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-% Title: The Theory of Heat Radiation %
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-% Author: Max Planck %
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-% Translator: Morton Masius %
-% %
-% Release Date: June 18, 2012 [EBook #40030] %
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-The Project Gutenberg EBook of The Theory of Heat Radiation, by Max Planck
-
-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: The Theory of Heat Radiation
-
-Author: Max Planck
-
-Translator: Morton Masius
-
-Release Date: June 18, 2012 [EBook #40030]
-
-Language: English
-
-Character set encoding: ISO-8859-1
-
-*** START OF THIS PROJECT GUTENBERG EBOOK THE THEORY OF HEAT RADIATION ***
-\end{PGtext}
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-%% -----File: 001.png---Folio xx-------
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-% PLANCK and MASIUS
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-\Huge THE THEORY \\
-\large OF \\
-\Huge HEAT RADIATION
-\vfill\vfill
-
-\normalsize BY \\
-\large DR. MAX PLANCK \\[4pt]
-\scriptsize PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF BERLIN
-\vfill
-
-\normalsize AUTHORISED TRANSLATION
-\vfill
-
-BY \\
-\large MORTON MASIUS, M. A., Ph.\ D. (Leipzig) \\[4pt]
-\scriptsize INSTRUCTOR IN PHYSICS IN THE WORCESTER POLYTECHNIC INSTITUTE
-\vfill\vfill\vfill
-
-\normalsize WITH 7 ILLUSTRATIONS
-\vfill\vfill\vfill\vfill
-
-PHILADELPHIA \\
-\Large P. BLAKISTON'S SON \& CO. \\
-\normalsize 1012 WALNUT STREET
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-%% -----File: 004.png---Folio xx-------
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-\vfill\vfill
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-\scriptsize THE ˇ MAPLE ˇ PRESS ˇ YORK ˇ PA
-\end{center}
-%% -----File: 005.png---Folio xx-------
-
-
-\Preface{Translator's Preface}
-
-The present volume is a translation of the second edition of
-Professor \Name{Planck's} \textsc{Waermestrahlung} (1913). The profoundly
-original ideas introduced by \Name{Planck} in the endeavor to reconcile
-the electromagnetic theory of radiation with experimental facts
-have proven to be of the greatest importance in many parts of
-physics. Probably no single book since the appearance of \Name{Clerk
-Maxwell's} \textsc{Electricity and Magnetism} has had a deeper influence
-on the development of physical theories. The great majority of
-English-speaking physicists are, of course, able to read the work
-in the language in which it was written, but I believe that many
-will welcome the opportunity offered by a translation to study the
-ideas set forth by \Name{Planck} without the difficulties that frequently
-arise in attempting to follow a new and somewhat difficult line
-of reasoning in a foreign language.
-
-Recent developments of physical theories have placed the quantum
-of action in the foreground of interest. Questions regarding
-the bearing of the quantum theory on the law of equipartition of
-energy, its application to the theory of specific heats and to
-photoelectric effects, attempts to form some concrete idea of
-the physical significance of the quantum, that is, to devise a
-``model'' for it, have created within the last few years a large and
-ever increasing literature. Professor \Name{Planck} has, however, in
-this book confined himself exclusively to radiation phenomena
-and it has seemed to me probable that a brief résumé of this
-literature might prove useful to the reader who wishes to pursue
-the subject further. I have, therefore, with Professor \Name{Planck's}
-permission, given in an appendix a list of the most important
-papers on the subjects treated of in this book and others closely
-related to them. I have also added a short note on one or two
-derivations of formulć where the treatment in the book seemed
-too brief or to present some difficulties.
-%% -----File: 006.png---Folio xx-------
-
-In preparing the translation I have been under obligation for
-advice and helpful suggestions to several friends and colleagues
-and especially to Professor A.~W.~Duff who has read the manuscript
-and the galley proof.
-
-\Signature{Morton Masius.}{Worcester, Mass.,}{February,}{1914.}
-%% -----File: 007.png---Folio xx-------
-
-\Preface{Preface to Second Edition}
-
-Recent advances in physical research have, on the whole, been
-favorable to the special theory outlined in this book, in particular
-to the hypothesis of an elementary quantity of action. My radiation
-formula especially has so far stood all tests satisfactorily,
-including even the refined systematic measurements which have
-been carried out in the Physikalisch-technische Reichsanstalt
-at Charlottenburg during the last year. Probably the most
-direct support for the fundamental idea of the hypothesis of
-quanta is supplied by the values of the elementary quanta of
-matter and electricity derived from it. When, twelve years ago,
-I made my first calculation of the value of the elementary electric
-charge and found it to be $4.69ˇ10^{-10}$ electrostatic units, the value
-of this quantity deduced by \Name{J.~J. Thomson} from his ingenious
-experiments on the condensation of water vapor on gas ions,
-namely $6.5ˇ10^{-10}$ was quite generally regarded as the most
-reliable value. This value exceeds the one given by me by $38$~per~cent.
