, history, and biography, and as he could remember +what he read or heard, his knowledge was wide in many directions. His +linguistic power was characteristic. He read many European languages. +I remember his translating a long Swedish paper for my father. And he +took pleasure in the Platt Deutsch stories of Fritz Reuter. + +The discomfort from which he suffered during the meeting at Cambridge +of the International Congress of Mathematicians in August~1912, was in fact +the beginning of his last illness. An exploratory operation showed that he +was suffering from malignant disease. Happily he was spared the pain that +gives its terror to this malady. His nature was, as we have seen, simple and +direct with a pleasant residue of the innocence and eagerness of childhood. +In the manner of his death these qualities were ennobled by an admirable +and most unselfish courage. As his vitality ebbed away his affection only +showed the stronger. He wished to live, and he felt that his power of work +and his enjoyment of life were as strong as ever, but his resignation to the +sudden end was complete and beautiful. He died on Dec.~7, 1912, and was +buried at Trumpington. +\DPPageSep{035}{xxxiii} + + +\Heading{Honours, Medals, Degrees, Societies, etc.} +\index{Darwin, Sir George, genealogy!honours}% + +\Subsection{Order. \upshape K.C.B. 1905.} + +\Subsection{Medals\footnotemarkN.} +\footnotetextN{Sir~George's medals are deposited in the Library of Trinity College, Cambridge.} + +1883. Telford Medal of the Institution of Civil Engineers. + +1884. Royal Medal\footnotemarkN. +\footnotetextN{Given by the Sovereign on the nomination of the Royal Society.} + +1892. Royal Astronomical Society's Medal. + +1911. Copley Medal of the Royal Society. + +1912. Royal Geographical Society's Medal. + +\Subsection{Offices.} + +Fellow of Trinity College, Cambridge, and Plumian Professor in the +University. + +Vice-President of the International Geodetic Association, Lowell Lecturer +at Boston U.S.~(1897). + +Member of the Meteorological and Solar Physics Committees. + +Past President of the Cambridge Philosophical Society\footnotemarkN, Royal Astronomical +\footnotetextN{Re-elected in 1912.} +Society, British Association. + +\Subsection{Doctorates, etc.\ of Universities.} + +Oxford, Dublin, Glasgow, Pennsylvania, Padua (Socio onorario), Göttingen, +Christiania, Cape of Good Hope, Moscow (honorary member). + +\Subsection{Foreign or Honorary Membership of Academies, etc.} + +Amsterdam (Netherlands Academy), Boston (American Academy), +Brussels (Royal Society), Calcutta (Math.\ Soc.), Dublin (Royal Irish +Academy), Edinburgh (Royal Society), Halle (K.~Leop.-Carol.\ Acad.), +Kharkov (Math.\ Soc.), Mexico (Soc.\ ``Antonio Alzate''), Moscow (Imperial +Society of the Friends of Science), New York, Padua, Philadelphia (Philosophical +Society), Rome (Lincei), Stockholm (Swedish Academy), Toronto +(Physical Society), Washington (National Academy), Wellington (New +Zealand Inst.). + +\Subsection{Correspondent of Academies, etc.\ at} + +Acireale (Zelanti), Berlin (Prussian Academy), Buda Pest (Hungarian +Academy), Frankfort (Senckenberg.\ Natur.\ Gesell.), Göttingen (Royal Society), +Paris, St~Petersburg, Turin, Istuto Veneto, Vienna\footnotemarkN. +\footnotetextN{The above list is principally taken from that compiled by Sir~George for the Year-Book of + the Royal Society,~1912, and may not be quite complete. + + It should be added that he especially valued the honour conferred on him in the publication + of his collected papers by the Syndics of the University Press.} +\DPPageSep{036}{xxxiv} + + +%[** TN: Changed the running heads; original splits the title] +\Chapter{The Scientific Work of Sir George Darwin} +\BY{Professor E. W. Brown} +\index{Brown, Prof.\ E. W., Sir George Darwin's Scientific Work}% +\index{Darwin, Sir George, genealogy!scientific work, by Prof.\ E. W. Brown}% +\index{Darwin, Sir George, genealogy!characteristics of his work}% + +The scientific work of Darwin possesses two characteristics which cannot +fail to strike the reader who glances over the titles of the eighty odd papers +which are gathered together in the four volumes which contain most of his +publications. The first of these characteristics is the homogeneous nature +of his investigations. After some early brief notes, on a variety of subjects, +he seems to have set himself definitely to the task of applying the tests of +mathematics to theories of cosmogony, and to have only departed from it +when pressed to undertake the solution of practical problems for which there +was an immediate need. His various papers on viscous spheroids concluding +with the effects of tidal friction, the series on rotating masses of fluids, even +those on periodic orbits, all have the idea, generally in the foreground, of +developing the consequences of old and new assumptions concerning the past +history of planetary and satellite systems. That he achieved so much, in +spite of indifferent health which did not permit long hours of work at his +desk, must have been largely due to this single aim. + +The second characteristic is the absence of investigations undertaken for +their mathematical interest alone; he was an applied mathematician in the +strict and older sense of the word. In the last few decades another school of +applied mathematicians, founded mainly by Poincaré, has arisen, but it differs +essentially from the older school. Its votaries have less interest in the +phenomena than in the mathematical processes which are used by the student +of the phenomena. They do not expect to examine or predict physical +events but rather to take up the special classes of functions, differential +equations or series which have been used by astronomers or physicists, to +examine their properties, the validity of the arguments and the limitations +which must be placed on the results. Occasionally theorems of great physical +importance will emerge, but from the primary point of view of the investigations +these are subsidiary results. Darwin belonged essentially to the school which +studies the phenomena by the most convenient mathematical methods. Strict +logic in the modern sense is not applied nor is it necessary, being replaced in +most cases by intuition which guides the investigator through the dangerous +places. That the new school has done great service to both pure and applied +mathematics can hardly be doubted, but the two points of view of the subject +\DPPageSep{037}{xxxv} +will but rarely be united in the same man if much progress in either direction +is to be made. Hence we do not find and do not expect to find in Darwin's +work developments from the newer point of view. + +At the same time, he never seems to have been affected by the problem-solving +habits which were prevalent in Cambridge during his undergraduate +days and for some time later. There was then a large number of mathematicians +brought up in the Cambridge school whose chief delight was the +discovery of a problem which admitted of a neat mathematical solution. +The chief leaders were, of course, never very seriously affected by this +attitude; they had larger objects in view, but the temptation to work out +a problem, even one of little physical importance, when it would yield to +known mathematical processes, was always present. Darwin kept his aim +fixed. If the problem would not yield to algebra he has recourse to +arithmetic; in either case he never seemed to hesitate to embark on the +most complicated computations if he saw a chance of attaining his end. +The papers on ellipsoidal harmonic analysis and periodic orbits are instructive +examples of the labour which he would undertake to obtain a knowledge of +physical phenomena. + +One cannot read any of his papers without also seeing another feature, +his preference for quantitative rather than qualitative results. If he saw +any possibility of obtaining a numerical estimate, even in his most speculative +work, he always made the necessary calculations. His conclusions +thus have sometimes an appearance of greater precision than is warranted +by the degree of accuracy of the data. But Darwin himself was never +misled by his numerical conclusions, and he is always careful to warn his +readers against laying too great a stress on the numbers he obtains. + +In devising processes to solve his problems, Darwin generally adopted +those which would lead in a straightforward manner to the end he had +in view. Few ``short cuts'' are to be found in his memoirs. He seems to +have felt that the longer processes often brought out details and points +of view which would otherwise have been concealed or neglected. This is +particularly evident in the papers on Periodic Orbits. In the absence of +general methods for the discovery and location of the curves, his arithmetic +showed classes of orbits which would have been difficult to find by analysis, +and it had a further advantage in indicating clearly the various changes +which the members of any class undergo when the parameter varies. Yet, +in spite of the large amount of numerical work which is involved in many +of his papers, he never seemed to have any special liking for either algebraic +or numerical computation; it was something which ``had to be done.'' Unlike +J.~C.~Adams and G.~W.~Hill, who would often carry their results to a large +number of places of decimals, Darwin would find out how high a degree of +accuracy was necessary and limit himself to it. +\DPPageSep{038}{xxxvi} + +The influence which Darwin exerted has been felt in many directions. +\index{Cosmogony, Sir George Darwin's influence on}% +\index{Darwin, Sir George, genealogy!his first papers}% +\index{Darwin, Sir George, genealogy!his influence on cosmogony}% +The exhibition of the necessity for quantitative and thorough analysis of the +problems of cosmogony and celestial mechanics has been perhaps one of his +chief contributions. It has extended far beyond the work of the pupils who +were directly inspired by him. While speculations and the framing of new +hypotheses must continue, but little weight is now attached to those which +are defended by general reasoning alone. Conviction fails, possibly because +it is recognised that the human mind cannot reason accurately in these +questions without the aids furnished by mathematical symbols, and in any +case language often fails to carry fully the argument of the writer as against +the exact implications of mathematics. If for no other reason, Darwin's work +marks an epoch in this respect. + +To the pupils who owed their first inspiration to him, he was a constant +\index{Darwin, Sir George, genealogy!his relationship with his pupils}% +\index{Pupils, Darwin's relationship with his}% +friend. First meeting them at his courses on some geophysical or astronomical +subject, he soon dropped the formality of the lecture-room, and they +found themselves before long going to see him continually in the study at +Newnham Grange. Who amongst those who knew him will fail to remember +the sight of him seated in an armchair with a writing board and papers +strewn about the table and floor, while through the window were seen +glimpses of the garden filled in summer time with flowers? While his +lectures in the class-room were always interesting and suggestive, the chief +incentive, at least to the writer who is proud to have been numbered amongst +his pupils and friends, was conveyed through his personality. To have spent +an hour or two with him, whether in discussion on ``shop'' or in general +conversation, was always a lasting inspiration. And the personal attachment +of his friends was strong; the gap caused by his death was felt to be far +more than a loss to scientific progress. Not only the solid achievements +contained in his published papers, but the spirit of his work and the example +of his life will live as an enduring memorial of him. + +\tb + +Darwin's first five papers, all published in~1875, are of some interest as +showing the mechanical turn of his mind and the desire, which he never lost, +for concrete illustrations of whatever problem might be interesting him. +A Peaucellier's cell is shown to be of use for changing a constant force into +one varying inversely as the square of the distance, and it is applied to the +description of equipotential lines. A method for describing graphically the +second elliptic integral and one for map projection on the face of a polyhedron +are also given. There are also a few other short papers of the same kind but +of no special importance, and Darwin says that he only included them in his +collected works for the sake of completeness. + +His first important contributions obviously arose through the study +of the works of his predecessors, and though of the nature of corrections to +\DPPageSep{039}{xxxvii} +previously accepted or erroneous ideas, they form definite additions to the +subject of cosmogony. The opening paragraph of the memoir ``On the +influence of geological changes in the earth's axis of rotation'' describes the +situation which prompted the work. ``The subject of the fixity or mobility +of the earth's axis of rotation in that body, and the possibility of variations +in the obliquity of the ecliptic, have from time to time attracted the notice +of mathematicians and geologists. The latter look anxiously for some grand +cause capable of producing such an enormous effect as the glacial period. +Impressed by the magnitude of the phenomenon, several geologists have +postulated a change of many degrees in the obliquity of the ecliptic and +a wide variability in the position of the poles on the earth; and this, again, +they have sought to refer back to the upheaval and subsidence of continents.'' +He therefore subjects the hypothesis to mathematical examination under +various assumptions which have either been put forward by geologists or +which he considers \textit{à~priori} probable. The conclusion, now well known to +astronomers, but frequently forgotten by geologists even at the present time, +is against any extensive wanderings of the pole during geological times. +``Geologists and biologists,'' writes Professor Barrell\footnotemarkN, ``may array facts +\footnotetextN{\textit{Science}, Sept.~4, 1914, p.~333.}% +\index{Barrell, Prof., Cosmogony as related to Geology and Biology}% +\index{Cosmogony, Sir George Darwin's influence on!as related to Geology and Biology, by Prof.\ Barrell}% +which suggest such hypotheses, but the testing of their possibility is really +a problem of mathematics, as much as are the movements of precession, +and orbital perturbations. Notwithstanding this, a number of hypotheses +concerning polar migration have been ingeniously elaborated and widely +promulgated without their authors submitting them to these final tests, or +in most cases even perceiving that an accordance with the known laws of +mechanics was necessary\ldots. A reexamination of these assumptions in the +light of forty added years of geological progress suggests that the actual +changes have been much less and more likely to be limited to a fraction +of the maximum limits set by Darwin. His paper seems to have checked +further speculation upon this subject in England, but, apparently unaware +of its strictures, a number of continental geologists and biologists have +carried forward these ideas of polar wandering to the present day. The +hypotheses have grown, each creator selecting facts and building up from +his particular assortment a fanciful hypothesis of polar migration unrestrained +even by the devious paths worked out by others.'' The methods +used by Darwin are familiar to those who investigate problems connected +with the figure of the earth, but the whole paper is characteristic of his style +in the careful arrangement of the assumptions, the conclusions deduced +therefrom, the frequent reduction to numbers and the summary giving the +main results. + +It is otherwise interesting because it was the means of bringing Darwin +\index{Darwin, Sir George, genealogy!association with Lord Kelvin}% +\index{Kelvin, associated with Sir George Darwin}% +into close connection with Lord Kelvin, then Sir~William Thomson. The +\DPPageSep{040}{xxxviii} +latter was one of the referees appointed by the Royal Society to report on it, +and, as Darwin says, ``He seemed to find that on these occasions the quickest +way of coming to a decision was to talk over the subject with the author +himself---at least this was frequently so as regards myself.'' Through his +whole life Darwin, like many others, prized highly this association, and he +considered that his whole work on cosmogony ``may be regarded as the +scientific outcome of our conversation of the year~1877; but,'' he adds, ``for +me at least science in this case takes the second place.'' + +Darwin at this time was thirty-two years old. In the three years since +he started publication fourteen memoirs and short notes, besides two statistical +papers on marriage between first cousins, form the evidence of his +activity. He seems to have reached maturity in his mathematical power +and insight into the problems which he attacked without the apprenticeship +which is necessary for most investigators. Probably the comparatively late +age at which he began to show his capacity in print may have something to +do with this. Henceforth development is rather in the direction of the full +working out of his ideas than growth of his powers. It seems better therefore +to describe his further scientific work in the manner in which he arranged +it himself, by subject instead of in chronological order. And here we have +the great advantage of his own comments, made towards the end of his +life when he scarcely hoped to undertake any new large piece of work. +Frequent quotation will be made from these remarks which occur in the +prefaces to the volumes, in footnotes and in his occasional addresses. + +The following account of the Earth-Moon series of papers is taken bodily +\index{Earth-Moon theory of Darwin, described by Mr S. S. Hough}% +from the Notice in the \textit{Proceedings of the Royal Society}\footnoteN + {Vol.~\Vol{89\;A}, p.~i.} +by Mr~S.~S. Hough, +who was himself one of Darwin's pupils. + +``The conclusions arrived at in the paper referred to above were based on +the assumption that throughout geological history, apart from slow geological +changes, the Earth would rotate sensibly as if it were rigid. It is shown that +a departure from this hypothesis might possibly account for considerable +excursions of the axis of rotation within the Earth itself, though these would +be improbable, unless, indeed, geologists were prepared to abandon the view +`that where the continents now stand they have always stood'; but no such +effect is possible with respect to the direction of the Earth's axis in space. +Thus the present condition of obliquity of the Earth's equator could in no +way be accounted for as a result of geological change, and a further cause +had to be sought. Darwin foresaw a possibility of obtaining an explanation +in the frictional resistance to which the tidal oscillations of the mobile parts +of a planet must be subject. The investigation of this hypothesis gave rise +to a remarkable series of papers of far-reaching consequence in theories of +cosmogony and of the present constitution of the Earth. +\DPPageSep{041}{xxxix} + +``In the first of these papers, which is of preparatory character, `On the +Bodily Tides of Viscous and Semi-elastic Spheroids, and on the Ocean Tides +on a Yielding Nucleus' (\textit{Phil.\ Trans.}, 1879, vol.~170), he adapts the analysis +of Sir~William Thomson, relating to the tidal deformations of an elastic +sphere, to the case of a sphere composed of a viscous liquid or, more generally, +of a material which partakes of the character either of a solid or a fluid +according to the nature of the strain to which it is subjected. For momentary +deformations it is assumed to be elastic in character, but the elasticity is +considered as breaking down with continuation of the strain in such a manner +that under very slow variations of the deforming forces it will behave sensibly +as if it were a viscous liquid. The exact law assumed by Darwin was dictated +rather by mathematical exigencies than by any experimental justification, but +the evidence afforded by the flow of rocks under continuous stress indicates +that it represents, at least in a rough manner, the mechanical properties +which characterise the solid parts of the Earth. + +``The chief practical result of this paper is summed up by Darwin himself +by saying that it is strongly confirmatory of the view already maintained by +Kelvin that the existence of ocean tides, which would otherwise be largely +masked by the yielding of the ocean bed to tidal deformation, points to +a high effective rigidity of the Earth as a whole. Its value, however, +lies further in the mathematical expressions derived for the reduction in +amplitude and retardation in phase of the tides resulting from viscosity +which form the starting-point for the further investigations to which the +author proceeded. + +``The retardation in phase or `lag' of the tide due to the viscosity +implies that a spheroid as tidally distorted will no longer present a +symmetrical aspect as if no such cause were operative. The attractive forces +on the nearer and more distant parts will consequently form a non-equilibrating +system with resultant couples tending to modify the state of +rotation of the spheroid about its centre of gravity. The action of these +couples, though exceedingly small, will be cumulative with lapse of time, +and it is their cumulative effects over long intervals which form the subject +of the next paper, `On the Precession of a Viscous Spheroid and on the +Remote History of the Earth' (\textit{Phil.\ Trans.}, 1879, vol.~170, Part~II, pp.~447--530). +The case of a single disturbing body (the Moon) is first considered, +but it is shown that if there are two such bodies raising tidal disturbances +(the Sun and Moon) the conditions will be materially modified from the +superposed results of the two disturbances considered separately. Under +certain conditions of viscosity and obliquity the obliquity of the ecliptic +will increase, and under others it will diminish, but the analysis further +yields `some remarkable results as to the dynamical stability or instability +of the system\ldots for moderate degrees of viscosity, the position of zero +\DPPageSep{042}{xl} +obliquity is unstable, but there is a position of stability at a high obliquity. +For large viscosities the position of zero obliquity becomes stable, and +(except for a very close approximation to rigidity) there is an unstable +position at a larger obliquity, and again a stable one at a still larger one.' + +``The reactions of the tidal disturbing force on the motion of the Moon +are next considered, and a relation derived connecting that portion of the +apparent secular acceleration of the Moon's mean motion, which cannot be +otherwise accounted for by theory, with the heights and retardations of the +several bodily tides in the Earth. Various hypotheses are discussed, but with +the conclusion that insufficient evidence is available to form `any estimate +having any pretension to accuracy\ldots as to the present rate of change due to +tidal friction.' + +``But though the time scale involved must remain uncertain, the nature +of the physical changes that are taking place at the present time is practically +free from obscurity. These involve a gradual increase in the length +of the day, of the month, and of the obliquity of the ecliptic, with a gradual +recession of the Moon from the Earth. The most striking result is that +these changes can be traced backwards in time until a state is reached when +the Moon's centre would be at a distance of only about $6000$~miles from the +Earth's surface, while the day and month would be of equal duration, +estimated at $5$~hours $36$~minutes. The minimum time which can have +elapsed since this condition obtained is further estimated at about $54$~million +years. This leads to the inevitable conclusion that the Moon and Earth at +one time formed parts of a common mass and raises the question of how and +why the planet broke up. The most probable hypothesis appeared to be +that, in accordance with Laplace's nebular hypothesis, the planet, being +partly or wholly fluid, contracted, and thus rotated faster and faster, until the +ellipticity became so great that the equilibrium was unstable. + +``The tentative theory put forward by Darwin, however, differs from the +nebular hypothesis of Laplace in the suggestion that instability might set +in by the rupture of the body into two parts rather than by casting off a +ring of matter, somewhat analogous to the rings of Saturn, to be afterwards +consolidated into the form of a satellite. + +``The mathematical investigation of this hypothesis forms a subject to +which Darwin frequently reverted later, but for the time he devoted himself +to following up more minutely the motions which would ensue after the +supposed planet, which originally consisted of the existing Earth and Moon +in combination, had become detached into two separate masses. In the +final section of a paper `On the Secular Changes in the Elements of the +Orbit of a Satellite revolving about a Tidally Distorted Planet' (\textit{Phil.\ +Trans.}, 1880, vol.~171), Darwin summarises the results derived in his +different memoirs. Various factors ignored in the earlier investigations, +\DPPageSep{043}{xli} +such as the eccentricity and inclination of the lunar orbit, the distribution +of the heat generated by tidal friction and the effects of inertia, were duly +considered and a complete history traced of the evolution resulting from +tidal friction of a system originating as two detached masses nearly in +contact with one another and rotating nearly as though they were parts +of one rigid body. Starting with the numerical data suggested by the +Earth-Moon System, `it is only necessary to postulate a sufficient lapse of +time, and that there is not enough matter diffused through space to resist +materially the motions of the Moon and Earth,' when `a system would +necessarily be developed which would bear a strong resemblance to our own.' +`A theory, reposing on \textit{verae causae}, which brings into quantitative correlation +the lengths of the present day and month, the obliquity of the ecliptic, +and the inclination and eccentricity of the lunar orbit, must, I think, have +strong claims to acceptance.' + +``Confirmation of the theory is sought and found, in part at least, in the +case of other members of the Solar System which are found to represent +various stages in the process of evolution indicated by the analysis. + +``The application of the theory of tidal friction to the evolution of the +Solar System and of planetary sub-systems other than the Earth-Moon +System is, however, reconsidered later, `On the Tidal Friction of a Planet +attended by Several Satellites, and on the Evolution of the Solar System' +(\textit{Phil.\ Trans.}, 1882, vol.~172). The conclusions drawn in this paper are +that the Earth-Moon System forms a unique example within the Solar +System of its particular mode of evolution. While tidal friction may +perhaps be invoked to throw light on the distribution of the satellites +among the several planets, it is very improbable that it has figured as the +dominant cause of change of the other planetary systems or in the Solar +System itself.'' + +For some years after this series of papers Darwin was busy with practical +tidal problems but he returned later ``to the problems arising in connection +with the genesis of the Moon, in accordance with the indications previously +arrived at from the theory of tidal friction. It appeared to be of interest to +trace back the changes which would result in the figures of the Earth and +Moon, owing to their mutual attraction, as they approached one another. +The analysis is confined to the consideration of two bodies supposed constituted +of homogeneous liquid. At considerable distances the solution of the +problem thus presented is that of the equilibrium theory of the tides, but, +as the masses are brought nearer and nearer together, the approximations +available for the latter problem cease to be sufficient. Here, as elsewhere, +when the methods of analysis could no longer yield algebraic results, Darwin +boldly proceeds to replace his symbols by numerical quantities, and thereby +succeeds in tracing, with considerable approximation, the forms which such +\DPPageSep{044}{xlii} +figures would assume when the two masses are nearly in contact. He even +carries the investigation farther, to a stage when the two masses in part +overlap. The forms obtained in this case can only he regarded as satisfying +the analytical, and not the true physical conditions of the problem, as, of +course, two different portions of matter cannot occupy the same space. +They, however, suggest that, by a very slight modification of conditions, +a new form could be found, which would fulfil all the conditions, in which +the two detached masses are united into a single mass, whose shape has been +variously described as resembling that of an hour-glass, a dumb-bell, or a pear. +This confirms the suggestion previously made that the origin of the Moon was +to be sought in the rupture of the parent planet into two parts, but the theory +was destined to receive a still more striking confirmation from another source. + +``While Darwin was still at work on the subject, there appeared the great +\index{Poincaré, reference to, by Sir George Darwin!on equilibrium of fluid mass in rotation}% +\index{Equilibrium of a rotating fluid}% +\index{Rotating fluid, equilibrium of}% +memoir by M.~Poincaré, `Sur l'équilibre d'une masse fluide animée d'un +mouvement de rotation' (\textit{Acta Math.}, vol.~7). + +``The figures of equilibrium known as Maclaurin's spheroid and Jacobi's +\index{Jacobi's ellipsoid}% +\index{Maclaurin's spheroid}% +ellipsoid were already familiar to mathematicians, though the conditions of +stability, at least of the latter form, were not established. By means of +analysis of a masterly character, Poincaré succeeded in enunciating and +applying to this problem the principle of exchange of stabilities. This principle +may be briefly indicated as follows: Imagine a dynamical system such as +a rotating liquid planet to be undergoing evolutionary change such as would +result from a gradual condensation of its mass through cooling. Whatever +be the varying element to which the evolutionary changes may be referred, +it may be possible to define certain relatively simple modes of motion, the +features associated with which will, however, undergo continuous evolution. +If the existence of such modes has been established, M.~Poincaré shows that +the investigation of their persistence or `stability' may be made to depend +on the evaluation of certain related quantities which he defines as coefficients +of stability. The latter quantities will be subject to evolutionary +change, and it may happen that in the course of such change one or more +of them assumes a zero value. Poincaré shows that such an occurrence +indicates that the particular mode of motion under consideration coalesces +at this stage with one other mode which likewise has a vanishing coefficient +of stability. Either mode will, as a rule, be possible before the change, but +whereas one will be stable the other will be unstable. The same will be +true after the change, but there will be an interchange of stabilities, whereby +that which was previously stable will become unstable, and \textit{vice versâ}. +An illustration of this principle was found in the case of the spheroids of +Maclaurin and the ellipsoids of Jacobi. The former in the earlier stages of +evolution will represent a stable condition, but as the ellipticity of surface +increases a stage is reached where it ceases to be stable and becomes unstable. +\DPPageSep{045}{xliii} +At this stage it is found to coalesce with Jacobi's form which involves in its +further development an ellipsoid with three unequal axes. Poincaré shows +that the latter form possesses in its earlier stages the requisite elements of +stability, but that these in their turn disappear in the later developments. +In accordance with the principle of exchange of stabilities laid down by +him, the loss of stability will occur at a stage where there is coalescence +with another form of figure, to which the stability will be transferred, and +this form he shows at its origin resembles the pear which had already been +indicated by Darwin's investigation. The supposed pear-shaped figure was +thus arrived at by two entirely different methods of research, that of Poincaré +tracing the processes of evolution forwards and that of Darwin proceeding +backwards in time. + +``The chain of evidence was all but complete; it remained, however, to +consider whether the pear-shaped figure indicated by Poincaré, stable in its +earlier forms, could retain its stability throughout the sequence of changes +necessary to fill the gap between these forms and the forms found by Darwin. + +``In later years Darwin devoted much time to the consideration of this +\index{Ellipsoidal harmonics}% +\index{Harmonics, ellipsoidal}% +problem. Undeterred by the formidable analysis which had to be faced, he +proceeded to adapt the intricate theory of Ellipsoidal Harmonics to a form in +which it would admit of numerical application, and his paper `Ellipsoid +Harmonic Analysis' (\textit{Phil.\ Trans.},~A, 1901, vol.~197), apart from the application +for which it was designed, in itself forms a valuable contribution +to this particular branch of analysis. With the aid of these preliminary +investigations he succeeded in tracing with greater accuracy the form of the +pear-shaped figure as established by Poincaré, `On the Pear-shaped Figure of +\index{Pear-shaped figure of equilibrium}% +Equilibrium of a Rotating Mass of Liquid' (\textit{Phil.\ Trans.},~A, 1901, vol.~198), +and, as he considered, in establishing its stability, at least in its earlier forms. +Some doubt, however, is expressed as to the conclusiveness of the argument +employed, as simultaneous investigations by M.~Lia\-pou\-noff pointed to an +\index{Liapounoff's work on rotating liquids}% +opposite conclusion. Darwin again reverts to this point in a further paper +`On the Figure and Stability of a Liquid Satellite' (\textit{Phil.\ Trans.},~A, 1906, +vol.