-Meanwhile the experimental methods, improved in
-an admirable way by the labors of \Name{E.~Rutherford}, \Name{E.~Regener},
-\Name{J.~Perrin}, \Name{R.~A. Millikan}, \Name{The Svedberg} and others, have without
-exception decided in favor of the value deduced from the theory
-of radiation which lies between the values of \Name{Perrin} and \Name{Millikan}.
-
-To the two mutually independent confirmations mentioned,
-there has been added, as a further strong support of the hypothesis
-of quanta, the heat theorem which has been in the meantime
-announced by \Name{W.~Nernst}, and which seems to point unmistakably
-to the fact that, not only the processes of radiation, but also the
-molecular processes take place in accordance with certain elementary
-quanta of a definite finite magnitude. For the hypothesis
-of quanta as well as the heat theorem of \Name{Nernst} may be reduced
-to the simple proposition that the thermodynamic probability
-(\Sec{120}) of a physical state is a definite integral number,
-or, what amounts to the same thing, that the entropy of a state
-has a quite definite, positive value, which, as a minimum, becomes
-%% -----File: 008.png---Folio xx-------
-zero, while in contrast therewith the entropy may, according to
-the classical thermodynamics, decrease without limit to minus
-infinity. For the present, I would consider this proposition as
-the very quintessence of the hypothesis of quanta.
-
-In spite of the satisfactory agreement of the results mentioned
-with one another as well as with experiment, the ideas from which
-they originated have met with wide interest but, so far as I am
-able to judge, with little general acceptance, the reason probably
-being that the hypothesis of quanta has not as yet been satisfactorily
-completed. While many physicists, through conservatism,
-reject the ideas developed by me, or, at any rate, maintain
-an expectant attitude, a few authors have attacked them for the
-opposite reason, namely, as being inadequate, and have felt compelled
-to supplement them by assumptions of a still more radical
-nature, for example, by the assumption that any radiant energy
-whatever, even though it travel freely in a vacuum, consists of
-indivisible quanta or cells. Since nothing probably is a greater
-drawback to the successful development of a new hypothesis
-than overstepping its boundaries, I have always stood for making
-as close a connection between the hypothesis of quanta and the
-classical dynamics as possible, and for not stepping outside of
-the boundaries of the latter until the experimental facts leave no
-other course open. I have attempted to keep to this standpoint
-in the revision of this treatise necessary for a new edition.
-
-The main fault of the original treatment was that it began with
-the classical electrodynamical laws of emission and absorption,
-whereas later on it became evident that, in order to meet the
-demand of experimental measurements, the assumption of finite
-energy elements must be introduced, an assumption which is in
-direct contradiction to the fundamental ideas of classical electrodynamics.
-It is true that this inconsistency is greatly reduced
-by the fact that, in reality, only mean values of energy are taken
-from classical electrodynamics, while, for the statistical calculation,
-the real values are used; nevertheless the treatment must,
-on the whole, have left the reader with the unsatisfactory feeling
-that it was not clearly to be seen, which of the assumptions made
-in the beginning could, and which could not, be finally retained.
-
-In contrast thereto I have now attempted to treat the subject
-from the very outset in such a way that none of the laws stated
-%% -----File: 009.png---Folio xx-------
-need, later on, be restricted or modified. This presents the
-advantage that the theory, so far as it is treated here, shows no
-contradiction in itself, though certainly I do not mean that it
-does not seem to call for improvements in many respects, as
-regards both its internal structure and its external form. To
-treat of the numerous applications, many of them very important,
-which the hypothesis of quanta has already found in other parts
-of physics, I have not regarded as part of my task, still less to
-discuss all differing opinions.
-
-Thus, while the new edition of this book may not claim to
-bring the theory of heat radiation to a conclusion that is satisfactory
-in all respects, this deficiency will not be of decisive
-importance in judging the theory. For any one who would make
-his attitude concerning the hypothesis of quanta depend on
-whether the significance of the quantum of action for the elementary
-physical processes is made clear in every respect or may
-be demonstrated by some simple dynamical model, misunderstands,
-I believe, the character and the meaning of the hypothesis
-of quanta. It is impossible to express a really new
-principle in terms of a model following old laws. And, as regards
-the final formulation of the hypothesis, we should not
-forget that, from the classical point of view, the physics of
-the atom really has always remained a very obscure, inaccessible
-region, into which the introduction of the elementary
-quantum of action promises to throw some light.
-
-Hence it follows from the nature of the case that it will require
-painstaking experimental and theoretical work for many years
-to come to make gradual advances in the new field. Any one
-who, at present, devotes his efforts to the hypothesis of quanta,
-must, for the time being, be content with the knowledge that the
-fruits of the labor spent will probably be gathered by a future
-generation.
-
-\Signature{The Author.}{Berlin,}{November,}{1912.}
-%% -----File: 010.png---Folio xx-------
-%[Blank Page]
-%% -----File: 011.png---Folio xx-------
-
-\Preface{Preface to First Edition}
-
-\enlargethispage{\baselineskip}
-In this book the main contents of the lectures which I gave at
-the University of Berlin during the winter semester 1906--07 are
-presented. My original intention was merely to put together
-in a connected account the results of my own investigations,
-begun ten years ago, on the theory of heat radiation; it soon became
-evident, however, that it was desirable to include also the
-foundation of this theory in the treatment, starting with Kirchhoff's
-Law on emitting and absorbing power; and so I attempted
-to write a treatise which should also be capable of serving as an
-introduction to the study of the entire theory of radiant heat on
-a consistent thermodynamic basis. Accordingly the treatment
-starts from the simple known experimental laws of optics and
-advances, by gradual extension and by the addition of the results
-of electrodynamics and thermodynamics, to the problems of the
-spectral distribution of energy and of irreversibility. In doing
-this I have deviated frequently from the customary methods of
-treatment, wherever the matter presented or considerations
-regarding the form of presentation seemed to call for it, especially
-in deriving Kirchhoff's laws, in calculating Maxwell's radiation
-pressure, in deriving Wien's displacement law, and in generalizing
-it for radiations of any spectral distribution of energy whatever.
-
-I have at the proper place introduced the results of my own
-investigations into the treatment. A list of these has been added
-at the end of the book to facilitate comparison and examination
-as regards special details.
-
-I wish, however, to emphasize here what has been stated more
-fully in the last paragraph of this book, namely, that the theory
-thus developed does not by any means claim to be perfect or
-complete, although I believe that it points out a possible way of
-accounting for the processes of radiant energy from the same
-point of view as for the processes of molecular motion.
-%% -----File: 012.png---Folio xx-------
-%[Blank Page]
-%% -----File: 013.png---Folio xx-------
-\TableOfContents
-\iffalse
-\Preface{TABLE OF CONTENTS}
-
-\begin{tabular}{rlr}
-\multicolumn{3}{c}{}\\
-\multicolumn{3}{c}{\textbf{PART I}}\\
-\multicolumn{3}{c}{\textbf{FUNDAMENTAL FACTS AND DEFINITIONS}}\\
-\textsc{Chap} & & \textsc{Page}\\
-I.& General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . &1\\
-II.& Radiation at Thermodynamic Equilibrium. Kirchhoff's Law. & \\
- & Black Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . &22\\
-\multicolumn{3}{c}{}\\
-\multicolumn{3}{c}{\textbf{PART II}}\\
-\multicolumn{3}{c}{\textbf{DEDUCTIONS FROM ELECTRODYNAMICS AND}}\\
-\multicolumn{3}{c}{\textbf{THERMODYNAMICS}}\\
-I. & Maxwell's Radiation Pressure . . . . . . . . . . . . . . . . . . . . . . & 49\\
-II. & Stefan-Boltzmann Law of Radiation . . . . . . . . . . . . . . . . . . & 59\\
-III. & Wien's Displacement Law . . . . . . . . . . . . . . . . . . . . . . . . & 69\\
-IV. & Radiation of any Arbitrary Spectral Distribution of Energy. . . . & \\
- & Entropy and Temperature of Monochromatic Radiation . . . . . . & 87\\
-V. & Electrodynamical Processes in a Stationary Field of Radiation & 103\\
-\multicolumn{3}{c}{}\\
-\multicolumn{3}{c}{\textbf{PART III}}\\
-\multicolumn{3}{c}{\textbf{ENTROPY AND PROBABILITY}}\\
-I. & Fundamental Definitions and Laws. Hypothesis of Quanta . . . . &113\\
-II. & Ideal Monatomic Gases . . . . . . . . . . . . . . . . . . . . . . . . . & 127\\
-III. & Ideal Linear Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . & 135\\
-IV. & Direct Calculation of the Entropy in the Case of Thermodynamic& \\
- & Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & 144\\
-\multicolumn{3}{c}{}\\
-\multicolumn{3}{c}{\textbf{PART IV}}\\
-\multicolumn{3}{c}{\textbf{A SYSTEM OF OSCILLATORS IN A STATIONARY}}\\
-\multicolumn{3}{c}{\textbf{FIELD OF RADIATION}}\\
-I. & The Elementary Dynamical Law for the Vibrations of an Ideal& \\
- & Oscillator. Hypothesis of Emission of Quanta . . . . . . . . . . . . . & 151\\
-II. & Absorbed Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & 155\\
-III. & Emitted Energy. Stationary State . . . . . . . . . . . . . . . . . . . . & 161\\
-IV. & The Law of the Normal Distribution of Energy. Elementary& \\
- & Quanta of Matter and of Electricity . . . . . . . . . . . . . . . . . . . & 167
-\end{tabular}
-\fi
-%% -----File: 014.png---Folio xx-------
-\iffalse
-\begin{tabular}{rlr}
-\multicolumn{3}{c}{\textbf{PART V}}\\
-\multicolumn{3}{c}{}\\
-\multicolumn{3}{c}{\textbf{IRREVERSIBLE RADIATION PROCESSES}}\\
-\multicolumn{3}{c}{}\\
-I. & Fields of Radiation in General . . . . . . . . . . . . . . . . . . . . & 189\\
-II. & One Oscillator in the Field of Radiation . . . . . . . . . . . . . . & 196\\
-III. & A System of Oscillators . . . . . . . . . . . . . . . . . . . . . . . . & 200\\
-IV. &Conservation of Energy and Increase of Entropy. Conclusion. & 205\\
- & List of Papers on Heat Radiation and the Hypothesis of Quanta & \\
- & by the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & 216\\
- & Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &218\\
- & Errata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &225
-\end{tabular}
-\fi
-%% -----File: 015.png---Folio xx-------
-
-
-\MainMatter
-\Part{I}{Fundamental Facts and Definitions}
-%% -----File: 016.png---Folio xx-------
-%[Blank Page]
-%% -----File: 017.png---Folio 1-------
-%[** TN: Chapter macro prints text below]
-% RADIATION OF HEAT
-
-\Chapter{I}{General Introduction}
-
-\Section{1.} Heat may be propagated in a stationary medium in two
-entirely different ways, namely, by conduction and by radiation.