~206), in which is considered the stability of two isolated liquid masses in +the stage at which they are in close proximity, i.e.,~the condition which would +obtain, in the Earth-Moon System, shortly after the Moon had been severed +from the Earth. The ellipsoidal harmonic analysis previously developed is +then applied to the determination of the approximately ellipsoidal forms +which had been indicated by Roche. The conclusions arrived at seem to +\index{Roche's ellipsoid}% +point, though not conclusively, to instability at the stage of incipient rupture, +but in contradistinction to this are quoted the results obtained by Jeans, who +\index{Jeans, J. H., on rotating liquids}% +considered the analogous problems of the equilibrium and rotation of infinite +rotating cylinders of liquid. This problem is the two-dimensional analogue +of the problems considered by Darwin and Poincaré, but involves far greater +\DPPageSep{046}{xliv} +simplicity of the conditions. Jeans finds solutions of his problem strictly +analogous to the spheroids of Maclaurin, the ellipsoids of Jacobi, and the +pear of Poincaré, and is able to follow the development of the latter until the +neck joining the two parts has become quite thin. He is able to establish +conclusively that the pear is stable in its early stages, while there is no +evidence of any break in the stability up to the stage when it divides itself +into two parts.'' + +Darwin's own final comments on this work next find a place here. +He is writing the preface to the second volume of his Collected Works in~1908, +after which time nothing new on the subject came from his pen. +``The observations of Dr~Hecker,'' he says, ``and of others do not afford +\index{Hecker's observations on retardation of tidal oscillations in the solid earth}% +evidence of any considerable amount of retardation in the tidal oscillations +of the solid earth, for, within the limits of error of observation, the +phase of the oscillation appears to be the same as if the earth were purely +elastic. Then again modern researches in the lunar theory show that the +secular acceleration of the moon's mean motion is so nearly explained by +means of pure gravitation as to leave but a small residue to be referred +to the effects of tidal friction. We are thus driven to believe that at present +\index{Tidal friction as a true cause of change}% +tidal friction is producing its inevitable effects with extreme slowness. But +we need not therefore hold that the march of events was always so leisurely, +and if the earth was ever wholly or in large part molten, it cannot have been +the case. + +``In any case frictional resistance, whether it be much or little and +whether applicable to the solid planet or to the superincumbent ocean, is +a true cause of change\ldots. + +``For the astronomer who is interested in cosmogony the important point +is the degree of applicability of the theory as a whole to celestial evolution. +To me it seems that the theory has rather gained than lost in the esteem of +men of science during the last 25~years, and I observe that several writers +are disposed to accept it as an established acquisition to our knowledge of +cosmogony. + +``Undue weight has sometimes been laid on the exact numerical values +assigned for defining the primitive configurations of the earth and moon. +In so speculative a matter close accuracy is unattainable, for a different +theory of frictionally retarded tides would inevitably load to a slight difference +in the conclusion; moreover such a real cause as the secular increase +in the masses of the earth and moon through the accumulation of meteoric +dust, and possibly other causes, are left out of consideration. + +``The exact nature of the process by which the moon was detached from +the earth must remain even more speculative. I suggested that the fission +of the primitive planet may have been brought about by the synchronism of +the solar tide with the period of the fundamental free oscillation of the +\DPPageSep{047}{xlv} +planet, and the suggestion has received a degree of attention which I never +anticipated. It may be that we shall never attain to a higher degree of +certainty in these obscure questions than we now possess, but I would +maintain that we may now hold with confidence that the moon originated +by a process of fission from the primitive planet, that at first she revolved in +an orbit close to the present surface of the earth, and that tidal friction +has been the principal agent which transformed the system to its present +configuration. + +``The theory for a long time seemed to lie open to attack on the ground +\index{Kelvin, associated with Sir George Darwin!cooling of earth}% +that it made too great demands on time, and this has always appeared to +me the greatest difficulty in the way of its acceptance. If we were still +compelled to assent to the justice of Lord Kelvin's views as to the period +of time which has elapsed since the earth solidified, and as to the age of the +solar system, we should also have to admit the theory of evolution under +tidal influence as inapplicable to its full extent. Lord Kelvin's contributions +to cosmogony have been of the first order of importance, but his arguments +on these points no longer carry conviction with them. Lord Kelvin contended +that the actual distribution of land and sea proves that the planet +solidified at a time when the day had nearly its present length. If this +were true the effects of tidal friction relate to a period antecedent to the +solidification. But I have always felt convinced that the earth would adjust +its ellipticity to its existing speed of rotation with close approximation.'' + +After some remarks concerning the effects of the discovery of radio-activity +and the energy resident in the atom on estimates of geological time, +he continues, ``On the whole then it may be maintained that deficiency +of time does not, according to our present state of knowledge, form a bar to +the full acceptability of the theory of terrestrial evolution under the influence +of tidal friction. + +``It is very improbable that tidal friction has been the dominant cause +of change in any of the other planetary sub-systems or in the solar system +itself, yet it seems to throw light on the distribution of the satellites amongst +the several planets. It explains the identity of the rotation of the moon +with her orbital motion, as was long ago pointed out by Kant and Laplace, +and it tends to confirm the correctness of the observations according to which +Venus always presents the same face to the sun.'' + +Since this was written much information bearing on the point has been +gathered from the stellar universe. The curious curves of light-changes in +certain classes of spectroscopic binaries have been well explained on the +assumption that the two stars are close together and under strong tidal +distortion. Some of these, investigated on the same hypothesis, even seem +to be in actual contact. In chap.~\Vol{XX} of the third edition~(1910) of his book +on the Tides, Darwin gives a popular summary of this evidence which had +\DPPageSep{048}{xlvi} +in the interval been greatly extended by the discovery and application of +the hypothesis to many other similar systems. In discussing the question +Darwin sets forth a warning. He points out that most of the densities +which result from the application of the tidal theory are very small compared +with that of the sun, and he concludes that these stars are neither homogeneous +nor incompressible. Hence the figures calculated for homogeneous +liquid can only be taken to afford a general indication of the kind of figure +which we might expect to find in the stellar universe. + +Perhaps Darwin's greatest service to cosmogony was the successful effort +\index{Numerical work on cosmogony}% +which he made to put hypotheses to the test of actual calculation. Even +though the mathematical difficulties of the subject compel the placing of +many limitations which can scarcely exist in nature, yet the solution of even +these limited problems places the speculator on a height which he cannot +hope to attain by doubtful processes of general reasoning. If the time +devoted to the framing and setting forth of cosmogonic hypotheses by various +writers had been devoted to the accurate solution of some few problems, the +newspapers and popular scientific magazines might have been less interesting +to their readers, but we should have had more certain knowledge of our +universe. Darwin himself engaged but little in speculations which were +not based on observations or precise conclusions from definitely stated +assumptions, and then only as suggestions for further problems to be +undertaken by himself or others. And this view of progress he communicated +to his pupils, one of whom, Mr~J.~H. Jeans, as mentioned above, is +continuing with success to solve those gravitational problems on similar +lines. + +The nebular hypothesis of Kant and Laplace has long held the field as +\index{Kant, Nebular Hypothesis}% +\index{Laplace, Nebular Hypothesis}% +the most probable mode of development of our solar system from a nebula. +At the present time it is difficult to say what are its chief features. Much +criticism has been directed towards every part of it, one writer changing +a detail here, another there, and still giving to it the name of the best known +exponent. The only salient point which seems to be left is the main hypothesis +that the sun, planets and satellites were somehow formed during the +process of contraction of a widely diffused mass of matter to the system as +we now see it. Some writers, including Darwin himself, regard a gaseous +nebula contracting under gravitation as the essence of Laplace's hypotheses, +distinguishing this condition from that which originates in the accretion +of small masses. Others believe that both kinds of matter may be present. +After all it is only a question of a name, but it is necessary in a discussion to +know what the name means. + +Darwin's paper, ``The mechanical conditions of a swarm of meteorites,'' +\index{Mechanical condition of a swarm of meteorites}% +is an attempt to show that, with reasonable hypotheses, the nebula and the +small masses under contraction by collisions may have led to the same result. +\DPPageSep{049}{xlvii} +In his preface to volume~\Vol{IV} he says with respect to this paper: ``Cosmogonists +are of course compelled to begin their survey of the solar system at some +arbitrary stage of its history, and they do not, in general, seek to explain +how the solar nebula, whether gaseous or meteoritic, came to exist. My +investigation starts from the meteoritic point of view, and I assume the +meteorites to be moving indiscriminately in all directions. But the doubt +naturally arises as to whether at any stage a purely chaotic motion of the +individual meteorites could have existed, and whether the assumed initial +condition ought not rather to have been an aggregate of flocks of meteorites +moving about some central condensation in orbits which intersect one another +at all sorts of angles. If this were so the chaos would not be one consisting +of individual stones which generate a quasi-gas by their collisions, but it +would be a chaos of orbits. But it is not very easy to form an exact picture +of this supposed initial condition, and the problem thus seems to elude +mathematical treatment. Then again have I succeeded in showing that a +pair of meteorites in collision will be endowed with an effective elasticity? +If it is held that the chaotic motion and the effective elasticity are quite +imaginary, the theory collapses. It should however be remarked that an +infinite gradation is possible between a chaos of individuals and a chaos +of orbits, and it cannot be doubted that in most impacts the colliding stones +would glance from one another. It seems to me possible, therefore, that my +two fundamental assumptions may possess such a rough resemblance to truth +as to produce some degree of similitude between the life-histories of gaseous +and meteoritic nebulae. If this be so the Planetesimal Hypothesis of +Chamberlain and Moulton is nearer akin to the Nebular Hypothesis than +\index{Chamberlain and Moulton, Planetesimal Hypothesis}% +\index{Moulton, Chamberlain and, Planetesimal Hypothesis}% +\index{Planetesimal Hypothesis of Chamberlain and Moulton}% +the authors of the former seem disposed to admit. + +``Even if the whole of the theory could be condemned as futile, yet the +paper contains an independent solution of the problem of Lane and Ritter; +and besides the attempt to discuss the boundary of an atmosphere, where +the collisions have become of vanishing rarity, may still perhaps be worth +something.'' + +In writing concerning the planetesimal hypothesis, Darwin seems to have +forgotten that one of its central assumptions is the close approach of two +stars which by violent tidal action drew off matter in spiral curves which +became condensed into the attendants of each. This is, in fact, one of the +most debatable parts of the hypothesis, but one on which it is possible to +get evidence from the distribution of such systems in the stellar system. +Controversy on the main issue is likely to exist for many years to come. + +Quite early in his career Darwin was drawn into practical tidal problems +\index{Tidal problems, practical}% +by being appointed on a Committee of the British Association with Adams, +to coordinate and revise previous reports drawn up by Lord Kelvin. He +evidently felt that the whole subject of practical analysis of tidal observations +\DPPageSep{050}{xlviii} +needed to be set forth in full and made clear. His first report consequently +contains a development of the equilibrium theory of the Tides, and later, +after a careful analysis of each harmonic component, it proceeds to outline in +detail the methods which should be adopted to obtain the constants of each +component from theory or observation, as the case needed. Schedules and +forms of reduction are given with examples to illustrate their use. + +There are in reality two principal practical problems to be considered. +The one is the case of a port with much traffic, where it is possible to obtain +tide heights at frequent intervals and extending over a long period. While +the accuracy needed usually corresponds to the number of observations, it is +always assumed that the ordinary methods of harmonic analysis by which all +other terms but that considered are practically eliminated can be applied; +the corrections when this is not the case are investigated and applied. The +other problem is that of a port infrequently visited, so that we have only +a short series of observations from which to obtain the data for the computation +of future tides. The possible accuracy here is of course lower than in +the former case but may be quite sufficient when the traffic is light. In his +third report Darwin takes up this question. The main difficulty is the +separation of tides which have nearly the same period and which could not +be disentangled by harmonic analysis of observations extending over a very +few weeks. Theory must therefore be used, not only to obtain the periods, +but also to give some information about the amplitudes and phases if this +separation is to be effected. The magnitude of the tide-generating force is +used for the purpose. Theoretically this should give correct results, but it is +often vitiated by the form of the coast line and other circumstances depending +on the irregular shape of the water boundary. Darwin shows however that +fair prediction can generally be obtained; the amount of numerical work is +of course much smaller than in the analysis of a year's observations. This +report was expanded by Darwin into an article on the Tides for the \textit{Admiralty +Scientific Manual}. + +Still another problem is the arrangement of the analysis when times and +heights of high and low water alone are obtainable; in the previous papers +the observations were supposed to be hourly or obtained from an automatically +recording tide-gauge. The methods to be used in this case are of course +well known from the mathematical side: the chief problem is to reduce the +arithmetical work and to put the instructions into such a form that the +ordinary computer may use them mechanically. The problem was worked +out by Darwin in~1890, and forms the subject of a long paper in the +\textit{Proceedings of the Royal Society}. + +A little later he published the description of his now well known abacus, +\index{Abacus}% +designed to avoid the frequent rewriting\DPnote{[** TN: Not hyphenated in original]} of the numbers when the harmonic +analysis for many different periods is needed. Much care was taken to obtain +\DPPageSep{051}{xlix} +the right materials. The real objection to this, and indeed to nearly all the +methods devised for the purpose, is that the arrangement and care of the +mechanism takes much longer time than the actual addition of the numbers +after the arrangement has been made. In this description however there +are more important computing devices which reduce the time of computation +to something like one-fifth of that required by the previous methods. +The principal of these is the one in which it is shown how a single set +of summations of $9000$~hourly values can be made to give a good many +terms, by dividing the sums into proper groups and suitably treating +them. + +Another practical problem was solved in his Bakerian Lecture ``On Tidal +\index{Bakerian lecture}\Pagelabel{xlix}% +Prediction.'' In a previous paper, referred to above, Darwin had shown how +the tidal constants of a port might be obtained with comparatively little +expense from a short series of high and low water observations. These, +however, are of little value unless the port can furnish the funds necessary +to predict the future times and heights of the tides. Little frequented ports +can scarcely afford this, and therefore the problem of replacing such predictions +by some other method is necessary for a complete solution. ``The +object then,'' says Darwin, ``of the present paper, is to show how a general +tide-table, applicable for all time, may be given in such a form that anyone, +with an elementary knowledge of the \textit{Nautical Almanac}, may, in a few +minutes, compute two or three tides for the days on which they are required. +The tables will also be such that a special tide-table for any year may be +computed with comparatively little trouble.'' + +This, with the exception of a short paper dealing with the Tides in the +Antarctic as shown by observations made on the \textit{Discovery}, concludes Darwin's +published work on practical tidal problems. But he was constantly in correspondence +about the subject, and devoted a good deal of time to government +work and to those who wrote for information. + +In connection with these investigations it was natural that he should +\index{Rigidity of earth, from fortnightly tides}% +\index{Tide, fortnightly}% +turn aside at times to questions of more scientific interest. Of these the +fortnightly tide is important because by it some estimate may be reached as +to the earth's rigidity. The equilibrium theory while effective in giving the +periods only for the short-period tides is much more nearly true for those of +long period. Hence, by a comparison of theory and observation, it is possible +to see how much the earth yields to distortion produced by the moon's +attraction. Two papers deal with this question. In the first an attempt is +made to evaluate the corrections to the equilibrium theory caused by the +continents; this involves an approximate division of the land and sea +surfaces into blocks to which calculation may be applied. In the second +tidal observations from various parts of the earth are gathered together for +comparison with the theoretical values. As a result, Darwin obtains the +\DPPageSep{052}{l} +oft-quoted expression for the rigidity of the earth's mass, namely, that it is +effectively about that of steel. An attempt made by George and Horace +Darwin to measure the lunar disturbance of gravity by means of the +pendulum is in reality another approach to the solution of the same problem. +The attempt failed mainly on account of the local tremors which were produced +by traffic and other causes. Nevertheless the two reports contain +much that is still interesting, and their value is enhanced by a historical +account of previous attempts on the same lines. Darwin had the satisfaction +of knowing that this method was later successful in the hands of Dr~Hecker +\index{Hecker's observations on retardation of tidal oscillations in the solid earth}% +whose results confirmed his first estimate. Since his death the remarkable +experiment of Michelson\footnoteN + {\textit{Astrophysical Journal}, March,~1914.} +\index{Michelson's experiment on rigidity of earth}% +\index{Rigidity of earth, from fortnightly tides!Michelson's experiment}% +with a pipe partly filled with water has given +a precision to the determination of this constant which much exceeds that +of the older methods; he concludes that the rigidity and viscosity are at least +equal to and perhaps exceed those of steel. + +It is here proper to refer to Darwin's more popular expositions of the +\index{Tides, The@\textit{Tides, The}}% +\index{Tides, articles on}\Pagelabel{l}% +work of himself and others. He wrote several articles on Tides, notably for +the \textit{Encyclopaedia Britannica} and for the \textit{Encyclopaedie der Mathematischen +Wissenschaften}, but he will be best remembered in this connection for his +volume \textit{The Tides} which reached its third edition not long before his +death. The origin of it was a course of lectures in~1897 before the Lowell +Institute of Boston, Massachusetts. An attempt to explain the foundations +and general developments of tidal theory is its main theme. It naturally +leads on to the subject of tidal friction and the origin of the moon, and +therewith are discussed numerous questions of cosmogony. From the point +of view of the mathematician, it is not only clear and accurate but gives the +impression, in one way, of a \textit{tour de force}. Although Darwin rarely has to +ask the reader to accept his conclusions without some description of the +nature of the argument by which they are reached, there is not a single +algebraic symbol in the whole volume, except in one short footnote where, on +a minor detail, a little algebra is used. The achievement of this, together +with a clear exposition, was no light task, and there are few examples to be +found in the history of mathematics since the first and most remarkable of all, +Newton's translation of the effects of gravitation into geometrical reasoning. +\textit{The Tides} has been translated into German (two editions), Hungarian, +Italian and Spanish. + +In 1877 the two classical memoirs of G.~W.~Hill on the motion of the +\index{Hill, G. W., Lunar Theory}% +moon were published. The first of these, \textit{Researches in the Lunar Theory}, +contains so much of a pioneer character that in writing of any later work on +celestial mechanics it is impossible to dismiss it with a mere notice. One +portion is directly concerned with a possible mode of development of the +lunar theory and the completion of the first step in the process. The usual +\DPPageSep{053}{li} +method of procedure has been to consider the problem of three bodies as an +extension of the case of two bodies in which the motion of one round the +other is elliptic. Hill, following a suggestion of Euler which had been +worked out by the latter in some detail, starts to treat the problem as a +very special particular case of the problem of three bodies. One of them, +the earth, is of finite mass; the second, the sun, is of infinite mass and at +an infinite distance but is revolving round the former with a finite and +constant angular velocity. The third, the moon, is of infinitesimal mass, but +moves at a finite distance from the earth. Stated in this way, the problem +of the moon's motion appears to bear no resemblance to reality. It is, +however, nothing but a limiting case where certain constants, which are +small in the case of the actual motion, have zero values. The sun is +actually of very great mass compared with the earth, it is very distant as +compared with the distance of the moon, its orbit round the earth (or \textit{vice +versâ}) is nearly circular, and the moon's mass is small compared with that +of the earth. The differential equations which express the motion of +the moon under these limitations are fairly simple and admit of many +transformations. + +Hill simplifies the equations still further, first by supposing the moon +so started that it always remains in the same fixed plane with the earth +and the sun (its actual motion outside this plane is small). He then uses +moving rectangular axes one of which always points in the direction of the +sun. Even with all these limitations, the differential equations possess many +classes of solutions, for there will be four arbitrary constants in the most +general values of the coordinates which are to be derived in the form of a +doubly infinite series of harmonic terms. His final simplification is the +choice of one of these classes obtained by giving a zero value to one of +the arbitrary constants; in the moon's motion this constant is small. The +orbit thus obtained is of a simple character but it possesses one important +property; relative to the moving axes it is closed and the body following +it will always return to the same point of it (relative to the moving axis) +after the lapse of a definite interval. In other words, the relative motion +is periodic. + +Hill develops this solution literally and numerically for the case of our +satellite with high accuracy. This accuracy is useful because the form of +the orbit depends solely on the ratio of the mean rates of motion of the sun +and moon round the earth, and these rates, determined from centuries of +observation, are not affected by the various limitations imposed at the outset. +The curve does not differ much from a circle to the eye but it includes the +principal part of one of the chief differences of the motion from that in a +circle with uniform velocity, namely, the inequality long known as the +``variation''; hence the name since given to it, ``the Variational Orbit.'' Hill, +\DPPageSep{054}{lii} +however, saw that it was of more general interest than its particular application +to our satellite. He proceeds to determine its form for other values +of the mean rates of motion of the two bodies. This gives a family of +periodic orbits whose form gradually varies as the ratio is changed; the +greater the ratio, the more the curve differs from a circle. + +It is this idea of Hill's that has so profoundly changed the whole outlook +of celestial mechanics. Poincaré took it up as the basis of his celebrated +prize essay of~1887 on the problem of three bodies and afterwards expanded +his work into the three volumes; \textit{Les méthodes nouvelles de la Mécanique +Céleste}. His treatment throughout is highly theoretical. He shows that +\index{Poincaré, reference to, by Sir George Darwin!\textit{Les Méthodes Nouvelles de la Mécanique Céleste}}% +there must be many families of periodic orbits even for specialised problems +in the case of three bodies, certain general properties are found, and much +information concerning them which is fundamental for future investigation +is obtained. + +It is doubtful if Darwin had paid any special attention to Hill's work +on the moon for at least ten years after its appearance. All this time he +was busy with the origin of the moon and with tidal work. Adams had +published a brief \textit{résumé} of his own work on lines similar to those of Hill +immediately after the memoirs of the latter appeared, but nothing further +on the subject came from his pen. The medal of the Royal Astronomical +Society was awarded to Hill in~1888, and Dr~Glaisher's address on his work +\index{Glaisher, Dr J. W. L., address on presenting the gold medal of the R.A.S. to G. W. Hill}% +\index{Hill, G. W., Lunar Theory!awarded gold medal of R.A.S.}% +\index{Hill, G. W., Lunar Theory!lectures by Darwin on Lunar Theory}% +contains an illuminating analysis of the methods employed and the ideas +which are put forward. Probably both Darwin and Adams had a considerable +share in making the recommendation. Darwin often spoke of his +difficulties in assimilating the work of others off his own beat and possibly +this address started him thinking about the subject, for it was at his recommendation +in the summer of 1888 that the writer took up the study of Hill's +papers. ``They seem to be very good,'' he said, ``but scarcely anyone knows +much about them.'' + +He lectured on Hill's work for the first time in the Michaelmas Term +of~1893, and writes of his difficulties in following parts of them, more +particularly that on the Moon's Perigee which contains the development of +the infinite determinant. He concludes, ``I can't get on with my own work +until these lectures are over---but Hill's papers are splendid.'' One of his +pupils on this occasion was Dr~P.~H. Cowell, now Director of the Nautical +Almanac office. The first paper of the latter was a direct result of these +lectures and it was followed later by a valuable series of memoirs in which +the constants of the lunar orbit and the coefficients of many of the periodic +terms were obtained with great precision. Soon after these lectures Darwin +started his own investigations on the subject. But they took a different +line. The applications to the motion of the moon were provided for and +Poincaré had gone to the foundations. Darwin felt, however, that the work of +\DPPageSep{055}{liii} +the latter was far too abstract to satisfy those who, like himself, frequently +needed more concrete results, either for application or for their own mental +satisfaction. In discussing periodic orbits he set himself the task of tracing +numbers of them in order, as far as possible, to get a more exact knowledge +of the various families which Poincaré's work had shown must exist. Some +of Hill's original limitations are dropped. Instead of taking a sun of infinite +mass and at an infinite distance, he took a mass ten times that of the +planet and at a finite distance from that body. The orbit of each round +the other is circular and of uniform motion, the third body being still of +infinitesimal mass. Any periodic orbit which may exist is grist to his mill +whether it circulate, about one body or both or neither. + +Darwin saw little hope of getting any extensive results by solutions of +\index{Numerical work, great labour of}% +\index{Periodic orbits, Darwin begins papers on}% +\index{Periodic orbits, Darwin begins papers on!great numerical difficulties of}% +\index{Periodic orbits, Darwin begins papers on!stability of}% +the differential equations in harmonic series. It was obvious that the slowness +of convergence or the divergence would render the work far too doubtful. +He adopted therefore the tedious process of mechanical quadratures, starting +at an arbitrary position on the $x$-axis with an arbitrary speed in a direction +parallel to the $y$-axis. Tracing the orbit step-by-step, he again reaches the +$x$-axis. If the final velocity there is perpendicular to the axis, the orbit is +periodic. If not, he starts again with a different speed and traces another +orbit. The process is continued, each new attempt being judged by the +results of the previous orbits, until one is obtained which is periodic. The +amount of labour involved is very great since the actual discovery of a +periodic orbit generally involved the tracing of from three to five or even +more non-periodic paths. Concerning one of the orbits he traced for his last +paper on the subject, he writes: ``You may judge of the work when I tell +you that I determined $75$~positions and each averaged $\frac{3}{4}$~hr.\ (allowing for +correction of small mistakes---which sometimes is tedious). You will see +that it is far from periodic\ldots. I have now got six orbits of this kind.'' And all +this to try and find only one periodic orbit belonging to a class of whose +existence he was quite doubtful. + +Darwin's previous work on figures of equilibrium of rotating fluids made +the question of the stability of the motion in these orbits a prominent factor +in his mind. He considered it an essential part in their classification. To +determine this property it was necessary, after a periodic orbit had been +obtained, to find the effect of a small variation of the conditions. For this +purpose, Hill's second paper of~1877, on the Perigee of the Moon, is used. +After finding the variation orbit in his first paper. Hill makes a start +towards a complete solution of his limited differential equations by finding +an orbit, not periodic and differing slightly from the periodic orbit already +obtained. The new portion, the difference between the two, when expressed +as a sum of harmonic terms, contains an angle whose uniform rate of change,~$c$, +depends only on the constants of the periodic orbit. The principal +\DPPageSep{056}{liv} +portion of Hill's paper is devoted to the determination of~$c$ with great +precision. For this purpose, the infinite determinant is introduced and +expanded into infinite series, the principal parts of which are expressed by +a finite number of well known functions; the operations Hill devised to +achieve this have always called forth a tribute to his skill. Darwin uses +this constant~$c$ in a different way. If it is real, the orbit is stable, if +imaginary, unstable. In the latter case, it may be a pure imaginary or a +complex number; hence the necessity for the two kinds of unstability. + +In order to use Hill's method, Darwin is obliged to analyse a certain +function of the coordinates in the periodic orbit into a Fourier series, and to +obtain the desired accuracy a large number of terms must be included. +For the discovery of~$c$ from the infinite determinant, he adopts a mode of +expansion of his own better suited to the purpose in hand. But in any case +the calculation is laborious. In a later paper, he investigates the stability +by a different method because Hill's method fails when the orbit has +sharp flexures. + +For the classification into families, Darwin follows the changes according +\index{Hough, S. S., Darwin's work on Earth-Moon Theory, xxxviii; Periodic Orbits}% +\index{Periodic orbits, Darwin begins papers on!classification of, by Jacobi's integral}% +to variations in the constant of relative energy,~$C$. The differential equations +referred to the moving axes admit a Jacobian integral, the constant of +which is~$C$. One property of this integral Hill had already developed, +namely, that the curve obtained by making the kinetic energy zero is one +which the body cannot cross. Darwin draws the curves for different values +of~$C$ with care. He is able to show in several cases the origin of the +families he has found and much use is made of Poincaré's proposition, that +all such families originate in pairs, for following the changes. But even +his material is sometimes insufficient, especially where two quite different +pairs of families originate near the same point on the $x$-axis, and some later +corrections of the classification partly by himself and partly by Mr~S.~S. Hough +were necessary. In volume~\Vol{IV} of his collected works these corrections are +fully explained. + +The long first memoir was published in~1896. Nothing further on the +subject appeared from his hand until 1909 when a shorter paper containing +a number of new orbits was printed in the Monthly Notices of the Royal +Astronomical Society. Besides some additions and corrections to his older +families he considers orbits of ejection and retrograde orbits. During the +interval others had been at work on similar lines while Darwin with +increasing duties thrust upon him only found occasional opportunities to +keep his calculations going. A final paper which appears in the present +volume was the outcome of a request by the writer that a trial should be +made to find a member of a librating class of orbits for the mass ratio~$1:10$ +which had been shown to exist and had been traced for the mass ratio~$1:1048$. +The latter arose in an attempt to consider the orbits of the Trojan group of +\DPPageSep{057}{lv} +asteroids. He failed to find one but in the course of his work discovered +another class of great interest, which shows the satellite ultimately falling +into the planet. He concludes, ``My attention was first drawn to periodic +orbits by the desire to discover how a Laplacian ring could coalesce into +a planet. With this object in view I tried to discover how a large planet +could affect the mean motion of a small one moving in a circular orbit at +the same mean distance. After various failures the investigation drifted +towards the work of Hill and Poincaré, so that the original point of view +was quite lost and it is not even mentioned in my paper on `Periodic Orbits.' +It is of interest, to me at least, to find that the original aspect of the problem +has emerged again.'' It is of even greater interest to one of his pupils to +find that after more than twenty years of work on different lines in celestial +mechanics, Darwin's last paper should be on the same part of the subject to +which both had been drawn from quite different points of view. + +Thus Darwin's work on what appeared to be a problem in celestial +mechanics of a somewhat unpractical nature sprang after all from and +finally tended towards the question which had occupied his thoughts nearly +all his life, the genesis and evolution of the solar system. +\DPPageSep{058}{lvi} +%[Blank Page] +\DPPageSep{059}{1} +\index{Orbits, periodic|see{Periodic orbits}} + + +\Chapter{Inaugural Lecture} +\index{Inaugural lecture}% +\index{Cambridge School of Mathematics}% +\index{Lecture, inaugural}% +\index{Mathematical School at Cambridge}% + +\Heading{(Delivered at Cambridge, in 1883, on election to the +Plumian Professorship)} + +\First{I propose} to take advantage of the circumstance that this is the first of +the lectures which I am to give, to say a few words on the Mathematical +School of this University, and especially of the position of a professor in +regard to teaching at the present time. + +There are here a number of branches of scientific study to which there +are attached laboratories, directed by professors, or by men who occupy the +position and do the duties of professors, but do not receive their pay from, +nor full recognition by, the University. Of these branches of science I have +comparatively little to say. + +You are of course aware of the enormous impulse which has been given +to experimental science in Cambridge during the last ten years. It would +indeed have been strange if the presence of such men as now stand at the +head of those departments had not created important Schools of Science. +And yet when we consider the strange constitution of our University, it +may be wondered that they have been able to accomplish this. I suspect +that there may be a considerable number of men who go through their +University course, whose acquaintance with the scientific activity of the place +is limited by the knowledge that there is a large building erected for some +obscure purpose in the neighbourhood of the Corn Exchange. Is it possible +that any student of Berlin should be heard to exclaim, ``Helmholtz, who is +Helmholtz?'' And yet some years ago I happened to mention the name of +one of the greatest living mathematicians, a professor in this University, +in the presence of a first class man and fellow of his College, and he made +just such an exclamation. + +This general state of apathy to the very existence of science here has +now almost vanished, but I do not think I have exaggerated what it was +some years ago. Is not there a feeling of admiration called for for\DPnote{[** TN: Double word OK]} those, who +by their energy and ability have raised up all the activity which we now see? +\DPPageSep{060}{2} + +For example, Foster arrived here, a stranger to the University, without +University post or laboratory. I believe that during his first term Balfour +and one other formed his whole class. And yet holding only that position +of a College lecturer which he holds at this minute, he has come to make +Cambridge the first Physiological School of Great Britain, and the range of +buildings which the University has put at his disposal has already proved +too small for his requirements\footnotemarkN. His pupil Balfour had perhaps a less +\footnotetextN{Sir Michael Foster was elected the first Professor of Physiology a few weeks after the + delivery of this lecture.