-Conduction of heat depends on the temperature of the medium
-in which it takes place, or more strictly speaking, on the non-uniform
-distribution of the temperature in space, as measured by
-the temperature gradient. In a region where the temperature
-of the medium is the same at all points there is no trace of heat
-conduction.
-
-Radiation of heat, however, is in itself entirely independent of
-the temperature of the medium through which it passes. It is
-possible, for example, to concentrate the solar rays at a focus by
-passing them through a converging lens of ice, the latter remaining
-at a constant temperature of~$0°$, and so to ignite an inflammable
-body. Generally speaking, radiation is a far more complicated
-phenomenon than conduction of heat. The reason for this is
-that the state of the radiation at a given instant and at a given
-point of the medium cannot be represented, as can the flow of
-heat by conduction, by a single vector (that is, a single directed
-quantity). All heat rays which at a given instant pass through
-the same point of the medium are perfectly independent of one
-another, and in order to specify completely the state of the
-radiation the intensity of radiation must be known in all the
-directions, infinite in number, which pass through the point in
-question; for this purpose two opposite directions must be
-considered as distinct, because the radiation in one of them is
-quite independent of the radiation in the other.
-%% -----File: 018.png---Folio 2-------
-
-\Section{2.} Putting aside for the present any special theory of heat
-radiation, we shall state for our further use a law supported by a
-large number of experimental facts. This law is that, so far as
-their physical properties are concerned, heat rays are identical
-with light rays of the same wave length. The term ``heat radiation,''
-then, will be applied to all physical phenomena of the
-same nature as light rays. Every light ray is simultaneously a
-heat ray. We shall also, for the sake of brevity, occasionally
-speak of the ``color'' of a heat ray in order to denote its wave
-length or period. As a further consequence of this law we shall
-apply to the radiation of heat all the well-known laws of experimental
-optics, especially those of reflection and refraction, as
-well as those relating to the propagation of light. Only the
-phenomena of diffraction, so far at least as they take place in
-space of considerable dimensions, we shall exclude on account of
-their rather complicated nature. We are therefore obliged to
-introduce right at the start a certain restriction with respect to
-the size of the parts of space to be considered. Throughout the
-following discussion it will be assumed that the linear dimensions
-of all parts of space considered, as well as the radii of curvature
-of all surfaces under consideration, are large compared with the
-wave lengths of the rays considered. With this assumption we
-may, without appreciable error, entirely neglect the influence of
-diffraction caused by the bounding surfaces, and everywhere
-apply the ordinary laws of reflection and refraction of light.
-To sum up: We distinguish once for all between two kinds of
-lengths of entirely different orders of magnitude---dimensions of
-bodies and wave lengths. Moreover, even the differentials of the
-former, \ie, elements of length, area and volume, will be regarded
-as large compared with the corresponding powers of wave lengths.
-The greater, therefore, the wave length of the rays we wish to
-consider, the larger must be the parts of space considered. But,
-inasmuch as there is no other restriction on our choice of size
-of the parts of space to be considered, this assumption will not
-give rise to any particular difficulty.
-
-\Section{3.} Even more essential for the whole theory of heat radiation
-than the distinction between large and small lengths, is the
-distinction between long and short intervals of time. For the
-definition of intensity of a heat ray, as being the energy transmitted
-%% -----File: 019.png---Folio 3-------
-by the ray per unit time, implies the assumption that the
-unit of time chosen is large compared with the period of vibration
-corresponding to the color of the ray. If this were not so, obviously
-the value of the intensity of the radiation would, in general,
-depend upon the particular phase of vibration at which the
-measurement of the energy of the ray was begun, and the intensity
-of a ray of constant period and amplitude would not be independent
-of the initial phase, unless by chance the unit of time
-were an integral multiple of the period. To avoid this difficulty,
-we are obliged to postulate quite generally that the unit of time,
-or rather that element of time used in defining the intensity, even
-if it appear in the form of a differential, must be large compared
-with the period of all colors contained in the ray in question.