}% +uphill game to play, for the germs of the School of Natural Science were +already laid when he began his work as a teacher. But he did not merely +aid in the further developments of what he found, for he struck out in a +new line---that line of study which his own original work has gone, I +believe, a very long way to transform and even create. He did not live +to see the full development of the important school and laboratory which +he had founded. But thanks to his impulse it is now flourishing, and will +doubtless prosper under the able hands into which the direction has fallen. +His name ought surely to live amongst us for what he did; for those who +had the fortune to be his friends the remembrance of him cannot die, for +what he was. + +I should be going too far astray were I to continue to expatiate on the +work of Rayleigh, Stuart, and the others who are carrying on the development +of practical work in various branches within these buildings. It must +suffice to say that each school has had its difficulties, and that those difficulties +have been overcome by the zeal of those concerned in the management. + +But now let us turn to the case of the scientific professors who have no +laboratories to direct, and I speak now of the mathematical professors. In +comparison with the prosperity of which I have been speaking, I think +it is not too much to say that there is no vitality. I belong to this class of +professors, and I am far from flattering myself that I can do much to impart +life to the system. But if I shall not succeed I may perhaps be pardoned +if I comfort myself by the reflection, that it may not be entirely my own fault. + +The University has however just entered on a new phase; I have the +honour to be the first professor elected under the new Statutes now in force. +A new scheme for the examinations in Mathematics is in operation, and it +may be that such an opportunity will now be afforded as has hitherto been +wanting. We can but try to avail ourselves of the chance. + +To what causes are we to assign the fact that our most eminent +teachers of mathematics have hitherto been very frequently almost without +classes? It surely cannot be that Stokes, Adams and Cayley have \textit{nothing} +to say worth hearing by students of mathematics. Granting the possibility +\DPPageSep{061}{3} +that a distinguished man may lack the power of exposition, yet it is inadmissible +that they are \textit{all} deficient in that respect. No, the cause is not far +to seek, it lies in the Mathematical Tripos. How far it is desirable that the +system should be so changed, that it will be advisable for students in their +own interest to attend professorial lectures, I am not certain; but it can +scarcely be doubted that if there were no Tripos, the attendance at such +lectures would be larger. + +In hearing the remarks which I am about to make on the Mathematical +\index{Mathematical School at Cambridge!Tripos}% +\index{Tripos, Mathematical}% +Tripos, you must bear in mind that I have hitherto taken no part in mathematical +teaching of any kind, and therefore must necessarily be a bad judge +of the possibilities of mathematical training, and of its effects on most minds. +A year and a half ago I took part as Additional Examiner in the Mathematical +Tripos, and I must confess that I was a good deal discouraged by what +I saw. Now do not imagine that I flatter myself I was one jot better in all +these respects than others, when I went through the mill. I too felt the +pressure of time, and scribbled down all I could in my three hours, and +doubtless presented to my examiners some very pretty muddles. I can only +congratulate myself that the men I examined were not my competitors. + +In order to determine whether anything can be done to improve this +state of things, let us consider the merits and demerits of our Mathematical +School. One of the most prominent evils is that our system of examination +has a strong tendency to make men regard the subjects more as a series of +isolated propositions than as a whole; and much attention has to be paid to a +point, which is really important for the examination, viz.~where to begin and +where to leave off in answering a question. The \textit{coup d'{\oe}il} of the whole +subject is much impaired; but this is to some extent inherent in any system +of examination. This result is, however, principally due to our custom of +setting the examinees to reproduce certain portions of the books which they +have studied; that is to say this evil arises from the ``bookwork'' questions. +I have a strong feeling that such questions should be largely curtailed, and +that the examinees should by preference be asked for transformations and +modifications of the results obtained in the books. I suppose a certain amount +of bookwork must be retained in order to permit patient workers, who are +not favoured by any mathematical ability, to exhibit to the examiners that +they have done their best. But for men with any mathematical power +there can be no doubt that such questions as I suggest would give a far +more searching test, and their knowledge of the subject would not have +to be acquired in short patches. + +I should myself like to see an examination in which the examinees were +allowed to take in with them any books they required, so that they need not +load their memories with formulae, which no original worker thinks of trying +\DPPageSep{062}{4} +to remember. A first step in this direction has been taken by the introduction +of logarithm tables into the Senate House; and I fancy that a +terrible amount of incompetence was exhibited in the result. I may remark +by the way that the art of computation is utterly untaught here, and that +readiness with figures is very useful in ordinary life. I have done a good +deal of such work myself, but I had to learn it by practice and from a few +useful hints from others who had mastered it. + +It is to be regretted that questions should be set in examinations which +are in fact mere conjuring tricks with symbols, a kind of double acrostic; +another objectionable class of question is the so-called physical question which +has no relation to actual physics. This kind of question was parodied once +by reference to ``a very small elephant, whose weight may be neglected,~etc.'' +Examiners have often hard work to find good questions, and their difficulties +are evidenced by such problems as I refer to. I think, however, that of late +this kind of exercise is much less frequent than formerly. + +I am afraid the impression is produced in the minds of many, that if +a problem cannot be solved in a few hours, it cannot be solved at all. At any +rate there seems to be no adequate realisation of the process by which most +original work is done, when a man keeps a problem before him for weeks, +months, years and gnaws away from time to time when any new light may +strike him. + +I think some of our text books are to blame in this; they impress the +\index{Mathematical School at Cambridge!text-books}% +\index{Text-books, mathematical}% +student in the same way that a high road must appear to a horse with +blinkers. The road stretches before him all finished and macadamised, +having existed for all he knows from all eternity, and he sees nothing of +by-ways and foot-paths. Now it is the fact that scarcely any subject is so +way worn that there are not numerous unexplored by-paths, which may lead +across to undiscovered countries. I do not advocate that the student should +be led along and made to examine all the cul-de-sacs and blind alleys, as he +goes; he would never got on if he did so, but I do protest against that tone +which I notice in many text books that mathematics is a spontaneously +growing fruit of the tree of knowledge, and that all the fruits along \textit{that} +road have been gathered years ago. Rather let him see that the whole +grand work is the result of the labours of an army of men, each exploring +his little bit, and that there are acres of untouched ground, where he too may +gather fruit: true, if he begins on original work, he may think that he has +discovered something new and may very likely find that someone has been +before him; but at least he \textit{too} will have had the enormous pleasure of +discovery. + +There is another fault in the system of examinations, but I hardly know +whether it can be appreciably improved. It is this:---the system gives very +\DPPageSep{063}{5} +little training in the really important problem both of practical life and of +mathematics, viz.~the determination of the exact nature of the question +which is to be attacked, the making up of your mind as to what you will do. +Everyone who has done original work knows that at first the subject generally +presents itself as a chaos of possible problems, and careful analysis +is necessary before that chaos is disentangled. The process is exactly that +of a barrister with his brief. A pile of papers is set before him, and from +that pile he has to extract the precise question of law or fact on which +the whole turns. When he has mastered the story and the precise point, +he has generally done the more difficult part of his work. In most cases, +it is exactly the same in mathematical work; and when the question has +been pared down until its characteristics are those of a Tripos question, of +however portentous a size, the battle is half won. It only remains to the +investigator then to avail himself of all the ``morbid aptitude for the +manipulation of symbols'' which he may happen to possess. + +In examination, however, the whole of this preparatory part of the work +is done by the examiner, and every examiner must call to mind the weary +threshing of the air which he has gone through in trying ``to get a question'' +out of a general idea. Now the limitation of time in an examination makes +this evil to a large extent irremediable; but it seems to me that some good +may be done by requesting men to write essays on particular topics, +because in this case their minds are not guided by a pair of rails carefully +prepared by an examiner. + +In the report on the Tripos for~1882, I spoke of the slovenliness of style +which characterised most of the answers. It appears to me that this is really +much more than a mere question of untidiness and annoyance to examiners. +The training here seems to be that form and style are matters of no moment, +and answers are accordingly sent up in examination which are little more +than rough notes of solutions. But I insist that a mathematical writer +should attend to style as much as a literary man. + +Some of our Cambridge writers on mathematics seem never to have +recovered from the ill effects of their early training, even when they devote +the rest of their life to original work. I wish some of you would look at the +artistic mode of presentation practised by Gauss, and compare it with the +standard of excellence which passes muster here. Such a comparison will +not prove gratifying to our national pride. + +Where there is slovenliness of style it is, I think, almost certain that +there will be wanting that minute attention to form on which the successful, +or at least easy, marshalling of a complex analytical development depends. +The art of carrying out such work has to be learnt by trial and error by +the men trained in our school, and yet the inculcation of a few maxims +\DPPageSep{064}{6} +would generally be of great service to students, provided they are made to +attend to them in their work. The following maxims contain the pith of +the matter, although they might be amplified with advantage if I were to +detain you over this point for some time. + +1st. Choose the notation with great care, and where possible use a +standard notation. + +2nd. Break up the analysis into a series of subsections, each of which +may be attended to in detail. + +3rd. Never attempt too many transformations in one operation. + +4th. Write neatly and not quickly, so that in passing from step to step +there may be no mistakes of copying. + +A man who undertakes any piece of work, and does not attend to some +such rules as these, doubles his chances of mistake; even to short pieces +of work such as examination questions the same applies, and I have little +doubt that many a score of questions have been wrongly worked out from +want of attention to these points. + +It is true that great mathematicians have done their work in very +various styles, but we may be sure that those who worked untidily gave +themselves much unnecessary trouble. Within my own knowledge I may +say that Thomson [Lord Kelvin] works in a copy-book, which is produced at +Railway Stations and other conveniently quiet places for studious pursuits; +Maxwell worked in part on the backs of envelopes and loose sheets of paper +crumpled up in his pocket\footnotemarkN; Adams' manuscript is as much a model of +\footnotetextN{I think that he must have been only saved from error by his almost miraculous physical + insight, and by a knowledge of the time when work must be done neatly. But his \textit{Electricity} + was crowded with errata, which have now been weeded out one by one.}% +neatness in mathematical writing as Porson's of Greek writing. There is, of +course, no infallibility in good writing, but believe me that untidiness surely +has its reward in mistakes. I have spoken only on the evils of slovenliness +in its bearing on the men as mathematicians---I cannot doubt that as a +matter of general education it is deleterious. + +I have dwelt long on the demerits of our scheme, because there is hope +of amending some of them, but of the merits there is less to be said because +they are already present. The great merit of our plan seems to me to be +reaped only by the very ablest men in the year. It is that the student is +enabled to get a wide view over a great extent of mathematical country, +and if he has not assimilated all his knowledge thoroughly, yet he knows +that it is so, and he has a fair introduction to many subjects. This +advantage he would have lost had he become a pure specialist and original +investigator very early in his career. But this advantage is all a matter +of degree, and even the ablest man cannot cover an indefinitely long course +\DPPageSep{065}{7} +in his three years. Year by year new subjects were being added to the +curriculum, and the limit seemed to have been exceeded; whilst the +disastrous effects on the weaker brethren were becoming more prominent. +I cannot but think that the new plan, by which a man shall be induced to +become a partial specialist, gives us better prospects. + +Another advantage we gain by our strict competition is that a man must +be bright and quick; he must not sit mooning over his papers; he is quickly +brought to the test,---either he can or he cannot do a definite problem in +a finite time---if he cannot he is found out. Then if our scheme checks +original investigation, it at least spares us a good many of those pests of +science, the man who churns out page after page of~$x, y, z,$ and thinks he +has done something in producing a mass of froth. That sort of man is +quickly found out here, both for his own good and the good of the world +at large. Lastly this place has the advantage of having been the training +school of nearly all the English mathematicians of eminence, and of having +always attracted---as it continues to attract---whatever of mathematical +ability is to be found in the country. These are great merits, and in the +endeavour to remove blemishes, we must see that we do not destroy them. + +A discussion of the Mathematical Tripos naturally brings us face to face +with a much abused word, namely ``Cram.'' + +The word connotes bad teaching, and accordingly teaching with reference +to examinations has been supposed to be bad because it has been called +cram. The whole system of private tuition commonly called coaching has +been nick-named cram, and condemned accordingly. I can only say for +myself that I went to a private tutor whose name is familiar to everyone +in Cambridge, and found the most excellent and thorough teaching; far +be it from me to pretend that I shall prove his equal as a teacher. Whatever +fault is to be found, it is not with the teaching, but it lies in the +system. It is obviously necessary that when a vast number of new subjects +are to be mastered the most rigorous economy in the partition of the student's +time must be practised, and he is on no account to be allowed to spend +more than the requisite minimum on any one subject, even if it proves +attractive to him. The private tutor must clearly, under the old regime, +act as director of studies for his pupils strictly in accordance with examination +requirements; for place in the Tripos meant pounds, shillings, and +pence to the pupil. The system is now a good deal changed, and we may +hope that it will be possible henceforth to keep the examination less +incessantly before the student, who may thus become a student of a subject, +instead of a student for a Tripos. + +And now I think you must see the peculiar difficulties of a professor of +mathematics; his vice has been that he tried to teach a subject \textit{only}, and +\DPPageSep{066}{8} +private tutors felt, and felt justly, that they could not, in justice to their +pupils' prospects, conscientiously recommend the attendance at more than +a very small number of professorial lectures. But we are now at the beginning +of a new regime and it may be that now the professors have their +chance. But I think it depends much more on the examiners than on the +professors. If examiners can and will conduct the examinations in such +a manner that it shall ``pay'' better to master something thoroughly, than +to have a smattering of much, we shall see a change in the manner of +learning. Otherwise there will not be much change. I do not know how +it will turn out, but I do know that it is the duty of professors to take such +a chance if it exists. + +My purpose is to try my best to lecture in such a way as will impart an +interest to the subject itself and to help those who wish to learn, so that +they may reap advantage in examinations---provided the examinations are +conducted wisely. +\DPPageSep{067}{9} + + +\Chapter{Introduction to Dynamical Astronomy} +\index{Introduction to Dynamical Astronomy}% +\index{Dynamical Astronomy, introduction to}% + +\First{The} field of dynamical astronomy is a wide one and it is obvious that +it will be impossible to consider even in the most elementary manner +all branches of it; for it embraces all those effects in the heavens which may +be attributed to the effects of gravitation. In the most extended sense of +the term it may be held to include theories of gravitation itself. Whether +or not gravitation is an ultimate fact beyond which we shall never penetrate +is as yet unknown, but Newton, whose insight into physical causation was +almost preternatural, regarded it as certain that some further explanation +was ultimately attainable. At any rate from the time of Newton down to +to-day men have always been striving towards such explanation---it must be +admitted without much success. The earliest theory of the kind was that +of Lesage, promulgated some $170$~years ago. He conceived all space to be +filled with what he called ultramundane corpuscles, moving with very great +velocities in all directions. They were so minute and so sparsely distributed +that their mutual collisions were of extreme rarity, whilst they bombarded +the grosser molecules of ordinary matter. Each molecule formed a partial +shield to its neighbours, and this shielding action was held to furnish an +explanation of the mutual attraction according to the law of the inverse +square of the distance, and the product of the areas of the sections of the +two molecules. Unfortunately for this theory it is necessary to assume that +there is a loss of energy at each collision, and accordingly there must be +a perpetual creation of kinetic energy of the motion of the ultramundane +corpuscles at infinity. The theory is further complicated by the fact that +the energy lost by the corpuscle at each collision must have been communicated +to the molecule of matter, and this must occur at such a rate as to +vaporize all matter in a small fraction of a second. Lord Kelvin has, however, +pointed out that there is a way out of this fundamental difficulty, for +if at each collision the ultramundane corpuscle should suffer no loss of total +kinetic energy but only a transformation of energy of translation into energy +of internal vibration, the system becomes conservative of energy and the +eternal creation of energy becomes unnecessary. On the other hand, gravitation +will not be transmitted to infinity, but only to a limited distance. +\DPPageSep{068}{10} +I will not refer further to this conception save to say that I believe that no +man of science is disposed to accept it as affording the true road. + +It may be proved that if space were an absolute plenum of incompressible +fluid, and that if in that fluid there were points towards which the fluid +streams from all sides and disappears, those points would be urged towards +one another with a force varying inversely as the square of the distance +and directly as the product of the intensities of the two inward streams. +Such points are called sinks and the converse, namely points from whence +the fluid streams, are called sources. Now two sources also attract one +another according to the same law; on the other hand a source and a sink +repel one another. If we could conceive matter to be all sources or all sinks +we should have a mechanical theory of gravitation, but no one has as yet +suggested any means by which this can be realised. Bjerknes of Christiania +has, however, suggested a mechanical means whereby something of the kind +may be realised. Imagine an elastic ball immersed in water to swell and +contract rhythmically, then whilst it is contracting the motion of the surrounding +water is the same as that due to a sink at its centre, and whilst +it is expanding the motion is that due to a source. Hence two balls which +expand and contract in exactly the same phase will attract according to the +law of gravitation on taking the average over a period of oscillation. If, +however, the pulsations are in opposite phases the resulting force is one of +repulsion. If then all matter should resemble in some way the pulsating +balls we should have an explanation, but the absolute synchronism of the +pulsations throughout all space imports a condition which does not commend +itself to physicists. I may mention that Bjerknes has actually realised these +conclusions by experiment. Although it is somewhat outside our subject +I may say that if a ball of invariable volume should execute a small +rectilinear oscillation, its advancing half gives rise to a source and the +receding half to a sink, so that the result is what is called a doublet. Two +oscillating balls will then exercise on one another forces analogous to that +of magnetic particles, but the forces of magnetism are curiously inverted. +This quasi-magnetism of oscillating balls has also been treated experimentally +by Bjerknes. However curious and interesting these speculations +and experiments may be, I do not think they can afford a working hypothesis +of gravitation. + +A new theory of gravitation which appears to be one of extraordinary +\index{Gravitation, theory of}% +ingenuity has lately been suggested by a man of great power, viz.~Osborne +Reynolds, but I do not understand it sufficiently to do more than point +out the direction towards which he tends. He postulates a molecular ether. +I conceive that the molecules of ether are all in oscillation describing orbits +in the neighbourhood of a given place. If the region of each molecule be +replaced by a sphere those spheres may be packed in a hexagonal arrangement +\DPPageSep{069}{11} +completely filling all space. We may, however, come to places where the +symmetrical piling is interrupted, and Reynolds calls this a region of misfit. + +Then, according to this theory, matter consists of misfit, so that matter is +the deficiency of molecules of ether. Reynolds claims to show that whilst +the particular molecules which don't fit are continually changing the amount +of misfit is indestructible, and that two misfits attract one another. The +theory is also said to explain electricity. Notwithstanding that Reynolds +is not a good exponent of his own views, his great achievements in science +are such that the theory must demand the closest scrutiny. + +The newer theories of electricity with which the name of Prof.~J.~J. +Thomson is associated indicate the possibility that mass is merely an electrodynamic +phenomenon. This view will perhaps necessitate a revision of all +our accepted laws of dynamics. At any rate it will be singular if we shall +have to regard electrodynamics as the fundamental science, and subsequently +descend from it to the ordinary laws of motion. How much these notions +are in the air is shown by the fact that at a congress of astronomers, held in +1902 at Göttingen, the greater part of one day's discussion was devoted +to the astronomical results which would follow from the new theory of +electrons. + +I have perhaps said too much about the theories of gravitation, but it +should be of interest to you to learn how it teems with possibilities and how +great is the present obscurity. + +Another important subject which has an intimate relationship with +Dynamical Astronomy is that of abstract dynamics. This includes the +general principles involved in systems in motion under the action of conservative +forces and the laws which govern the stability of systems. Perhaps +the most important investigators in this field are Lagrange and Hamilton, +and in more recent times Lord Kelvin and Poincaré. + +Two leading divisions of dynamical astronomy are the planetary theory +\index{Lunar and planetary theories compared}% +\index{Planetary and lunar theories compared}% +and the theory of the motion of the moon and of other satellites. A first +approximation in all these cases is afforded by the case of simple elliptic +motion, and if we are to consider the case of comets we must include +parabolic and hyperbolic motion round a centre. Such a first approximation +is, however, insufficient for the prediction of the positions of any of the bodies +in our solar system for any great length of time, and it becomes necessary +to include the effects of the disturbing action of one or more other bodies. +The problem of disturbed revolution may be regarded as a single problem +in all its cases, but the defects of our analysis are such that in effect its +several branches become very distinct from one another. It is usual to +speak of the problem of disturbed revolution as the problem of three bodies, +for if it were possible to solve the case where there are three bodies we +\DPPageSep{070}{12} +should already have gone a long way towards the solution of that more +complex case where there are any number of bodies. + +Owing to the defects of our analysis it is at present only possible to +obtain accurate results of a general character by means of tedious expansions. +All the planets and all the satellites have their motions represented with +more or less accuracy by ellipses, but this first approximation ceases to be +satisfactory for satellites much more rapidly than is the case for planets. +The eccentricities of the ellipses and the inclinations of the orbits are in most +cases inconsiderable. It is assumed then that it is possible to effect the +requisite expansions in powers of the eccentricities and of suitable functions +of the inclinations. Further than this it is found necessary to expand in +powers of the ratios of the mean distances of the disturbed and disturbing +bodies from the centre. It is at this point that the first marked separation +of the lunar and planetary theories takes place. In the lunar theory the +distance of the sun (disturber) from the earth is very great compared with +that of the moon, and we naturally expand in this ratio in order to start +with as few terms as possible. In the planetary theory the ratio of the +distances of the disturbed and disturbing bodies---two planets---from the sun +may be a large fraction. For example, the mean distances of Venus and the +earth are approximately in the ratio~$7:10$, and in order to secure sufficient +accuracy a large number of terms is needed. In the case of the planetary +theory the expansion is delayed as long as possible. + +Again, in the lunar theory the mass of the disturbing body is very +great compared with that of the primary, a ratio on which it is evident that +the amount of perturbation greatly depends. On the other hand, in the +planetary theory the disturbing body has a very small mass compared with +that of the primary, the sun. From these facts we are led to expect that +large terms will be present in the expressions for the motion of the moon +due to the action of the sun, and that the later terms in the expansion will +rapidly decrease; and in the planetary theory we expect large numbers of +terms all of about equal magnitude and none of them very great. This +expectation is, however, largely modified by some further remarks to be made. + +You know that a dynamical system may have various modes of free +oscillation of various periods. If then a disturbing force with a period differing +but little from that of one of the modes of free oscillation acts on the +system for a long time it will generate an oscillation of large amplitude. + +A familiar instance of this is in the roll of a ship at sea. If the incidence +of the waves on the ship is such that the succession of impulses is very +nearly identical in period with the natural period of the ship, the roll becomes +large. In analysis this physical fact is associated with a division by a small +divisor on integration. +\DPPageSep{071}{13} +As an illustration of the simplest kind suppose that the equation of motion +of a system under no forces were +\[ +\frac{d^{2}x}{dt^{2}} + n^{2}x = 0. +\] +Then we know that the solution is +\[ +x = A \cos nt + B \sin nt, +\] +that is to say the free period is~$\dfrac{2 \pi}{n}$. Suppose then such a system be acted on +by a perturbing force $F\cos(n - \epsilon)t$, where $\epsilon$~is small; the equation of motion is +\[ +\frac{d^{2}x}{dt^{2}} + n^{2}x = F\cos(n - \epsilon)t, +\] +and the solution corresponding to such a disturbing force is +\[ +x = \frac{F}{-(n - \epsilon)^{2} + n^{2}} \cos(n - \epsilon)t + = \frac{F}{2n\epsilon - \epsilon^{2}} \cos(n - \epsilon)t. +\] +If $\epsilon$~is small the amplitude becomes great, and this arises, as has been said, by +a division by a small divisor. + +Now in both lunar and planetary theories the coefficients of the periodic +terms become frequently much greater than might have been expected +\textit{à~priori}. In the lunar theory before this can happen in such a way as to +cause much trouble the coefficients have previously become so small that it +is not necessary to consider them. But suppose in the planetary theory $n, n'$ +are the mean motions of two planets round the primary. Then coefficients +will continually be having multipliers of the forms +\[ +\frac{n'}{in ± i'n'} \text{ and } \left(\frac{n'}{in ± i'n'}\right)^{2}, +\] +where $i, i'$ are small positive integers. In general the larger $i, i'$ the smaller is +the coefficient to begin with, but owing to the fact that the ratio~$n : n'$ may +very nearly approach that of two small integers a coefficient may become very +great; e.g.~$5$~Jovian years nearly equal $2$~of Saturn, while the ratio of +the mean distances is~$6 : 11$. The result is a large long inequality with a +period of $913$~years in the motions of those two planets. The periods of the +principal terms in the moon's motion are generally short, but some have +large coefficients, so that the deviation from elliptic motion is well marked. + +The general problem of three bodies is in its infancy, and as yet but little +is known as to the possibilities in the way of orbits and as to their stabilities. + +Another branch of our subject is afforded by the precession and nutation +of the earth, or any other planet, under the influence of the attractions of +disturbing bodies. This is the problem of disturbed rotation and it presents +a strong analogy with the problem of disturbed elliptic motion. When a top +\DPPageSep{072}{14} +spins with absolute steadiness we say that it is asleep. Now the earth in its +rotation may be asleep or it may not be so---there is nothing but observation +which is capable of deciding whether it is so or not. This is equally true +whether the rotation takes place under external perturbation or not. If the +earth is asleep its motion presents a perfect analogy with circular orbital +motion; if it wobbles the analogy is with elliptic motion. The analogy is +such that the magnitude of the wobble corresponds with the eccentricity of +orbit and the position of greatest departure with the longitude of pericentre. +Until the last $20$~years it has always been supposed that the earth is asleep +in its rotation, but the extreme accuracy of modern observation, when subjected +to the most searching analysis by Chandler and others, has shown +that there is actually a small wobble. This is such that the earth's axis of +rotation describes a small circle about the pole of figure. The theory of +precession indicated that this circle should be described in a period of +$305$~days, and all the earlier astronomers scrutinised the observations with +the view of detecting such an inequality. It was this preconception, apparently +well founded, which prevented the detection of the small inequality +in question. It was Chandler who first searched for an inequality of unknown +period and found a clearly marked one with a period of $428$~days. +He found also other smaller inequalities with a period of a year. This +wandering of the pole betrays itself most easily to the observer by changes +in the latitude of the place of observation. + +The leading period in the inequality of latitude is then one of $428$~days. +\index{Latitude, variation of}% +\index{Variation, the!of latitude}% +The theoretical period of $305$~days was, as I have said, apparently well +established, but after the actual period was found to be $428$~days Newcomb +pointed out that if the earth is not absolutely rigid, but slightly changes +its shape as the axis of rotation wanders, such a prolongation of period +would result. Thus these purely astronomical observations end by affording +a measure of the effective rigidity of the earth's mass. + +The theory of the earth's figure and the variation of gravity as we vary +\index{Earth's figure, theory of}% +our position on the surface or the law of variation of gravity as we descend +into mines are to be classified as branches of dynamical astronomy, although +in these cases the velocities happen to be zero. This theory is intimately +connected with that of precession, for it is from this that we conclude that +the free wobble of the perfectly rigid earth should have a period of $305$~days. +The ellipticity of the earth's figure also has an important influence on the +motion of the moon, and the determination of a certain inequality in the +moon's motion affords the means of finding the amount of ellipticity of the +earth's figure with perhaps as great an accuracy as by any other means. +Indeed in the case of Jupiter, Saturn, Mars, Uranus and Neptune the +ellipticity is most accurately determined in this way. The masses also of the +planets may be best determined by the periods of their satellites. +\DPPageSep{073}{15} + +The theory of Saturn's rings is another branch. The older and now +\index{Saturn's rings}% +obsolete views that the rings are solid or liquid gave the subject various +curious and difficult mathematical investigations. The modern view---now +well established---that they consist of an indefinite number of meteorites +which collide together from time to time presents a number of problems of +great difficulty. These were ably treated by Maxwell, and there does not +seem any immediate prospect of further extension in this direction. + +Then the theory of the tides is linked to astronomy through the fact that +it is the moon and sun which cause the tides, so that any inequality in their +motions is reflected in the ocean. + +On the fringe of our subject lies the whole theory of figures of equilibrium +of rotating liquids with the discussion of the stability of the various +possible forms and the theory of the equilibrium of gaseous planets. In this +field there is yet much to discover. + +This subject leads on immediately to theories of the origin of planetary +systems and to cosmogony. Tidal theory, on the hypothesis that the tides +are resisted by friction, leads to a whole series of investigations in speculative +astronomy whose applications to cosmogony are of great interest. + +Up to a recent date there was little evidence that gravitation held good +\index{Gravitation, theory of!universal}% +outside the solar system, but recent investigations, carried out largely by +means of the spectroscopic determinations of velocities of stars in the line of +sight, have shewn that there are many other systems, differing very widely +from our own, where the motions seem to be susceptible of perfect explanation +by the theory of gravitation. These new extensions of gravitation +outside our system are leading to many new problems of great difficulty +and we may hope in time to acquire wider views as to the possibilities of +motion in the heavens. + +This hurried sketch of our subject will show how vast it is, and I cannot +hope in these lectures to do more than touch on some of the leading topics. +\DPPageSep{074}{16} + + +\Chapter{Hill's Lunar Theory} +\index{Hill, G. W., Lunar Theory!lectures by Darwin on Lunar Theory}% +\index{Hill, G. W., Lunar Theory!characteristics of his Lunar Theory}% +\index{Lunar Theory, lecture on}% + +\Section{§ 1. }{Introduction\footnotemark.} + +\footnotetext{The references in this section are to Hill's ``Researches in the Lunar Theory'' first published + (1878) in the \textit{American Journal of Mathematics}, vol.~\Vol{I.} pp.~5--26, 129--147 and reprinted in + \textit{Collected Mathematical Works}, vol.~\Vol{I.} pp.~284--335. Hill's other paper connected with these + lectures is entitled ``On the Part of the Motion of the Lunar Perigee which is a function of the + Mean Motions of the Sun and Moon,'' published separately in 1877 by John Wilson and~Son, + Cambridge, Mass., and reprinted in \textit{Acta Mathematica}, vol.~\Vol{VIII.} pp.~1--36, 1886 and in \textit{Collected + Mathematical Works}, vol.~\Vol{I.} pp.~243--270.} + +\First{An} account of Hill's \textit{Lunar Theory} can best be prefaced by a few +quotations from Hill's original papers. These will indicate the peculiarities +which mark off his treatment from that of earlier writers and also, to some +extent, the reasons for the changes he introduced. Referring to the well-known +expressions which give, for undisturbed elliptic motion, the rectangular +coordinates as explicit functions of the time---expressions involving nothing +more complicated than Bessel's functions of integral order---Hill writes: + +``Here the law of series is manifest, and the approximation can easily be +carried as far as we wish. But the longitude and latitude, variables employed +by nearly all lunar theorists, are far from having such simple expressions; in +fact their coefficients cannot be finitely expressed in terms of Besselian +functions. And if this is true in the elliptic theory how much more likely is +a similar thing to be true when the complexity of the problem is increased +by the consideration of disturbing forces?\ldots\ There is also another advantage +in employing coordinates of the former kind (rectangular): the differential +equations are expressed in purely algebraic functions, while with the latter +(polar) circular functions immediately present themselves.'' + +In connection with the parameters to be used in the expansions Hill +argues thus: + +``Again as to parameters all those who have given literal developments, +Laplace setting the example, have used the parameter~$\m$, the ratio of the +sidereal month to the sidereal year. But a slight examination, even of the +results obtained, ought to convince anyone that this is a most unfortunate +selection in regard to convergence. Yet nothing seems to render the +parameter desirable, indeed the ratio of the synodic month to the sidereal +year would appear to be more naturally suggested as a parameter.'' +\DPPageSep{075}{17} + +When considering the order of the differential equations and the method +of integration, Hill wrote: + +``Again the method of integration by undetermined coefficients is most +likely to give us the nearest approach to the law of series; and in this +method it is as easy to integrate a differential equation of the second order +as one of the first, while the labour is increased by augmenting the number +of variables and equations. But Delaunay's method doubles the number of +variables in order that the differential equations may be all of the first order. +Hence in this disquisition I have preferred to use the equations expressed in +terms of the coordinates rather than those in terms of the elements; and, in +general, always to diminish the number of unknown quantities and equations +by augmenting the order of the latter. In this way the labour of making a +preliminary development of~$R$ in terms of the elliptic elements is avoided.'' + +We may therefore note the characteristics of Hill's method as follows: + +(1) Use of rectangular coordinates. + +(2) Expansion of series in powers of the ratio of the synodic month to +the sidereal year. + +(3) Use of differential equations of the second order which are solved by +assuming series of a definite type and equating coefficients. + +In these lectures we shall obtain only the first approximation to the +solution of Hill's differential equations. The method here followed is not +that given by Hill, although it is based on the same principles as his method. +Our work only involves simple algebra, and probably will be more easily +understood than Hill's. If followed in detail to further approximations, it +would prove rather tedious, but it leads to the results we require without too +much labour. If it is desired to follow out the method further, reference +should be made to Hill's own writings. + +\Section{§ 2. }{Differential Equations of Motion and Jacobi's Integral.} +\index{Differential Equations of Motion}% +\index{Equations of motion}% + +Let $E, M, \m'$ denote the masses or positions of the earth, moon, and sun, +and let $G$~be the centre of inertia of $E$~and~$M$. Let $x, y, z$ be the rectangular +coordinates of~$M$ with $E$~as origin, and let $x', y', z'$ be the coordinates +of~$\m'$ referred to parallel axes through~$G$. The coordinates of~$M$ relative to +the axes through~$G$ are clearly~$\dfrac{E}{E + M} x$, $\dfrac{E}{E + M} y$, $\dfrac{E}{E + M} z$; those of~$E$ are +$-\dfrac{E}{E + M} x$, $-\dfrac{E}{E + M} y$, $-\dfrac{E}{E + M} z$. The distances $EM, E\m', M\m'$\DPnote{** TN: Inconsistent overlines in original} are denoted +\DPPageSep{076}{18} +by $r, r_1, \Delta$ respectively. It is assumed that $G$~describes a Keplerian ellipse +round~$\m'$ so that $x', y', z'$ are known functions of the time. The accelerations +of~$M$ relative to~$E$ are shewn in the diagram. +\begin{figure}[hbt!] +\centering +\Input[0.75\textwidth]{p018} +\caption{Fig.~1.} +\end{figure} + +We have +\begin{gather*} +r^{2} = x^{2} + y^{2} + z^{2}, \\ +\begin{aligned} +r_{1}^{2} + &= \left(x' + \frac{Mx}{E + M}\right)^{2} + + \left(y' + \frac{My}{E + M}\right)^{2} + + \left(z' + \frac{Mz}{E + M}\right)^{2}, \\ +\Delta^{2} + &= \left(x' - \frac{Ex}{E + M}\right)^{2} + + \left(y' - \frac{Ey}{E + M}\right)^{2} + + \left(z' - \frac{Ez}{E + M}\right)^{2}. +\end{aligned} +\end{gather*} + +Hence +\begin{gather*} +\frac{\dd r}{\dd x} = \frac{x}{r}, \\ +\begin{aligned} +\frac{E + M}{M}\, \frac{\dd r_{1}}{\dd x} + &= \frac{x' + \dfrac{Mx}{E + M}}{r_{1}}, \\ +-\frac{E + M}{M}\, \frac{\dd \Delta}{\dd x} + &= \frac{x' - \dfrac{Ex}{E + M}}{\Delta}; +\end{aligned} +\end{gather*} +\begin{alignat*}{3} +\text{$\therefore$ the direction cosines of }& EM &&\text{ are }&& + \frac{\dd r}{\dd x},\ \frac{\dd r}{\dd y},\ \frac{\dd r}{\dd z},\\ +% +\Ditto[the ]\Ditto[direction ]\Ditto[cosines of ]& E\m' &&\text{ are }&& + \Neg\frac{E+M}{M}\left(\frac{\dd r_{1}}{\dd x},\ \frac{\dd r_{1}}{\dd y},\ \frac{\dd r_{1}}{\dd z}\right),\\ +\Ditto[the ]\Ditto[direction ]\Ditto[cosines of ]& M\m' &&\text{ are }&& + -\frac{E+M}{M}\left(\frac{\dd \Delta}{\dd x},\: \frac{\dd \Delta}{\dd y},\: \frac{\dd \Delta}{\dd z}\right). +\end{alignat*} + +If $X, Y, Z$ denote the components of acceleration of~$M$ relative to axes +through~$E$, +\DPPageSep{077}{19} +\[ +\left. +\begin{aligned} + X &= -\frac{E+M}{r^{2}}\, \frac{\partial r}{\partial x} + - \frac{\m'}{\Delta^{2}}\, \frac{E + M}{E} + \frac{\partial\Delta}{\partial x} - \frac{\m'}{r_{1}^{2}}\, \frac{E + M}{M}\, + \frac{\partial r_{1}}{\partial x}\\ + &= \frac{\partial F}{\partial x},\\ +& +\lintertext{where}{\phantom{= -\frac{E+M}{r^{2}} \frac{\partial r}{\partial x} + - \frac{\m'}{\Delta^{2}} \frac{E+M}{E} + \frac{\partial\Delta}{\partial x} - \frac{\m'}{r_{1}^{2}} \frac{E+M}{M} + \frac{\partial r_{1}}{\partial x}}} \\ +F &= \frac{E+M}{r} + \frac{\m'}{\Delta}\, \frac{E+M}{E} + + \frac{\m'}{r_{1}}\, \frac{E + M}{M}. \\ +&\lintertext{\indent Similarly,}{\phantom{= -\frac{E+M}{r^{2}} \frac{\partial r}{\partial x} + - \frac{\m'}{\Delta^{2}} \frac{E+M}{E} + \frac{\partial\Delta}{\partial x} - \frac{\m'}{r_{1}^{2}} \frac{E+M}{M} + \frac{\partial r_{1}}{\partial x}}} \\ +Y &= \frac{\partial F}{\partial y},\ +Z =\frac{\partial F}{\partial z}. +\end{aligned} +\right\} +\Tag{(1)} +\] + +Let $r'$~be the distance between $G$~and~$\m'$, and let $\theta$~be the angle~$\m'GM$; +then +\begin{align*} +r'^{2} &= x'^{2} + y'^{2} + z'^{2} \text{ and } + \cos\theta = \frac{xx' + yy' + zz'}{rr'}, \\ +r_{1}^{2} &= r'^{2} + \frac{2M}{E + M}\, rr' \cos\theta + \left(\frac{Mr}{E + M}\right)^{2}, \\ +\Delta^{2} &= r'^{2} - \frac{2E}{E + M}\, rr' \cos\theta + \left(\frac{Er}{E + M}\right)^{2}. +\end{align*} + +Since $r$~is very small compared with~$r'$, +\begin{gather*} +\begin{aligned} +\frac{1}{r_{1}} + &= \frac{1}{r'} \left\{1 - \frac{M}{E + M}\, \frac{r}{r'} \cos\theta + + \left(\frac{M}{E + M} · \frac{r}{r'} \right)^{2} + (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots\right\}, \\ +% +\frac{1}{\Delta} + &= \frac{1}{r'} \left\{1 + \frac{E}{E + M}\, \frac{r}{r'} \cos\theta + + \left(\frac{E}{E + M} · \frac{r}{r'} \right)^{2} + (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots\right\}. +\end{aligned} \\ +% +\therefore \frac{1}{E\Delta} + \frac{1}{Mr_{1}} + = \frac{E + M}{EM} · \frac{1}{r'} + + \frac{1}{E + M} · \frac{r^{2}}{r'^{3}} + (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots. +\end{gather*} + +Hence +\[ +F = \frac{E + M}{r} + \frac{\m'(E + M)^{2}}{EMr'} + + \frac{\m'r^{2}}{r'^{3}} (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2})\ldots. +\] + +But the second term does not involve $x, y, z$, and may be dropped. +\[ +\therefore +F = \frac{E + M}{r} + + \frac{\m'r^{2}}{r'^{3}} (\tfrac{3}{2} \cos^{2} \theta - \tfrac{1}{2}), +\Tag{(2)} +\] +neglecting terms in~$\dfrac{r^{3}}{r'^{4}}$. + +We will now find an approximate expression for~$F$, paying attention to +the magnitude of the various terms in the actual earth-moon-sun system. +As a first rough approximation, $r'$~is a constant~$\a'$, and $G\m'$~rotates with +uniform angular velocity~$n'$. This neglects the effect on the sun of the earth +and moon not being collected at~$G$ (this effect is very small), and it neglects +the eccentricity of the solar orbit. In order that the coordinates of the sun +relative to the earth might be nearly constant, we introduce axes $x, y$ +\DPPageSep{078}{20} +rotating with angular velocity~$n'$ in the plane of the sun's orbit round the +earth; the $x$-axis being so chosen that it passes through the sun. When +required, a $z$-axis is taken perpendicular to the plane of~$x, y$. As before, let +$x, y, z$ be the coordinates of the moon; the sun's coordinates will be approximately +$\a', 0, 0$. In this approximation $r\cos\theta = x$ and +\[ +F = \frac{E + M}{r} + + \tfrac{3}{2}\, \frac{\m'}{\a'^{3}} x^{2} + - \tfrac{1}{2} \m' \frac{r^{2}}{\a'^{3}}. +\] + +This suggests the following general form for~$F$, instead of that given in +equation~\Eqref{(2)}: +\begin{align*} +F = \frac{E + M}{r} + &+ \tfrac{3}{2}\, \frac{\m'}{\a'^{3}} x^{2} + + \tfrac{3}{2} \m' \left( \frac{r^{2} \cos^{2}\theta}{r'^{3}} - \frac{x^{2}}{\a'^{3}} \right) \\ + &- \tfrac{1}{2}\, \frac{\m'}{\a'^{3}} (x^{2} + y^{2}) + - \tfrac{1}{2}\, \frac{\m'}{\a'^{3}} z^{2}\DPnote{** Why aren't previous terms combined?} + + \tfrac{1}{2} \m' r^{2} \left(\frac{1}{\a'^{3}} - \frac{1}{r'^{3}}\right). +\end{align*} + +For the sake of future developments, we now introduce a new notation. +Let $\nu$~be the moon's synodic mean motion and put $m = \dfrac{n'}{\nu} = \dfrac{n'}{n - n'}$\footnotemark. In the +\footnotetext{In the lunar theory $n'$~is supposed to be a known constant, while $n$ (or~$m$) is one of the + constants of integration the value of which is not yet determined and can only be determined + from the observations. So far $n$ (or~$m$) is quite arbitrary.}% +case of our moon, $m$~is approximately~$\frac{1}{12}$: this is a small quantity in +powers of which our expressions will be obtained. If we neglect $E$~and~$M$ +compared with~$\m'$, we have $\m' = n'^{2} \a'^{3}$, whence $\dfrac{\m'}{\a'^{3}} = n'^{2} = \nu^{2} m^{2}$. Let us also +write $E + M = \kappa \nu^{2}$, and then we get +\begin{align*}%[** TN: Re-broken] +F &+ \tfrac{1}{2} n'^{2} (x^{2} + y^{2}) \\ + &= \nu^{2} \biggl[\frac{\kappa}{r} + \tfrac{1}{2} m^{2} (3x^{2} - z^{2}) + + \tfrac{3}{2} m^{2} \left(\frac{\a'^{3}}{r'^{3}}\, r^{2} \cos^{2}\theta - x^{2}\right) + + \tfrac{1}{2} m^{2} r^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right)\biggr]. +\end{align*} + +For convenience we write +\Pagelabel{20} +\[ +\Omega + = \tfrac{3}{2} m^{2} \left(\frac{\a'^{3}}{r'^{3}} r^{2} \cos^{2}\theta - x^{2}\right) + + \tfrac{1}{2} m^{2} r^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right), +\] +and then +\[ +F + \tfrac{1}{2} n'^{2} (x^{2} + y^{2}) + = \nu^{2} \left[\frac{\kappa}{r} + \tfrac{1}{2} m^{2} (3x^{2} - z^{2}) + \Omega\right]. +\] + +The equations of motion for uniformly rotating axes\footnote + {See any standard treatise on Dynamics.} +are +\[ +\left. +\begin{alignedat}{3} +\frac{d^{2}x}{dt^{2}} &- 2n' \frac{dy}{dt} &&- n'^{2} x + &&= \frac{\dd F}{\dd x}\Add{,} \\ +\frac{d^{2}y}{dt^{2}} &- 2n' \frac{dx}{dt} &&- \DPtypo{n'}{n'^{2}} y + &&= \frac{\dd F}{\dd y}\Add{,} \\ +\frac{d^{2}z}{dt^{2}} & && + &&= \frac{\dd F}{\dd z}\Add{,} +\end{alignedat} +\right\} +\] +\DPPageSep{079}{21} +\index{Jacobi's ellipsoid!integral}% +which give +\begin{alignat*}{5} +&\frac{d^{2}x}{dt^{2}}-2n'\,\frac{dy}{dt} + &&=\frac{\dd}{\dd x}\left[F + \tfrac{1}{2}n'^{2}(x^{2} + y^{2})\right] + &&=\nu^{2}\biggl[-\frac{\kappa x}{r^{3}} &+{}&& 3m^{2}x &+ \frac{\dd \Omega}{\dd x}\biggr],\\ +% +&\frac{d^{2}y}{dt^{2}}+2n'\,\frac{dx}{dt} + &&=\frac{\dd}{\dd y}\left[F + \tfrac{1}{2}n'^{2}(x^{2} + y^{2})\right] + &&=\nu^{2}\biggl[-\frac{\kappa y}{r^{3}} &&&&+\frac{\dd \Omega}{\dd y}\biggr],\\ +% +&\frac{d^{2}z}{dt^{2}} + &&=\frac{\dd}{\dd z}\left[F + \tfrac{1}{2}n'^{2}(x^{2} + y^{2})\right] + &&=\nu^{2}\biggl[-\frac{\kappa z}{r^{3}} &-{}&& m^{2}z &+ \frac{\dd \Omega}{\dd z}\biggr]. +\end{alignat*} + +We might write $\tau = \nu t$ and on dividing the equations by~$\nu^2$ use $\tau$~henceforth +as equivalent to time; or we might choose a special unit of time such +that $\nu$~is unity. In either case our equations become +\[ +\left. +\begin{alignedat}{4} +\frac{d^{2}x}{d\tau^{2}} + & - 2m\frac{dy}{d\tau} + &&+ \frac{\kappa x}{r^{3}} + &&-& 3m^{2}x + =& \frac{\dd \Omega}{\dd x}\Add{,} \\ +% +\frac{d^{2}y}{d\tau^{2}} + & + 2m\frac{dx}{d\tau} + &&+ \frac{\kappa y}{r^{3}} && + &=& \frac{\dd \Omega}{\dd y}\Add{,} \\ +% +\frac{d^{2}z}{d\tau^{2}} & + &&+ \frac{\kappa z}{r^{3}} + &&+& m^{2}z + =& \frac{\dd \Omega}{\dd z}\Add{.} +\end{alignedat} +\right\} +\Tag{(3)} +\] + +If we multiply these equations respectively by $2\dfrac{dx}{d\tau}$, $2\dfrac{dy}{d\tau}$, $2\dfrac{dz}{d\tau}$ and add +them, we have +\begin{multline*}%[** TN: Slightly wide] +\frac{d}{d\tau}\Biggl\{ + \left(\frac{dx}{d\tau}\right)^{2} + + \left(\frac{dy}{d\tau}\right)^{2} + + \left(\frac{dz}{d\tau}\right)^{2}\Biggr\} + - 2\kappa \frac{d}{d\tau}\left(\frac{1}{r}\right) + - 3m^{2} \frac{d}{d\tau}(x^{2}) + + m^{2} \frac{d}{d\tau}(z^{2})\\ + =2\left(\frac{\dd \Omega}{\dd x}\,\frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\,\frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\,\frac{dz}{d\tau}\right). +\end{multline*} + +The whole of the left-hand side is a complete differential; the right-hand +side needs the addition of the term $2\dfrac{\dd \Omega}{\dd \tau}$. + +Let us put for brevity +\[ +V^{2} + = \left(\frac{dx}{d\tau}\right)^{2} + + \left(\frac{dy}{d\tau}\right)^{2} + + \left(\frac{dz}{d\tau}\right)^{2}. +\] + +Then +\[ +V^{2} = \frac{2\kappa}{r} + 3m^{2}x^{2} - m^{2}z^{2} + + 2\int_{0}^{\tau} \left[ + \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau} \right]d\tau + C. +\Tag{(4)} +\] + +If the earth moved round the sun with uniform angular velocity~$n'$, the +axis of~$x$ would always pass through the sun, and therefore we should have +\[ +x' = r' = \a',\quad +y' = z' = 0\Add{,} +\] +and +\[ +r\cos\theta = \frac{xx' + yy' + zz'}{r'} = x, +\] +\DPPageSep{080}{22} +giving +\[ +\frac{\a'^{3}}{r'^{3}} r^{2} \cos^{2}\theta - x^{2} = 0. +\] + +In this case $\Omega$~would vanish. It follows that $\Omega$~must involve as a factor +the eccentricity of the solar orbit. + +It is proposed as a first approximation to neglect that eccentricity, and +this being the case, our equations become +\[ +\left. +\begin{alignedat}{5} +\frac{d^{2}x}{d\tau^{2}} + &- 2m \frac{dy}{d\tau} &+ \frac{\kappa x}{r^{3}} &-& 3m^{2} x &= 0\Add{,} \\ +\frac{d^{2}y}{d\tau^{2}} + &+ 2m \frac{dx}{d\tau} &+ \frac{\kappa y}{r^{3}} && &= 0\Add{,} \\ +\frac{d^{2}z}{d\tau^{2}} + & &+ \frac{\kappa z}{r^{3}} &+& m^{2} z &= 0\Add{.} +\end{alignedat} +\right\} +\Tag{(5)} +\] + +Of these equations one integral is known, viz.\ Jacobi's integral, +\[ +V^{2} = 2\frac{\kappa}{r} + 3m^{2} x^{2} - m^{2} z^{2} + C. +\] + +\Section{§ 3. }{The Variational Curve.} +\index{Variational curve, defined}% + +In ordinary theories the position of a satellite is determined by the +departure from a simple ellipse---fixed or moving. The moving ellipse is +preferred to the fixed one, because it is found that the departures of the +actual body from the moving ellipse are almost of a periodic nature. But +the moving ellipse is not the solution of any of the equations of motion +occurring in the theory. Instead of referring the true orbit to an ellipse, +Hill introduced as the orbit of reference, or intermediate orbit, a curve +suggested by his differential equations, called the ``variational curve.'' + +We have already neglected the eccentricity of the solar orbit, and will +now go one step further and neglect the inclination of the lunar orbit to the +ecliptic, so that $z$~disappears. If the path of a body whose motion satisfies +\[ +\left. +\begin{alignedat}{2} +\frac{d^{2}x}{d\tau^{2}} - 2m \frac{dy}{d\tau} + &+ \left(\frac{\kappa}{r^{3}} - 3m^{2} \right) x &&= 0\\ +\frac{d^{2}y}{d\tau^{2}} + 2m \frac{dx}{d\tau} + &+ \frac{\kappa y}{r^{3}} &&= 0 +\end{alignedat} +\right\} +\Tag{(6)} +\] +intersects the $x$-axis at right angles, the circumstances of the motion before +and after intersection are identical, but in reverse order. Thus, if time +be counted from the intersection, $x = f(\tau^{2})$, $y = \tau f(\tau^{2})$; for if in the differential +equations the signs of $y$~and~$\tau$ are reversed, but $x$~left unchanged, +the equations are unchanged. + +A similar result holds if the path intersects~$y$ at right angles, for if +$x$~and~$\tau$ have signs changed, but $y$~is unaltered, the equations are unaltered. +\DPPageSep{081}{23} + +Now it is evident that the body may start from a given point on the +$x$-axis, and at right angles to it, with different velocities, and that within +certain limits it may reach the axis of~$y$ and cross it at correspondingly +different angles. If the right angle lie between some of these, we judge +from the principle of continuity that there is some intermediate velocity with +which the body would arrive at and cross the $y$-axis at right angles. + +If the body move from one axis to the other, crossing both at right +\index{Variational curve, defined!determined}% +angles, it is plain that the orbit is a closed curve symmetrical to both axes. +Thus is obtained a particular solution of the differential equations. This +solution is the ``variational curve.'' While the general integrals involve four +arbitrary constants, the variational curve has but two, which may be taken to +be the distance from the origin at the $x$~crossing and the time of crossing. + +For the sake of brevity, we may measure time from the instant of +crossing~$x$. + +Then since $x$~is an even function of~$\tau$ and $y$~an odd one, both of +period~$2\pi$, it must be possible to expand $x$~and~$y$ by Fourier Series---thus +\begin{alignat*}{4} +x &= A_{0} \cos \tau &&+ A_{1} \cos 3\tau &&+ A_{2} \cos 5\tau &&+ \ldots\ldots, \\ +y &= B_{0} \sin \tau &&+ B_{1} \sin 3\tau &&+ B_{2} \sin 5\tau &&+ \ldots\ldots. +\end{alignat*} + +When $\tau$~is a multiple of~$\pi$, $y = 0$; and when it is an odd multiple +of~$\dfrac{\pi}{2}$, $x = 0$: also in the first case $\dfrac{dx}{d\tau} = 0$ and in the second $\dfrac{dy}{d\tau} = 0$. Thus +these conditions give us the kind of curve we want. It will be noted that +there are no terms with even multiples of~$\tau$; such terms have to be omitted +if $x, \dfrac{dx}{d\tau}$ are to vanish at $\tau = \pi/2$,~etc.\DPnote{** Slant fraction} + +We do not propose to follow Hill throughout the arduous analysis by +which he determines the nature of this curve with the highest degree of +accuracy, but will obtain only the first rough approximation to its form---thereby +merely illustrating the principles involved. + +Accordingly we shall neglect all terms higher than those in~$3\tau$. It is +also convenient to change the constants into another form. Thus we write +\begin{align*} +A_{0} &= a_{0} + a_{-1},\quad A_{1} = a_{1}, \\ +B_{0} &= a_{0} - a_{-1},\quad B_{1} = a_{1}. +\end{align*} +We have one constant less than before, but it will be seen that this is +sufficient, for in fact $A_{1}$~and~$B_{1}$ only differ by terms of an order which we +are going to neglect. We assume $a_{1}, a_{-1}$ to be small quantities. + +Hence +\begin{align*} +x &= (a_{0} + a_{-1}) \cos\tau + a_{1} \cos 3\tau, \\ +y &= (a_{0} - a_{-1}) \sin\tau + a_{1} \sin 3\tau. +\end{align*} +\DPPageSep{082}{24} + +Since +\begin{alignat*}{4} +\cos 3\tau &= && 4\cos^{3}\tau - 3\cos\tau &&= &&\cos\tau(1 - 4\sin^{2}\tau), \\ +\sin 3\tau &= -&& 4\sin^{3}\tau + 3\sin\tau &&= -&&\sin\tau(1 - 4\cos^{2}\tau), +\end{alignat*} +we have +\[ +\left. +\begin{aligned} +x = a_{0} \cos\tau &\left[1 + \frac{a_{1} + a_{-1}}{a_{0}} + - \frac{4a_{1}}{a_{0}} \sin^{2}\tau\right]\Add{,} \\ +y = a_{0} \sin\tau &\left[1 - \frac{a_{1} + a_{-1}}{a_{0}} + + \frac{4a_{1}}{a_{0}} \cos^{2}\tau\right]\Add{.} +\end{aligned} +\right\} +\] + +Neglecting powers of $a_{1}, a_{-1}$ higher than the first, we deduce +\begin{align*} +r^{2} &= a_{0}^{2} \left[1 + 2\frac{a_{1} + a_{-1}}{a_{0}} \cos2\tau\right], +\Allowbreak +\frac{1}{r^{3}} + &= \frac{1}{a_{0}^{3}} \left[1 - 3\frac{a_{1} + a_{-1}}{a_{0}} \cos2\tau\right] \\ + &= \frac{1}{a_{0}^{3}} \left[1 - 3\frac{a_{1} + a_{-1}}{a_{0}} + 6\frac{a_{1} + a_{-1}}{a_{0}} \sin^{2}\tau\right] \\ + &= \frac{1}{a_{0}^{3}} \left[1 + 3\frac{a_{1} + a_{-1}}{a_{0}} - 6\frac{a_{1} + a_{-1}}{a_{0}} \cos^{2}\tau\right]; +\Allowbreak +\frac{\kappa x}{r^{3}} + &= \frac{\kappa}{a_{0}^{2}} \cos\tau + \left[1 - \frac{2a_{1} + 2a_{-1}}{a_{0}} + + \frac{2a_{1} + 6a_{-1}}{a_{0}} \sin^{2}\tau\right], \\ +\frac{\kappa y}{r^{3}} + &= \frac{\kappa}{a_{0}^{2}} \sin\tau + \left[1 + \frac{2a_{1} + 2a_{-1}}{a_{0}} + - \frac{2a_{1} + 6a_{-1}}{a_{0}} \cos^{2}\tau\right], +\Allowbreak +%[** TN: Added breaks at second equalities] +\frac{d^{2} x}{d\tau^{2}} + &= -\left[\left(a_{0} + a_{-1}\right) \cos\tau + 9a_{1} \cos3\tau\right] \\ + &= -\cos\tau \left[a_{0} + 9a_{1} + a_{-1} - 36a_{1} \sin^{2}\tau\right], +\Allowbreak +\frac{d^{2} y}{d\tau^{2}} + &= -\left[\left(a_{0} - a_{-1}\right) \sin\tau + 9a_{1} \sin3\tau\right] \\ + &= -\sin\tau \left[a_{0} - 9a_{1} + a_{-1} - 36a_{1} \cos^{2}\tau\right]. +\end{align*} + +With the required accuracy +\[ +-2m \frac{dy}{d\tau} = -2m a_{0}\cos\tau,\ + 2m \frac{dx}{d\tau} = -2m a_{0} \sin\tau, \text{ and } + 3m^{2} x = 3m^{2} a_{0} \cos\tau. +\] + +Substituting these results in the differential equations,~\Eqref{(6)}, we get +\begin{multline*} +a_{0}\cos\tau + \biggl[-1 - \frac{9a_{1} + a_{-1}}{a_{0}} + \frac{36a_{1}}{a_{0}}\sin^{2}\tau - 2m \\ + + \frac{\kappa}{a_{0}^{3}} \left(1 - \frac{2a_{1} + 2a_{-1}}{a_{0}} + + \frac{2a_{1} + 6a_{-1}}{a_{0}} \sin^{2}\tau\right) - 3m^{2}\biggr] = 0, +\end{multline*} +\begin{multline*} +a_{0}\sin\tau + \biggl[-1 + \frac{9a_{1} + a_{-1}}{a_{0}} - \frac{36a_{1}}{a_{0}}\cos^{2}\tau - 2m \\ + + \frac{\kappa}{a_{0}^{3}} \left(1 + \frac{2a_{1} + 2a_{-1}}{a_{0}} + - \frac{2a_{1} + 6a_{-1}}{a_{0}} \cos^{2}\tau\right)\biggr] = 0. +\end{multline*} +\DPPageSep{083}{25} + +Equating to zero the coefficients of $\cos\tau$, $\cos\tau \sin^{2}\tau$, $\sin\tau$, $\sin\tau \cos^{2}\tau$, +we get +\[ +\left. +\begin{gathered} +\begin{alignedat}{2} +&-1 - \frac{9a_{1} + a_{-1}}{a_{0}} - 2m + + \frac{\kappa}{a_{0}^{3}} \left(1 - \frac{2a_{1} + 2a_{-1}}{a_{0}}\right) + & -3m^{2} &= 0\Add{,} \\ +&-1 + \frac{9a_{1} + a_{-1}}{a_{0}} - 2m + + \frac{\kappa}{a_{0}^{3}} \left(1 + \frac{2a_{1} + 2a_{-1}}{a_{0}}\right) + &&= 0\Add{,} +\end{alignedat} +\\ +% +\frac{36a_{1}}{a_{0}} + + \frac{\kappa}{a_{0}^{2}} \left(\frac{2a_{1} + 6a_{-1}}{a_{0}}\right) = 0\Add{.} +\end{gathered} +\right\} +\Tag{(7)} +\] + +As there are only three equations for the determination of $\dfrac{\kappa}{a_{0}^{3}}$, $\dfrac{a_{1}}{a_{0}}$, $\dfrac{a_{-1}}{a_{0}}$ +our assumption that $A_{1} = B_{1} = a_{1}$ is justified to the order of small quantities +considered. + +Half the sum and difference of the first two give +\begin{gather*} +-1 - 2m - \tfrac{3}{2} m^{2} + \frac{\kappa}{a_{0}^{3}} = 0, \\ +\frac{9a_{1} + a_{-1}}{a_{0}} + \frac{2\kappa}{a_{0}^{3}}\, \frac{a_{1} + a_{-1}}{a_{0}} + + \tfrac{3}{2} m^{2} = 0. +\end{gather*} + +Therefore +\begin{align*} +&\frac{\kappa}{a_{0}^{3}} = 1 + 2m + \tfrac{3}{2} m^{2}, \\ +&\frac{11a_{1}}{a_{0}} + \frac{3a_{-1}}{a_{0}} = -\tfrac{3}{2}m^{2}, + \text{ to our order of accuracy, viz.~$m^{2}$}; \\ +\intertext{also} +&\frac{19a_{1}}{a_{0}} + \frac{3a_{-1}}{a_{0}} = 0, + \text{ from the third equation;} +\end{align*} +\begin{gather*} +\therefore \frac{8a_{1}}{a_{0}} = \tfrac{3}{2} m^{2}, \\ +\left. +\begin{aligned} +\frac{a_{1}}{a_{0}} + &= \tfrac{3}{16} m^{2},\quad \frac{a_{-1}}{a_{0}} + = -\tfrac{19}{16} m^{2}\Add{,} \\ +\frac{\kappa}{a_{0}^{3}} + &= 1 + 2m + \tfrac{3}{2} m^{2}\Add{.} +\end{aligned} +\right\} +\Tag{(8)} +\end{gather*} + +Hence +\begin{align*} +x &= a_{0}\left[(1 - \tfrac{19}{16} m^{2}) \cos\tau + + \tfrac{3}{16} m^{2} \cos 3\tau\right], \\ +y &= a_{0}\left[(1 + \tfrac{19}{16} m^{2}) \sin\tau + + \tfrac{3}{16} m^{2} \sin 3\tau\right], +\end{align*} +or perhaps more conveniently for future work +\[ +\left. +\begin{aligned} +x &= a_{0}\cos\tau + \left[1 - m^{2} - \tfrac{3}{4} m^{2} \sin^{2}\tau \right]\Add{,} \\ +y &= a_{0}\sin\tau + \left[1 + m^{2} + \tfrac{3}{4} m^{2} \cos^{2}\tau \right]\Add{.} +\end{aligned} +\right\} +\Tag{(9)} +\] + +It will be seen that those are the equations to an oval curve, the semi-axes +of which are $a_{0}(1 - m^{2})$, $a_{0}(1 + m^{2})$ along and perpendicular to the line +joining the earth and sun. If $r, \theta$~be the polar coordinates of a point on the +curve, +\begin{align*} +r^{2} &= a_{0}^{2}[1 - 2m^{2} \cos 2\tau], \\ +\intertext{giving} +r &= a_{0}[1 - m^{2} \cos 2\tau]. +\Tag{(10)} +\end{align*} +\DPPageSep{084}{26} +Also +\begin{gather*} +\begin{aligned} +\tan\theta &= \frac{y}{x} = \tan\tau \bigl[1 + 2m^{2} + \tfrac{3}{4} m^{2}\bigr] \\ + &= \bigl(1 + \tfrac{11}{4}\bigr) \tan\tau. +\end{aligned} \\ +\therefore \tan(\theta - \tau) + = \frac{\tan\tau}{1 + \tan^{2}\tau} · \tfrac{11}{4} m^{2} + = \tfrac{11}{8} \sin 2\tau, +\end{gather*} +giving +\[ +\theta = \tau + \tfrac{11}{8} m^{2} \sin 2\tau. +\Tag{(11)} +\] + +If $\a$~be the mean distance corresponding to a mean motion~$n$ in an +undisturbed orbit, Kepler's third law gives +\[ +n^{2}\a^{3} = E + M = \kappa \nu^{2}. +\Tag{(12)} +\] + +But +\[ +\frac{n}{\nu} = \frac{n - n' + n'}{n - n'} = 1 + m. +\] +Hence +\begin{gather*} +(1 + m)^{2} \a^{3} = \kappa = a_{0}^{3} (1 + 2m + \tfrac{3}{2} m^{2}), \\ +\frac{a_{0}^{3}}{\a^{3}} = \frac{1 + 2m + m^{2}}{1 + 2m + m^{2} + \tfrac{1}{2} m^{2}}, \\ +\intertext{and} +a_{0} = \a(1 - \tfrac{1}{6} m^{2}). +\Tag{(13)} +\end{gather*} + +This is a relation between $a_{0}$ and the undisturbed mean distance. + + +\Section{§ 4. }{Differential Equations \texorpdfstring{\protect\\}{} +for Small Displacements from the Variational Curve.} +\index{Small displacements from variational curve}% +\index{Variational curve, defined!small displacements from}% + +If the solar perturbations were to vanish, $m$~would be zero and we should +have $x = a_{0}\cos\tau$, $y = a_{0}\sin\tau$ so that the orbit would be a circle. We may +therefore consider the orbit already found as a circular orbit distorted by solar +influence. [We have indeed put $\Omega = 0$, but the terms neglected are small +and need not be considered at present.] As the circular orbit is only a +special solution of the problem of two bodies, we should not expect the +variational curve to give the actual motion of the moon. In fact it is known +that the moon moves rather in an ellipse of eccentricity~$\frac{1}{20}$ than in a circle or +variational curve. The latter therefore will only serve as an approximation +to the real orbit in the same way as a circle serves as an approximation to an +ellipse. An ellipse of small eccentricity can be obtained by ``free oscillations'' +about a circle, and what we proceed to do is to determine free oscillations +about the variational curve. We thus introduce two new arbitrary constants---determining +the amplitude and phase of the oscillations---and so get the +general solution of our differential equations~\Eqref{(6)}. The procedure is exactly +similar to that used in dynamics for the discussion of small oscillations about +a steady state, i.e.,~the moon is initially supposed to lie near the variational +curve, and its subsequent motion is determined relatively to this curve. At +first only first powers of the small quantities will be used---an approximation +\DPPageSep{085}{27} +which corresponds to the first powers of the eccentricity in the elliptic theory. +If required, further approximations can be made. + +Suppose then that $x, y$ are the coordinates of a point on the variational +curve which we have found to satisfy the differential equations of motion and +that $x + \delta x$, $y + \delta y$ are the coordinates of the moon in her actual orbit, then +since $x, y$~satisfy the equations it is clear that the equations to be satisfied +by~$\delta x, \delta y$ are +\[ +\left. +\begin{alignedat}{2} +&\frac{d^{2}}{d\tau^{2}}\, \delta x - 2m \frac{d}{d\tau}\, \delta y + + \kappa \delta \left(\frac{x}{r^{3}}\right) &- 3m^{2}\, \delta x &= 0\Add{,} \\ +% +&\frac{d^{2}}{d\tau^{2}}\, \delta y + 2m \frac{d}{d\tau}\, \delta x + + \kappa \delta \left(\frac{y}{r^{3}}\right) &&= 0\Add{.} +\end{alignedat} +\right\} +\Tag{(14)} +\] + +\begin{wrapfigure}[14]{r}{1.75in} + \centering + \Input[1.75in]{p027} + \caption{Fig.~2.} +\end{wrapfigure} +Now it is not convenient to proceed immediately +from these equations as you may see by +considering how you would proceed if the orbit of +reference were a simple undisturbed circle. The +obvious course is to replace~$\delta x, \delta y$ by normal +and tangential displacements~$\delta p, \delta s$. + +Suppose then that $\phi$~denotes the inclination +of the outward normal of the variational curve to +the $x$-axis. Then we have +\[ +\left. +\begin{aligned} +\delta x &= \delta p \cos\phi - \delta s \sin\phi\Add{,} \\ +\delta y &= \delta p \sin\phi + \delta s \cos\phi\Add{.} +\end{aligned} +\right\} +\Tag{(15)} +\] + +Multiply the first differential equation~\Eqref{(14)} by~$\cos\phi$ and the second by~$\sin\phi$ +and add; and again multiply the first by~$\sin\phi$ and the second by~$\cos\phi$ +and subtract. We have +\[ +\left. +\begin{aligned} +\cos\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}} + &+ \sin\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} + - 2m \left[\cos\phi\, \frac{d\, \delta y}{d\tau} + - \sin\phi\, \frac{d\, \delta x}{d\tau}\right] \\ +% + &+ \kappa \cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa \sin\phi\, \delta\left(\frac{y}{r^{3}}\right) + - 3m^{2}\cos\phi\, \delta x = 0, \\ +% +-\sin\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}} + &+ \cos\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} + + 2m \left[\sin\phi\, \frac{d\, \delta y}{d\tau} + + \cos\phi\, \frac{d\, \delta x}{d\tau}\right] \\ +% + &- \kappa \sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + + 3m^{2}\sin\phi\, \delta x = 0. +\end{aligned} +\right\} +\Tag{(16)} +\] + +Now we have from~\Eqref{(15)} +\[ +\delta p = \delta x \cos\phi + \delta y \sin\phi,\quad +\delta s = -\delta x \sin\phi + \delta y \cos\phi. +\] + +Therefore +\begin{align*} +\frac{d\, \delta p}{d\tau} + &= \Neg\cos\phi\, \frac{d\, \delta x}{d\tau} + + \sin\phi\, \frac{d\, \delta y}{d\tau} + + (-\delta x \sin\phi + \delta y \cos\phi)\, \frac{d\phi}{d\tau}, \\ + % +\frac{d\, \delta s}{d\tau} + &= -\sin\phi\, \frac{d\, \delta x}{d\tau} + + \cos\phi\, \frac{d\, \delta y}{d\tau} + - (\Neg\delta x \cos\phi + \delta y \sin\phi)\, \frac{d\phi}{d\tau}. +\end{align*} +\DPPageSep{086}{28} + +Hence the two expressions which occur in the second group of terms of~\Eqref{(16)} +are +\begin{align*} +\cos\phi\, \frac{d\, \delta y}{d\tau} - \sin\phi\, \frac{d\, \delta x}{d\tau} + &= \frac{d\, \delta s}{d\tau} + \delta p\, \frac{d\phi}{d\tau}, \\ +% +\sin\phi\, \frac{d\, \delta y}{d\tau} + \cos\phi\, \frac{d\, \delta x}{d\tau} + &= \frac{d\, \delta p}{d\tau} - \delta s\, \frac{d\phi}{d\tau}. +\end{align*} + +When we differentiate these again, we obtain the first group of terms in~\Eqref{(16)}. +Inverting the order of the equations we have +\begin{align*} +\cos\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}} + &+ \sin\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} \\ + &= \frac{d^{2}\, \delta p}{d\tau^{2}} + - \frac{d\, \delta s}{d\tau}\, \frac{d\phi}{d\tau} + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} + - \left(\cos\phi\, \frac{d\, \delta y}{d\tau} + - \sin\phi\, \frac{d\, \delta x}{d\tau}\right) \frac{d\phi}{d\tau} \\ + &= \frac{d^{2}\, \delta p}{d\tau^{2}} + - 2\frac{d\, \delta s}{d\tau}\, \frac{d\phi}{d\tau} + - \delta p\, \left(\frac{d\phi}{d\tau}\right)^{2} + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}}, +\Allowbreak +-\sin\phi\, \frac{d^{2}\, \delta x}{d\tau^{2}} + &+ \cos\phi\, \frac{d^{2}\, \delta y}{d\tau^{2}} \\ + &= \frac{d^{2}\, \delta s}{d\tau^{2}} + + \frac{d\, \delta p}{d\tau}\, \frac{d\phi}{d\tau} + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} + + \left(\sin\phi\, \frac{d\, \delta y}{d\tau} + + \cos\phi\, \frac{d\, \delta x}{d\tau}\right) \frac{d\phi}{d\tau} \\ + &= \frac{d^{2}\, \delta s}{d\tau^{2}} + + 2\frac{d\, \delta p}{d\tau}\, \frac{d\phi}{d\tau} + - \delta s\, \left(\frac{d\phi}{d\tau}\right)^{2} + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}}. +\end{align*} + +Substituting in~\Eqref{(16)}, we have as our equations +\[ +\left. +\begin{aligned} +\frac{d^{2}\, \delta p}{d\tau^{2}} + &- \delta p \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m \frac{d\phi}{d\tau}\right] + - 2\frac{d\, \delta s}{d\tau} \left(\frac{d\phi}{d\tau} + m\right) + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} \\ +&\qquad + + \kappa\cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa\sin\phi\, \delta\left(\frac{y}{r^{3}}\right) + - 3m^{2}\cos\phi\, \delta x = 0\Add{,} \\ +% +\frac{d^{2}\, \delta s}{d\tau^{2}} + &- \delta s \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m \frac{d\phi}{d\tau}\right] + + 2\frac{d\, \delta p}{d\tau} \left(\frac{d\phi}{d\tau} + m\right) + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} \\ +&\qquad + - \kappa\sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa\cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + + 3m^{2}\sin\phi\, \delta x = 0\Add{.} +\end{aligned} +\right\} +\Tag{(17)} +\] + +Variation of the Jacobian integral +\[ +V^{2} + = \left(\frac{dx}{d\tau}\right)^{2} + + \left(\frac{dy}{d\tau}\right)^{2} + = \frac{2\kappa}{r} + 3m^{2}x^{2} + C +\] +gives +\[ +\frac{dx}{d\tau}\, \frac{d\, \delta x}{d\tau} + +\frac{dy}{d\tau}\, \frac{d\, \delta y}{d\tau} + = -\frac{\kappa}{r^{3}}\, \delta r + + 3m^{2}x\, \delta x.\footnotemark%[** TN: Moved mark after period] +\] +\footnotetext{We could introduce a term~$\delta C$, but the variation of the orbit which we are introducing + is one for which $C$~is unaltered.