-
-The last statement leads to an important conclusion as to
-radiation of variable intensity. If, using an acoustic analogy,
-we speak of ``beats'' in the case of intensities undergoing periodic
-changes, the ``unit'' of time required for a definition of
-the instantaneous intensity of radiation must necessarily be small
-compared with the period of the beats. Now, since from the
-previous statement our unit must be large compared with a period
-of vibration, it follows that the period of the beats must be large
-compared with that of a vibration. Without this restriction it
-would be impossible to distinguish properly between ``beats''
-and simple ``vibrations.'' Similarly, in the general case of an
-arbitrarily variable intensity of radiation, the vibrations must
-take place very rapidly as compared with the relatively slower
-changes in intensity. These statements imply, of course, a certain
-far-reaching restriction as to the generality of the radiation
-phenomena to be considered.
-
-It might be added that a very similar and equally essential
-restriction is made in the kinetic theory of gases by dividing the
-motions of a chemically simple gas into two classes: visible,
-coarse, or molar, and invisible, fine, or molecular. For, since the
-velocity of a single molecule is a perfectly unambiguous quantity,
-this distinction cannot be drawn unless the assumption be made
-that the velocity-components of the molecules contained in sufficiently
-small volumes have certain mean values, independent of
-the size of the volumes. This in general need not by any means be
-the case. If such a mean value, including the value zero, does not
-%% -----File: 020.png---Folio 4-------
-exist, the distinction between motion of the gas as a whole and
-random undirected heat motion cannot be made.
-
-Turning now to the investigation of the laws in accordance with
-which the phenomena of radiation take place in a medium supposed
-to be at rest, the problem may be approached in two ways:
-We must either select a certain point in space and investigate the
-different rays passing through this one point as time goes on, or
-we must select one distinct ray and inquire into its history, that
-is, into the way in which it was created, propagated, and finally
-destroyed. For the following discussion, it will be advisable to
-start with the second method of treatment and to consider
-first the three processes just mentioned.
-
-\Section[4.]{4. Emission.}---The creation of a heat ray is generally denoted
-by the word emission. According to the principle of the conservation
-of energy, emission always takes place at the expense of
-other forms of energy (heat,\footnote
- {Here as in the following the German ``Körperwärme'' will be rendered simply as
- ``heat.''~(Tr.)}
-chemical or electric energy,~etc.)\
-and hence it follows that only material particles, not geometrical
-volumes or surfaces, can emit heat rays. It is true that for the
-sake of brevity we frequently speak of the surface of a body as
-radiating heat to the surroundings, but this form of expression
-does not imply that the surface actually emits heat rays. Strictly
-speaking, the surface of a body never emits rays, but rather it
-allows part of the rays coming from the interior to pass through.
-The other part is reflected inward and according as the fraction
-transmitted is larger or smaller the surface seems to emit more or
-less intense radiations.
-
-We shall now consider the interior of an emitting substance
-assumed to be physically homogeneous, and in it we shall select
-any volume-element~$d\tau$ of not too small size. Then the energy
-which is emitted by radiation in unit time by all particles in this
-volume-element will be proportional to~$d\tau$. Should we attempt
-a closer analysis of the process of emission and resolve it into its
-elements, we should undoubtedly meet very complicated conditions,
-for then it would be necessary to consider elements of
-space of such small size that it would no longer be admissible to
-think of the substance as homogeneous, and we would have to
-allow for the atomic constitution. Hence the finite quantity
-%% -----File: 021.png---Folio 5-------
-obtained by dividing the radiation emitted by a volume-element~$d\tau$
-by this element~$d\tau$ is to be considered only as a certain mean
-value. Nevertheless, we shall as a rule be able to treat the phenomenon
-of emission as if all points of the volume-element~$d\tau$
-took part in the emission in a uniform manner, thereby greatly
-simplifying our calculation. Every point of~$d\tau$ will then be the
-vertex of a pencil of rays diverging in all directions. Such a
-pencil coming from one single point of course does not represent
-a finite amount of energy, because a finite amount is emitted
-only by a finite though possibly small volume, not by a single
-point.
-
-We shall next assume our substance to be isotropic. Hence
-the radiation of the volume-element~$d\tau$ is emitted uniformly in
-all directions of space. Draw a cone in an arbitrary direction,
-having any point of the radiating element as vertex, and describe
-around the vertex as center a sphere of unit radius. This sphere
-intersects the cone in what is known as the solid angle of the cone,
-and from the isotropy of the medium it follows that the radiation
-in any such conical element will be proportional to its solid angle.