} + +Now +\[ +\frac{dx}{d\tau} = -V\sin\phi,\quad +\frac{dy}{d\tau} = V\cos\phi, +\] +\DPPageSep{087}{29} +and +\begin{alignat*}{4} + \frac{d\, \delta x}{d\tau} + &= \cos\phi\, \frac{d\, \delta p}{d\tau} + &&- \delta s \cos\phi\, \frac{d\phi}{d\tau} + &&- \sin\phi\, \frac{d\, \delta s}{d\tau} + &&- \sin\phi\, \delta p\, \frac{d\phi}{d\tau}, \\ +% + \frac{d\, \delta y}{d\tau} + &= \sin\phi\, \frac{d\, \delta p}{d\tau} + &&- \delta s \sin\phi\, \frac{d\phi}{d\tau} + &&+ \cos\phi \frac{d\, \delta s}{d\tau} + &&+ \cos\phi\, \delta p\, \frac{d\phi}{d\tau}. +\end{alignat*} + +Hence +\[ +\frac{dx}{d\tau}\, \frac{d\, \delta x}{d\tau} + +\frac{dy}{d\tau}\, \frac{d\, \delta y}{d\tau} + = V \left(\frac{d\, \delta s}{d\tau} + + \delta p\, \frac{d\phi}{d\tau}\right). +\] + +Also +\begin{align*} +-\frac{\kappa\, \delta r}{r^{2}} + &= -\frac{\kappa}{r^{3}}(x\, \delta x + y\, \delta y) \\ %[** TN: Added break] + &= -\frac{\kappa x}{r^{3}}(\delta p \cos\phi - \delta s \sin\phi) + -\frac{\kappa y}{r^{3}}(\delta p \sin\phi + \delta s \cos\phi) \\ + &= -\frac{\kappa}{r^{3}} \bigl[\delta p\, (x \cos\phi + y \sin\phi) + + \delta s\, (-x \sin\phi + y \cos\phi)\bigr]. +\end{align*} + +Thus, retaining the term $3m^{2} x\, \delta x$ in its original form, the varied Jacobian +integral becomes +\Pagelabel{29} +\begin{multline*} +V\left(\frac{d\, \delta s}{d\tau} + \delta p\, \frac{d\phi}{d\tau}\right) \\ + = -\frac{\kappa}{r^{3}} \bigl[\delta p\, (x \cos\phi + y \sin\phi) + + \delta s\, (-x \sin\phi + y \cos\phi)\bigr] + 3m^{2} x\, \delta x. +\Tag{(18)} +\end{multline*} + +Before we can solve the differential equations~\Eqref{(17)} for $\delta p, \delta s$ we require to +express all the other variables occurring in them, in terms of~$\tau$ by means of +the equations obtained in~\SecRef{3}. + + +\Section{§ 5. }{Transformation of the equations in \SecRef{4}.} + +We desire to transform the differential equations~\Eqref{(17)} so that the only +variables involved will be $\delta p, \delta s, \tau$. We shall then be in a position to solve +for $\delta p, \delta s$ in terms of~$\tau$. + +We have +\[ +r\, \delta r = x\, \delta x + y\, \delta y + = ( x \cos\phi + y \sin\phi)\, \delta p + + (-x \sin\phi + y \cos\phi)\, \delta s. +\] + +Hence +\begin{align*} +\cos\phi\, &\delta\left(\frac{x}{r^{3}}\right) + +\sin\phi\, \delta\left(\frac{y}{r^{3}}\right) \\ + &= \frac{1}{r^{3}} (\delta x \cos\phi + \delta y \sin\phi) + - \frac{3}{r^{5}} (x \cos\phi + y \sin\phi) r\, \delta r +\Allowbreak + &= \frac{\delta p}{r^{3}} - \frac{3}{r^{5}} + \biggl[(x^{2} \cos^{2} \phi + y^{2} \sin^{2} \phi + + 2xy \sin\phi \cos\phi)\, \delta p \\ + &\qquad \rlap{$\displaystyle + + (- x^{2} \sin\phi \cos\phi + + xy \cos^{2}\phi + - xy \sin^{2}\phi + + y^{2} \sin\phi \cos\phi)\, \delta s\biggr]$} +\Allowbreak + &= \frac{\delta p}{r^{3}} - \frac{3}{r^{5}} \biggl[ + \bigl\{\tfrac{1}{2}(x^{2} + y^{2}) + + \tfrac{1}{2}(x^{2} - y^{2}) \cos 2\phi + + xy \sin 2\phi\bigr\}\, \delta p \\ + &\qquad\qquad\qquad + + \bigl\{-\tfrac{1}{2}(x^{2} - y^{2}) \sin 2\phi + + xy \cos 2\phi\bigr\}\, \delta s \biggr] +\Allowbreak + &= \frac{\delta p}{r^{3}} \left[ + -\tfrac{1}{2} - \tfrac{3}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \cos 2\phi + - \frac{3xy}{r^{2}} \sin 2\phi + \right] \\ + &\qquad\qquad\qquad + - \frac{3\delta s}{r^{3}} \left[ + \frac{xy}{r^{2}} \cos 2\phi + - \tfrac{1}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \sin 2\phi + \right], +\Tag{(19)} +\Allowbreak +\DPPageSep{088}{30} +-\sin\phi\, &\delta\left(\frac{x}{r^{3}}\right) + + \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) \\ + &= \frac{1}{r^{3}} (-\delta x \sin\phi + \delta y \cos\phi) + - \frac{3}{r^{3}} (-x \sin\phi + y \cos\phi)r\, \delta r +\Allowbreak + &= \frac{\delta s}{r^{3}} - \frac{3}{r^{5}} \biggl[ + (-x^{2} \sin\phi \cos\phi + - xy \sin^{2}\phi + xy \cos^{2}\phi + + y^{2} \sin\phi \cos\phi)\, \rlap{$\delta p$} \\ + &\qquad\qquad\qquad + + (x^{2} \sin^{2}\phi + y^{2} \cos^{2}\phi + - 2xy \sin\phi \cos\phi)\, \delta s\biggr] +\Allowbreak + &= \frac{\delta s}{r^{3}} + - \frac{3}{r^{5}} \biggl[ + \bigl\{-\tfrac{1}{2}(x^{2} - y^{2}) \sin 2\phi + xy\cos 2\phi\bigr\}\, \delta p \\ + &\qquad\qquad\qquad + + \bigl\{\tfrac{1}{2}(x^{2} + y^{2}) - \tfrac{1}{2}(x^{2} - y^{2})\cos 2\phi + - xy\sin 2\phi\bigr\}\, \delta s \biggr] +\Allowbreak + &= -\frac{3\, \delta p}{r^{3}} \biggl[\frac{xy}{r^{2}}\cos 2\phi + - \tfrac{1}{2} \frac{x^{2} - y^{2}}{r^{2}} \sin 2\phi\biggr] \\ + &\qquad\qquad\qquad + + \frac{\delta s}{r^{3}} \biggl[ + -\tfrac{1}{2} + \tfrac{3}{2} \frac{x^{2} - y^{2}}{r^{2}} \cos 2\phi + + \frac{3xy}{r^{2}} \sin 2\phi \biggr]. +\Tag{(20)} +\end{align*} + +We shall consider the terms $3m^{2}\, \delta x \begin{array}{@{\,}c@{\,}}\cos\\ \sin\end{array} \phi$ later (\Pageref{33}). + +The next step is to substitute throughout the differential equations~\Eqref{(17)} +the values of~$x, y$ and~$\phi$ which correspond to the undisturbed orbit. For +simplicity in writing we drop the linear factor~$a_{0}$. It can be easily +introduced when required. + +We have already found, in~\Eqref{(9)}, +\begin{alignat*}{2} +x &= \cos\tau (1 - \tfrac{19}{16} m^{2}) + \tfrac{3}{16} m^{2}\cos 3\tau + &&= \cos\tau (1 - m^{2} - \tfrac{3}{4} m^{2}\sin^{2}\tau), \\ +x &= \sin\tau (1 + \tfrac{19}{16} m^{2}) + \tfrac{3}{16} m^{2}\sin 3\tau + &&= \sin\tau (1 + m^{2} + \tfrac{3}{4} m^{2}\cos^{2}\tau). +\end{alignat*} + +Then +\begin{align*} +\frac{dx}{d\tau} + &= -\sin\tau(1 - \tfrac{7}{4} m^{2} + \tfrac{9}{4} m^{2}\cos^{2} \tau) + = -\sin\tau(1 + \tfrac{1}{2} m^{2} - \tfrac{9}{4} m^{2}\sin^{2} \tau), \\ +% +\frac{dy}{d\tau} + &= \Neg\cos\tau(1 + \tfrac{7}{4} m^{2} - \tfrac{9}{4} m^{2}\sin^{2} \tau) + = \Neg\cos\tau(1 - \tfrac{1}{2} m^{2} + \tfrac{9}{4} m^{2}\cos^{2} \tau). +\end{align*} + +Whence +\begin{align*} +V^{2} + &= \left(\frac{dx}{d\tau}\right)^{2} + \left(\frac{dy}{d\tau}\right)^{2} \\ +%[** TN: Added break] + &= \sin^{2}\tau (1 + m^{2} - \tfrac{9}{2} m^{2}\sin^{2}\tau) + + \cos^{2}\tau (1 - m^{2} + \tfrac{9}{2} m^{2}\cos^{2}\tau) \\ +% + &= 1 - m^{2} \cos 2\tau + \tfrac{9}{2} m^{2}\cos 2\tau + = 1 + \tfrac{7}{2} m^{2}\cos 2\tau \\ +% + &= 1 + \tfrac{7}{2} m^{2} - 7 m^{2}\sin^{2}\tau + = 1 - \tfrac{7}{2} m^{2} + 7 m^{2}\cos^{2}\tau. +\end{align*} + +Therefore +\[ +\frac{1}{V} + = 1 + \tfrac{7}{4} m^{2} - \tfrac{7}{2} m^{2}\cos^{2}\tau + = 1 - \tfrac{7}{4} m^{2} + \tfrac{7}{2} m^{2}\sin^{2}\tau + = 1 - \tfrac{7}{4} m^{2} \cos 2\tau. +\] +\DPPageSep{089}{31} + +Now +\[ +\sin\phi = -\frac{1}{V}\, \frac{dx}{d\tau},\quad +\cos\phi = \frac{1}{V}\, \frac{dy}{d\tau}. +\] + +Therefore +\begin{align*} +\sin\phi + &= \sin\tau(1 + \tfrac{1}{2} m^{2} - \tfrac{9}{4} m^{2}\sin^{2}\tau + - \tfrac{7}{4} m^{2} + \tfrac{7}{2} m^{2}\sin^{2}\tau) \\ + &= \sin\tau(1 - \tfrac{5}{4} m^{2} + \tfrac{5}{4} m^{2}\sin^{2}\tau) + = \sin\tau(1 - \tfrac{5}{4} m^{2}\cos^{2}\tau), +\Allowbreak +\cos\phi + &= \cos\tau(1 - \tfrac{1}{2} m^{2} + \tfrac{9}{4} m^{2}\cos^{2}\tau + + \tfrac{7}{4} m^{2} - \tfrac{7}{2} m^{2}\cos^{2}\tau) \\ + &= \cos\tau(1 + \tfrac{5}{4} m^{2} - \tfrac{5}{4} m^{2}\cos^{2}\tau) + = \cos\tau(1 + \tfrac{5}{4} m^{2}\sin^{2}\tau); +\Allowbreak +\sin2\phi + &= \sin2\tau(1 - \tfrac{5}{4} m^{2}\cos2\tau), \\ +% +\cos2\phi + &= \cos2\tau + \tfrac{5}{4} m^{2}\sin^{2}2\tau); +\Allowbreak +\cos\phi\, \frac{d\phi}{d\tau} + &= \Neg\cos\tau(1 - \tfrac{5}{4} m^{2} + \tfrac{15}{4} m^{2} \sin^{2}\tau), \\ +% +\sin\phi\, \frac{d\phi}{d\tau} + &= -\sin\tau(1 + \tfrac{5}{4} m^{2} - \tfrac{15}{4} m^{2} \cos^{2}\tau). +\end{align*} + +Summing the squares of these, +\begin{align*} +\left(\frac{d\phi}{d\tau}\right)^{2} + &= \cos^{2}\tau(1 - \tfrac{5}{2} m^{2} + \tfrac{15}{2} m^{2} \sin^{2}\tau) + + \sin^{2}\tau(1 + \tfrac{5}{2} m^{2} - \tfrac{15}{2} m^{2} \cos^{2}\tau) \\ + &= 1 - \tfrac{5}{2} m^{2} \cos2\tau, +\end{align*} +and thence +\[ +\frac{d\phi}{d\tau} = 1 - \tfrac{5}{4} m^{2} \cos2\tau. +\Tag{(21)} +\] + +Differentiating again +\[ +\frac{d^{2}\phi}{d\tau^{2}} = \tfrac{5}{2} m^{2} \sin 2\tau. +\] + +We are now in a position to evaluate all the earlier terms in the +differential equations~\Eqref{(17)}. + +Thus +\[ +\left. +\begin{aligned}%[** TN: Re-broken] +\frac{d^{2}\, \delta p}{d\tau^{2}} + &- \delta p \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m\frac{d\phi}{d\tau}\right] + - 2\frac{d\, \delta s}{d\tau}\left(\frac{d\phi}{d\tau} + m\right) + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} \\ +% +&= \frac{d^{2}\, \delta p}{d^{2}} + \delta p \bigl[-1 + \tfrac{5}{2} m^{2}\cos 2\tau - 2m\bigr] \\ + &\qquad\qquad + - 2\frac{d\, \delta s}{d\tau}(1 + m - \tfrac{5}{4} m^{2}\cos2\tau) + - \tfrac{5}{2} m^{2}\sin2\tau\, \delta s\Add{,} \\ +% +\frac{d^{2}\, \delta s}{d\tau^{2}} + &- \delta s \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m\frac{d\phi}{d\tau}\right] + + 2\frac{d\, \delta p}{d\tau}\left(\frac{d\phi}{d\tau} + m\right) + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} \\ +% +&= \frac{d^{2}\, \delta s}{d^{2}} + \delta s \bigl[-1 + \tfrac{5}{2} m^{2}\cos 2\tau - 2m\bigr] \\ + &\qquad\qquad + + 2\frac{d\, \delta p}{d\tau}(1 + m - \tfrac{5}{4} m^{2}\cos2\tau) + + \tfrac{5}{2} m^{2}\sin2\tau\, \delta p\Add{.} +\end{aligned} +\right\} +\Tag{(22)} +\] +\DPPageSep{090}{32} + +We now have to evaluate the several terms involving $x$~and~$y$ in \Eqref{(18)},~\Eqref{(19)},~\Eqref{(20)}. + +\begin{align*} +x \cos\phi + y \sin\phi + &= \cos^{2}\tau (1 - m^{2} - \tfrac{3}{4} m^{2} \sin^{2}\tau + \tfrac{5}{4} m^{2} \sin^{2}\tau) \\ + &\,+ \sin^{2}\tau (1 + m^{2} + \tfrac{3}{4} m^{2} \cos^{2}\tau - \tfrac{5}{4} m^{2} \cos^{2}\tau) \\ + &= 1 - m^{2} \cos 2\tau, +\Allowbreak +% +-x \sin\phi + y \cos\phi + &= -\sin\tau \cos\tau (1 - m^{2} - \tfrac{3}{4} m^{2} \sin^{2}\tau - \tfrac{5}{4} m^{2} \cos^{2}\tau) \\ + &\quad+ \sin\tau \cos\tau (1 + m^{2} + \tfrac{3}{4} m^{2} \cos^{2}\tau + \tfrac{5}{4} m^{2} \sin^{2}\tau) \\ + &= 2m^{2} \sin 2\tau; +\Allowbreak +% +r^{2} = x^{2} + y^{2} &= 1 - 2m^{2} \cos 2\tau, +\Allowbreak +% +x^{2} - y^{2} &= \cos^{2}\tau(1 - 2m^{2} - \tfrac{3}{2} m^{2}\sin^{2}\tau) \\ + &\,- \sin^{2}\tau (1 + 2m^{2} + \tfrac{3}{2} m^{2}\cos^{2}\tau) \\ + &= \cos 2\tau - 2m^{2} - \tfrac{3}{4} m^{2}\sin^{2} 2\tau, +\Allowbreak +% +xy &= \tfrac{1}{2}\sin 2\tau(1 + \tfrac{3}{4} m^{2}\cos 2\tau); +\Allowbreak +% +(x^{2} - y^{2}) \cos 2\phi + &= \begin{aligned}[t] + \cos^{2}2\tau - 2m^{2} \cos 2\tau + &- \tfrac{3}{4} m^{2} \sin^{2}2\tau \cos 2\tau \\ + &+ \tfrac{5}{4} m^{2} \sin^{2}2\tau \cos 2\tau + \end{aligned} \\ + &= \cos 2\tau (\cos 2\tau - 2m^{2} + \tfrac{1}{2} m^{2} \sin^{2}2\tau), +\Allowbreak +% +(x^{2} - y^{2}) \sin 2\phi + &= \sin 2\tau (\cos 2\tau - 2m^{2} - \tfrac{3}{4} m^{2} \sin^{2}2\tau - \tfrac{5}{4} m^{2} \cos^{2}2\tau) \\ + &= \sin 2\tau (\cos 2\tau - \tfrac{11}{4} m^{2} - \tfrac{1}{2} m^{2} \cos^{2}2\tau); +\Allowbreak +% +xy \cos 2\phi + &= \tfrac{1}{2} \sin 2\tau (\cos 2\tau + \tfrac{5}{4} m^{2} \sin^{2}2\tau + \tfrac{3}{4} m^{2} \cos^{2}2\tau) \\ + &= \tfrac{1}{2} \sin 2\tau (\cos 2\tau + \tfrac{5}{4} m^{2} - \tfrac{1}{2} m^{2} \cos^{2}2\tau), \\ +% +xy \sin 2\phi + &= \tfrac{1}{2}\sin^{2}2\tau (1 - \tfrac{5}{4} m^{2}\cos 2\tau + \tfrac{3}{4} m^{2}\cos 2\tau) \\ + &= \tfrac{1}{2}\sin^{2}2\tau (1 - \tfrac{1}{2} m^{2}\cos 2\tau). +\end{align*} + +Therefore +\begin{gather*} +\begin{aligned} +&\tfrac{1}{2}(x^{2} - y^{2})\cos 2\phi + xy \sin 2\phi \\ +% + &= \tfrac{1}{2}\cos^{2}2\tau - m^{2}\cos 2\tau + \tfrac{1}{4} m^{2}\sin^{2}2\tau \cos 2\tau + + \tfrac{1}{2}\sin^{2}2\tau - \tfrac{1}{4} m^{2}\sin^{2}2\tau \cos 2\tau \\ +% + &= \tfrac{1}{2}(1 - 2m^{2}\cos 2\tau) = \tfrac{1}{2}r^{2}, +\end{aligned} \\ +% + \therefore + -\tfrac{1}{2} \mp \tfrac{3}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \cos 2\phi \mp \frac{3xy}{r^{2}}\sin 2\phi + = -\tfrac{1}{2} \mp \tfrac{3}{2} = -2 \text{ or } +1. +\end{gather*} + +These are the coefficients of~$\dfrac{\delta p}{r^{3}}$ in the expression~\Eqref{(19)} for +\[ +\cos\phi\, \delta \left(\frac{x}{r^{3}}\right) + +\sin\phi\, \delta \left(\frac{y}{r^{3}}\right), +\] +and of~$\dfrac{\delta s}{r^{3}}$ in the expression~\Eqref{(20)} for $-\sin\phi\, \delta\left(\dfrac{x}{r^{3}}\right) + \cos\phi\, \delta\left(\dfrac{y}{r^{3}}\right)$. +\DPPageSep{091}{33} + +Again +\begin{align*} +-\tfrac{1}{2}(x^{2} - y^{2}) \sin 2\phi &+ xy \cos 2\phi \\ + &= + \begin{alignedat}[t]{3} + -\tfrac{1}{2} \sin 2\tau + &(\cos 2\tau &&- \tfrac{11}{4} m^{2} &&- \tfrac{1}{2} m^{2} \cos^{2} 2\tau) \\ + +\tfrac{1}{2} \sin 2\tau + &(\cos 2\tau &&+ \tfrac{5}{4} m^{2} &&- \tfrac{1}{2} m^{2} \cos^{2} 2\tau) + \end{alignedat} \\ + &= 2m^{2} \sin 2\tau. +\end{align*} + +Then since to the order zero, $r^{3} = 1$, we have +\[ +3\left(\frac{xy}{r^{2}} \cos 2\phi - \tfrac{1}{2}\, \frac{x^{2} - y^{2}}{r^{2}} \sin 2\phi\right) + = 6m^{2} \sin 2\tau. +\] + +This is the coefficient of~$-\dfrac{\delta s}{r^{3}}$ in $\cos\phi\, \delta\left(\dfrac{x}{r^{3}}\right) + \sin\phi\, \delta\left(\dfrac{y}{r^{3}}\right)$ and of~$-\dfrac{\delta p}{r^{3}}$ in +$-\sin\phi\, \delta\left(\dfrac{x}{r^{3}}\right) + \cos\phi\, \delta\left(\dfrac{y}{r^{3}}\right)$. + +Hence we have +\[ +\left. +\begin{aligned} +\cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + +\sin\phi\, \delta\left(\frac{y}{r^{3}}\right) + &= -2\frac{\delta p}{r^{3}} - \frac{6m^{2}}{r^{3}}\, \delta s \sin 2\tau \\ + &= -2\delta p\, (1 + 3m^{2} \cos 2\tau) + - 6m^{2}\, \delta s \sin 2\tau\Add{,} \\ +% +-\sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + &= -\frac{\delta p}{r^{3}} · 6m^{2} \sin 2\tau + \frac{\delta s}{r^{3}} \\ + &= -6m^{2}\, \delta p \sin 2\tau + \delta s\, (1 + 3m^{2} \cos 2\tau)\Add{.} +\end{aligned} +\right\} +\Tag{(23)} +\] + +These two expressions are to be multiplied by~$\kappa$ in the differential +equations~\Eqref{(17)}. + +{\stretchyspace +The other terms which occur in the differential equations are $-3m^{2}\cos\phi\, \delta x$ +and~$+3m^{2}\sin\phi\, \delta x$.\Pagelabel{33}} + +Since $m^{2}$~occurs in the coefficient we need only go to the order zero of +small quantities in $\cos\phi\, \delta x$ and~$\sin\phi\, \delta x$. + +Thus +\begin{align*}%[** TN: Added two breaks] +3m^{2}\, \delta x \cos\phi + &= 3m^{2} (\delta p \cos\tau - \delta s \sin\tau) \cos\tau \\ + &= \tfrac{3}{2} m^{2}\, \delta p\, (1 + \cos 2\tau) + - \tfrac{3}{2} m^{2}\, \delta s \sin 2\tau, \\ +% +3m^{2}\, \delta x \sin\phi + &= 3m^{2} (\delta p \cos\tau - \delta s \sin\tau) \sin\tau \\ + &= \tfrac{3}{2} m^{2}\, \delta p \sin 2\tau + - \tfrac{3}{2} m^{2}\, \delta s\, (1 - \cos 2\tau). +\end{align*} + +Now $\kappa = 1 + 2m + \frac{3}{2} m^{2}$, and hence +\begin{align*} + \kappa \cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + &+ \kappa \sin\phi\, \delta\left(\frac{y}{r^{3}}\right) - 3m^{2}\, \delta x \cos\phi \\ + &= -2\delta p\, (1 + 3m^{2} \cos 2\tau + 2m + \tfrac{3}{2} m^{2}) + - 6m^{2}\, \delta s \sin 2\tau \\ + &\quad -\tfrac{3}{2} m^{2}\, \delta p\, (1 + \cos 2\tau) + \tfrac{3}{2} m^{2}\, \delta s \sin 2\tau \\ + &= -2\delta p\, [1 + 2m + \tfrac{9}{4} m^{2} + \tfrac{15}{4} m^{2} \cos 2\tau] + - \tfrac{9}{2} m^{2}\, \delta s \sin 2\tau, +\Allowbreak +\DPPageSep{092}{34} +-\kappa \sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + &+ \kappa \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + 3m^{2}\, \delta x \sin\phi \\ +% + &= -6m^{2}\, \delta p \sin 2\tau + + \delta s\, (1 + 2m + \tfrac{3}{2} m^{2} + 3m^{2} \cos2\tau) \\ + &\quad + \tfrac{3}{2} m^{2}\, \delta p \sin 2\tau + - \delta s\, (\tfrac{3}{2} m^{2} - \tfrac{3}{2} m^{2} \cos2\tau) \\ +% + &= -\tfrac{9}{2} m^{2}\, \delta p \sin2\tau + + \delta s\, (1 + 2m + \tfrac{9}{2} m^{2} \cos2\tau). +\end{align*} + +Hence +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + - \delta p \left[\left(\frac{d\phi}{d\tau}\right)^{2} + + 2m\left(\frac{d\phi}{d\tau}\right)\right] + - 2 \frac{d\, \delta s}{d\tau} \left(\frac{d\phi}{d\tau} + m\right) + - \delta s\, \frac{d^{2}\phi}{d\tau^{2}} \\ +% + + \kappa \cos\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa \sin\phi\, \delta\left(\frac{y}{r^{3}}\right) + - 3m^{2} \cos\phi\, \delta x = 0 +\end{multline*} +becomes +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + - \delta p\, [1 + 2m - \tfrac{5}{2} m^{2} \cos 2\tau] + - 2 \frac{d\, \delta s}{d\tau}(1 + m - \tfrac{5}{4} m^{2} \cos 2\tau) + - \tfrac{5}{2} m^{2}\, \delta s \sin 2\tau \\ +% + - 2\delta p\, [1 + 2m + \tfrac{9}{4} m^{2} + \tfrac{15}{4} m^{2} \cos 2\tau] + - \tfrac{9}{2} m^{2}\, \delta s \sin 2\tau = 0 +\end{multline*} +or +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + - \delta p\, [3 + 6m + \tfrac{9}{2} m^{2} + 5m^{2} \cos 2\tau] + - 2 \frac{d\, \delta s}{d\tau} (1 + m - \tfrac{5}{4} m^{2} \cos 2\tau) \\ +% + - 7m^{2}\, \delta s \sin 2\tau = 0. +\Tag{(24)} +\end{multline*} + +This is the first of our equations transformed. + +Again the second equation is +\begin{multline*} +\frac{d^{2}\, \delta s}{d\tau^{2}} + - \delta s \left[\left(\frac{d\phi}{d\tau}\right)^{2} + 2m \frac{d\phi}{d\tau}\right] + + 2 \frac{d\, \delta p}{d\tau} \left(\frac{d\phi}{d\tau} + m \right) + + \delta p\, \frac{d^{2}\phi}{d\tau^{2}} \\ +% + - \kappa \sin\phi\, \delta\left(\frac{x}{r^{3}}\right) + + \kappa \cos\phi\, \delta\left(\frac{y}{r^{3}}\right) + + 3m^{2} \sin\phi\, \delta x = 0, +\end{multline*} +and it becomes +\begin{multline*} +\frac{d^{2}\, \delta s}{d\tau^{2}} + + \delta s\, (-1 - 2m + \tfrac{5}{2} m^{2} \cos 2\tau) + + 2 \frac{d\, \delta p}{d\tau}(1 + m - \tfrac{5}{4} m^{2} \cos 2\tau) + + \tfrac{5}{2} m^{2}\, \delta p \sin 2\tau \\ +% + - \tfrac{9}{2} m^{2}\, \delta p \sin 2\tau + + \delta s\, (1 + 2m + \tfrac{9}{2} m^{2} \cos 2\tau) = 0. +\end{multline*} + +Whence +\[ +\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos 2\tau + + 2 \frac{d\, \delta p}{d\tau} (1 + m -\tfrac{5}{4} m^{2} \cos 2\tau) + - 2m^{2}\, \delta p \sin 2\tau = 0. +\Tag{(25)} +\] + +This is the second of our equations transformed. + +The Jacobian integral gives +\begin{align*}%[** TN: Rebroken] +\frac{d\, \delta s}{d\tau} &+ \delta p\, \frac{d\phi}{d\tau} \\ + &= \frac{3m^{2} x\, \delta x}{V} + - \frac{\kappa}{V r^{3}} \bigl[\delta p\, (x \cos\phi + y \sin\phi) + \delta s\, (-x \sin\phi + y \cos\phi)\bigr] +\Allowbreak + &= 3m^{2} \cos\tau (\delta p \cos\tau - \delta s \sin\tau) \\ + &\qquad\qquad + - (1 + 2m + \tfrac{3}{2} m^{2} - \tfrac{7}{4} m^{2} \cos2\tau + + 3m^{2} \cos2\tau) \\ + &\qquad\qquad\qquad\qquad\Add{·} + \bigl[\delta p\, (1 - m^{2} \cos2\tau) + 2m^{2}\, \delta s \sin2\tau\bigr] +\Allowbreak +\DPPageSep{093}{35} + &= \frac{3m^{2}}{2}\, \delta p\, (1 + \cos 2\tau) + - \frac{3m^{2}}{2}\, \delta s \sin 2\tau \\ + &\qquad -\delta p\, (1 + 2m + \tfrac{3}{2} m^{2} + + \tfrac{5}{4} m^{2} \cos2\tau - m^{2} \cos2\tau) - 2m^{2}\, \delta s \sin2\tau +\Allowbreak + &= -\delta p\, (1 + 2m - \tfrac{5}{4} m^{2} \cos2\tau) + - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau. +\end{align*} + +Substituting for~$\dfrac{d\phi}{d\tau}$ its value from~\Eqref{(21)} +\begin{align*} +\frac{d\, \delta s}{d\tau} + &= -\delta p\, (1 + 2m - \tfrac{5}{4} m^{2} \cos2\tau) + - \delta p\, (1 - \tfrac{5}{4} m^{2} \cos2\tau) + - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau \\ +% + &= -\delta p\, (2 + 2m - \tfrac{5}{2} m^{2} \cos2\tau) + - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau \\ +% +\frac{2d\, \delta s}{d\tau} + &= -4\delta p\, (1 + m - \tfrac{5}{4} m^{2} \cos2\tau) + - 7m^{2}\, \delta s \sin2\tau \\ +\frac{2d\, \delta s}{d\tau} &(1 + m - \tfrac{5}{4} m^{2} \cos2\tau) + = -4\delta p\, (1 + 2m + m^{2} - \tfrac{5}{2} m^{2} \cos2\tau) + - 7m^{2}\, \delta s \sin2\tau \\ +% +\frac{2d\, \delta s}{d\tau} &(1 + m - \tfrac{5}{4} m^{2} \cos2\tau) + + 7m^{2}\, \delta s \sin2\tau + = -4\delta p\, (1 + 2m + m^{2} - \tfrac{5}{2} m^{2} \cos2\tau). +\Tag{(26)} +\end{align*} + +This expression occurs in~\Eqref{(24)}, and therefore can be used to eliminate +$\dfrac{d\, \delta s}{d\tau}$ from it. + +Substituting we get +\begin{gather*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + + \delta p\, \bigl[-3 - 6m - \tfrac{9}{2} m^{2} - 5m^{2} \cos2\tau + + 4 + 8m + 4m^{2} - 10 m^{2} \cos2\tau\bigr] = 0, +\Allowbreak +\left. +\begin{gathered} +\lintertext{i.e.} +{\frac{d^{2}\, \delta p}{d\tau^{2}} + \delta p\, \bigl[1 + 2m - \tfrac{1}{2} m^{2} - 15m^{2} \cos 2\tau\bigr] = 0.} \\ +\lintertext{And}{\frac{d\, \delta s}{d\tau} + = -2\delta p\, (1 + m - \tfrac{5}{4} m^{2} \cos2\tau) + - \tfrac{7}{2} m^{2}\, \delta s \sin2\tau.} +\end{gathered} +\right\} +\Tag{(27)} +\end{gather*} + +If we differentiate the second of these equations, which it is to be +remembered was derived from Jacobi's integral and therefore involves our +second differential equation, we get +\Pagelabel{35} +\begin{align*}%[** TN: Rebroken] +\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos2\tau + &+ \tfrac{7}{2} m^{2} \sin 2\tau\, \frac{d\, \delta s}{d\tau} \\ + &+ 2 (1 + m - \tfrac{5}{4} m^{2} \cos 2\tau)\, \frac{d\, \delta p}{d\tau} + + 5 m^{2}\, \delta p \sin 2\tau = 0, +\end{align*} +and eliminating~$\dfrac{d\, \delta s}{d\tau}$ +\begin{align*} +\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos2\tau + &- 7m^{2}\, \delta p \sin 2\tau \\ + &+ 2 (1 + m - \tfrac{5}{4} m^{2} \cos2\tau)\, \frac{d\, \delta p}{d\tau} + + 5m^{2}\, \delta p \sin 2\tau = 0, +\end{align*} +\DPPageSep{094}{36} +or +\[ +\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos 2\tau + + 2(1 + m - \tfrac{5}{4} m^{2} \cos 2\tau)\, \frac{d\, \delta p}{d\tau} + - 2m^{2}\, \delta p \sin 2\tau = 0, +\] +and this is as might be expected our second differential equation which was +found above. Hence we only require to consider the equations~\Eqref{(27)}. + +\Section{§ 6. }{Integration of an important type of Differential Equation.} +\index{Differential Equation, Hill's}% +\index{Hill, G. W., Lunar Theory!Special Differential Equation}% + +The differential equation for~$\delta p$ belongs to a type of great importance +in mathematical physics. We may write the typical equation in the form +\[ +\frac{d^{2}x}{dt^{2}} + + (\Theta_{0} + 2\Theta_{1} \cos 2t + 2\Theta_{2} \cos 4t + \dots) x = 0, +\] +where $\Theta_{0}, \Theta_{1}, \Theta_{2}, \dots$ are constants depending on increasing powers of a small +quantity~$m$. It is required to find a solution such that $x$~remains small for +all values of~$t$. + +Let us attempt the apparently obvious process of solution by successive +approximations. + +Neglecting $\Theta_{1}, \Theta_{2}, \dots$, we get as a first approximation +\[ +x = A \cos(t \sqrt{\Theta_{0}} + \epsilon). +\] + +Using this value for~$x$ in the term multiplied by~$\Theta_{1}$, and neglecting $\Theta_{2}, +\Theta_{3}, \dots$, we get +\[ +\frac{d^{2}x}{dt^{2}} + + \Theta_{0} x + A\Theta_{1} \left\{ + \cos\bigl[t(\sqrt{\Theta_{0}} + 2) + \epsilon\bigr] + + \cos\bigl[t(\sqrt{\Theta_{0}} - 2) + \epsilon\bigr]\right\} = 0. +\] + +Solving this by the usual rules we get the second approximation +\begin{align*}%[** TN: Rebroken] +x = A\biggl\{\cos\left[t\sqrt{\Theta_{0}} + \epsilon\right] + &+ \frac{\Theta_{1} \cos\left[t(\sqrt{\Theta_{0}} + 2) + \epsilon\right]} + {4(\sqrt{\Theta_{0}} + 1)} \\ + &- \frac{\Theta_{1} \cos\left[t(\sqrt{\Theta_{0}} - 2) + \epsilon\right]} + {4(\sqrt{\Theta_{0}} - 1)} + \biggr\}. +\end{align*} + +Again using this we have the differential equation +\[ +\begin{split} +\frac{d^{2}x}{dt^{2}} + &+ \Theta_{0} x + A\Theta_{1}\left\{ + \cos\bigl[t(\sqrt{\Theta_{0}} + 2) + \epsilon\bigr] + + \cos\bigl[t(\sqrt{\Theta_{0}} - 2) - \epsilon\bigr] + \right\} \\ +% + &+ \frac{A\Theta_{1}^{2}}{4(\sqrt{\Theta_{0}} + 1)} \left\{ + \cos\bigl[t(\sqrt{\Theta_{0}} + 4) + \epsilon\bigr] + + \cos(t\sqrt{\Theta_{0}} + \epsilon) + \right\} \\ +% + &- \frac{A\Theta_{1}^{2}}{4(\sqrt{\Theta_{0}} - 1)} \left\{ + \cos(t\sqrt{\Theta_{0}} + \epsilon) + + \cos\bigl[t(\sqrt{\Theta_{0}} - 4) + \epsilon\bigr] + \right\} \\ +% + &+ A\Theta_{2} \left\{ + \cos\bigl[t(\sqrt{\Theta_{0}} + 4) + \epsilon\bigr] + + \cos\bigl[t(\sqrt{\Theta_{0}} - 4) + \epsilon\bigr] + \right\} = 0. +\end{split} +\] + +Now this equation involves terms of the form~$B \cos(t\sqrt{\Theta_{0}} + \epsilon)$; on +integration terms of the form~$Ct\sin(t\sqrt{\Theta_{0}} + \epsilon)$ will arise. But these terms +are not periodic and do not remain small when $t$~increases. $x$~will therefore +not remain small and the argument will fail. The assumption on which these +approximations have been made is that the period of the principal term of~$x$ +can be determined from $\Theta_{0}$~alone and is independent of~$\Theta_{1}, \Theta_{2}, \dots$. But the +\DPPageSep{095}{37} +appearance of secular terms leads us to revise this assumption and to take as +a first approximation +\[ +x = A \cos (ct \sqrt{\Theta_{0}} + \epsilon), +\] +where $c$~is nearly equal to~$1$ and will be determined, if possible, to prevent +secular terms arising. + +It will, however, be more convenient to write as a first approximation +\[ +x = A \cos (ct + \epsilon), +\] +where $c$~is nearly equal to~$\Surd{\Theta_{0}}$. + +Using this value of~$x$ in the term involving~$\Theta_{1}$, our equation becomes +\[ +\frac{d^{2}x}{dt^{2}} + + \Theta_{0} x + A\Theta_{1}\left\{ + \cos\bigl[(c + 2)t + \epsilon\bigr] + \cos\bigl[(c - 2)t + \epsilon\bigr] + \right\} = 0, +\] +and the second approximation is +\begin{align*} +x = A \cos (ct + \epsilon) + &+ \frac{A\Theta_{1}}{(c + 2)^{2} - \Theta_{0}} \cos\bigl[(c + 2)t + \epsilon\bigr] \\ + &+ \frac{A\Theta_{1}}{(c - 2)^{2} - \Theta_{0}} \cos\bigl[(c - 2)t + \epsilon\bigr].\footnotemark +\end{align*} +\footnotetext{This is not a solution of the previous equation, unless we actually put $c=\sqrt{\Theta_{0}}$ in the + first term.}% + +Proceeding to another approximation with this value of~$x$, we get +\[ +\begin{split} +\frac{d^{2}x}{dt^{2}} + &+ \Theta_{0}x + A\Theta_{1}\left\{ + \cos\bigl[(c + 2)t + \epsilon\bigr] + \cos\bigl[(c - 2)t + \epsilon\bigr] + \right\} \\ +% + &+ \frac{A\Theta_{1}^{2}}{(c + 2)^{2} - \Theta_{0}} \left\{ + \cos\bigl[(c + 4)t + \epsilon\bigr] + \cos(ct + \epsilon)\right\} \\ +% + &+ \frac{A\Theta_{1}^{2}}{(c - 2)^{2} - \Theta_{0}} \left\{ + \cos(ct + \epsilon) + \cos\bigl[(c - 4)t + \epsilon\bigr]\right\} \\ +% + &+ A\Theta_{2}\left\{ + \cos\bigl[(c + 4)t + \epsilon\bigr] + \cos\bigl[(c - 4)t + \epsilon\bigr] + \right\} =0. +\end{split} +\] + +We might now proceed to further approximations but just as a term in +$\cos (ct + \epsilon)$ generates in the solution terms in +\[ +\cos\bigl[(c ± 2)t + \epsilon\bigr]\quad \text{and}\quad +\cos\bigl[(c ± 4)t + \epsilon\bigr], +\] +terms in +\[ +\cos\bigl[(c ± 2)t + \epsilon\bigr]\quad \text{and}\quad +\cos\bigl[(c ± 4)t + \epsilon\bigr] +\] +will generate new terms in~$\cos(ct + \epsilon)$, i.e.~terms of exactly the same nature +as the term initially assumed. Hence to get our result it will be best to +begin by assuming a series containing all the terms which will arise. + +Various writers have found it convenient to introduce exponential instead +of trigonometric functions. Following their example we shall therefore write +the differential equation in the form +\[ +\frac{d^{2}x}{dt^{2}} + + x\sum_{-\infty}^{+\infty} \Theta_{i} e^{2it\Surd{-1}} = 0, +\Tag{(28)} +\] +where +\[ +\Theta_{-i} = \Theta_{i}, +\] +\DPPageSep{096}{38} +and the solution is assumed to be +\[ +x = \sum_{-\infty}^{+\infty} A_{j} e^{(c + 2j)t \sqrt{-1}}, +\] +where the ratios of all the coefficients~$A_{j}$, and~$c$, are to be determined by +equating coefficients of different powers of~$e^{t\sqrt{-1}}$. + +Substituting this expression for~$x$ in the differential equation, we get +\[ +-\sum_{-\infty}^{+\infty} (c + 2j)^{2} A_{j} e^{(c + 2j)t\sqrt{-1}} + + \sum_{-\infty}^{+\infty} A_{j} e^{(c + 2j)t \sqrt{-1}} + \sum_{-\infty}^{+\infty} \Theta_{i} e^{2i t\sqrt{-1}} = 0, +\] +and equating to zero the coefficient of~$e^{(c + 2j)t \sqrt{-1}}$, +\begin{multline*} +-(c + 2j)^{2}A_{j} + A_{j}\Theta_{0} + + A_{j-1}\Theta_{1} + A_{j-2}\Theta_{2} + A_{j-3}\Theta_{3} + \dots \\ + + A_{j+1}\Theta_{-1} + A_{j+2}\Theta_{-2} + A_{j+3}\Theta_{-3} + \dots = 0. +\end{multline*} + +Hence the succession of equations is +\index{Hill, G. W., Lunar Theory!infinite determinant}% +\index{Infinite determinant, Hill's}% +\iffalse +\begin{align*} +\dots &+ \bigl[\Theta_{0} - (c-4)^2\bigr]A_{-2} + \Theta_{-1}A_{-1} + \Theta_{-2}A_{0} + \Theta_{-3}A_{1} + \Theta_{-4}A_{2} + \dots = 0, \\ +\dots &+ \Theta_{1}A_{-2} + \bigl[\Theta_{0} - (c-2)^2\bigr]A_{-1} + \Theta_{-1}A_{0} + \Theta_{-2}A_{1} + \Theta_{-3}A_{2} + \dots = 0, \\ +\dots &+ \Theta_{2}A_{-2} + \Theta_{1}A_{-1} + (\Theta_{0} - c^2)A_{0} + \Theta_{-1}A_{1} + \Theta_{-2}A_{2} + \dots = 0, \\ +\dots &+ \Theta_3A_{-2} + \Theta_{2}A_{-1} + \Theta_{1}A_{0} + \bigl[\Theta_{0} - (c+2)^2\bigr]A_{1} + \Theta_{-1}A_{2} + \dots = 0, \\ +\dots &+ \Theta_4A_{-2} + \Theta_3A_{-1} + \Theta_{2}A_{0} + \Theta_{1}A_{1} + \bigl[\Theta_{0} - (c+4)^2\bigr]A_{2} + \dots = 0. +\end{align*} +\fi +{\small +\[ +\begin{array}{@{\,}*{17}{c@{\,}}} +\hdotsfor{17} \\ +\dots &+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c-4)^2\bigr]A_{-2}} &+& \Theta_{-1}A_{-1} &+& \Theta_{-2}A_{0} &+& \Theta_{-3}A_{1} &+& \Theta_{-4}A_{2} &+& \dots &=& 0, \\ +\dots &+& \Theta_{1}A_{-2}&+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c-2)^2\bigr]A_{-1}} &+& \Theta_{-1}A_{0} &+& \Theta_{-2}A_{1} &+& \Theta_{-3}A_{2} &+& \dots &=& 0, \\ +\dots &+& \Theta_{2}A_{-2}&+& \Theta_{1}A_{-1} &+& \multicolumn{3}{c}{(\Theta_{0} - c^2)A_{0}} &+& \Theta_{-1}A_{1} &+& \Theta_{-2}A_{2} &+& \dots &=& 0, \\ +\dots &+& \Theta_3A_{-2} &+& \Theta_{2}A_{-1} &+& \Theta_{1}A_{0} &+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c+2)^2\bigr]A_{1}} &+& \Theta_{-1}A_{2} &+& \dots &=& 0, \\ +\dots &+& \Theta_4A_{-2} &+& \Theta_3A_{-1} &+& \Theta_{2}A_{0} &+& \Theta_{1}A_{1} &+& \multicolumn{3}{c}{\bigl[\Theta_{0} - (c+4)^2\bigr]A_{2}} &+& \dots &=& 0. \\ +\hdotsfor{17} +\end{array} +\]} + +We clearly have an infinite determinantal equation for~$c$. + +If we take only three columns and rows, we get +\begin{multline*} +\bigl[\Theta_{0} - (c - 2)^{2}\bigr] \bigl[\Theta_{0} - c^{2}\bigr] \bigl[\Theta_{0} - (c + 2)^{2}\bigr] + - \Theta_{1}^{2} \bigl[\Theta_{0} - (c - 2)^{2}\bigr] - \Theta_{1}^{2} \bigl[\Theta_{0} - (c + 2)^{2}\bigr] \\ +% + - \Theta_{2}^{2}(\Theta_{0} - c^{2}) + 2\Theta_{1}^{2} \Theta_{2} = 0, \\ +% +\bigl[(\Theta_{0} - c^{2} - 4)^{2} - 16c^{2}\bigr] \bigl[\Theta_{0} - c^{2}\bigr] + - 2\Theta_{1}^{2}(\Theta_{0} - c^{2} - 4) + - \Theta_{2}^{2}(\Theta_{0} - c^{2}) + 2\Theta_{1}^{2} \Theta_{2} = 0. +\end{multline*} + +If we neglect $(\Theta_{0} - c^{2})^{3}$ which is certainly small +\begin{multline*} +\bigl[-8(\Theta_{0} - c^{2}) + 16 + 16(\Theta_{0} - c^{2}) - 16\Theta_{0}\bigr] \bigl[\Theta_{0} - c^{2}\bigr] \\ +% + \shoveright{ -(\Theta_{0} - c^{2}) \bigl[2\Theta_{1}^{2} + \Theta_{2}^{2}\bigr] + 8\Theta_{1}^{2} + 2\Theta_{1}^{2} \Theta_{2} = 0,} \\ +% + \shoveright{8(\Theta_{0} - c^{2})^{2} + (\Theta_{0} - c^{2})(16 - 16\Theta_{0} - 2\Theta_{1}^{2} - \Theta_{2}^{2}) + 8\Theta_{1}^{2} + 2\Theta_{1}^{2} \Theta_{2} = 0,} \\ +% +(\Theta_{0} - c^{2})^2 + 2(\Theta_{0} - c^{2})(1 - \Theta_{0} - \tfrac{1}{8}\Theta_{1}^{2} - \tfrac{1}{16}\Theta_{2}^{2}) + \Theta_{1}^{2} + \tfrac{1}{4}\Theta_{1}^{2} \Theta_{2} = 0. +\end{multline*} + +Since $\Theta_{1}^{2}, \Theta_{2}^{2}$ are small compared with~$1 - \Theta_{0}$, and $\Theta_{2}$~compared with~$1$, we +have as a rougher approximation +\[ +(c^{2} - \Theta_{0})^{2} + 2(\Theta_{0} - 1) (c^{2} - \Theta_{0}) = -\Theta_{1}^{2}, +\] +\DPPageSep{097}{39} +whence +\begin{gather*} +c^{2} - \Theta_{0} + = -(\Theta_{0} - 1) ± \sqrt{(\Theta_{0} - 1)^{2} - \Theta_{1}^{2}}, \\ +% +c^{2} = 1 ± \sqrt{(\Theta_{0} - 1)^{2} - \Theta_{1}^{2}}. +\end{gather*} + +Now $c^{2} = \Theta_{0}$ when $\Theta_{1} = 0$. Hence we take the positive sign and get +\[ +c = \sqrt{1 + \sqrt{(\Theta_{0} - 1)^{2} - \Theta_{1}^{2}}}, +\Tag{(29)} +\] +which is wonderfully nearly correct. + +For further discussion of the equation for~$c$, see Notes~1,~2, pp.~\Pgref{note:1},~\Pgref{note:2}. %[** TN: pp 53, 55 in original] + +\Section{§ 7. }{Integration of the Equation for $\delta p$.} + +We now return to the Lunar Theory and consider the solution of our +differential equation. Assume it to be +\[ +\delta p = A_{-1}\cos\bigl[(c - 2)\tau + \epsilon\bigr] + + A_{0}\cos(c\tau + \epsilon) + + A_{1}\cos\bigl[(c + 2)\tau + \epsilon\bigr]. +\] + +On substitution in~\Eqref{(27)} we get +\begin{align*} + A_{-1} \bigl[(1 + 2m - \tfrac{1}{2} m^{2} - 15 m^{2}\cos2\tau) &- (c - 2)^{2}\bigr]\cos\bigl[(c - 2)\tau + \epsilon\bigr] \\ +% ++ A_{0} \bigl[(1 + 2m - \tfrac{1}{2} m^{2} - 15 m^{2}\cos2\tau) &- c^{2}\bigr]\cos(c\tau + \epsilon) \\ +% ++ A_{1} \bigl[(1 + 2m - \tfrac{1}{2} m^{2} - 15 m^{2}\cos2\tau) &- (c + 2)^{2}\bigr]\cos\big[(c + 2)\tau + \epsilon\bigr] = 0. +\end{align*} + +Then we equate to zero the coefficients of the several cosines. + +1st~$\cos(c\tau + \epsilon)$ gives +\[ +-\tfrac{15}{2} m^{2}A_{-1} + + A_{0}(1 + 2m - \tfrac{1}{2} m^{2} - c^{2}) + - \tfrac{15}{2} m^{2}A_{1} = 0. +\] + +2nd~$\cos \bigl[(c - 2)\tau + \epsilon\bigr]$ gives +\[ +A_{-1} \bigl[1 + 2m - \tfrac{1}{2} m^{2} - (c - 2)^{2}\bigr] + - \tfrac{15}{2} m^{2}A_{0} = 0. +\] + +3rd~$\cos \bigl[(c + 2)\DPtypo{t}{\tau}\bigr] + \epsilon]$ gives +\[ +-\tfrac{15}{2} m^{2}A_{0} + A_{1}\bigl[1 + 2m - \tfrac{1}{2} m^{2} - (c + 2)^{2}] = 0. +\] + +If we neglect terms in~$m^{2}$ the first equation gives us $c^{2} = 1 + 2m$, and +\Pagelabel{39} +therefore $c = 1 + m$, $c - 2 = -(1 - m)$, $c + 2 = 3 + m$. + +The second and third equations then reduce to +\[ +4m A_{-1} = 0;\quad A_{1}(-8 - 4m) = 0. +\] + +From this it follows that $A_{-1}$~is at least of order~$m$ and $A_{1}$~at least of +order~$m^{2}$. + +Then since we are neglecting higher powers than~$m^{2}$, the first equation +reduces to +\[ +A_{0}(1 + 2m - \tfrac{1}{2} m^{2} - c^{2}) = 0, +\] +so that +\[ +c^{2} = 1 + 2m - \tfrac{1}{2} m^{2}\quad \text{or}\quad +c = 1 + m - \tfrac{3}{4} m^{2}. +\] + +Thus +\[ +(c - 2)^{2} = (1 - m + \tfrac{3}{4} m^{2})^{2} + = 1 - 2m + \tfrac{5}{2} m^{2}, +\] +and +\[ +1 + 2m - \tfrac{1}{2} m^{2} - (c - 2)^2 + = 4m - 3m^{2}. +\] +\DPPageSep{098}{40} + +Hence the second equation becomes +\[ +A_{-1}(4m - 3m^{2}) = \tfrac{15}{2} m^{2}A_{0}; +\] +and since $A_{-1}$~is of order~$m$, the term~$-3m^{2}A_{-1}$ is of order~$m^{3}$ and therefore +negligible. Hence +\[ +4m A_{-1} = \tfrac{15}{2} m^{2} A_{0} \quad \text{or}\quad +A_{-1} = \tfrac{15}{8} m A_{0}, +\] +and we cannot obtain $A_{-1}$~to an order higher than the first. + +The third equation is +\[ +-\tfrac{15}{2} m^{2} A_{0} + A_{1}[1 - 9] = 0, +\] +or +\[ +A_{1} = -\tfrac{15}{16} m^{2} A_{0}. +\] + +We have seen that $A_{-1}$~can only be obtained to the first order; so it is +useless to retain terms of a higher order in~$A_{1}$. Hence our solution is +\[ +A_{-1} = \tfrac{15}{8} m A_{0},\quad +A_{1} = 0. +\] + +Hence +\[ +\delta p = A_0 \left\{\cos(c\tau + \epsilon) + \tfrac{15}{8} m \cos\bigl[(c - 2)\tau + \epsilon\bigr]\right\}. +\Tag{(30)} +\] + +In order that the solution may agree with the more ordinary notation we +write $A_{0} = -a_{0}e$, and obtain +\[ +\left. +\begin{gathered} +\delta p = -a_{0}e \cos(c\tau + \epsilon) - \tfrac{15}{8} m a_{0} e \cos\bigl[(c - 2)\tau + \epsilon\bigr]\Add{,} \\ +\lintertext{where} +{c = 1 + m - \tfrac{3}{4} m^{2}\Add{.}} +\end{gathered} +\right\} +\Tag{(31)} +\] + +To the first order of small quantities the equation~\Eqref{(27)} for~$\delta s$ was +\begin{align*} +\frac{d\, \delta s}{d\tau} + &= -2(1 + m)\, \delta p \\ + &= 2(1 + m)a_{0}e \cos(c\tau + \epsilon) + + \tfrac{15}{4} m a_{0} e \cos\bigl[(c - 2)\tau + \epsilon\bigr]. +\end{align*} + +If we integrate and note that $c = 1 + m$ so that $c - 2 = -(1 - m)$, we have +\Pagelabel{40} +\[ +\delta s = 2a_{0} e \sin(c\tau + \epsilon) + - \tfrac{15}{4} m a_{0} e \sin\bigl[(c - 2)\tau + \epsilon\bigr]. +\Tag{(32)} +\] + +We take the constant of integration zero because $e = 0$ will then correspond +to no displacement along the variational curve. + +In order to understand the physical meaning of the results let us consider +the solution when~$m = 0$, i.e.~when the solar perturbation vanishes. + +Then +\[ +\delta p = -a_{0} e \cos (c\tau + \epsilon),\quad +\delta s = 2a_{0} e \sin (c\tau + \epsilon). +\] + +In the undisturbed orbit +\[ +x = a_{0} \cos\tau,\quad +y = a_{0} \sin\tau \quad \text{so that}\quad +\phi = \tau, +\] +and +\begin{gather*} +\begin{aligned} +\delta x &= \delta p \cos\phi - \delta s \sin\phi, \\ +\delta y &= \delta p \sin\phi + \delta s \cos\phi; +\end{aligned} \\ +\begin{aligned} +\delta x &= -a_{0} e \cos(c\tau + \epsilon)\cos\tau - 2a_{0} e \sin(c\tau + \epsilon)\sin\tau, \\ +\delta y &= -a_{0} e \cos(c\tau + \epsilon)\sin\tau + 2a_{0} e \sin(c\tau + \epsilon)\cos\tau. +\end{aligned} +\end{gather*} +\DPPageSep{099}{41} + +Therefore writing $X = x + \delta x$, $Y = y + \delta y$, $X = R \cos\Theta$, $Y = R \sin\Theta$, +\begin{alignat*}{3} +X &= a_{0}\bigl[\cos\tau &&- e \cos(c\tau + \epsilon)\cos\tau + &&- 2e \sin(c\tau + \epsilon)\sin\tau\bigr], \\ +% +Y &= a_{0}\bigl[\sin\tau &&- e \cos(c\tau + \epsilon)\sin\tau + &&+ 2e \sin(c\tau + \epsilon)\cos\tau\bigr]. +\end{alignat*} + +Therefore +\[ +R^{2} = a_{0}^{2} \bigl[1 - 2e \cos(c\tau + \epsilon)\bigr] +\] +or +\[ +R = a_{0} \bigl[1 - e \cos(c\tau + \epsilon)\bigr] + = \frac{a_{0}}{1 + e \cos(c\tau + \epsilon)}. +\Tag{(33)} +\] + +Again +\begin{alignat*}{2} +\cos\Theta &= \cos\tau &&- 2e \sin (c\tau + \epsilon)\sin\tau, \\ +\sin\Theta &= \sin\tau &&+ 2e \sin (c\tau + \epsilon)\cos\tau. +\end{alignat*} + +Hence +\[ +\sin(\Theta - \tau) = 2e \sin(c\tau + \epsilon), +\] +giving +\[ +\Theta = \tau + 2e \sin(c\tau + \epsilon). +\Tag{(34)} +\] + +It will be noted that the equations for $R, \Theta$ are of the same form as the +first approximation to the radius vector and true longitude in undisturbed +elliptic motion. When we neglect the solar perturbation by putting $m = 0$ +we see that $e$~is to be identified with the eccentricity and $c\tau + \epsilon$~with the +mean anomaly. + +\footnotemark~We can interpret~$c$ in terms of the symbols of the ordinary lunar theories. +%[** TN: Minor rewording coded using \DPtypo] +\footnotetext{\DPtypo{From here till the foot of this page} + {In the next three paragraphs} a slight knowledge of ordinary lunar theory is + supposed. The results given are not required for the further development of Hill's theory.