-This holds for cones of any size. If we take the solid angle as infinitely
-small and of size~$d\Omega$ we may speak of the radiation emitted
-in a certain direction, but always in the sense that for the emission
-of a finite amount of energy an infinite number of directions
-are necessary and these form a finite solid angle.
-
-\Section{5.} The distribution of energy in the radiation is in general
-quite arbitrary; that is, the different colors of a certain radiation
-may have quite different intensities. The color of a ray in experimental
-physics is usually denoted by its wave length, because
-this quantity is measured directly. For the theoretical treatment,
-however, it is usually preferable to use the frequency~$\nu$ instead,
-since the characteristic of color is not so much the wave length,
-which changes from one medium to another, as the frequency,
-which remains unchanged in a light or heat ray passing through
-stationary media. We shall, therefore, hereafter denote a certain
-color by the corresponding value of~$\nu$, and a certain interval
-of color by the limits of the interval $\nu$~and~$\nu'$, where $\nu' > \nu$. The
-radiation lying in a certain interval of color divided by the magnitude
-$\nu' - \nu$ of the interval, we shall call the mean radiation in the
-interval $\nu$~to~$\nu'$. We shall then assume that if, keeping $\nu$ constant,
-%% -----File: 022.png---Folio 6-------
-we take the interval $\nu' - \nu$ sufficiently small and denote it by~$d\nu$
-the value of the mean radiation approaches a definite limiting
-value, independent of the size of~$d\nu$, and this we shall briefly call
-the ``radiation of frequency~$\nu$.'' To produce a finite intensity
-of radiation, the frequency interval, though perhaps small, must
-also be finite.
-
-We have finally to allow for the polarization of the emitted
-radiation. Since the medium was assumed to be isotropic the
-emitted rays are unpolarized. Hence every ray has just twice
-the intensity of one of its plane polarized components, which
-could, \eg, be obtained by passing the ray through a \Name{Nicol's}
-prism.
-
-\Section{6.} Summing up everything said so far, we may equate the total
-energy in a range of frequency from $\nu$ to~$\nu + d\nu$ emitted in the
-time~$dt$ in the direction of the conical element~$d\Omega$ by a \DPchg{volume
-element}{volume-element}~$d\tau$ to
-\[
-dt ˇ d\tau ˇ d\Omega ˇ d\nu ˇ 2\epsilon_{\nu}.
-\Tag{(1)}
-\]
-The finite quantity~$\epsilon_{\nu}$ is called the coefficient of emission of the
-medium for the frequency~$\nu$. It is a positive function of~$\nu$ and
-refers to a plane polarized ray of definite color and direction. The
-total emission of the volume-element~$d\tau$ may be obtained from
-this by integrating over all directions and all frequencies. Since
-$\epsilon_{\nu}$~is independent of the direction, and since the integral over all
-conical elements~$d\Omega$ is~$4\pi$, we get:
-\[
-dt ˇ d\tau ˇ 8\pi \int_{0}^{\infty} \epsilon_{\nu}\, d\nu.
-\Tag{(2)}
-\]
-
-\Section{7.} The coefficient of emission~$\epsilon$ depends, not only on the frequency~$\nu$,
-but also on the condition of the emitting substance
-contained in the volume-element~$d\tau$, and, generally speaking,
-in a very complicated way, according to the physical and chemical
-processes which take place in the elements of time and volume in
-question. But the empirical law that the emission of any volume-element
-depends entirely on what takes place inside of this element
-holds true in all cases (\Name{Prevost's} principle). A body~$A$
-at $100° \Celsius$ emits toward a body~$B$ at $0° \Celsius$ exactly the same
-amount of radiation as toward an equally large and similarly
-situated body~$B'$ at $1000°$~C\@. The fact that the body~$A$ is cooled
-%% -----File: 023.png---Folio 7-------
-by~$B$ and heated by~$B'$ is due entirely to the fact that $B$~is a
-weaker, $B'$~a stronger emitter than~$A$.
-
-We shall now introduce the further simplifying assumption
-that the physical and chemical condition of the emitting substance
-depends on but a single variable, namely, on its absolute
-temperature~$T$. A necessary consequence of this is that the
-coefficient of emission~$\epsilon$ depends, apart from the frequency~$\nu$
-and the nature of the medium, only on the temperature~$T$.
-The last statement excludes from our consideration a number
-of radiation phenomena, such as fluorescence, phosphorescence,
-electrical and chemical luminosity, to which \Name{E.~Wiedemann} has
-given the common name ``phenomena of luminescence.'' We
-shall deal with pure ``temperature radiation'' exclusively.
-
-A special case of temperature radiation is the case of the
-chemical nature of the emitting substance being invariable. In
-this case the emission takes place entirely at the expense of the
-heat of the body. Nevertheless, it is possible, according to what
-has been said, to have temperature radiation while chemical
-changes are taking place, provided the chemical condition is completely
-determined by the temperature.
-
-\Section[8.]{8. Propagation.}---The propagation of the radiation in a medium
-assumed to be homogeneous, isotropic, and at rest takes place in
-straight lines and with the same velocity in all directions, diffraction
-phenomena being entirely excluded. Yet, in general, each
-ray suffers during its propagation a certain weakening, because
-a certain fraction of its energy is continuously deviated from its
-original direction and scattered in all directions. This phenomenon
-of ``scattering,'' which means neither a creation nor a
-destruction of radiant energy but simply a change in distribution,
-takes place, generally speaking, in all media differing from an
-absolute vacuum, even in substances which are perfectly pure
-chemically.\footnote
- {See, \eg, \Name{Lobry de~Bruyn} and \Name{L.~K. Wolff}, Rec.\ des Trav.\ Chim.\ des Pays-Bas~\textbf{23},
- p.~155, 1904.}
-The cause of this is that no substance is homogeneous
-in the absolute sense of the word. The smallest elements of
-space always exhibit some discontinuities on account of their
-atomic structure. Small impurities, as, for instance, particles of
-dust, increase the influence of scattering without, however, appreciably
-affecting its general character. Hence, so-called ``turbid''
-%% -----File: 024.png---Folio 8-------
-media, \ie, such as contain foreign particles, may be quite properly
-regarded as optically homogeneous,\footnote
- {To restrict the word homogeneous to its absolute sense would mean that it could not be
- applied to any material substance.}
-provided only that the
-linear dimensions of the foreign particles as well as the distances
-of neighboring particles are sufficiently small compared with the
-wave lengths of the rays considered. As regards optical phenomena,
-then, there is no fundamental distinction between chemically
-pure substances and the turbid media just described. No space
-is optically void in the absolute sense except a vacuum. Hence
-a chemically pure substance may be spoken of as a vacuum made
-turbid by the presence of molecules.
-
-A typical example of scattering is offered by the behavior of
-sunlight in the atmosphere. When, with a clear sky, the sun
-stands in the zenith, only about two-thirds of the direct radiation
-of the sun reaches the surface of the earth. The remainder is
-intercepted by the atmosphere, being partly absorbed and
-changed into heat of the air, partly, however, scattered and
-changed into diffuse skylight. This phenomenon is produced
-probably not so much by the particles suspended in the atmosphere
-as by the air molecules themselves.
-
-Whether the scattering depends on reflection, on diffraction, or
-on a resonance effect on the molecules or particles is a point that
-we may leave entirely aside. We only take account of the fact
-that every ray on its path through any medium loses a certain
-fraction of its intensity. For a very small distance,~$s$, this fraction
-is proportional to~$s$, say
-\[
-\beta_{\nu} s
-\Tag{(3)}
-\]
-where the positive quantity~$\beta_{\nu}$ is independent of the intensity of
-radiation and is called the ``coefficient of scattering'' of the medium.
-Inasmuch as the medium is assumed to be isotropic, $\beta_{\nu}$~is
-also independent of the direction of propagation and polarization
-of the ray. It depends, however, as indicated by the
-subscript~$\nu$, not only on the physical and chemical constitution of
-the body but also to a very marked degree on the frequency.
-For certain values of~$\nu$, $\beta_{\nu}$~may be so large that the straight-line
-propagation of the rays is virtually destroyed. For other values
-of~$\nu$, however, $\beta_{\nu}$~may become so small that the scattering can
-%% -----File: 025.png---Folio 9-------
-be entirely neglected. For generality we shall assume a mean
-value of~$\beta_{\nu}$. In the cases of most importance $\beta_{\nu}$~increases quite
-appreciably as $\nu$~increases, \ie, the scattering is noticeably larger
-for rays of shorter wave length;\footnote
- {\Name{Lord Rayleigh}, Phil.\ Mag., \textbf{47}, p.~379, 1899.}
-hence the blue color of diffuse skylight.
-
-The scattered radiation energy is propagated from the place
-where the scattering occurs in a way similar to that in which the
-emitted energy is propagated from the place of emission, since
-it travels in all directions in space. It does not, however, have
-the same intensity in all directions, and moreover is polarized
-in some special directions, depending to a large extent on the
-direction of the original ray. We need not, however, enter into
-any further discussion of these questions.
-
-\Section{9.} While the phenomenon of scattering means a continuous
-modification in the interior of the medium, a discontinuous
-change in both the direction and the intensity of a ray occurs
-when it reaches the boundary of a medium and meets the surface
-of a second medium. The latter, like the former, will be assumed
-to be homogeneous and isotropic. In this case, the ray is in
-general partly reflected and partly transmitted. The reflection
-and refraction may be ``regular,'' there being a single reflected
-ray according to the simple law of reflection and a single transmitted
-ray, according to \Name{Snell's} law of refraction, or, they may be
-``diffuse,'' which means that from the point of incidence on the
-surface the radiation spreads out into the two media with intensities
-that are different in different directions. We accordingly
-describe the surface of the second medium as ``smooth'' or
-``rough'' respectively. Diffuse reflection occurring at a rough
-surface should be carefully distinguished from reflection at a
-smooth surface of a turbid medium. In both cases part of the
-incident ray goes back to the first medium as diffuse radiation.