}% +When no perturbations are considered the moon moves in an ellipse. The +\index{Apse, motion of}% +perturbations cause the moon to deviate from this simple path. If a fixed +ellipse is taken, these deviations increase with the time. It is found, +however, that if we consider the ellipse to be fixed in shape and size but with +the line of apses moving with uniform angular velocity, the actual motion of +the moon differs from this modified elliptic motion only by small periodic +quantities. If $n$~denote as before the mean sidereal motion of the moon and +$\dfrac{d\varpi}{dt}$~the mean motion of the line of apses, the argument entering into the +elliptic inequalities is~$\left(n - \dfrac{d\varpi}{dt}\right)t + \epsilon$. This must be the same as~$c\tau + \epsilon$, i.e.~as +$c(n - n')t + \epsilon$. + +Hence +\[ +n -\frac{d\varpi}{dt} = c(n - n'), +\] +giving +\begin{align*} +\frac{d\varpi}{n\, dt} + &= 1 - c \frac{n - n'}{n} \\ + &= 1 - \frac{c}{1 + m}\quad \text{since} \quad +m = \frac{n'}{ n - n'}. +\end{align*} + +A determination of~$c$ is therefore equivalent to a determination of the rate +of change of perigee; the value of~$c$ we have already obtained gives +\index{Perigee, motion of}% +\[ +\frac{d\varpi}{n\, dt} = \tfrac{3}{4} m^{2}. +\] +\DPPageSep{100}{42} + +Returning to our solution, and for simplicity again dropping the factor~$a_{0}$, +we have from \Eqref{(31)},~\Eqref{(32)} +\begin{align*} +\delta p &= -\tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr] - e \cos(c\tau + \epsilon), \\ +% +\delta s &= -\tfrac{15}{4} m e \sin\bigl[(c - 2)\tau + \epsilon\bigr] + 2e \sin(c\tau + \epsilon). +\end{align*} + +Also $\cos\phi = \cos\tau$, $\sin\phi = \sin\tau$ to the first order of small quantities, and +\[ +\delta x = \delta p \cos\phi - \delta s \sin\phi,\quad +\delta y = \delta p \sin\phi + \delta s \cos\phi. +\] +Therefore +\begin{multline*} +\delta x + = -\tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr]\cos\tau + - e \cos(c\tau + \epsilon) \cos\tau \\ +% + + \tfrac{15}{4} m e \sin\bigl[(c - 2)\tau + \epsilon\bigr]\sin\tau + - 2e \sin(c\tau + \epsilon) \sin\tau, +\end{multline*} +\begin{multline*} +\delta y + = -\tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr] \sin\tau + - e \cos(c\tau + \epsilon) \sin\tau \\ +% + - \tfrac{15}{4} m e \sin\bigl[(c - 2)\tau + \epsilon\bigr] \cos\tau + + 2e \sin(c\tau + \epsilon) \cos\tau. +\end{multline*} + +Now let $X = x + \delta x$, $Y = y + \delta y$ and we have by means of the values of $x, +y$ in the variational curve +\begin{align*} +X &= \cos\tau \bigl[1 - m^{2} + - \tfrac{3}{4} m^{2} \sin^{2}\tau + - \tfrac{15}{8} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - e \cos(c\tau + \epsilon)\bigr] \\ + &\qquad\qquad\qquad\qquad + + \sin\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr], \\ +% +Y &= \sin\tau \bigl[1 + m^{2} + + \tfrac{3}{4} m^{2} \cos^{2}\tau + - \tfrac{15}{8} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - e \cos(c\tau + \epsilon)\bigr] \\ + &\qquad\qquad\qquad\qquad + - \cos\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr]. +\end{align*} + +Writing $R^{2} = X^{2} + Y^{2}$, we obtain to the requisite degree of approximation +\begin{align*} +R^{2} &= \cos^{2}\tau \bigl[1 - 2m^{2} + - \tfrac{3}{2} m^{2} \sin^{2}\tau + - \tfrac{15}{4} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \cos(c\tau + \epsilon)\bigr] \\ +% + &+ \sin^{2}\tau \bigl[1 + 2m^{2} + + \tfrac{3}{2} m^{2} \cos^{2}\tau + - \tfrac{15}{4} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \cos(c\tau + \epsilon)\bigr] \\ +% + &+ \sin 2\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr] \\ +% + &- \sin 2\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr], \\ +% +R^{2} &= 1 - 2m^{2} \cos 2\tau + - \tfrac{15}{4} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \cos(c\tau + \epsilon). +\end{align*} + +Hence reintroducing the factor~$a_{0}$ which was omitted for the sake of brevity +\[ +R = a_{0}\bigl[1 - e \cos(c\tau + \epsilon) + - \tfrac{15}{8} m e \cos\bigl\{(c - 2)\tau + \epsilon\bigr\} + - m^{2} \cos 2\tau\bigr]. +\Tag{(35)} +\] + +This gives the radius vector; it remains to find the longitude. + +We multiply the expressions for $X, Y$ by~$1/R$,\DPnote{** Slant fraction} i.e.~by +\[ +1 + e \cos(c\tau + \epsilon) + + \tfrac{15}{8} m e \cos\bigl[(c - 2)\tau + \epsilon\bigr] + + m^{2} \cos 2\tau, +\] +and remembering that +\[ +m^{2} \cos 2\tau + = m^{2} - 2m^{2} \sin^{2}\tau + = 2m^{2} \cos^{2}\tau - m^{2}, +\] +we get +\begin{align*} +\cos\Theta + &= \cos\tau \bigl[1 - \tfrac{11}{4} m^{2} \sin^{2}\tau\bigr] + - \sin\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr], \\ +% +\sin\Theta + &= \sin\tau \bigl[1 + \tfrac{11}{4} m^{2} \cos^{2}\tau\bigr] + - \cos\tau \bigl[\tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + - 2e \sin(c\tau + \epsilon)\bigr]. +\end{align*} + +Whence +\[ +\sin(\Theta - \tau) + = \tfrac{11}{8} m^{2} \sin 2\tau + - \tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + + 2e \sin(c\tau + \epsilon), +\] +\DPPageSep{101}{43} +or to our degree of approximation +\[ +\Theta = \tau + \tfrac{11}{8} m^{2} \sin 2\tau + - \tfrac{15}{4} m e \sin\bigl\{(c - 2)\tau + \epsilon\bigr\} + + 2e \sin(c\tau + \epsilon). +\Tag{(36)} +\] + +We now transform these results into the ordinary notation. +\index{Equation, annual!of the centre}% +\index{Latitude of the moon}% + +\footnotemark~Let $l, v$ be the moon's mean and true longitudes, and $l'$~the sun's mean +\footnotetext{From here till the end of this paragraph is not a part of Hill's theory, it is merely a + comparison with ordinary lunar theories.}% +longitude. Then $\Theta$~being the moon's true longitude relatively to the moving +axes, we have +\[ +v = \Theta + l'. +\] + +Also +\begin{gather*} +\tau + l' = (n - n')t + n't =l, \\ +\therefore \tau = l - l'. +\end{gather*} + +We have seen that $c\tau + \epsilon$ is the moon's mean anomaly, or~$l - \varpi$, +\[ +\therefore (c - 2)\tau + \epsilon = l - \varpi - 2(l - l') = -(l + \varpi - 2l'). +\] + +Then substituting these values in the expressions for $R$~and~$\Theta$ and +adding~$l'$ to the latter we have on noting that $a_{0} = \a(1 - \frac{1}{6} m^{2})$ +\index{Evection}% +\[ +\left. +\begin{aligned} +R &= \a\bigl[1 - \tfrac{1}{6} m^{2} + - \UnderNote{e \cos(l - \varpi)}{equation of centre} + - \UnderNote{\tfrac{15}{8} m e \cos(l - 2l' + \varpi)}{evection} + - \UnderNote{m^{2} \cos 2(l - l')\bigr]}{variation}\Add{,} \\ +% +v &= l + \UnderNote{2e \sin (l - \varpi)}{equation of centre} + + \UnderNote{\tfrac{15}{4} m e \sin(l - 2l' + \varpi)}{evection} + + \UnderNote{\tfrac{11}{8} m^{2} \sin 2(l - l')}{variation}\Add{.} +\end{aligned} +\right\} +\Tag{(37)} +\] + +The names of the inequalities in radius vector and longitude are written +below, and the values of course agree with those found in ordinary lunar +theories. + +\Section{§ 8. }{Introduction of the Third Coordinate.} +\index{Third coordinate introduced}% +\index{Variation, the}% + +Still keeping $\Omega=0$, consider the differential equation for~$z$ in~\Eqref{(5)} +\[ +\frac{d^{2}z}{d\tau^{2}} + \frac{\kappa z}{r^{3}} + m^{2}z = 0. +\] + +From~\Eqref{(8)} +\[ +\frac{\kappa}{a_{0}^{3}} = 1 + 2m + \tfrac{3}{2} m^{2}, +\] +and from~\Eqref{(10)} +\[ +\frac{a_{0}^{3}}{r^{3}} = 1 + 3m^{2} \cos 2\tau. +\] + +The equation may therefore be written +\[ +\frac{d^{2}z}{d\tau^{2}} + z(1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau) = 0. +\] + +This is an equation of the type considered in~\SecRef{6} and therefore we +assume +\[ +z = B_{-1} \cos\bigl\{(g - 2)\tau + \zeta\bigr\} + + B_{0} \cos(g\tau + \zeta) + + B_{1} \cos\bigl\{(g + 2)\tau + \zeta\bigr\}. +\] +\DPPageSep{102}{44} + +On substitution we get +\begin{align*} +B_{-1} &\bigl[-(g - 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau \bigr] \cos\bigl[(g - 2)\tau + \zeta \bigr] \\ +% ++ B_{0} &\bigl[-g^{2} + 1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau \bigr] \cos(g\tau + \zeta) \\ +% ++ B_{1} &\bigl[-(g + 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2} + 3m^{2} \cos 2\tau \bigr] \cos \bigl[(g + 2)\tau + \zeta \bigr] = 0. +\end{align*} + +The coefficients of $\cos(g\tau + \zeta)$, $\cos \bigl[(g - 2)\tau + \zeta\bigr]$, $\cos \bigl[(g + 2)\tau + \zeta\bigr]$ give +respectively +\[ +\left. +\begin{alignedat}{2} +&\tfrac{3}{2} m^{2} B_{-1} + B_{0} \bigl[-g^{2} + 1 + 2m + \tfrac{5}{2} m^{2}\bigr] + \tfrac{3}{2} m^{2} B_{1} &&= 0\Add{,} \\ +% +&B_{-1} \bigl[-(g - 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2} ] + \tfrac{3}{2} m^{2} B_{0} &&= 0\Add{,} \\ +% +&\tfrac{3}{2} m^{2} B_{0} + B_{1} \bigl[-(g + 2)^{2} + 1 + 2m + \tfrac{5}{2} m^{2}\bigr] &&= 0\Add{.} +\end{alignedat} +\right\} +\Tag{(38)} +\] + +As a first approximation drop the terms in~$m^{2}$. The first of these equations +then gives $g^{2} = 1 + 2m$. The third equation then shews that $\dfrac{B_{1}}{B_{0}}$~is of +order~$m^{2}$. But a factor~$m$ can be removed from the second equation shewing +that $\dfrac{B_{-1}}{B_{0}}$~is of order~$m$ and can only be determined to this order. Hence +$B_{1}$~can be dropped. [Cf.~pp.~\Pgref{39},~\Pgref{40}.] + +Considering terms in~$m^{2}$ we now get from the first equation +\[ +g^{2} = 1 + 2m + \tfrac{5}{2} m^{2}. +\] + +Therefore +\begin{gather*} +g = 1 + m + \tfrac{5}{4} m^{2} - \tfrac{1}{2} m^{2} + = 1 + m + \tfrac{3}{4} m^{2}, \\ +(g - 2)^{2} = (1 - m)^{2} = 1 - 2m, \text{ neglecting terms in~$m^{2}$}. +\end{gather*} + +The second equation then gives +\[ +B_{-1} = -\tfrac{3}{8} m B_{0}, +\] +and the solution is +\[ +z = B_{0} \bigl[\cos(g\tau + \zeta) - \tfrac{3}{8} m \cos\bigl\{(g - 2)\tau + \zeta\bigr\}\bigr]. +\Tag{(39)} +\] + +We shall now interpret this equation geometrically. To do so we neglect +the solar perturbation and we get +\[ +z = B_{0} \cos(g\tau + \zeta). +\Tag{(40)} +\] + +\begin{wrapfigure}{r}{1.5in} + \centering + \Input[1.5in]{p044} + \caption{Fig.~3.} +\end{wrapfigure} +Now consider the moon to move in a plane orbit inclined at angle~$i$ to +the ecliptic and let $\Omega$~be the longitude of the lunar +node, $l$~the longitude of the moon, $\beta$~the latitude. + +The right-angled spherical triangle gives +\[ +\tan\beta = \tan i \sin(l - \Omega) +\] +and therefore +\[ +z = r \tan\beta = r \tan i \sin (l - \Omega). +\] +\DPPageSep{103}{45} + +As we are only dealing with a first approximation we may put $r = a_{0}$ and +so we interpret +\begin{gather*} +B_{0} = a_{0} \tan i, \\ +g\tau + \zeta = l - \Omega -\tfrac{1}{2}\pi. +\end{gather*} + +\footnotemark~We can easily find the significance of~$g$, for differentiating this equation +\footnotetext{From here till end of paragraph is a comparison with ordinary lunar theories.}% +with respect to the time we get +\begin{gather*} +g(n - n') = n - \frac{d\Omega}{dt}, \\ +\begin{aligned} +\therefore \frac{d\Omega}{n\, dt} + &= 1 - \frac{g(n - n')}{n} \\ + &= 1 + \frac{g}{1 + m} \\ + &= -\tfrac{3}{4} m^{2} \text{ to our approximation.} +\end{aligned} +\end{gather*} +Thus we find that the node has a retrograde motion. + +We have +\begin{align*} +g\tau + \zeta + &= l - \Omega - \tfrac{1}{2}\pi, \\ +% +(g - 2)\tau + \zeta + &= l - \Omega - \tfrac{1}{2}\pi - 2(l - l') \\ +% + &= -(l - 2l' + \Omega) - \tfrac{1}{2}\pi. +\end{align*} + +If we write $s = \tan\beta$, $k = \tan i$, we find +\[ +s = k \sin(l - \Omega) + \tfrac{3}{8} m k \sin(l - 2l' + \Omega). +\Tag{(41)} +\] + +The last term in this equation is called the evection in latitude. +\index{Evection!in latitude}% + +\Section{§ 9. }{Results obtained.} + +We shall now shortly consider the progress we have made towards the +actual solution of the moon's motion. We obtained first of all a special +solution of the differential equations assuming the motion to be in the ecliptic +and neglecting certain terms in the force function denoted by~$\Omega$\footnotemark. This gave +\footnotetext{The $\Omega$~of \Pageref{20}, not that of the preceding paragraph.}% +us a disturbed circular orbit in the plane of the ecliptic. We have since +introduced the first approximation to two free oscillations about this motion, +the one corresponding to eccentricity of the orbit, the other to an inclination +of the orbit to the ecliptic. + +It is found to be convenient to refer the motion of the moon to the projection +on the ecliptic. We will denote by~$r_{1}$ the curtate radius vector, so +that $r_{1}^{2} = x^{2} + y^{2}$, $r^{2} = r_{1}^{2} + z^{2}$; the $x, y$~axes rotating as before with angular +velocity~$n'$ in the plane of the ecliptic. In determining the variational curve,~\SecRef{3}, +we put $\Omega = 0$, $r = r_{1}$. It will appear therefore that in finding the actual +motion of the moon we shall require to consider not only~$\Omega$ but new terms in~$z^{2}$. +In the next section we shall discuss the actual motion of the moon, making +use of the approximations we have already obtained. +\DPPageSep{104}{46} + +\Section{§ 10. }{General Equations of Motion and their solution.} +\index{Equations of motion}% + +We have +\[ +r_{1}^{2} = x^{2} + y^{2} \text{ and } +r^{2} = r_{1}^{2} + z^{2}. +\] + +Hence +\[ +\frac{1}{r^{3}} + = \frac{1}{r_{1}^{3}} \left(1 - \frac{3}{2}\, \frac{z^{2}}{r_{1}^{2}}\right); + \text{ and } +\frac{1}{r} + = \frac{1}{r_{1}} \left(1 - \frac{1}{2}\, \frac{z^{2}}{r_{1}^{2}}\right), +\] +to our order of accuracy. + +The original equations~\Eqref{(3)} may now be written +\[ +\left. +\begin{alignedat}{4} +\frac{d^{2}x}{d\tau^{2}} + &- 2m\, \frac{dy}{d\tau} &&+ \frac{\kappa x}{r_{1}^{3}} &&- 3m^{2}x + &&= \frac{\dd \Omega}{\dd x} + \frac{3}{2}\, \frac{\kappa z^{2}x}{r_{1}^{5}}\Add{,} \\ +% +\frac{d^{2}y}{d\tau^{2}} + &+ 2m\, \frac{dx}{d\tau} &&+ \frac{\kappa y}{r_{1}^{3}} && + &&= \frac{\dd \Omega}{\dd y} + \frac{3}{2}\, \frac{\kappa z^{2}y}{r_{1}^{5}}\Add{,} \\ +% +\frac{d^{2}z}{d\tau^{2}} + & &&+ \frac{\kappa z}{r_{1}^{3}} &&+ m^{2}z + &&= \frac{\dd \Omega}{\dd z} + \frac{3}{2}\, \frac{\kappa z^{2}}{r_{1}^{5}}\Add{.} +\end{alignedat} +\right\} +\Tag{(42)} +\] + +If we multiply by $2\dfrac{dx}{d\tau}$, $2\dfrac{dy}{d\tau}$, $2\dfrac{dz}{d\tau}$ and add, we find that the Jacobian +integral becomes +\[ +V^{2} = 2\frac{\kappa}{r_{1}} + m^{2}(3x^{2} - z^{2}) + - \frac{\kappa z^{2}}{r_{1}^{3}} + + 2\int_{0}^{\tau} \left( + \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dz}{d\tau} + \right) d\tau + C, +\Tag{(43)} +\] +where +\[ +V^{2} = V_{1}^{2} + \left(\frac{dz}{d\tau}\right)^{2} + = \left(\frac{dx}{d\tau}\right)^{2} + + \left(\frac{dy}{d\tau}\right)^{2} + + \left(\frac{dz}{d\tau}\right)^{2}. +\] + +Now +\[ +\Omega = \tfrac{3}{2} m^{2} \left(\frac{\a'^{3}}{r'^{3}}\, r^{2}\cos^{2} - x^{2}\right) + + \tfrac{1}{2} m^{2} r^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right), +\] +and +\[ +\cos\theta = \frac{xx' + yy' + zz'}{rr'} + = \frac{xx' + yy'}{rr'}, \text{ since $z' = 0$}. +\] + +Hence +\[ +\Omega = \tfrac{3}{2} m^{2} \left\{\frac{\a'^{3}}{r'^{3}}(xx' + yy')^{2} - x^{2}\right\} + + \tfrac{1}{2} m^{2} (x^{2} + y^{2}) \left(1 - \frac{\a'^{3}}{r'^{3}}\right) + + \tfrac{1}{2} m^{2} z^{2} \left(1 - \frac{\a'^{3}}{r'^{3}}\right). +\] + +When we neglected $\Omega$~and~$z$, we found the solution +\begin{alignat*}{2} +x &= a_{0}\bigl[(1 - \tfrac{19}{16} m^{2})\cos\tau + &&+ \tfrac{3}{16} m^{2}\cos 3\tau\bigr], \\ +y &= a_{0}\bigl[(1 + \tfrac{19}{16} m^{2})\sin\tau + &&+ \tfrac{3}{16} m^{2}\sin 3\tau\bigr]. +\end{alignat*} + +We now require to determine the effect of the terms introduced on the +right, and for brevity we write +\[ +X = \frac{\dd \Omega}{\dd x} + \frac{3}{2}\, \frac{\kappa z^{2}x}{r_{1}^{5}},\quad +Y = \frac{\dd \Omega}{\dd y} + \frac{3}{2}\, \frac{\kappa z^{2}y}{r_{1}^{5}}. +\] + +When we refer to~\SecRef{4} and consider how the differential equations for~$\delta p, \delta s$ +were formed from those for~$\delta x, \delta y$, we see that the new terms~$X, Y$ on +the right-hand sides of the differential equations for~$\delta x, \delta y$ will lead to new +terms $X\cos\phi - Y\sin\phi$, $-X\sin\phi + Y\cos\phi$ on the right-hand sides of those +for~$\delta p, \delta s$. +\DPPageSep{105}{47} + +Hence taking the equations \Eqref{(24)}~and~\Eqref{(25)} for $\delta p$~and~$\delta s$ and introducing +these new terms, we find +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + + \delta p\, \bigl[-3 - 6m - \tfrac{9}{2}m^{2} - 5m^{2}\cos 2\tau\bigr] + - 2\frac{d\, \delta s}{d\tau}\, (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) \\ +\shoveright{-7m^{2}\, \delta s \sin 2\tau = X\cos\phi + Y\sin\phi,} \\ +% +\shoveleft{\frac{d^{2}\, \delta s}{d\tau^{2}} + + 7m^{2}\, \delta s \cos 2\tau + + 2\frac{d\, \delta p}{d\tau}\, (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) + - 2m^{2}\, \delta p \sin 2\tau} \\ + = -X\sin\phi + Y\cos\phi. +\end{multline*} + +In this analysis we shall include all terms to the order~$m k^{2}$, where $k$~is the +small quantity in the expression for~$z$. Terms involving~$m^{2}z^{2}$ will therefore +be neglected. In the variation of the Jacobian integral the term~$\dfrac{dz}{d\tau}\, \dfrac{d\, \delta z}{d\tau}$ can +obviously be neglected. The variation of the Jacobian integral therefore +gives (cf.~pp.~\Pgref{29},~\Pgref{35}) +\begin{multline*} +\frac{d\, \delta s}{d\tau} + = -2\delta p\, (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) + - \tfrac{7}{2}m^{2}\, \delta s \sin 2\tau \\ +% + + \frac{1}{V_{1}} \biggl[\int_{0}^{\tau}\!\! + \left(\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau}\right) d\tau + + \tfrac{1}{2} \biggl\{\delta C - \frac{\kappa z^{2}}{r_{1}^{3}} + - \left(\frac{dz}{d\tau}\right)^{2}\biggr\} + \biggr], +\Tag{(44)} +\end{multline*} +where $\delta C$~will be chosen as is found most convenient. [In the previous work +we chose $\delta C = 0$.] + +By means of this equation we can eliminate~$\delta s$ from the differential +equation for~$\delta p$. For +\begin{align*} +2\frac{d\, \delta s}{d\tau}\, (1 &+ m - \tfrac{5}{4}m^{2}\cos 2\tau) + 7m^{2}\, \delta s \sin 2\tau \\ +% + &= -4\delta p\, (1 + 2m + m^{2} - \tfrac{5}{2}m^{2} \cos 2\tau) \\ +% + &+ \frac{2}{V_{1}} (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) + \biggl[\int_{0}^{\tau}\left( + \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau} \right) d\tau \\ +% + &+ \tfrac{1}{2} \left\{\delta C - \frac{\kappa z^{2}}{r_{1}^{3}} - \left(\frac{dz}{d\tau}\right)^{2}\right\} + \biggr], +\end{align*} +and therefore +\begin{align*} +\frac{d^{2}\delta p}{d\tau^{2}} + &+ \delta p\, (1 + 2m - \tfrac{1}{2}m^{2} - 15m^{2}\cos 2\tau) + = X\cos\phi + Y\sin\phi \\ +% + &+ \frac{2}{V_{1}} (1 + m - \tfrac{5}{4}m^{2}\cos 2\tau) + \biggl[\int_{0}^{\tau} \left( + \frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + + \frac{\dd \Omega}{\dd z}\, \frac{dz}{d\tau}\right) d\tau \\ +% + &+ \tfrac{1}{2}\left\{\delta C - \frac{\kappa z^{2}}{r_{1}^{3}} - \left(\frac{dz}{d\tau}\right)^{2}\right\} + \biggr]. +\Tag{(45)} +\end{align*} + +We first neglect~$\Omega$ and consider $X, Y$~as arising only from terms +in~$z^{2}$, i.e.\ +\begin{gather*} +X = \frac{3}{2}\, \frac{\kappa z^{2}x}{r_{1}^{5}},\quad +Y = \frac{3}{2}\, \frac{\kappa z^{2}y}{r_{1}^{5}}. \\ +% +\therefore X\cos\phi + Y\sin\phi + = \frac{3}{2}\, \frac{\kappa z^{2}}{r_{1}^{5}}(x\cos\phi + y\sin\phi). +\end{gather*} +\DPPageSep{106}{48} + +To the required order of accuracy. +\begin{gather*} +z = ka_{0} \cos(g\tau + \zeta),\quad \frac{\kappa}{a_{0}^{3}} = 1 + 2m, \\ +% +r_{1} = a_{0},\quad \phi = \tau,\quad x = a_{0}\cos\tau,\quad y = a_{0}\sin\tau. \\ +% +\therefore X \cos\phi + Y \sin\phi + = \tfrac{3}{4}(1 + 2m)k^{2}a_{0} \bigl[1 + \cos 2(g\tau + \zeta)\bigr]. +\end{gather*} + +Also to order~$m$ +\begin{align*} +\frac{\kappa z^{2}}{r_{1}^{3}} + \left(\frac{dz}{d\tau}\right)^{2} + &= (1 + 2m) k^{2}a_{0}^{2} \cos^{2}(g\tau + \zeta) + + g^{2}k^{2}a_{0}^{2} \sin^{2}(g\tau + \zeta) \\ +% + &= (1 + 2m) k^{2}a_{0}^{2}, +\end{align*} +since $g^{2} = 1 + 2m$. + +The equation for~$\delta p$ becomes therefore, as far as regards the new terms +now introduced, +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + \delta p\, (1 + 2m) + = \tfrac{3}{4}(1 + 2m) k^{2}a_{0} \bigl[1 + \cos 2(g\tau + \zeta)\bigr] \\ + + \frac{(1 + m)}{a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr]. +\end{multline*} + +Hence +\[ +\delta p - \tfrac{3}{4} k^{2}a_{0} + - \frac{(1 - m)}{a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr] + = \tfrac{3}{4}\frac{1 + 2m}{1 + 2m + 4g^{2}} k^{2}a_{0} \cos 2(g\tau + \zeta), \footnotemark +\] +\footnotetext{It is of course only the special integral we require. The general integral when the right-hand + side is zero has already been dealt with,~\SecRef{7}.}% +but +\begin{gather*} +g^{2} = 1 + 2m, \text{ and therefore } +1 + 2m - 4g^{2} = -3(1 + 2m), \\ +% +\therefore \delta p = \tfrac{3}{4} k^{2}a_{0} + + \frac{(1 - m)}{a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr] + - \tfrac{1}{4} k^{2}a_{0} \cos 2(g\tau + \zeta). +\end{gather*} + +Again the varied Jacobian integral is +\begin{align*} +\frac{d\, \delta s}{d\tau} + &= -2(1 + m)\, \delta p + + \frac{1}{2a_{0}} \bigl[\delta C - (1 - 2m) k^{2}a_{0}^{2}\bigr] \\ +% + &= -\tfrac{3}{2}(1 + m) k^{2}a_{0} + - \frac{3}{2a_{0}} \bigl[\delta C - (1 + 2m) k^{2}a_{0}^{2}\bigr] + + \tfrac{1}{2}(1 + m) k^{2}a_{0} \cos 2(g\tau + \zeta). +\end{align*} + +In order that $\delta s$~may not increase with the time we choose~$\delta C$ so that the +constant term is zero, +\begin{align*} +\therefore \delta C &= m k^{2}a_{0}, +\intertext{and} +\frac{d\, \delta s}{d\tau} + &= \tfrac{1}{2}(1 - m) k^{2}a_{0} \cos 2(g\tau + \zeta), \\ +% +\intertext{giving} +\delta s &= \tfrac{1}{4} k^{2}a_{0} \sin 2(g\tau + \zeta), +\Tag{(46)} +\end{align*} +as there is no need to introduce a new constant\footnotemark. Using the value of~$\delta C$ just +\footnotetext{Cf.\ same point in connection with equation~\Eqref{(32)}.}% +found we get +\[ +\delta p = -\tfrac{1}{4} k^{2}a_{0} + - \tfrac{1}{4} k^{2}a_{0} \cos 2(g\tau + \zeta). +\Tag{(47)} +\] + +Having obtained $\delta p$~and~$\delta s$, we now require~$\delta x, \delta y$. These are +\begin{align*} +\delta x &= \delta p \cos\phi - \delta s \sin\phi, \\ +\delta y &= \delta p \sin\phi + \delta s \cos\phi. +\end{align*} +\DPPageSep{107}{49} + +In this case with sufficient accuracy $\phi = \tau$, +\begin{alignat*}{3} +\delta x + &= - \tfrac{1}{4} a_{0}k^{2} \cos\tau + &&- \tfrac{1}{4} a_{0}k^{2} \cos\tau \cos 2(g\tau + \zeta) + &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \sin 2(g\tau + \zeta), \\ +% +\delta y + &= - \tfrac{1}{4} a_{0}k^{2} \sin\tau + &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \cos 2(g\tau + \zeta) + &&+ \tfrac{1}{4} a_{0}k^{2} \cos\tau \sin 2(g\tau + \zeta). +\end{alignat*} + +Dropping the recent use of~$X, Y$ in connection with the forces and using +as before $X = x + \delta x$, $Y = y + \delta y$ we have +\begin{alignat*}{3} +X &= a_{0}\cos\tau(1 - \tfrac{1}{4}k^{2}) + &&- \tfrac{1}{4} a_{0}k^{2} \cos\tau \cos 2(g\tau + \zeta) + &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \sin 2(g\tau + \zeta), \\ +% +Y &= a_{0}\sin\tau(1 - \tfrac{1}{4}k^{2}) + &&- \tfrac{1}{4} a_{0}k^{2} \sin\tau \cos 2(g\tau + \zeta) + &&+ \tfrac{1}{4} a_{0}k^{2} \cos\tau \sin 2(g\tau + \zeta), \\ +% +R^{2} &= \rlap{$X^{2} + Y^{2} + = a_{0}^{2}(1 - \tfrac{1}{2}k^{2}) + - \tfrac{1}{2} a_{0}^{2}k^{2} \cos 2(g\tau + \zeta)$,}&&&& \\ +% +R &= \rlap{$a_{0}\bigl[1 - \tfrac{1}{4}k^{2} + - \tfrac{1}{4}k^{2} \cos 2(g\tau + \zeta)\bigr]$.}&&&& +\Tag{(48)} +\end{alignat*} + +We thus get corrected result in radius vector as projected on to the ecliptic. + +Again +\begin{alignat*}{2} +\cos\Theta &= \frac{X}{R} + &&= \cos\tau - \tfrac{1}{4} k^{2} \sin\tau \sin 2(g\tau + \zeta), \\ +% +\sin\Theta &= \frac{Y}{R} + &&= \sin\tau + \tfrac{1}{4} k^{2} \cos\tau \sin 2(g\tau + \zeta), \\ +% +\Theta - \tau + &= \rlap{$\sin(\Theta - \tau) = \tfrac{1}{4} k^{2} \sin 2(g\tau + \zeta)$.}&& +\Tag{(49)} +\end{alignat*} + +Hence we have as a term in the moon's longitude $\frac{1}{4}k^{2}\sin 2(g\tau + \zeta)$. Terms +\index{Reduction, the}% +of this type are called the reduction; they result from referring the moon's +orbit to the ecliptic. + +We have now only to consider the terms depending on~$\Omega$. We have seen +that $\Omega$~vanishes when the solar eccentricity,~$e'$, is put equal to zero. We shall +only develop~$\Omega$ as far as first power of~$e'$. + +The radius vector~$r'$, and the true longitude~$v'$, of the sun are given to the +required approximation by +\begin{align*} +r' &= \a' \bigl\{1 - e'\cos(n't - \varpi')\bigr\}, \\ +v' &= n't + 2e'\sin(n't - \varpi'). +\end{align*} + +Hence +\begin{alignat*}{2} +x' &= r'\cos(v' - n't) = r' &&= \a' \bigl\{1 - e'\cos(n't - \varpi')\bigr\}, \\ +y' &= r'\sin(v' - n't) &&= 2\a'e' \sin(n't - \varpi'). +\end{alignat*} +And +\begin{gather*} +n't = m\tau; \\ +\begin{aligned} +\therefore \frac{xx' + yy'}{\a'} + &= x - e'x \cos(m\tau - \varpi) + 2e'y \sin(m\tau - \varpi), \\ +\left(\frac{xx' + yy'}{\a'}\right)^{2} + &= x^{2} - 2e'x^{2} \cos(m\tau - \varpi) + 4e'xy \sin(m\tau - \varpi), \\ +\frac{\a'^{5}}{r'^{5}} + &= 1 + 5e' \cos(m\tau - \varpi), +\end{aligned} +\Allowbreak +\DPPageSep{108}{50} +\frac{3m^{2}}{2} \left\{\frac{\a'^{3}}{r'^{5}} (xx' + yy')^{2} - x^{2}\right\} + = \frac{9m^{2}}{2} e' x^{2} \cos(m\tau - \varpi') + + 6m^{2} e'xy \sin(m\tau - \varpi'), \\ +% +\tfrac{1}{2} m^{2} (x^{2} + y^{2} + z^{2}) \left(1 - \frac{\a'^{3}}{r'^{3}}\right) + = -\tfrac{3}{2} m^{2} (x^{2} + y^{2} + z^{2}) e' \cos(m\tau - \varpi'), \\ +\Omega + = m^{2} e' \bigl[3x^{2} \cos(m\tau - \varpi') + + 6xy \sin(m\tau - \varpi') - \tfrac{3}{2} y^{2} \cos(m\tau - \varpi') \bigr], +\end{gather*} +for we neglect~$m^{2}z^{2}$ when multiplied by~$e'$, +\begin{align*} +\frac{\dd \Omega}{\dd x} + &= 6m^{2}e' \bigl[x \cos(m\tau - \varpi') + y \sin(m\tau - \varpi')\bigr], \\ +% +\frac{\dd \Omega}{\dd y} + &= 6m^{2}e' \bigl[x \sin(m\tau - \varpi') - \tfrac{1}{2} y \cos(m\tau - \varpi')\bigr]. +\end{align*} + +It is sufficiently accurate for us to take +\begin{align*} +x &= a_{0} \cos \tau,\quad +y = a_{0} \sin \tau, \\ +\phi &= \tau; +\end{align*} +\begin{multline*} +\therefore +\frac{\dd \Omega}{\dd x} \cos\phi + +\frac{\dd \Omega}{\dd y} \sin\phi + = 6m^{2} e' a_{0} \bigl[\cos^{2}\tau \cos(m\tau - \varpi') + + \cos\tau \sin\tau \sin(m\tau - \varpi') \\ +% +\shoveright{+ \cos\tau \sin\tau \sin(m\tau - \varpi') + - \tfrac{1}{2} \sin^{2}\tau \cos(m\tau - \varpi') \bigr]} \\ +% +\shoveleft{= 3m^{2} e'a_{0} \bigl[\cos(m\tau - \varpi') + + \cos 2\tau \cos(m\tau - \varpi') + 2\sin 2\tau \sin(m\tau - \varpi') \bigr]} \\ +% +\shoveright{- \tfrac{1}{2} \cos(m\tau - \varpi') + \tfrac{1}{2} \cos2\tau \cos(m\tau - \varpi')} \\ +% +\shoveleft{= 3m^{2} e' a_{0} \bigl[\tfrac{1}{2} \cos(m\tau - \varpi') + + \tfrac{3}{4} \cos \bigl\{(2 + m)\tau - \varpi' \bigr\} + + \tfrac{3}{4} \cos \bigl\{(2 - m)\tau + \varpi' \bigr\}} \\ +% +\shoveright{+ \cos \bigl\{(2 - m)\tau + \varpi' \bigr\} + - \cos \bigl\{(2 + m)\tau - \varpi' \bigr\} \bigr]} \\ +% +\shoveleft{= \tfrac{3}{2} m^{2} e'a_{0} \bigl[\cos(m\tau - \varpi') + - \tfrac{1}{2} \cos \bigl\{(2 + m)\tau - \varpi' \bigr\} + + \tfrac{7}{2} \cos \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr].} +\end{multline*} + +Again +\begin{multline*} +\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + +\frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau} + = 6m^{2} e'a_{0} \bigl[-\sin\tau \cos\tau \cos(m\tau - \varpi') + - \sin^{2} \tau \sin(m\tau - \varpi') \\ +% +\shoveright{+ \cos^{2} \tau \sin(m\tau - \varpi') + - \tfrac{1}{2} \sin\tau \cos\tau \cos(m\tau - \varpi') \bigr]} \\ +% +\shoveleft{= 3m^{2} e'a_{0} \bigl[-\tfrac{3}{2} \sin 2\tau \cos(m\tau - \varpi') + + 2 \cos 2\tau \sin(m\tau - \varpi') \bigr]} \\ +% +\shoveleft{= \tfrac{3}{2} m^{2} e'a_{0} \bigl[-\tfrac{3}{2} \sin \bigl\{(2 + m)\tau - \varpi' \bigr\} + - \tfrac{3}{2} \sin \bigl\{(2 - m)\tau + \varpi' \bigr\}} \\ +% +\shoveright{+ 2\sin \bigl\{(2 + m)\tau - \varpi' \bigr\} + - 2\sin \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr]} \\ +% +\shoveleft{= \tfrac{3}{2} m^{2} e'a_{0} \bigl[\tfrac{1}{2} \sin \bigl\{(2 + m)\tau - \varpi' \bigr\} + - \tfrac{7}{2} \sin \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr],} \\ +% +\shoveleft{2 \int \left(\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}\right) d\tau + = -\tfrac{3}{2} m^{2} e'a_{0} \bigl[\tfrac{1}{2} \cos \bigl\{(2 + m)\tau - \varpi' \bigr\}} \\ +% +\shoveright{- \tfrac{7}{2} \cos \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr];} \\ +% +\shoveleft{\therefore + \frac{\dd \Omega}{\dd x} \cos\phi + + \frac{\dd \Omega}{\dd y} \sin\phi + + 2 \int \left(\frac{\dd \Omega}{\dd x}\, \frac{dx}{d\tau} + + \frac{\dd \Omega}{\dd y}\, \frac{dy}{d\tau}\right) d\tau + = \tfrac{3}{2} m^{2} e'a_{0} \bigl[\cos(m\tau - \varpi')} \\ +% + - \cos \bigl\{(2 + m)\tau - \varpi' \bigr\} + + 7\cos \bigl\{(2 - m)\tau + \varpi' \bigr\} \bigr]. +\end{multline*} +\DPPageSep{109}{51} + +Hence to the order required +\begin{multline*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + (1 + 2m)\, \delta p = \tfrac{3}{2} m^{2} e'a_{0} +\bigl[ +\cos(m\tau - \varpi') - \cos \left\{(2 + m) \tau - \varpi'\right\} \\ + + 7 \cos \left\{(2 - m)\tau + \varpi'\right\}\bigr], +\end{multline*} +\[ +\begin{aligned} +\delta p &= \tfrac{3}{2} m^{2} e'a_{0} + \left[\frac{\cos(m\tau - \varpi')}{-m^{2} + 1 + 2m} + - \frac{ \cos\left\{(2 + m)\tau - \varpi'\right\}}{-(4 + 4m) + 1 + 2m} + + \frac{7\cos\left\{(2 - m)\tau + \varpi'\right\}}{-(4 - 4m) + 1 + 2m}\right] \\ +% + &= \tfrac{3}{2} m^{2} e'a_{0} + \left[\cos(m\tau - \varpi') + + \tfrac{1}{3} \cos \left\{(2 + m)\tau - \varpi'\right\} + - \tfrac{7}{3} \cos \left\{(2 - m)\tau + \varpi'\right\}\right]\Add{,} +\end{aligned} +\Tag{(50)} +\] +{\setlength{\abovedisplayskip}{0pt}% +\setlength{\belowdisplayskip}{0pt}% +\begin{multline*} +\frac{d\, \delta s}{d\tau} + = -2\delta p\, (1 + m) + + \frac{1}{V}\int \left(\frac{d\Omega}{dx}\, \frac{dx}{d\tau} + +\frac{d\Omega}{dy}\, \frac{dy}{d\tau}\right) d\tau \\ +% +\shoveleft{= -3m^{2} e'a_{0} \left[\cos(m\tau - \varpi') + + \tfrac{1}{3}\cos\left\{(2 + m)\tau - \varpi'\right\} + - \tfrac{7}{3}\cos\left\{(2 - m)\tau + \varpi'\right\}\right]} \\ +% +\shoveright{- \tfrac{3}{4} m^{2} e'\left[\tfrac{1}{2} \cos\left\{(2 + m)\tau - \varpi'\right\} + - \tfrac{7}{2} \cos \left\{(2 - m)\tau + \varpi'\right\}\right]} \\ +% +\shoveleft{= -3m^{2}e'a_0 \bigl[\cos(m\tau - \varpi') + + \tfrac{11}{24} \cos\left\{(2 + m)\tau - \varpi'\right\}} \\ +% +\shoveright{-\tfrac{77}{24} \cos\left\{(2 - m)\tau + \varpi'\right\}\bigr];} \\ +\end{multline*} +\begin{multline*} +\therefore \delta s = - 3m e'a_{0} \sin(m\tau - \varpi') + - 3m^{2} e'a_{0} \bigl[\tfrac{11}{48} \sin \left\{(2 + m) \tau - \varpi'\right\} \\ + - \tfrac{77}{48} \sin\left\{(2 - m)\tau + \varpi'\right\}\bigr]\Add{.} +\Tag{(51)} +\end{multline*}} + +Hence to order~$m e'$, to which order only our result is correct, +\[ +\delta p = 0, \quad +\delta s = -3m e'a_{0} \sin (m\tau - \varpi'). +\] + +And following our usual method for obtaining new terms in radius vector +and longitude +\begin{align*} +\delta x &= \delta p \cos \phi - \delta s \sin \phi, \quad +\delta y = \delta p \sin \phi + \delta s \cos \phi, \\ +\delta x &= +%[** TN: Hack to align second equation with previous second equation] + \settowidth{\TmpLen}{$\delta p \cos \phi - \delta s \sin \phi$,\quad} + \makebox[\TmpLen][l]{$- \delta s \sin \tau$,}\, +\delta y = \delta s \cos \tau, \\ +X &= a_{0} \left[\cos \tau + 3m e' \sin \tau \sin (m\tau - \varpi')\right], \\ +Y &= a_{0} \left[\sin \tau - 3m e' \cos \tau \sin (m\tau - \varpi')\right], \\ +R^{2} &= a_{0}^{2} \left[1 + 3m e' \sin 2\tau \sin (m\tau - \varpi') + - 3m e' \sin 2\tau \sin (m\tau - \varpi')\right] = a_{0}^{2}, \\ +\Tag{(52)} +\end{align*} +and to the order required there is no term in radius vector +\begin{align*} +\cos \Theta &= \cos \tau + 3m e' \sin \tau \sin (m\tau - \varpi'),\\ +\sin \Theta &= \sin \tau - 3m e' \cos \tau \sin (m\tau - \varpi'),\\ +\sin (\Theta - \tau) &= - 3m e' \sin (m\tau - \varpi'),\\ +\Theta &= \tau - 3m e' \sin(m\tau - \varpi'). +\Tag{(53)} +\end{align*} + +The new term in the longitude is~$-3m e' \sin (l' - \varpi')$. This term is called +the annual equation. +\index{Annual Equation}% +\index{Equation, annual}% +\DPPageSep{110}{52} + +\Section{§ 11. }{Compilation of Results.} + +Let $v$~be the longitude, $s$~the tangent of the latitude (or to our order +simply the latitude). When we collect our results we find +\begin{align*} +v &= \settowidth{\TmpLen}{longitude}% + \UnderNote{\makebox[\TmpLen][c]{$l$}}{% + \parbox[c]{\TmpLen}{\centering(mean\\ longitude\\ ${}= nt + \epsilon$)}} + + \UnderNote{2e \sin (l - \varpi)}{% + \settowidth{\TmpLen}{equation to}% + \parbox[c]{\TmpLen}{\centering equation to\\ the centre}} + + \UnderNote{\tfrac{15}{4} m e \sin(l - 2l' + \varpi)}{evection} + + \UnderNote{\tfrac{11}{8} m^2 \sin2(l - l')}{variation} \\ +% + &\qquad\qquad\qquad + \UnderNote{-\tfrac{1}{4} k^{2} \sin 2(l - \Omega)}{reduction} + - \UnderNote{3m e' \sin(l' - \varpi')}{annual equation}, \\ +% +s &= k \sin(l - \Omega) + + \UnderNote{\tfrac{3}{8} m k \sin(l - 2l' + \Omega)}{evection in latitude}. +\end{align*} + +For~$R$, the projection of the radius vector on the ecliptic, we get +\begin{multline*} +R = \a\bigl[1 - \tfrac{1}{6} m^{2} - \tfrac{1}{4} k^{2} + - \UnderNote{e \cos(l - \varpi)}{% + \settowidth{\TmpLen}{equation to the}% + \parbox[c]{\TmpLen}{\centering equation to the\\ centre}} + - \UnderNote{\tfrac{15}{8} m e \cos(l - 2l' - \varpi)}{evection} + - \UnderNote{m^{2} \cos 2(l - l')}{variation} \\ +% + + \UnderNote{\tfrac{1}{4} k^{2} \cos 2(l - \Omega)}{reduction}\bigr]. +\Tag{(54)} +\end{multline*} + +To get the actual radius vector we require to multiply by~$\sec\beta$, i.e.~by +\[ +1 + \tfrac{1}{2} k^{2} \sin^{2}(l - \Omega) \text{ or } +1 + \tfrac{1}{4} k^{2} - \tfrac{1}{4} k^{2} \cos 2(l - \Omega). +\] + +This amounts to removing the terms $-\frac{1}{4}k^{2} + \frac{1}{4}k^{2}\cos2(l - \Omega)$. The radius +vector then is +\[ +\a \bigl[1 - \tfrac{1}{6} m^{2} - e \cos(l - \varpi) + - \tfrac{15}{8} m e \cos(l - 2l' + \varpi) - m^{2} \cos2(l - l')\bigr]. +\] + +This is independent of~$k$, but $k$~will enter into product terms of higher +order than we have considered. The perturbations are excluded by putting +$m = 0$ and the value of the radius vector is then independent of~$k$ as it +should be. The quantity of practical importance is not the radius vector but +its reciprocal. To our degree of approximation it is +\[ +\frac{1}{\a}\bigl[1 + \tfrac{1}{6} m^{2} + e \cos(l - \varpi) + + \tfrac{15}{8} m e \cos(l - 2l' + \varpi) + m^{2}\cos2(l - l')\bigr]. +\] + +It may be noted in conclusion that the terms involving only~$e$ in the +coefficient, and designated the equation to the centre, are not perturbations +but the ordinary elliptic inequalities. There are terms in~$e^{2}$ but these have +not been included in our work. +\DPPageSep{111}{53} + +\Note{1.}{On the Infinite Determinant of \SecRef{5}.} +\index{Hill, G. W., Lunar Theory!infinite determinant}% +\index{Infinite determinant, Hill's}% + +We assume (as has been justified by Poincaré) that we may treat the +infinite determinant as though it were a finite one. + +For every row corresponding to~$+i$ there is another corresponding to~$-i$, +and there is one for~$i =0$. + +If we write~$-c$ for~$c$ the determinant is simply turned upside down. +Hence the roots occur in pairs and if $c_{0}$~is a root $-c_{0}$~is also a root. + +If for $c$ we write~$c ± 2j$, where $j$~is an integer, we simply shift the centre +of the determinant. + +Hence if $c_{0}$~is a root, $± c_{0} ± 2j$~are also roots. + +But these are the roots of $\cos \pi c = \cos \pi c_{0}$. + +Therefore the determinant must be equal to +\[ +k(\cos \pi c - \cos \pi c_{0}). +\] + +If all the roots have been enumerated, $k$~is independent of~$c$. + +Now the number of roots cannot be affected by the values assigned to +the~$\Theta$'s. Let us put $\Theta_{1} = \Theta_{2} = \Theta_{3} = \dots = 0$. + +The determinant then becomes equal to the product of the diagonal terms +and the equation is +\[ +\dots \bigl[\Theta_{0} - (c - 2)^{2}\bigr] + \bigl[\Theta_{0} - c^{2}\bigr] + \bigl[\Theta_{0} - (c + 2)^{2}\bigr] \dots = 0. +\] + +$c_{0} = ±\Surd{\Theta_{0}}$ is one pair of roots, and all the others are given by~$c_{0} ± 2i$. + +Hence there are no more roots and $k$~is independent of~$c$. + +The determinant which we have obtained is inconvenient because the +diagonal elements increase as we pass away from the centre while the non-diagonal +elements are of the same order of magnitude for all the rows. But +the roots of the determinant are not affected if the rows are multiplied by +numerical constants and we can therefore introduce such numerical multipliers +as we may find convenient. + +The following considerations indicate what multipliers may prove useful. +If we take a finite determinant from the centre of the infinite one it can be +completely expanded by the ordinary processes. Each of the terms in the +expansion will only involve~$c$ through elements from the principal diagonal +and the term obtained by multiplying all the elements of this diagonal will +contain the highest power of~$c$. When the determinant has $(2i + 1)$ rows +and columns, the highest power of~$c$ will be~$-c^{2(2i + 1)}$. We wish to associate +the infinite determinant with~$\cos \pi c$. Now +\[ +\cos \pi c + = \left(1 - \frac{4c^{2}}{1}\right) + \left(1 - \frac{4c^{2}}{9}\right) + \left(1 - \frac{4c^{2}}{25}\right) \dots. +\] +\DPPageSep{112}{54} + +The first $2i + 1$~terms of this product may be written +\[ +\left(1 - \frac{2c}{4i + 1}\right) +\left(1 - \frac{2c}{4i - 1}\right) \dots +\left(1 + \frac{2c}{4i - 1}\right) +\left(1 + \frac{2c}{4i + 1}\right), +\] +and the highest power of~$c$ in this product is +\[ +\frac{4c^{2}}{(4i)^{2} - 1} · \frac{4c^{2}}{\bigl\{4(i - 1)\bigr\}^{2} - 1} \dots \frac{4c^{2}}{(4i)^{2} - 1}. +\] + +Hence we multiply the $i$th~row below or above the central row by~$\dfrac{-4}{(4i)^{2} - 1}$. +The $i$th~diagonal term below the central term will now be~$\dfrac{4\bigl[(2i + c)^{2} - \Theta_{0}\bigr]}{(4i)^{2} - 1}$ +and will be denoted by~$\{i\}$. It clearly tends to unity as $i$~tends to infinity by +positive or negative values. The $i$th~row below the central row will now +read +\[ +\dots +\frac{-4\Theta_{2}}{(4i)^{2} - 1},\quad +\frac{-4\Theta_{1}}{(4i)^{2} - 1},\quad \{i\},\quad +\frac{-4\Theta_{1}}{(4i)^{2} - 1},\quad +\frac{-4\Theta_{2}}{(4i)^{2} - 1},\dots. +\] + +The new determinant which we will denote by~$\nabla (c)$ has the same roots +as the original one and so we may write +\[ +\nabla (c) = k' \{\cos \pi c - \cos \pi c_{0}\}, +\] +where $k'$~is a new numerical constant. But it is easy to see that~$k' = 1$. +This was the object of introducing the multipliers and that it is true is easily +proved by taking the case of $\Theta_{1} = \Theta_{2} = \dots = 0$ and $\Theta_{0} = \frac{1}{4}$, in which case the +determinant reduces to~$\cos \pi c$. We thus have the equation +\[ +\nabla (c) = \cos \pi c - \cos \pi c_{0}, +\] +which can be considered as an identity in~$c$. + +Putting $c = 0$ we get +\[ +\nabla (0) = 1 - \cos \pi c_{0}. +\] + +$\nabla (0)$~depends only on the~$\Theta$'s; written so as to shew the principal elements +it is +\[ +\left\lvert +\begin{array}{@{}c *{5}{r} c@{}} +\multicolumn{7}{c}{\dotfill} \\ +\dots & \tfrac{4}{63}(16-\Theta_{0}),& -\tfrac{4}{63}\Theta_{1},& -\tfrac{4}{63}\Theta_{2},& -\tfrac{4}{63}\Theta_{3},& -\tfrac{4}{63}\Theta_{4},& \dots \\ +\dots & -\tfrac{4}{15}\Theta_{1},& \tfrac{4}{15}(4-\Theta_{0}),& -\tfrac{4}{15}\Theta_{1},& -\tfrac{4}{15}\Theta_{2},& -\tfrac{4}{15}\Theta_{3},& \dots \\ +\dots & 4\Theta_{2},& 4\Theta_{1},& 4\Theta_{0},& 4\Theta_{1},& 4\Theta_{2},& \dots \\ +\dots & -\tfrac{4}{15}\Theta_{3},& -\tfrac{4}{15}\Theta_{2},& -\tfrac{4}{15}\Theta_{1},& \tfrac{4}{15}(4-\Theta_{0}),& -\tfrac{4}{15}\Theta_{1},& \dots \\ +\dots & -\tfrac{4}{63}\Theta_{4},& -\tfrac{4}{63}\Theta_{3},& -\tfrac{4}{63}\Theta_{2},& -\tfrac{4}{63}\Theta_{1},& \tfrac{4}{63}(16-\Theta_{0}),& \dots \\ +\multicolumn{7}{c}{\dotfill} +\end{array} +\right\rvert +\] + +{\stretchyspace +If $\Theta_{1}, \Theta_{2}$,~etc.\ vanish, the solution of the differential equation is $\cos(\Surd{\Theta_{0}} + \epsilon)$ +or~$c = \Surd{\Theta_{0}}$. But in this case the determinant has only diagonal terms and +the product of the diagonal terms of~$\nabla (0)$ is~$1 - \cos \pi \Surd{\Theta_{0}}$ or~$2 \sin^{2} \frac{1}{2}\pi\Surd{\Theta_{0}}$.} +\DPPageSep{113}{55} + +Hence we may divide each row by its diagonal member and put +$2 \sin^{2} \frac{1}{2} \pi \Surd{\Theta_{0}}$ outside. + +If therefore +{\small +\begin{align*} +\Delta(0) &= \left\lvert +\begin{array}{@{}c *{5}{>{\ }c@{,\ }} c} +\multicolumn{7}{c}{\dotfill} \\ +\dots & 1 & -\dfrac{\Theta_{1}}{16-\Theta_{0}}& -\dfrac{\Theta_{2}}{16-\Theta_{0}}& -\dfrac{\Theta_{3}}{16-\Theta_{0}}& -\dfrac{\Theta_{4}}{16-\Theta_{0}} & \dots \\ +\dots & -\dfrac{\Theta_{1}}{4-\Theta_{0}} & 1 & -\dfrac{\Theta_{1}}{4-\Theta_{0}} & -\dfrac{\Theta_{2}}{4-\Theta_{0}} & -\dfrac{\Theta_{3}}{4-\Theta_{0}} & \dots \\ +\dots & \dfrac{\Theta_{2}}{\Theta_{0}} & \dfrac{\Theta_{1}}{\Theta_{0}} & 1 & \dfrac{\Theta_{1}}{\Theta_{0}} & \dfrac{\Theta_{2}}{\Theta_{0}} & \dots \\ +\dots & -\dfrac{\Theta_{3}}{4-\Theta_{0}} & -\dfrac{\Theta_{2}}{4-\Theta_{0}} & -\dfrac{\Theta_{1}}{4-\Theta_{0}} & 1 & -\dfrac{\Theta_{1}}{4-\Theta_{0}} & \dots \\ +\multicolumn{7}{c}{\dotfill} +\end{array} +\right\rvert +\\ +\nabla(0) &= 2 \sin^{2} \tfrac{1}{2} \pi\Surd{\Theta_{0}} \Delta(0). +\end{align*}} + +Now since +\[ +\cos \pi c_{0} = 1 - \nabla (0) + = 1 - 2 \sin^{2} \frac{1}{2} \pi \Surd{\Theta_{0}} \Delta(0), +\] +we have +\Pagelabel{55} +\[ +\frac{\sin^{2} \frac{1}{2} \pi c_{0}}{\sin^{2} \frac{1}{2} \pi \Surd{\Theta_{0}}} + = \Delta(0), +\] +an equation to be solved for~$c_{0}$ (or~$c$). + +Clearly for stability $\Delta(0)$~must be positive and $\Delta(0) < \cosec^2 \frac{1}{2} \pi \Surd{\Theta_{0}}$. +Hill gives other transformations. + +\Note{2\footnotemark.}{On the periodicity of the integrals of the equation +\footnotetext{This treatment of the subject was pointed out to Sir~George Darwin by Mr~S.~S. Hough.} +\begin{gather*} +\frac{d^{2}\, \delta p}{d\tau^{2}} + \Theta\, \delta p = 0, \\ +\lintertext{where} +{\Theta = \Theta_{0} + + \Theta_{1} \cos 2\tau + + \Theta_{2} \cos 4\tau + \dots.} +\end{gather*}} +\index{Differential Equation, Hill's!periodicity of integrals of}% +\index{Hill, G. W., Lunar Theory!periodicity of integrals of}% +\index{Periodicity of integrals of Hill's Differential Equation}% + +Since the equation remains unchanged when $\tau$ becomes~$\tau + \pi$, it follows +that if $\delta p = F(\tau)$ is a solution $F(\tau + \pi)$ is also a solution. + +Let $\phi(\tau)$~be a solution subject to the conditions that when +\[ +\tau=0,\quad +\delta p = 1,\quad +\frac{d\, \delta p}{d\tau} = 0; \text{ i.e.\ } \phi(0) = 1,\quad +\phi'(0) = 0. +\] + +Let $\psi(\tau)$~be a second solution subject to the conditions that when +\[ +\tau=0,\quad +\delta p = 0,\quad +\frac{d\, \delta p}{d\tau} = 1; \text{ i.e.\ } \psi(0) = 0,\quad +\psi'(0) = 1. +\] +\DPPageSep{114}{56} + +It is clear that $\phi(\tau)$ is an even function of~$\tau$, and $\psi(\tau)$~an odd one, so +that +\begin{alignat*}{2} +\phi (-\tau) &= \Neg\phi(\tau),&\qquad \psi(-\tau)&= -\psi(\tau),\\ +\phi'(-\tau) &= -\phi(\tau),&\qquad \psi'(-\tau)&= \Neg\psi(\tau). +\end{alignat*} +Then the general solution of the equation is +\[ +\delta p = F(\tau) = A\phi(\tau) + B\psi(\tau), +\] +where $A$~and~$B$ are two arbitrary constants. + +Since $\phi(\tau + \pi)$, $\psi(\tau + \pi)$ are also solutions of the equation, it follows +that +\[ +\left. +\begin{aligned} +\phi(\tau + \pi) &= \alpha\phi(\tau) + \beta \psi(\tau)\Add{,} \\ +\psi(\tau + \pi) &= \gamma\phi(\tau) + \delta\psi(\tau)\Add{,} +\end{aligned} +\right\} +\Tag{(55)} +\] +where $\alpha, \beta, \gamma, \delta$ are definite constants. + +If possible let $A : B$ be so chosen that +\[ +F(\tau + \pi) = \nu F(\tau), +\] +where $\nu$~is a numerical constant. + +When we substitute for~$F$ its values in terms of $\phi$~and~$\psi$, we obtain +\[ +A\phi(\tau + \pi) + B\psi(\tau + \pi) = \nu\bigl[A\phi(\tau) + B\psi(\tau)\bigr]. +\] + +Further, substituting for $\phi(\tau + \pi)$, $\psi(\tau + \pi)$ their values, we have +\[ +A\bigl[\alpha\phi(\tau) + \beta \psi(\tau)\bigr] + +B\bigl[\gamma\phi(\tau) + \delta\psi(\tau)\bigr] + = \nu\bigl[A\phi(\tau) + B\psi(\tau)\bigr], +\] +whence +\[ +\bigl[A(\alpha - \nu) + B\gamma\bigr] \phi(\tau) + + \bigl[A\beta + B(\delta - \nu)\bigr] \psi(\tau) = 0. +\] + +Since this is satisfied for all values of~$\tau$, +\begin{align*} +A(\alpha - \nu) + B\gamma &= 0,\\ +A\beta + B(\delta - \nu) &= 0,\\ +\therefore(\alpha - \nu)(\delta - \nu) - \beta\gamma &= 0,\\ +\text{i.e.}\quad +\nu^{2} - (\alpha + \delta)\nu + \alpha\delta - \beta\gamma &= 0, +\end{align*} +an equation for $\nu$ in terms of the constants $\alpha, \beta, \gamma, \delta$. This equation can be +simplified. + +Since +\[ +\frac{d^{2}\phi}{d\tau^{2}} + \Theta\phi = 0,\qquad +\frac{d^{2}\psi}{d\tau^{2}} + \Theta\psi = 0, +\] +we have +\[ +\phi \frac{d^{2}\psi}{d\tau^{2}} - \psi \frac{d^{2}\phi}{d\tau^{2}} = 0. +\] +On integration of which +\[ +\phi\psi' - \psi\phi' = \text{const.} +\] +But +\[ +\phi(0) = 1,\quad +\psi'(0) = 1,\quad +\psi(0) = 0,\quad +\phi'(0) = 0. +\] + +Therefore the constant is unity; and +\[ +\phi(\tau)\psi'(\tau) - \psi(\tau)\phi'(\tau) = 1. +\Tag{(56)} +\] +\DPPageSep{115}{57} +But putting $\tau = 0$ in the equations~\Eqref{(55)}, and in the equations obtained by +differentiating them, +\begin{alignat*}{3} +\phi(\pi) &= \alpha\,\phi\,(0) &&+ \beta\,\psi(0) &&= \alpha,\\ +\psi(\pi) &= \gamma\,\phi\,(0) &&+ \delta\,\psi\,(0) &&= \gamma,\\ +\phi'(\pi) &= \alpha\phi'(0) &&+ \beta\psi'(0) &&= \beta,\\ +\psi'(\pi) &= \gamma\phi'(0) &&+ \delta\,\psi'(0) &&= \delta. +\end{alignat*} + +Therefore by~\Eqref{(56)}, +\[ +\alpha\delta - \beta\gamma = 1. +\] +Accordingly our equation for~$\nu$ is +\[ +\nu^{2} - (\alpha + \delta)\nu + 1 = 0 +\] +or +\[ +\tfrac{1}{2} \left(\nu + \frac{1}{\nu)}\right) = \tfrac{1}{2} (\alpha + \delta). +\] + +If now we put $\tau = -\frac{1}{2}\pi$ in~\Eqref{(55)} and the equations obtained by +differentiating them, +\begin{align*} +&\begin{alignedat}{4} +\phi(\tfrac{1}{2}\pi) + &= \alpha\phi(-\tfrac{1}{2}\pi) &&+ \beta\psi(-\tfrac{1}{2}\pi) + &&= \Neg\alpha\phi(\tfrac{1}{2}\pi) &&- \beta\psi(\tfrac{1}{2}\pi), \\ +% +\psi(\tfrac{1}{2}\pi) + &= \gamma\phi(-\tfrac{1}{2}\pi) &&+ \delta\psi(-\tfrac{1}{2}\pi) + &&= \Neg\gamma\phi(\tfrac{1}{2}\pi) &&- \delta\psi(\tfrac{1}{2}\pi), \\ +% +\phi'(\tfrac{1}{2}\pi) + &= \alpha\phi'(-\tfrac{1}{2}\pi) &&+ \beta\psi'(-\tfrac{1}{2}\pi) + &&= -\alpha\phi'(\tfrac{1}{2}\pi) &&+ \beta\psi'(\tfrac{1}{2}\pi), \\ +% +\psi'(\tfrac{1}{2}\pi) + &= \gamma\phi'(-\tfrac{1}{2}\pi) &&+ \delta\psi'(-\tfrac{1}{2}\pi) + &&= -\gamma\phi'(\tfrac{1}{2}\pi) &&+ \delta\psi'(\tfrac{1}{2}\pi),\\ +\end{alignedat} +\Allowbreak +&\frac{\phi(\tfrac{1}{2}\pi)}{\psi(\tfrac{1}{2}\pi)} + = \frac{\beta}{\alpha - 1} + = \frac{\delta + 1}{\gamma},\quad +\frac{\psi'(\tfrac{1}{2}\pi)}{\phi'(\tfrac{1}{2}\pi)} + = \frac{\alpha + 1}{\beta} + = \frac{\gamma}{\delta - 1}, \\ +% +&\frac{\phi(\tfrac{1}{2}\pi) \psi'(\tfrac{1}{2}\pi)} + {\psi(\tfrac{1}{2}\pi) \phi'(\tfrac{1}{2}\pi)} + = \frac{\alpha + 1}{\alpha - 1} = \frac{\delta + 1 }{\delta - 1}. +\end{align*} + +But since $\phi(\frac{1}{2}\pi)\psi'(\frac{1}{2}\pi) - \phi'(\frac{1}{2}\pi)\psi(\frac{1}{2}\pi) = 1$ we have +\[ +\alpha = \delta = \tfrac{1}{2}(\alpha + \delta) + = \phi (\tfrac{1}{2}\pi) \psi'(\tfrac{1}{2}\pi) + + \phi'(\tfrac{1}{2}\pi) \psi (\tfrac{1}{2}\pi). +\] + +Hence the equation for~$\nu$ may be written in five different forms, viz.\ +\begin{align*} +\tfrac{1}{2}\left(\nu + \frac{1}{\nu}\right) + &= \phi(\pi) = \psi'(\pi) + = \phi (\tfrac{1}{2}\pi)\psi'(\tfrac{1}{2}\pi) + + \phi'(\tfrac{1}{2}\pi)\psi (\tfrac{1}{2}\pi) \\ + &= 1 + 2\phi'(\tfrac{1}{2}\pi)\psi(\tfrac{1}{2}\pi) + = 2\phi(\tfrac{1}{2}\pi) \psi'(\tfrac{1}{2}\pi) - 1. +\Tag{(57)} +\end{align*} + +It remains to determine the meaning of~$\nu$ in terms of the~$c$ introduced in +the solution by means of the infinite determinant. + +The former solution was +\[ +\delta p = \sum_{-\infty}^{+\infty} + \bigl\{A_{j} \cos(c + 2j)\tau + B_{j} \sin(c + 2j)\tau\bigr\}, +\] +where +\[ +A_{j} : B_{j} \text{ as } -\cos\epsilon : \sin\epsilon. +\] +In the solution $\phi(\tau)$ we have $\phi(0) = 1$, $\phi'(0) = 0$, and $\phi(\tau)$~is an even +function of~$\tau$. Hence to get~$\phi(\tau)$ from~$\delta p$ we require to put $\sum A_{j} = 1$, and +$B_{j} = 0$ for all values of~$j$. +\DPPageSep{116}{58} + +This gives +\begin{align*} +\phi(\pi) &= \sum \bigl\{A_{j} \cos(c + 2j)\pi\bigr\} \\ + &=\cos\pi c \sum A_{j} = \cos\pi c. +\end{align*} +Similarly we may shew that $\psi'(\pi) = \cos\pi c$. + +It follows from equations~\Eqref{(57)} that +\begin{align*} +\cos\pi c &= \phi(\pi) = \psi'(\pi),\\ +\cos^{2} \tfrac{1}{2}\pi c + &= \phi(\tfrac{1}{2}\pi)\psi'(\tfrac{1}{2}\pi);\quad +\sin^{2} \tfrac{1}{2}\pi c + = -\phi'(\tfrac{1}{2}\pi)\psi(\tfrac{1}{2}\pi). +\end{align*} + +We found on \Pageref{55} that $\sin^{2} \frac{1}{2}\pi c = \sin^{2} \frac{1}{2}\pi \sqrt{\Theta_{0}} · \Delta(0)$, where $\Delta(0)$~is a +certain determinant. Hence the last solution being of this form, we have +the value of the determinant~$\Delta(0)$ in terms of $\phi$~and~$\psi$, viz.\ +\[ +\Delta(0) = - \frac{\phi'(\frac{1}{2}\pi)\psi(\frac{1}{2}\pi)} + {\sin^{2} \frac{1}{2}\pi \sqrt{\Theta_{0}}}. +\] + +From this new way of looking at the matter it appears that the value of~$c$ +may be found by means of the two special solutions $\phi$~and~$\psi$. +\DPPageSep{117}{59} + + +\Chapter{On Librating Planets and on a New Family +of Periodic Orbits} +\SetRunningHeads{On Librating Planets}{and on a New Family of Periodic Orbits} + +\Section{§ 1. }{Librating Planets.} +\index{Brown, Prof.\ E. W., Sir George Darwin's Scientific Work!new family of periodic orbits}% +\index{Librating planets}% +\index{Periodic orbits, Darwin begins papers on!new family of}% + +\First{In} Professor Ernest Brown's interesting paper on ``A New Family of +Periodic Orbits'' (\textit{M.N.}, \textit{R.A.S.}, vol.~\Vol{LXXI.}, 1911, p.~438) he shews how to +obtain the orbit of a planet which makes large oscillations about the vertex +of the Lagrangian equilateral triangle. In discussing this paper I shall +depart slightly from his notation, and use that of my own paper on ``Periodic +Orbits'' (\textit{Scientific Papers}, vol.~\Vol{IV.}, or \textit{Acta Math.}, vol.~\Vol{LI.}). ``Jove,''~J, of +mass~$1$, revolves at distance~$1$ about the ``Sun,''~S, of mass~$\nu$, and the orbital +angular velocity is~$n$, where~$n^{2} = \nu + 1$. + +{\stretchyspace +The axes of reference revolve with SJ~as axis of~$x$, and the heliocentric +and jovicentric rectangular coordinates of the third body are $x, y$ and +$x - 1, y$ respectively. The heliocentric and jovicentric polar co-ordinates\DPnote{[** TN: Hyphenated in original]} are +respectively $r, \theta$ and $\rho, \psi$. The potential function for relative energy is~$\Omega$.} + +The equations of motion and Jacobian integral, from which Brown +proceeds, are +\[ +\left. +\begin{gathered} +\begin{aligned} +\frac{d^{2}r}{dt^{2}} + - r \frac{d\theta}{dt} \left(\frac{d\theta}{dt} + 2n\right) + &= \frac{\dd \Omega}{\dd r}\Add{,} \\ +% +\frac{d}{dt} \left[r^{2} \left(\frac{d\theta}{dt} + n\right)\right] + &= \frac{\dd \Omega}{\dd \theta}\Add{,} \\ +% +\left(\frac{dr}{dt}\right)^{2} + + \left(r \frac{d\theta}{dt}\right)^{2} &= 2\Omega - C\Add{,} +\end{aligned} \\ +\lintertext{where}{2\Omega + = \nu\left(r^{2} + \frac{2}{r}\right) + \left(\rho^{2} + \frac{2}{\rho}\right)\Add{,}} +\end{gathered} +\right\} +\Tag{(1)} +\] + +The following are rigorous transformations derived from those equations, +virtually given by Brown in approximate forms in equation~(13), and at the +foot of p.~443:--- +\DPPageSep{118}{60} +\begin{align*} +\left(\frac{d\theta}{dt} + n\right)^{2} + &= A + \frac{1}{r}\, \frac{d^{2}r}{dt^{2}}, +\Tag{(2)} +\Allowbreak +% +\frac{dr}{dt} \left(L + 3 \frac{d^{2}r}{dt^{2}}\right) + &= B + D \left(\frac{d\theta}{dt} + n\right) - r \frac{d^{3}r}{dt^{3}}, +\Tag{(3)} +\Allowbreak +% +\frac{d^{2}r}{dt^{2}} \left(L + 3 \frac{d^{2}r}{dt^{2}}\right) + &= E \left(\frac{dr}{dt}\right)^{2} + + F \frac{dr}{dt}\, \frac{d\theta}{dt} + + G \left(\frac{d\theta}{dt}\right)^{2} + + H \frac{dr}{dt} + J \frac{d\theta}{dt} + K \\ + &\qquad\qquad\qquad\qquad + - 4 \frac{dr}{dt}\, \frac{d^{3}r}{dt^{3}} - r \frac{d^{4}r}{dt^{4}}, +\Tag{(4)} +\end{align*} +where +\begin{align*} +A &= n^{2} - \frac{\dd \Omega}{r\, \dd r} + = \frac{\nu}{r^{3}} + 1 + - \frac{1}{r} \left(\rho - \frac{1}{\rho^{2}}\right)\cos(\theta-\psi), +\Allowbreak +% +B &= -nr \frac{\dd^{2}\Omega}{\dd r\, \dd \theta} + = -n \sin\psi \left[\left(\rho - \frac{1}{\rho^{2}}\right) + \frac{3r}{\rho^{3}} \cos(\theta - \psi)\right], +\Allowbreak +% +D &= r \frac{\dd^{2}\Omega}{\dd r\, \dd \theta} + 2 \frac{\dd \Omega}{\dd \theta} + = 3 \sin\psi \left[\left(\rho - \frac{1}{\rho^{2}}\right) + \frac{r}{\rho^{3}} \cos(\theta - \psi)\right], +\Allowbreak +% +%[** TN: Added break] +L &= 4n^2r - r \frac{\dd^{2} \Omega}{\dd r^{2}} - 3 \frac{\dd \Omega}{\dd r} \\ + &= \frac{\nu}{r^{2}} + 3r + \frac{r}{\rho^{3}} + - 3\left(\rho - \frac{1}{\rho^{2}}\right) \cos(\theta - \psi) + - \frac{3r}{\rho^{3}} \cos^{2}(\theta - \psi), +\Allowbreak +% +E &= r \frac{\dd^{3} \Omega}{\dd r^{3}} + 4 \frac{\dd^{2} \Omega}{\dd r^{2}} - 4n^{2} \\ + &= \frac{2\nu}{r^{3}} + \frac{4}{\rho^{3}}\bigl[3 \cos^{2}(\theta - \psi) - 1\bigr] % \\ +% ++ \frac{3r}{\rho^{4}} \cos(\theta - \psi) \bigl[3 - 5\cos^{2}(\theta - \psi) \bigr], \\ +% +F &= 2r \frac{\dd^{3} \Omega}{\dd r^{2}\, \dd \theta} + + 4\frac{\dd^{2} \Omega}{\dd r\, \dd \theta} - 4 \frac{\dd \Omega}{r\, \dd \theta} + = \frac{6}{\rho^{4}} \sin\psi \bigl[5r \sin^{2}(\theta - \psi) - 4\cos\theta\bigr], \\ +% +G &= r \frac{\dd^{3} \Omega}{\dd r\, \dd \theta^{2}} + 2\frac{\dd^{2} \Omega}{\dd \theta^{2}} + = \frac{3r}{\rho} \left[\left(\rho - \frac{1}{\rho^{2}}\right)\cos\theta + - \frac{r}{\rho^{3}} \sin\psi(5 \sin^{2}(\theta - \psi) - 1)\right], +\Allowbreak +% +H &= -\frac{4n}{r}\, \frac{\dd \Omega}{\dd \theta} + = 4n\left(\rho - \frac{1}{\rho^{2}}\right) \sin(\theta - \psi), \\ +% +J &= 2n \frac{\dd^{2} \Omega}{\dd \theta^{2}} + = \frac{2nr}{\rho} \left[\left(\rho - \frac{1}{\rho^{2}}\right) \cos\theta + - \frac{3}{\rho^{2}} \sin\psi \sin(\theta - \psi)\right], \\ +% +K &= \frac{\dd \Omega}{r^{2}\, \dd \theta} \left(r\frac{\dd^{2} \Omega}{\dd r\, \dd \theta} + 2\frac{\dd \Omega}{\dd \theta}\right) + = \frac{3}{r} \left(\rho - \frac{1}{\rho^{2}}\right) + \sin\theta \sin\psi \left(1 + \frac{1}{\rho^{4}}\cos\psi\right). +\end{align*} +A great diversity of forms might be given to these functions, but the foregoing +seemed to be as convenient for computations as I could devise. + +It is known that when $\nu$~is less than~$24.9599$\footnote + {``Periodic Orbits,'' \textit{Scientific Papers}, vol.~\Vol{IV.}, p.~73.} +the vertex of the equilateral +triangle is an unstable solution of the problem, and if the body is +displaced from the vertex it will move away in a spiral orbit. Hence for +small values of~$\nu$ there are no small closed periodic orbits of the kind +considered by Brown. But certain considerations led him to conjecture that +\DPPageSep{119}{61} +there might still exist large oscillations of this kind. The verification of +such a conjecture would be interesting, and in my attempt to test his idea +I took $\nu$~equal to~$10$. This value was chosen because the results will thus +form a contribution towards that survey of periodic orbits which I have made +in previous papers for $\nu$~equal to~$10$. + +Brown's system of approximation, which he justifies for large values of~$\nu$, +may be described, as far as it is material for my present object, as follows:--- + +We begin the operation at any given point~$r, \theta$, such that $\rho$~is greater +than unity. + +Then in \Eqref{(2)}~and~\Eqref{(3)} $\dfrac{d^{2}r}{dt^{2}}$ and $\dfrac{d^{3}r}{dt^{3}}$ are neglected, and we thence find +$\dfrac{dr}{dt}$,~$\dfrac{d\theta}{dt}$. + +By means of these values of the first differentials, and neglecting $\dfrac{d^{3}r}{dt^{3}}$ +and $\dfrac{d^{4}r}{dt^{4}}$ in~\Eqref{(4)}, we find~$\dfrac{d^{2}r}{dt^{2}}$ from~\Eqref{(4)}. + +Returning to \Eqref{(2)}~and~\Eqref{(3)} and using this value of~$\dfrac{d^{2}r}{dt^{2}}$, we re-determine the +first differentials, and repeat the process until the final values of $\dfrac{dr}{dt}$ and $\dfrac{d\theta}{dt}$ +remain unchanged. We thus obtain the velocity at this point~$r, \theta$ on the +supposition that $\dfrac{d^{3}r}{dt^{3}}$, $\dfrac{d^{4}r}{dt^{4}}$ are negligible, and on substitution in the last of~\Eqref{(1)} +we obtain the value of~$C$ corresponding to the orbit which passes through the +chosen point. + +Brown then shews how the remainder of the orbit may be traced with all +desirable accuracy in the case where $\nu$~is large. It does not concern me to +follow him here, since his process could scarcely be applicable for small values +of~$\nu$. But if his scheme should still lead to the required result, the remainder +of the orbit might be traced by quadratures, and this is the plan which +I have adopted. If the orbit as so determined proves to be clearly non-periodic, +it seems safe to conclude that no widely librating planets can exist +for small values of~$\nu$. + +I had already become fairly confident from a number of trials, which will +be referred to hereafter, that such orbits do not exist; but it seemed worth +while to make one more attempt by Brown's procedure, and the result appears +to be of sufficient interest to be worthy of record. + +For certain reasons I chose as my starting-point +\begin{alignat*}{2} +x_{0} &= -.36200,\quad& y_{0} &= .93441, \\ +\intertext{which give} +r_{0} &= 1.00205,& \rho_0 &= 1.65173. +\end{alignat*} +\DPPageSep{120}{62} +The successive approximations to~$C$ were found to be +\[ +33.6977,\quad 33.7285,\quad 33.7237,\quad 33.7246,\quad 33.7243. +\] +I therefore took the last value as that of~$C$, and found also that the direction +of motion was given by $\phi_{0} = 2°\,21'$. These values of $x_{0}, y_{0}, \phi_{0}$, and~$C$ then +furnish the values from which to begin the quadratures. + +\FigRef[Fig.]{1} shews the result, the starting-point being at~B. The curve was +traced backwards to~A and onwards to~C, and the computed positions are +shewn by dots connected into a sweeping curve by dashes. +\begin{figure}[hbt!] + \centering + \Input{p062} + \caption{Fig.~1. Results derived from Professor Brown's Method.} + \Figlabel{1} +\end{figure} + +From~A back to perijove and from~C on to~J the orbit was computed as +undisturbed by the Sun\footnotemark. Within the limits of accuracy adopted the body +\footnotetext{When the body has been traced to the neighbourhood of~J, let it be required to determine + its future position on the supposition that the solar perturbation is negligible. Since the axes + of reference are rotating, the solution needs care, and it may save the reader some trouble if I set + down how it may be done conveniently. + + Let the coordinates, direction of motion, and velocity, at the moment $t = 0$ when solar + perturbation is to be neglected, be given by $x_{0}, y_{0}$ (or $r_{0}, \theta_{0}$, and $\rho_{0}, \psi_{0}$), $\phi_{0}, V_{0}$; and generally + let the suffix~$0$ to any symbol denote its value at this epoch. Then the mean distance~$\a$, mean + motion~$\mu$, and eccentricity~$e$ are found from + \begin{gather*} + \frac{1}{\a} + = \frac{2}{\rho_{0}} + - \bigl[V_{0}^{2} + 2\pi \rho_{0} V_{0} \cos(\phi_{0} + - \psi_{0}) + n^{2} \rho_{0}^{2}\bigr],\quad + \mu^{2} \a^{3} = 1, \\ + % + \a (1 - e^{2}) + = \bigl[V_{0} \rho_{0} \cos(\phi_{0} - \psi_{0}) + n \rho_0^{2}\bigr]^{2}. + \end{gather*} + Let $t = \tau$ be the time of passage of perijove, so that when $\tau$~is positive perijove is later than the + epoch $t = 0$. + + At any time~$t$ let $\rho, v, E$ be radius vector, true and eccentric anomalies; then + \begin{align*} + \rho &= \a(1 - e \cos E), \\ + \rho^{\frac{1}{2}} \cos \tfrac{1}{2} v + &= \a^{\frac{1}{2}}(1 - e)^{\frac{1}{2}}\cos \tfrac{1}{2} E, \\ +% + \rho^{\frac{1}{2}} \sin \tfrac{1}{2} v + &= \a^{\frac{1}{2}}(1 + e)^{\frac{1}{2}}\sin \tfrac{1}{2} E, \\ +% + \mu(t - \tau) &= E - e \sin E, \\ + \psi &= \psi_{0}- v_{0} + v - nt. + \end{align*} + + On putting $t = 0$, $E_{0}$~and~$\tau$ may be computed from these formulae, and it must be noted that + when $\tau$~is positive $E_{0}$~and~$v_{0}$ are to be taken as negative. + + The position of the body as it passes perijove is clearly given by + \[ + x - 1 = \a(1 - e)\cos(\psi_{0} - v_{0} - n\tau),\quad + y = \a(1 - e)\sin(\psi_{0} - v_{0} - n\tau). + \] + Any other position is to be found by assuming a value for~$E$, computing $\rho, v, t, \psi$, and using the + formulae + \[ + x - 1 = \rho \cos\psi,\quad y = \rho \sin\psi. + \] + + In order to find $V$~and~$\phi$ we require the formulae + \[ + \frac{1}{\rho}\, \frac{d\rho}{dt} = \frac{\a e\sin E}{\rho} · \frac{\mu \a}{\rho};\quad + \frac{dv}{dt} = \frac{\bigl[\a(1 - e^{2})\bigr]^{\frac{1}{2}}}{\rho} · \frac{\a^{\frac{1}{2}}}{\rho} · \frac{\mu \a}{\rho}, \\ + \] + and + \begin{align*} + V\sin \phi + &= -\frac{(x - 1)}{\rho}\, \frac{d\rho}{dt} + + y\left(\frac{dv}{dt} - n\right), \\ + % + V\cos \phi &= \Neg\frac{y}{\rho}\, \frac{d\rho}{dt} + + (x - 1) \left(\frac{dv}{dt} - n\right). + \end{align*} + + The value of~$V$ as computed from these should be compared with that derived from + \[ + V^{2} = \nu\left(r^{2} + \frac{2}{r}\right) + + \left(\rho^{2} + \frac{2}{\rho}\right) - C, + \] + and if the two agree pretty closely, the assumption as to the insignificance of solar perturbation + is justified. + + If the orbit is retrograde about~J, care has to be taken to use the signs correctly, for $v$~and~$E$ + will be measured in a retrograde direction, whereas $\psi$~will be measured in the positive direction. + + A similar investigation is applicable, \textit{mutatis mutandis}, when the body passes very close to~S\@.}% +collides with~J\@. +\DPPageSep{121}{63} + +Since the curve comes down on to the negative side of the line of syzygy~SJ +it differs much from Brown's orbits, and it is clear that it is not periodic. +Thus his method fails, and there is good reason to believe that his conjecture +is unfounded. + +After this work had been done Professor Brown pointed out to me in +a letter that if his process be translated into rectangular coordinates, the +approximate expressions for $dx/dt$~and~$dy/dt$\DPnote{** slant fractions} will have as a divisor the +function +\[ +Q = \left(4n^{2} - \frac{\dd^{2} \Omega}{\dd x^{2}}\right) + \left(4n^{2} - \frac{\dd^{2} \Omega}{\dd y^{2}}\right) + - \left(\frac{\dd^{2} \Omega}{\dd x\, \dd y}\right)^{2}. +\] +The method will then fail if~$Q$ vanishes or is small. +\DPPageSep{122}{64} + +I find that if we write $\Gamma = \dfrac{\nu}{r^{3}} + \dfrac{1}{\rho^{3}}$, the divisor may be written in the +form +\[ +Q = (3n^{2} + \Gamma)(3n^{2} - 2\Gamma) + \frac{\rho \nu}{r^{5}\rho^{5}} \sin\theta \sin\psi. +\] + +Now, Mr~T.~H. Brown, Professor Brown's pupil, has traced one portion of +the curve $Q = 0$, corresponding to $\nu = 10$, and he finds that it passes rather +near to the orbit I have traced. This confirms the failure of the method +which I had concluded otherwise. + +\Section{§ 2. }{Variation of an Orbit.} +\index{Orbit, variation of an}% +\index{Variation, the!of an orbit}% + +A great difficulty in determining the orbits of librating planets by +quadratures arises from the fact that these orbits do not cut the line of +syzygies at right angles, and therefore the direction of motion is quite indeterminate +at every point. I endeavoured to meet this difficulty by a method +of variation which is certainly feasible, but, unfortunately, very laborious. +In my earlier attempts I had drawn certain orbits, and I attempted to utilise +the work by the method which will now be described. + +The stability of a periodic orbit is determined by varying the orbit. The +form of the differential equation which the variation must satisfy does not +depend on the fact that the orbit is periodic, and thus the investigation in +§§~8,~9 of my paper on ``Periodic Orbits'' remains equally true when the +varied orbit is not periodic. + +Suppose, then, that the body is displaced from a given point of a non-periodic +orbit through small distances $\delta q\, V^{-\frac{1}{2}}$ along the outward normal and +$\delta s$~along the positive tangent, then we must have +\begin{gather*} +\frac{d^{2}\, \delta q}{ds^{2}} + \Psi\, \delta q = 0, \\ +% +\frac{d}{ds}\left(\frac{\delta s}{V}\right) + = -\frac{2\delta q}{V^{\frac{3}{2}}} \left(\frac{1}{R} + \frac{n}{V}\right), +\end{gather*} +where +\[ +\Psi = \frac{5}{2} \left(\frac{1}{R} + \frac{n}{V}\right)^{2} + - \frac{3}{2V^{2}} \left[\frac{\nu}{r^{3}}\cos^{2}(\phi - \theta) + + \frac{1}{\rho^{3}}\cos^{2}(\phi - \psi)\right] + + \frac{3}{4} \left(\frac{dV}{V\, ds}\right)^{2}, +\] +and +\[ +\frac{dV}{V\, ds} + = \frac{\nu}{V^{2}} \left(\frac{1}{r^{2}} - r\right)\sin(\phi - \theta) + + \frac{1}{V^{2}} \left(\frac{1}{\rho^{2}} - \rho\right) \sin(\phi - \psi). +\] +Also +\[ +\delta \phi = -\frac{1}{V^{\frac{1}{2}}} + \left[\frac{d\, \delta q}{ds} + - \tfrac{1}{2}\, \delta q \left(\frac{dV}{V\, ds}\right)\right] + + \frac{\delta s}{R}. +\] +\DPPageSep{123}{65} + +Since it is supposed that the coordinates, direction of motion, and radius +of curvature~$R$ have been found at a number of points equally distributed +along the orbit, it is clear that $\Psi$~may be computed for each of those +points. + +At the point chosen as the starting-point the variation may be of two +kinds:--- +\begin{alignat*}{2} +(1)\quad \delta q_0 &= \a, \qquad + \frac{d\delta q_{0}}{ds} &&= 0, \text{ where $\a$ is a constant}, \\ +% +(2)\quad \delta q_0 &=0, \qquad + \frac{d\delta q_{0}}{ds} &&= b, \text{ where $b$ is a constant}. +\end{alignat*} +Each of these will give rise to an independent solution, and if in either of +them $\a$~or~$b$ is multiplied by any factor, that factor will multiply all the +succeeding results. It follows, therefore, that we need not concern ourselves +with the exact numerical values of $\a$~or~$b$, but the two solutions will give us +all the variations possible. In the first solution we start parallel with the +original curve at the chosen point on either side of it, and at any arbitrarily +chosen small distance. In the second we start from the chosen point, but at +any arbitrary small inclination on either side of the original tangent. + +The solution of the equations for $\delta q$~and~$\delta s$ have to be carried out step by +step along the curve, and it may be worth while to indicate how the work +may be arranged. + +The length of arc from point to point of the unvaried orbit may be +denoted by~$\Delta s$, and we may take four successive values of~$\Psi$, say $\Psi_{n-1}, +\Psi_{n}, \Psi_{n+1}, \Psi_{n+2}$, as affording a sufficient representation of the march +of the function~$\Psi$ throughout the arc~$\Delta s$ between the points indicated by +$n$~to~$n+1$. + +If the differential equation for~$\delta q$ be multiplied by~$(\Delta s)^{2}$, and if we +introduce a new independent variable~$z$ such that~$dz = ds/\Delta s$,\DPnote{** slant fractions} and write +$X = \Psi(\Delta s)^{2}$, the equation becomes +\[ +\frac{d^{2}\, \delta q}{dz^{2}} = -X\, \delta q, +\] +and $z$~increases by unity as the arc increases by~$\Delta s$. + +Suppose that the integration has been carried as far as the point~$n$, and +that $\delta q_{0}, d\, \delta q_{0}/dz$ are the values at that point; then it is required to find $\delta q_{1}, +d\, \delta q_{1}/dz$ at the point~$n + 1$. + +If the four adjacent values of~$X$ are $X_{-1}, X_{0}, X_{1}, X_{2}$, and if +\[ +\delta_{1} = X_{1} - X_{0},\quad +\delta_{2} = \tfrac{1}{2} \bigl[(X_{2} - 2X_{1} + X_{0}) + (X_{1} - 2X_{0} + X_{-1})\bigr], +\] +Bessel's formula for the function~$X$ is +\[ +X = X_{0} + (\delta_{1} - \tfrac{1}{2}\delta_{2})z + + \tfrac{1}{2}\delta_{2}z^{2}\DPtypo{}{.} +\] +\DPPageSep{124}{66} +We now assume that throughout the arc $n$~to~$n + 1$, +\[ +\delta q = \delta q_{0} + \frac{d\, \delta q_{0}}{dz} z + + Q_{2} z^{2} + Q_{3} z^{3} + Q_{4} z^{4}, +\] +where $Q_{2}, Q_{3}, Q_{4}$ have to be determined so as to satisfy the differential +equation. + +On forming the product~$X\, \delta q$, integrating, and equating coefficients, we +find $Q_{2} = -\frac{1}{2} X_{0}\, \delta q_{0}$, and the values of~$Q_{3}, Q_{4}$ are easily found. In carrying out +this work I neglect all terms of the second order except~$X_{0}^{2}$. + +\pagebreak[1] +The result may be arranged as follows:---\pagebreak[0] \\ +Let +\begin{align*} +A &= 1 - \tfrac{1}{2} X_{0} - \tfrac{1}{6}\delta_{1} + + \tfrac{1}{24} (\delta_{2} + X_{0}^{2}), \\ +% +B &= 1 - \tfrac{1}{6} X_{0} - \tfrac{1}{12} \delta_{1} + \tfrac{1}{24} \delta_{2}, \\ +% +C &= X_{0} + \tfrac{1}{2} \delta_{1} + \tfrac{1}{12} \delta_{2} - \tfrac{1}{6} (\delta_{2} + X_{0}^{2}), \\ +% +D &= 1 - \tfrac{1}{2} X_{0} - \tfrac{1}{8} \delta_{1} + \tfrac{1}{6} \delta_{2}; +\end{align*} +then, on putting $z =1$, we find +\begin{align*} +\delta q_{1} &= \Neg A\, \delta q_{0} + B \frac{d\, \delta q_{0}}{dz}, \\ +\frac{d\, \delta q_{1}}{dz} &= -C\, \delta q_{0} + D \frac{d\, \delta q_{0}}{dz}. +\end{align*} + +When the~$\Psi$'s have been computed, the~$X$'s and $A, B, C, D$ are easily +found at each point of the unvaried orbit. We then begin the two solutions +from the chosen starting-point, and thus trace $\delta q$~and~$d\, \delta q/dz$ from point to +point both backwards and forwards. The necessary change of procedure when +$\Delta s$~changes in magnitude is obvious. + +The procedure is tedious although easy, but the work is enormously +increased when we pass on further to obtain an intelligible result from the +integration. When $\delta q$~and~$d\, \delta q/dz$ have been found at each point, a further +integration has to be made to determine~$\delta s$, and this has, of course, to be done +for each of the solutions. Next, we have to find the normal displacement~$\delta p$ +(equal to~$\delta q\, V^{-\frac{1}{2}}$), and, finally, $\delta p, \delta s$~have to be converted into rectangular +displacements~$\delta x, \delta y$. + +The whole process is certainly very laborious; but when the result is +attained it does furnish a great deal of information as to the character of the +orbits adjacent to the orbit chosen for variation. I only carried the work +through in one case, because I had gained enough information by this single +instance. However, it does not seem worth while to record the numerical +results in that case. + +In the variation which has been described, $C$~is maintained unchanged, +\DPPageSep{125}{67} +but it is also possible to vary~$C$. If $C$~becomes $C + \delta C$, it will be found that +the equations assume the form +\begin{align*} +\frac{d^{2}\, \delta q}{ds^{2}} + \Psi\, \delta q + + \frac{\delta C}{V^{\frac{3}{2}}} \left(\frac{1}{R} + \frac{n}{V}\right) &= 0, \\ +% +\frac{d}{ds}\left(\frac{\delta s}{V}\right) + + \frac{2\delta q}{V^{\frac{3}{2}}} \left(\frac{1}{R} + \frac{n}{V}\right) + + \frac{\delta C}{2V^{2}} &= 0. +\end{align*} + +But this kind of variation cannot be used with much advantage, for +although it is possible to evaluate $\delta q$~and~$\delta s$ for specific initial values of~$\delta C, +\delta q, d\, \delta q/ds$ at a specific initial point, only one single varied orbit is so deducible. +In the previous case we may assign any arbitrary values, either positive or +negative, to the constants denoted by $\a$~and~$b$, and thus find a group of varied +orbits. + +\Section{§ 3. }{A New Family of Periodic Orbits.} +\index{Periodic orbits, Darwin begins papers on!new family of}% + +In attempting to discover an example of an orbit of the kind suspected +by Brown, I traced a number of orbits. Amongst these was that one which +was varied as explained in~\SecRef{2}, although when the variation was effected I did +not suspect it to be in reality periodic in a new way. It was clear that it +could not be one of Brown's orbits, and I therefore put the work aside and +made a fresh attempt, as explained in~\SecRef{1}. Finally, for my own satisfaction, +I completed the circuit of this discarded orbit, and found to my surprise that +it belonged to a new and unsuspected class of periodic. The orbit in question +is that marked~$33.5$ in \FigRef{3}, where only the half of it is drawn which lies on +the positive side of~SJ\@. + +It will be convenient to use certain terms to indicate the various parts +of the orbits under discussion, and these will now be explained. Periodic +orbits have in reality neither beginning nor end; but, as it will be convenient +to follow them in the direction traversed from an orthogonal crossing of the +line of syzygies, I shall describe the first crossing as the ``beginning'' and the +second orthogonal crossing of~SJ as the ``end.'' I shall call the large curve +surrounding the apex of the Lagrangian equilateral triangle the ``loop,'' and +this is always described in the clockwise or negative direction. The portions +of the orbit near~J will be called the ``circuit,'' or the ``half-'' or ``quarter-circuit,'' +as the case may be. The ``half-circuits'' about~J are described +counter-clockwise or positively, but where there is a complete ``circuit'' it is +clockwise or negative. For example, in \FigRef{3} the orbit~$33.5$ ``begins'' with +a positive quarter-circuit, passes on to a negative ``loop,'' and ``ends'' in a +positive quarter-circuit. Since the initial and final quarter-circuits both cut~SJ +at right angles, the orbit is periodic, and would be completed by a similar +curve on the negative side of~SJ\@. In the completed orbit positively described +\DPPageSep{126}{68} +half-circuits are interposed between negative loops described alternately on +the positive and negative sides of~SJ\@. + +%[** TN: Moved up two paragraphs to accommodate pagination] +\begin{figure}[hbt!] + \centering + \Input{p068} + \caption{Fig.~2. Orbits computed for the Case of $C = 33.25$.} + \Figlabel{2} +\end{figure} +Having found this orbit almost by accident, it was desirable to find other +orbits of this kind; but the work was too heavy to obtain as many as is +desirable. There seems at present no way of proceeding except by conjecture, +and bad luck attended the attempts to draw the curve when $C$~is~$33.25$. The +various curves are shewn in \FigRef{2}, from which this orbit may be constructed +with substantial accuracy. + +In \FigRef{2} the firm line of the external loop was computed backwards, +starting at right angles to~SJ from $x = .95$, $y=0$, the point to which $480°$~is +attached. After the completion of the loop, the curve failed to come down +close to~J as was hoped, but came to the points marked $10°$~and~$0°$. The +``beginnings'' of two positively described quarter-circuits about~J are shewn +as dotted lines, and an orbit of ejection, also dotted, is carried somewhat +further. Then there is an orbit, shewn in firm line, ``beginning'' with a +negative half-circuit about~J, and when this orbit had been traced half-way +through its loop it appeared that the body was drawing too near to the curve +of zero velocity, from which it would rebound, as one may say. This orbit is +continued in a sense by a detached portion starting from a horizontal tangent +at $x = .2$, $y = 1.3$. It became clear ultimately that the horizontal tangent +ought to have been chosen with a somewhat larger value for~$y$. From these +\DPPageSep{127}{69} +attempts it may be concluded that the periodic orbit must resemble the +broken line marked as conjectural, and as such it is transferred to \FigRef{3} and +shewn there as a dotted curve. I shall return hereafter to the explanation +of the degrees written along these curves. + +Much better fortune attended the construction of the orbit~$33.75$ shewn +in \FigRef{3}, for, although the final perijove does not fall quite on the line of +syzygies, yet the true periodic orbit can differ but little from that shewn. +It will be noticed that in this case the orbit ``ends'' with a negative half-circuit, +and it is thus clear that if we were to watch the march of these +\begin{figure}[hbt!] + \centering + \Input{p069} + \caption{Fig.~3. Three Periodic Orbits.} + \Figlabel{3} +\end{figure} +orbits as $C$~falls from~$33.75$ to~$33.5$ we should see the negative half-circuit +shrink, pass through the ejectional stage, and emerge as a positive quarter-circuit +when $C$~is~$33.5$. + +The three orbits shewn in \FigRef{3} are the only members of this family that +I have traced. It will be noticed that they do not exhibit that regular +progress from member to member which might have been expected from the +fact that the values of~$C$ are equidistant from one another. It might be +suspected that they are really members of different families presenting similar +characteristics, but I do not think this furnishes the explanation. +\DPPageSep{128}{70} + +In describing the loop throughout most of its course the body moves +roughly parallel to the curve of zero velocity. For the values of~$C$ involved +here that curve is half of the broken horse-shoe described in my paper on +``Periodic Orbits'' (\textit{Scientific Papers}, vol.~\Vol{IV.}, p.~11, or \textit{Acta Math.}, vol.~\Vol{XXI.} +(1897)). Now, for $\nu = 10$ the horse-shoe breaks when $C$~has fallen to~$34.91$, +and below that value each half of the broken horse-shoe, which delimits the +forbidden space, shrinks. Now, since the orbits follow the contour of the +horse-shoe, it might be supposed that the orbits would also shrink as $C$~falls +in magnitude. On the other hand, as $C$~falls from~$33.5$ to~$33.25$, our figures +shew that the loop undoubtedly increases in size. This latter consideration +would lead us to conjecture that the loop for~$33.75$ should be smaller than +that for~$33.5$. Thus, looking at the matter from one point of view, we should +expect the orbits to shrink, and from another to swell as $C$~falls in value. +It thus becomes intelligible that neither conjecture can be wholly correct, +and we may thus find an explanation of the interlacing of the orbits as shewn +in my \FigRef{3}. + +It is certain from general considerations that families of orbits must +originate in pairs, and we must therefore examine the origin of these orbits, +and consider the fate of the other member of the pair. + +It may be that for values of~$C$ greater than~$33.75$ the initial positive +quarter-circuit about~J is replaced by a negative half-circuit; but it is +unnecessary for the present discussion to determine whether this is so or not, +and it will suffice to assume that when $C$~is greater than~$33.75$ the ``beginning'' +is as shewn in my figure. The ``end'' of~$33.75$ is a clearly marked negative +half-circuit, and this shews that the family originates from a coalescent pair of +orbits ``ending'' in such a negative half-circuit, with identical final orthogonal +crossing of~SJ in which the body passes from the negative to the positive +side of~SJ\@. + +This coalescence must occur for some critical value of~$C$ between $34.91$ +and~$33.75$, and it is clear that as $C$~falls below that critical value one +of the ``final'' orthogonal intersections will move towards~S and the other +towards~J. + +In that one of the pair for which the intersection moves towards~S the +negative circuit increases in size; in the other in which it moves towards~J +the circuit diminishes in size, and these are clearly the orbits which have +been traced. We next see that the negative circuit vanishes, the orbit +becomes ejectional, and the motion about~J both at ``beginning'' and ``end'' +has become positive. + +It may be suspected that when $C$~falls below~$33.25$ the half-circuits +round~J increase in magnitude, and that the orbit tends to assume the +form of a sort of asymmetrical double figure-of-8, something like the figure +\DPPageSep{129}{71} +which Lord Kelvin drew as an illustration of his graphical method of curve-tracing\footnotemark. +\footnotetext{\textit{Popular Lectures}, vol.~\Vol{I.}, 2nd~ed., p.~31; \textit{Phil.\ Mag.}, vol.~\Vol{XXXIV.}, 1892, p.~443.}% + +In the neighbourhood of Jove the motion of the body is rapid, but the +loops are described very slowly. The number of degrees written along the +curves in \FigRef{2} represent the angles turned through by Jove about the Sun +since the moment corresponding to the position marked~$0°$. Thus the firm +line which lies externally throughout most of the loop terminates with~$480°$. +Since this orbit cuts~SJ orthogonally, it may be continued symmetrically on +the negative side of~SJ, and therefore while the body moves from the point~$0°$ +to a symmetrical one on the negative side Jove has turned through~$960°$ round +the Sun, that is to say, through $2\frac{2}{3}$~revolutions. + +Again, in the case of the orbit beginning with a negative half-circuit, +shewn as a firm line, Jove has revolved through~$280°$ by the time the point +so marked is reached. We may regard this as continued in a sense by the +detached portion of an orbit marked with~$0°, 113°, 203°$; and since $280° + 203°$ +is equal to~$483°$, we again see that the period of the periodic orbit must be +about~$960°$, or perhaps a little more. + +In the cases of the other orbits more precise values may be assigned. For +$C = 33.5$, the angle~$nT$ (where $T$~is the period) is~$1115°$ or $3.1$~revolutions of +Jove; and for $C = 33.75$, $nT$~is~$1235°$ or $3.4$~revolutions. + +It did not seem practicable to investigate the stability of these orbits, but +we may suspect them to be unstable. + +The numerical values for drawing the orbits $C = 33.5$ and~$33.75$ are given +in an appendix, but those for the various orbits from which the conjectural +orbit $C = 33.25$ is constructed are omitted. I estimate that it is as laborious +to trace one of these orbits as to determine fully half a dozen of the simpler +orbits shewn in my earlier paper. + +Although the present contribution to our knowledge is very imperfect, +yet it may be hoped that it will furnish the mathematician with an +intimation worth having as to the orbits towards which his researches must +lead him. + +The librating planets were first recognised as small oscillations about the +triangular positions of Lagrange, and they have now received a very remarkable +extension at the hands of Professor Brown. It appears to me that the +family of orbits here investigated possesses an interesting relationship to +these librating planets, for there must be orbits describing double, triple, +and multiple loops in the intervals between successive half-circuits about +Jove. Now, a body which describes its loop an infinite number of times, +\DPPageSep{130}{72} +before it ceases to circulate round the triangular point, is in fact a librating +planet. It may be conjectured that when the Sun's mass~$\nu$ is yet smaller +than~$10$, no such orbit as those traced is possible. When $\nu$~has increased +to~$10$, probably only a single loop is possible; for a larger value a double loop +may be described, and then successively more frequently described multiple +loops will be reached. When $\nu$~has reached~$24.9599$ a loop described an +infinite number of times must have become possible, since this is the smallest +value of~$\nu$ which permits oscillation about the triangular point. If this idea +is correct, and if $\mathrm{N}$~denotes the number expressing the multiplicity of the +loop, then as $\nu$~increases $d\mathrm{N}/d\nu$~must tend to infinity; and I do not see why +this should not be the case. + +These orbits throw some light on cosmogony, for we see how small planets +with the same mean motion as Jove in the course of their vicissitudes tend +to pass close to Jove, ultimately to be absorbed into its mass. We thus see +something of the machinery whereby a large planet generates for itself a clear +space in which to circulate about the Sun. + +My attention was first drawn to periodic orbits by the desire to discover +how a Laplacian ring could coalesce into a planet. With that object in view +I tried to discover how a large planet would affect the motion of a small one +moving in a circular orbit at the same mean distance. After various failures +the investigation drifted towards the work of Hill and Poincaré, so that the +original point of view was quite lost and it is not even mentioned in my paper +on ``Periodic Orbits.'' It is of interest, to me at least, to find that the original +aspect of the problem has emerged again. + +\Appendix{Numerical results of Quadratures.} + +\Heading{$C = 33.5$.} + +\noindent\begin{minipage}{\textwidth} +\centering\footnotesize +\settowidth{\TmpLen}{Time from perijove up to $s=-2.1$ is given by $nt=9°\, 25'$.}% +\parbox{\TmpLen}{Perijove $x_0=1.0171$, $y_0=-.0034$, taken as zero. \\ +Time from perijove up to $s=-2.1$ is given by $nt=9°\, 25'$.} +\end{minipage} +\begin{longtable}{.{1,3}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} ,{6,2}!{\extracolsep{\fill}} .{2,3}}\\ +\ColHead{s} & \ColHead{x} & \ColHead{y} & \ColHead{\Z\Z\Z\Z\phi} & \ColHead{2n/V}\\ + & & & \ColHead{\AngleHeading} & \\ +\endhead +-2.1 &+\Z.8282&+\Z.0980& {}+66, 10 & 2.408 \\ + 2.0 & .7409 & .1467 & 55, 53 & 2.829 \\ + 1.9 & .6625 & .2084 & 48, 36 & 2.876 \\ + 1.8 & .5894 & .2766 & 46,\Z3 & 2.768 \\ + 1.7 & .5171 & .3457 & 46, 55 & 2.655 \\ + 1.6 & .4425 & .4124 & 49, 46 & 2.584 \\ + 1.5 & .3641 & .4744 & 53, 39 & 2.568 \\ +-1.4 &+\Z.2814&+\Z.5306& {}+57, 56 & 2.613 \\ +\DPPageSep{131}{73} +-1.3 &+\Z.1948&+\Z.5805& {}+62,\Z8 & 2.728 \\ + 1.2 & .1049 & .6243 & 65, 51 & 2.930 \\ + 1.1 &+\Z.0126& .6628 & 68, 38 & 3.251 \\ + 1.0 &-\Z.0810& .6979 & 69, 46 & 3.760 \\ + .9 & .1747 & .7330 & 68,\Z7 & 4.598 \\ + .85 & .2207 & .7526 & 65, 13 & 5.240 \\ + .8 & .2653 & .7754 & 60,\Z1 & 6.133 \\ + .75 & .3068 & .8035 & 50, 51 & 7.377 \\ + .725& .3252 & .8203 & 44,\Z2 & 8.139 \\ + .7 & .3412 & .8395 & 35, 17 & 8.944 \\ + .675& .3537 & .8611 & 24, 33 & 9.664 \\ + .65 & .3617 & .8848 & 12, 27 & 10.129 \\ + .625& .3644 & .9096 & {}+\Z0, 13 & 10.224 \\ + .6 & .3620 & .9344 & {}-10, 56 & 10.009 \\ + .575& .3552 & .9584 & 20, 31 & 9.655 \\ + .55 & .3448 & .9811 & 28, 30 & 9.205 \\ + .5 & .3161 & 1.0220 & 40, 48 & 8.448 \\ + .45 & .2806 & 1.0571 & 49, 38 & 7.872 \\ + .4 & .2405 & 1.0869 & 56, 51 & 7.460 \\ + .3 & .1518 & 1.1326 & 68,\Z4 & 6.961 \\ + .2 &-\Z.0565& 1.1626 & 76, 47 & 6.730 \\ +-\Z.1 &+\Z.0421& 1.1791 & 83, 58 & 6.647 \\ + .0 & .1419 & 1.1842 & {}-90,\Z0 & 6.633 \\ ++\Z.05& .1919 & 1.1830 & 180°+87, 21 & 6.630 \\ + .1 & .2418 & 1.1797 & 84, 54 & 6.626 \\ + .15 & .2915 & 1.1742 & 82, 38 & 6.609 \\ + .2 & .3410 & 1.1669 & 80, 31 & 6.572 \\ + .3 & .4389 & 1.1470 & 76, 31 & 6.432 \\ + .4 & .5353 & 1.1203 & 72, 33 & 6.201 \\ + .5 & .6295 & 1.0869 & 68, 16 & 5.912 \\ + .6 & .7208 & 1.0461 & 63, 29 & 5.605 \\ + .7 & .8081 & .9974 & 58,\Z8 & 5.313 \\ + .8 & .8902 & .9404 & 52, 12 & 5.055 \\ + .9 & .9656 & .8748 & 45, 39 & 4.842 \\ + 1.0 & 1.0326 & .8006 & 38, 22 & 4.671 \\ + 1.1 & 1.0889 & .7181 & 30, 11 & 4.540 \\ + 1.2 & 1.1321 & .6280 & 20, 46 & 4.435 \\ + 1.3 & 1.1585 & .5318 & \Z9, 38 & 4.326 \\ + 1.35 & 1.1642 & .4821 &180°+\Z3, 16 & 4.250 \\ + 1.4 & 1.1641 & .4322 &180°-\Z3, 40 & 4.141 \\ + 1.45 & 1.1577 & .3826 & 11,\Z5 & 3.983 \\ + 1.5 & 1.1448 & .3343 & 18, 44 & 3.758 \\ + 1.55 & 1.1257 & .2881 & 26,\Z8 & 3.460 \\ + 1.6 & 1.1011 & .2446 & 32, 39 & 3.100 \\ + 1.65 & 1.0723 & .2038 & 37, 33 & 2.701 \\ + 1.7 & 1.0408 & .1650 & 40,\Z4 & 2.291 \\ ++1.75 &+1.0087 &+\Z.1267& 180°-39, 12 & 1.893 \\ +\end{longtable} +\noindent\begin{minipage}{\textwidth} +\centering\footnotesize +\settowidth{\TmpLen}{Time from $s = 1.75$ to perijove given by $nt = 5°58'$.}% +\parbox{\TmpLen}{Time from $s = 1.75$ to perijove given by $nt = 5°58'$. \\ +Coordinates of perijove $x = .9501$, $y = -.0029$.} +\end{minipage} +\DPPageSep{132}{74} + +The following additional positions were calculated backwards from a perijove at +$x = .95$, $y = 0$, $\phi = 180°$. + +\[ +\begin{array}{.{1,4} c<{\qquad} .{1,4} c<{\qquad} ,{6,2}} +\ColHead{x} && \ColHead{y} && \ColHead{\Z\Z\Z\Z\phi} \\ + && && \ColHead{\AngleHeading} \\ ++\Z.9500 && +.0000 && 180°+\Z0, \Z0 \\ + .9512 && .0531 && 180°- 22, 30 \\ + .9647 && .0797 && 30, 52 \\ + .9756 && .0966 && 34, 48 \\ + .9874 && .1127 && 37, 37 \\ + 1.0128 && .1436 && 40, 37 \\ + 1.0390 && .1738 && 40, 56 \\ + 1.0649 && .2043 && 39, 12 \\ + 1.0893 && .2360 && 35, 51 \\ + 1.1114 && .2693 && 31, 16 \\ + 1.1463 && .3412 && 20, 10 \\ ++ 1.1661 && +.4186 && 180°-\Z8, 40 \\ +\end{array} +\] + +This supplementary orbit becomes indistinguishable in a figure of moderate size from +the preceding orbit, which is therefore accepted as being periodic. The period is given by +$nT = 1115°.4 = 3.1$ revolutions of Jove. + +\Heading{$C = 33.75$.} + +This orbit was computed from a conjectural starting-point which seemed likely to lead +to the desired result; the computation was finally carried backwards from the starting-point. +The coordinates of perijove were found to be $x_{0} = 1.0106$, $y_{0} = .0006$, which may be +taken as virtually on the line of syzygies. The motion from perijove is direct. + +\begin{longtable}{.{1,3}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} .{1,4}!{\extracolsep{\fill}} ,{6,2}!{\extracolsep{\fill}} .{2,3}}\\ +\ColHead{s} & \ColHead{x} & \ColHead{y} & \ColHead{\Z\Z\Z\Z\phi} & \ColHead{2n/V}\\ + & & & \ColHead{\AngleHeading} & \\ +\endhead +\ColHead{\text{perijove}} + &+1.0106 &+\Z.0006& \Z0, \Z0& \ColHead{\text{very nearly}} \\ +-\Z.3 & .9652 & .0403 & 66, 38 & 1.140 \\ +-\Z.3 & .9184 & .0578 & 71, \Z6& 1.635 \\ +-\Z.2 & .8713 & .0744 & 69, 27 & 2.075 \\ +-\Z.2 & .8251 & .0936 & 65, \Z3& 2.447 \\ +-\Z.1 & .7391 & .1444 & 54, 15 & 2.882 \\ + 0.0 & .6625 & .2084 & 47, \Z0& 2.946 \\ + .1 & .5911 & .2785 & 44, 44 & 2.850 \\ + .2 & .5202 & .3490 & 46, \Z0& 2.749 \\ + .3 & .4465 & .4165 & 49, 13 & 2.686 \\ + .4 & .3685 & .4791 & 53, 29 & 2.675 \\ + .5 & .2858 & .5352 & 58, 10 & 2.723 \\ + .6 & .1987 & .5844 & 62, 52 & 2.838 \\ + .7 & .1081 & .6265 & 67, 13 & 3.036 \\ + .8 &+\Z.0147& .6622 & 70, 49 & 3.348 \\ + .9 &-\Z.0805& .6929 & 73, 11 & 3.834 \\ + 1.0 & .1764 & .7213 & 73, 25 & 4.631 \\ + 1.1 & .2713 & .7525 & 69, 17 & 6.090 \\ + 1.15 & .3173 & .7721 & 63, 50 & 7.333 \\ + 1.2 & .3601 & .7977 & 53, 25 & 9.236 \\ + 1.225 & .3791 & .8140 & 45, \Z6& 10.360 \\ + 1.25 &-\Z.3951&+\Z.8332& 33, 54 & 11.840 \\ +\DPPageSep{133}{75} + 1.275 &-\Z.4064&+\Z.8553& {}+19, 53 & 12.955 \\ + 1.3 & .4118 & .8796 & {}+\Z4, 42& 13.412 \\ + 1.325 & .4108 & .9046 & {}-\Z9, 14& 13.174 \\ + 1.35 & .4043 & .9287 & 20, 35 & 12.599 \\ + 1.375 & .3936 & .9513 & 29, 25 & 11.945 \\ + 1.4 & .3800 & .9723 & 36, 21 & 11.364 \\ + 1.45 & .3466 & 1.0096 & 46, 23 & 10.471 \\ + 1.5 & .3082 & 1.0416 & 53, 25 & 9.849 \\ + 1.6 & .2227 & 1.0940 & 62, 21 & 9.034 \\ + 1.7 & .1317 & 1.1356 & 67, 59 & 8.347 \\ + 1.8 &-\Z.0377& 1.1696 & 72, \Z2& 7.618 \\ + 2.0 &+\Z.1563& 1.2184 & 79, 17 & 6.140 \\ + 2.2 & .3547 & 1.2407 & {}-88, 13 & 4.966 \\ + 2.4 & .5541 & 1.2300 & 180°+81, 54 & 4.182 \\ + 2.6 & .7487 & 1.1845 & 71, 49 & 3.665 \\ + 2.8 & .9322 & 1.1057 & 61, 40 & 3.305 \\ + 3.0 & 1.0989 & .9956 & 51, 24 & 3.052 \\ + 3.2 & 1.2429 & .8573 & 40, 54 & 2.873 \\ + 3.4 & 1.3588 & .6946 & 29, 55 & 2.751 \\ + 3.6 & 1.4402 & .5123 & 18, \Z1& 2.682 \\ + 3.8 & 1.4797 & .3168 & 180°+\Z4, 28& 2.670 \\ + 4.0 & 1.4674 & .1181 & 180°-12, 14 & 2.733 \\ + 4.1 & 1.4377 &+\Z.0227& 23, 43 & 2.806 \\ + 4.2 & 1.3894 &-\Z.0646& 35, 38 & 2.910 \\ + 4.3 & 1.3208 & .1366 & 52, 23 & 3.027 \\ + 4.35 & 1.2787 & .1635 & 62, 47 & 3.068 \\ + 4.4 & 1.2322 & .1817 & 74, 47 & 3.063 \\ + 4.45 & 1.1829 & .1892 & 180°-88, 15 & 2.983 \\ + 4.5 & 1.1332 & .1845 & {}+77, 25 & 2.780 \\ + 4.55 & 1.0863 & .1676 & 63, \Z8& 2.477 \\ + 4.6 & 1.0448 & .1399 & 49, 32 & 2.101 \\ + 4.65 & 1.0108 & .1034 & 36, 18 & 1.683 \\ + 4.7 & .9867 &-\Z.0598& 21, \Z1& 1.234 \\ +\ColHead{\text{perijove}} + & +\Z.990&+\Z.011 & \llap{\text{about }} 49, & \\ +\end{longtable} + +The orbit is not vigorously periodic, but an extremely small change at the beginning +would make it so. The period is given by $nT = 1234°.6 = 3.43$ revolutions of Jove. + +\normalsize +\DPPageSep{134}{76} + + +\Chapter{Address} +\index{Address to the International Congress of Mathematicians in Cambridge, 1912}% +\index{Cambridge School of Mathematics}% +\index{Congress, International, of Mathematicians at Cambridge, 1912}% +\index{Mathematical School at Cambridge}% +\index{Mathematicians, International Congress of, Cambridge, 1912}% + +\Heading{(Delivered before the International Congress of Mathematicians +at Cambridge in 1912)} + +\First{Four} years ago at our Conference at Rome the Cambridge Philosophical +Society did itself the honour of inviting the International Congress of +Mathematicians to hold its next meeting at Cambridge. And now I, as +President of the Society, have the pleasure of making you welcome here. +I shall leave it to the Vice-Chancellor, who will speak after me, to express +the feeling of the University as a whole on this occasion, and I shall +confine myself to my proper duty as the representative of our Scientific +Society. + +The Science of Mathematics is now so wide and is already so much +\index{Specialisation in Mathematics}% +specialised that it may be doubted whether there exists to-day any man +fully competent to understand mathematical research in all its many diverse +branches. I, at least, feel how profoundly ill-equipped I am to represent +our Society as regards all that vast field of knowledge which we classify as +pure mathematics. I must tell you frankly that when I gaze on some of the +papers written by men in this room I feel myself much in the same position +as if they were written in Sanskrit. + +But if there is any place in the world in which so one-sided a President +of the body which has the honour to bid you welcome is not wholly out of +place it is perhaps Cambridge. It is true that there have been in the past +at Cambridge great pure mathematicians such as Cayley and Sylvester, but +we surely may claim without undue boasting that our University has played +a conspicuous part in the advance of applied mathematics. Newton was +a glory to all mankind, yet we Cambridge men are proud that fate ordained +that he should have been Lucasian Professor here. But as regards the part +played by Cambridge I refer rather to the men of the last hundred years, +such as Airy, Adams, Maxwell, Stokes, Kelvin, and other lesser lights, who +have marked out the lines of research in applied mathematics as studied in +this University. Then too there are others such as our Chancellor, Lord +Rayleigh, who are happily still with us. +\DPPageSep{135}{77} + +Up to a few weeks ago there was one man who alone of all mathematicians +\index{Poincaré, reference to, by Sir George Darwin}% +might have occupied the place which I hold without misgivings as to his +fitness; I mean Henri Poincaré. It was at Rome just four years ago that +the first dark shadow fell on us of that illness which has now terminated so +fatally. You all remember the dismay which fell on us when the word passed +from man to man ``Poincaré is ill.'' We had hoped that we might again +have heard from his mouth some such luminous address as that which he +gave at Rome; but it was not to be, and the loss of France in his death +affects the whole world. + +It was in 1900 that, as president of the Royal Astronomical Society, +I had the privilege of handing to Poincaré the medal of the Society, and +I then attempted to give an appreciation of his work on the theory of the +tides, on figures of equilibrium of rotating fluid and on the problem of the +three bodies. Again in the preface to the third volume of my collected +papers I ventured to describe him as my patron Saint as regards the papers +contained in that volume. It brings vividly home to me how great a man +he was when I reflect that to one incompetent to appreciate fully one half of +his work yet he appears as a star of the first magnitude. + +It affords an interesting study to attempt to analyze the difference in the +\index{Galton, Sir Francis!analysis of difference in texture of different minds}% +textures of the minds of pure and applied mathematicians. I think that +I shall not be doing wrong to the reputation of the psychologists of half +a century ago when I say that they thought that when they had successfully +analyzed the way in which their own minds work they had solved the problem +before them. But it was Sir~Francis Galton who shewed that such a view is +erroneous. He pointed out that for many men visual images form the most +potent apparatus of thought, but that for others this is not the case. Such +visual images are often quaint and illogical, being probably often founded on +infantile impressions, but they form the wheels of the clockwork\DPnote{[** TN: Not hyphenated in original]} of many +minds. The pure geometrician must be a man who is endowed with great +powers of visualisation, and this view is confirmed by my recollection of the +difficulty of attaining to clear conceptions of the geometry of space until +practice in the art of visualisation had enabled one to picture clearly the +relationship of lines and surfaces to one another. The pure analyst probably +relies far less on visual images, or at least his pictures are not of a geometrical +character. I suspect that the mathematician will drift naturally to one branch +or another of our science according to the texture of his mind and the nature +of the mechanism by which he works. + +I wish Galton, who died but recently, could have been here to collect +from the great mathematicians now assembled an introspective account +of the way in which their minds work. One would like to know whether +students of the theory of groups picture to themselves little groups of dots; +or are they sheep grazing in a field? Do those who work at the theory +\DPPageSep{136}{78} +of numbers associate colour, or good or bad characters with the lower +ordinal numbers, and what are the shapes of the curves in which the +successive numbers are arranged? What I have just said will appear pure +nonsense to some in this room, others will be recalling what they see, and +perhaps some will now for the first time be conscious of their own visual +images. + +The minds of pure and applied mathematicians probably also tend to +differ from one another in the sense of aesthetic beauty. Poincaré has well +remarked in his \textit{Science et Méthode} (p.~57): +\index{Poincaré, reference to, by Sir George Darwin!\textit{Science et Méthode}, quoted}% + +``On peut s'étonner de voir invoquer la sensibilité apropos de démon\-stra\-tions +mathématiques qui, semble-t-il, ne peuvent intéresser que l'intelligence. +Ce serait oublier le sentiment de la beauté mathématique, de +l'harmonie des nombres et des formes, de l'élégance géometrique. C'est un +vrai sentiment esthétique que tous les vrais mathématiciens connaissent. +Et c'est bien là de la sensibilité.'' + +And again he writes: + +``Les combinaisons utiles, ce sont précisément les plus belles, je veux dire +celles qui peuvent le mieux charmer cette sensibilité spéciale que tous les +mathématiciens connaissent, mais que les profanes ignorent au point qu'ils +sont souvent tentés d'en sourire.'' + +Of course there is every gradation from one class of mind to the other, +and in some the aesthetic sense is dominant and in others subordinate. + +In this connection I would remark on the extraordinary psychological +interest of Poincaré's account, in the chapter from which I have already +quoted, of the manner in which he proceeded in attacking a mathematical +problem. He describes the unconscious working of the mind, so that his +conclusions appeared to his conscious self as revelations from another world. +I suspect that we have all been aware of something of the same sort, and +like Poincaré have also found that the revelations were not always to be +trusted. + +Both the pure and the applied mathematician are in search of truth, but +the former seeks truth in itself and the latter truths about the universe in +which we live. To some men abstract truth has the greater charm, to others +the interest in our universe is dominant. In both fields there is room for +indefinite advance; but while in pure mathematics every new discovery +is a gain, in applied mathematics it is not always easy to find the direction +in which progress can be made, because the selection of the conditions +essential to the problem presents a preliminary task, and afterwards there +arise the purely mathematical difficulties. Thus it appears to me at least, +that it is easier to find a field for advantageous research in pure than in +\DPPageSep{137}{79} +applied mathematics. Of course if we regard an investigation in applied +mathematics as an exercise in analysis, the correct selection of the essential +conditions is immaterial; but if the choice has been wrong the results lose +almost all their interest. I may illustrate what I mean by reference to +\index{Kelvin, associated with Sir George Darwin!cooling of earth}% +Lord Kelvin's celebrated investigation as to the cooling of the earth. He +was not and could not be aware of the radio-activity of the materials of which +the earth is formed, and I think it is now generally acknowledged that the +conclusions which he deduced as to the age of the earth cannot be maintained; +yet the mathematical investigation remains intact. + +The appropriate formulation of the problem to be solved is one of the +\index{Darwin, Sir George, genealogy!on his own work}% +greatest difficulties which beset the applied mathematician, and when he +has attained to a true insight but too often there remains the fact that +his problem is beyond the reach of mathematical solution. To the layman +the problem of the three bodies seems so simple that he is surprised to learn +that it cannot be solved completely, and yet we know what prodigies of +mathematical skill have been bestowed on it. My own work on the subject +cannot be said to involve any such skill at all, unless indeed you describe as +skill the procedure of a housebreaker who blows in a safe-door with dynamite +instead of picking the lock. It is thus by brute force that this tantalising +problem has been compelled to give up some few of its secrets, and great as +has been the labour involved I think it has been worth while. Perhaps this +work too has done something to encourage others such as Störmer\footnote + {\textit{Videnskabs Selskab}, Christiania, 1904.} +to similar +tasks as in the computation of the orbits of electrons in the neighbourhood +of the earth, thus affording an explanation of some of the phenomena of the +aurora borealis. To put at their lowest the claims of this clumsy method, +which may almost excite the derision of the pure mathematician, it +has served to throw light on the celebrated generalisations of Hill and +Poincaré. + +I appeal then for mercy to the applied mathematician and would ask +you to consider in a kindly spirit the difficulties under which he labours. +If our methods are often wanting in elegance and do but little to satisfy that +aesthetic sense of which I spoke before, yet they are honest attempts to +unravel the secrets of the universe in which we live. + +We are met here to consider mathematical science in all its branches. +Specialisation has become a necessity of modern work and the intercourse +which will take place between us in the course of this week will serve to +promote some measure of comprehension of the work which is being carried +on in other fields than our own. The papers and lectures which you will +hear will serve towards this end, but perhaps the personal conversations +outside the regular meetings may prove even more useful. +\DPPageSep{138}{80} +\backmatter +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} + +\Pagelabel{indexpage} + +\printindex + +\iffalse +%INDEX TO VOLUME V + +%A + +Abacus xlviii + +Address to the International Congress of Mathematicians in Cambridge, 1912#Address 76 + +Annual Equation 51 + +Apse, motion of 41 + +%B + +Bakerian lecture xlix + +Bakhuyzen, Dr Van d.\ Sande, Sir George Darwin's connection with the International Geodetic Association xxviii + +Barrell, Prof., Cosmogony as related to Geology and Biology xxxvii + +British Association, South African Meeting, 1905#British xxvi + +Brown, Prof.\ E. W., Sir George Darwin's Scientific Work xxxiv + new family of periodic orbits 59 + +%C + +Cambridge School of Mathematics 1, 76 + +Chamberlain and Moulton, Planetesimal Hypothesis xlvii + +Committees, Sir George Darwin on xxii + +Congress, International, of Mathematicians at Cambridge, 1912#Congress 76 + note by Sir Joseph Larmor xxix + +Cosmogony, Sir George Darwin's influence on xxxvi + as related to Geology and Biology, by Prof.\ Barrell xxxvii + +%D + +Darwin, Charles, ix; letters of xiii, xv + +Darwin, Sir Francis, Memoir of Sir George Darwin by ix + +Darwin, Sir George, genealogy ix + boyhood x + interested in heraldry xi + education xi + at Cambridge xii, xvi + friendships xiii, xvi + ill health xiv + marriage xix + children xx + house at Cambridge xix + games and pastimes xxi + personal characteristics xxx + energy xxxii + honours xxxiii + university work, described by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall xvii, xviii + work on scientific committees xxii + association with Lord Kelvin xv, xxxvii + scientific work, by Prof.\ E. W. Brown xxxiv + his first papers xxxvi + characteristics of his work xxxiv + his influence on cosmogony xxxvi + his relationship with his pupils xxxvi + on his own work 79 + +Darwin, Margaret, on Sir George Darwin's personal characteristics xxx + +Differential Equation, Hill's 36 + periodicity of integrals of 55 + +Differential Equations of Motion 17 + +Dynamical Astronomy, introduction to 9 + +%E + +Earth-Moon theory of Darwin, described by Mr S. S. Hough xxxviii + +Earth's figure, theory of 14 + +Ellipsoidal harmonics xliii + +Equation, annual 51 + of the centre 43 + +Equations of motion 17, 46 + +Equilibrium of a rotating fluid xlii + +Evection 43 + in latitude 45 + +%G + +Galton, Sir Francis ix + analysis of difference in texture of different minds 77 + +Geodetic Association, International xxvii, xxviii + +Glaisher, Dr J. W. L., address on presenting the gold medal of the R.A.S. to G. W. Hill lii + +Gravitation, theory of 9 + universal 15 + +%H + +Harmonics, ellipsoidal xliii + +Hecker's observations on retardation of tidal oscillations in the solid earth xliv, l + +Hill, G. W., Lunar Theory l + awarded gold medal of R.A.S. lii + lectures by Darwin on Lunar Theory lii, 16 + characteristics of his Lunar Theory 16 + Special Differential Equation 36 + periodicity of integrals of 55 + infinite determinant 38, 53 + +Hough, S. S., Darwin's work on Earth-Moon Theory, xxxviii; Periodic Orbits liv + +%I + +Inaugural lecture 1 + +Infinite determinant, Hill's 38, 53 + +Introduction to Dynamical Astronomy 9 +%\DPPageSep{139}{81} + +Jacobi's ellipsoid xlii + integral 21 + +Jeans, J. H., on rotating liquids xliii + +%K + +Kant, Nebular Hypothesis xlvi + +Kelvin, associated with Sir George Darwin xv, xxxvii + cooling of earth xlv, 79 + +%L + +Laplace, Nebular Hypothesis xlvi + +Larmor, Sir Joseph, Sir George Darwin's work on university committees xvii + International Geodetic Association xxvii + International Congress of Mathematicians at Cambridge 1912#Cambridge xxix + +Latitude of the moon 43 + +Latitude, variation of 14 + +Lecture, inaugural 1 + +Liapounoff's work on rotating liquids xliii + +Librating planets 59 + +Lunar and planetary theories compared 11 + +Lunar Theory, lecture on 16 + +%M + +Maclaurin's spheroid xlii + +Master of Christ's, Sir George Darwin's work on university committees xviii + +Mathematical School at Cambridge 1, 76 + text-books 4 + Tripos 3 + +Mathematicians, International Congress of, Cambridge, 1912#Cambridge xxix, 76 + +Mechanical condition of a swarm of meteorites xlvi + +Meteorological Council, by Sir Napier Shaw xxii + +Michelson's experiment on rigidity of earth l + +Moulton, Chamberlain and, Planetesimal Hypothesis xlvii + +%N + +Newall, Prof., Sir George Darwin's work on university committees xviii + +Numerical work on cosmogony xlvi + +Numerical work, great labour of liii + +%O + +Orbit, variation of an 64 + +Orbits, periodic, |see{Periodic} + +%P + +Pear-shaped figure of equilibrium xliii + +Perigee, motion of 41 + +Periodic orbits, Darwin begins papers on liii + great numerical difficulties of liii + stability of liii + classification of, by Jacobi's integral liv + new family of 59, 67 + +Periodicity of integrals of Hill's Differential Equation 55 + +Planetary and lunar theories compared 11 + +Planetesimal Hypothesis of Chamberlain and Moulton xlvii + +Poincaré, reference to, by Sir George Darwin 77 + on equilibrium of fluid mass in rotation xlii + \textit{Les Méthodes Nouvelles de la Mécanique Céleste} lii + \textit{Science et Méthode}, quoted 78 + +Pupils, Darwin's relationship with his xxxvi + +%R + +Raverat, Madame, on Sir George Darwin's personal characteristics xxx + +Reduction, the 49 + +Rigidity of earth, from fortnightly tides xlix + Michelson's experiment l + +Roche's ellipsoid xliii + +Rotating fluid, equilibrium of xlii + +%S + +Saturn's rings 15 + +Shaw, Sir Napier, Meteorological Council xxii + +Small displacements from variational curve 26 + +South African Meeting of the British Association, 1905#British xxvi + +Specialisation in Mathematics 76 + +%T + +Text-books, mathematical 4 + +Third coordinate introduced 43 + +Tidal friction as a true cause of change xliv + +Tidal problems, practical xlvii + +Tide, fortnightly xlix + +\textit{Tides, The} xxvii, l + +Tides, articles on l + +Tripos, Mathematical 3 + +%U + +University committees, Sir George Darwin on, by Sir Joseph Larmor, the Master of Christ's, and Prof.\ Newall xvii, xviii + +%V + +Variation, the 43 + of an orbit 64 + of latitude 14 + +Variational curve, defined 22 + determined 23 + small displacements from 26 +\fi +\DPPageSep{140}{82} +\newpage +\null\vfill +\begin{center} +\scriptsize +\textgoth{Cambridge}: \\[4pt] +PRINTED BY JOHN CLAY, M.A. \\[4pt] +AT THE UNIVERSITY PRESS +\end{center} +\vfill +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\cleardoublepage +\phantomsection +\pdfbookmark[0]{PG License}{Project Gutenberg License} +\SetRunningHeads{Licensing}{Licensing} +\pagenumbering{Roman} + +\begin{PGtext} +End of the Project Gutenberg EBook of Scientific Papers by Sir George Howard +Darwin, by George Darwin + +*** END OF THIS PROJECT GUTENBERG EBOOK SCIENTIFIC PAPERS *** + +***** This file should be named 35588-pdf.pdf or 35588-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/5/5/8/35588/ + +Produced by Andrew D. 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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..34c026f --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #35588 (https://www.gutenberg.org/ebooks/35588) |