-But in the first case the scattering occurs on the surface, in the
-second in more or less thick layers entirely inside of the second
-medium.
-
-\Section{10.} When a smooth surface completely reflects all incident
-rays, as is approximately the case with many metallic surfaces,
-it is termed ``reflecting.'' When a rough surface reflects all
-incident rays completely and uniformly in all directions, it is
-%% -----File: 026.png---Folio 10-------
-called ``white.'' The other extreme, namely, complete transmission
-of all incident rays through the surface never occurs with
-smooth surfaces, at least if the two contiguous media are at all
-optically different. A rough surface having the property of
-completely transmitting the incident radiation is described as
-``black.''
-
-In addition to ``black surfaces'' the term ``black body'' is also
-used. According to \Name{G.~Kirchhoff}\footnote
- {\Name{G.~Kirchhoff}, Pogg.\ Ann., \textbf{109}, p.~275, 1860. Gesammelte Abhandlungen, J.~A. Barth,
- Leipzig, 1882, p.~573. In defining a black body \Name{Kirchhoff} also assumes that the absorption
- of incident rays takes place in a layer ``infinitely thin.'' We do not include this in our
- definition.}
-it denotes a body which has
-the property of allowing all incident rays to enter without surface
-reflection and not allowing them to leave again. Hence it is
-seen that a black body must satisfy three independent conditions.
-First, the body must have a black surface in order to allow the
-incident rays to enter without reflection. Since, in general, the
-properties of a surface depend on both of the bodies which are in
-contact, this condition shows that the property of blackness as
-applied to a body depends not only on the nature of the body
-but also on that of the contiguous medium. A body which is
-black relatively to air need not be so relatively to glass, and \textit{vice
-versa}. Second, the black body must have a certain minimum
-thickness depending on its absorbing power, in order to insure
-that the rays after passing into the body shall not be able to
-leave it again at a different point of the surface. The more absorbing
-a body is, the smaller the value of this minimum thickness,
-while in the case of bodies with vanishingly small absorbing
-power only a layer of infinite thickness may be regarded as black.
-Third, the black body must have a vanishingly small coefficient of
-scattering (\Sec{8}). Otherwise the rays received by it would be
-partly scattered in the interior and might leave again through
-the surface.\footnote
- {For this point see especially \Name{A.~Schuster}, Astrophysical Journal, \textbf{21}, p.~1, 1905, who has
- pointed out that an infinite layer of gas with a black surface need by no means be a black
- body.}
-
-\Section{11.} All the distinctions and definitions mentioned in the two
-preceding paragraphs refer to rays of one definite color only.
-It might very well happen that, \eg, a surface which is rough for a
-certain kind of rays must be regarded as smooth for a different
-kind of rays. It is readily seen that, in general, a surface shows
-%% -----File: 027.png---Folio 11-------
-decreasing degrees of roughness for increasing wave lengths\DPtypo{}{.}
-Now, since smooth non-reflecting surfaces do not exist (\Sec{10}), it
-follows that all approximately black surfaces which may be realized
-in practice (lamp black, platinum black) show appreciable
-reflection for rays of sufficiently long wave lengths.
-
-\Section[12.]{12. Absorption.}---Heat rays are destroyed by ``absorption.''
-According to the principle of the conservation of energy the
-energy of heat radiation is thereby changed into other forms of
-energy (heat, chemical energy). Thus only material particles
-can absorb heat rays, not elements of surfaces, although sometimes
-for the sake of brevity the expression absorbing surfaces
-is used.
-
-Whenever absorption takes place, the heat ray passing through
-the medium under consideration is weakened by a certain fraction
-of its intensity for every element of path traversed. For a
-sufficiently small distance~$s$ this fraction is proportional to~$s$,
-and may be written
-\[
-\alpha_{\nu}s\Add{.}
-\Tag{(4)}
-\]
-Here $\alpha_{\nu}$~is known as the ``coefficient of absorption'' of the medium
-for a ray of frequency~$\nu$. We assume this coefficient to be
-independent of the intensity; it will, however, depend in general
-in non-homogeneous and anisotropic media on the position of~$s$
-and on the direction of propagation and polarization of the ray
-(example: tourmaline). We shall, however, consider only homogeneous
-isotropic substances, and shall therefore suppose that
-$\alpha_{\nu}$~has the same value at all points and in all directions in the
-medium, and depends on nothing but the frequency~$\nu$, the temperature~$T$,
-and the nature of the medium.
-
-Whenever $\alpha_{\nu}$~does not differ from zero except for a limited range
-of the spectrum, the medium shows ``selective'' absorption. For
-those colors for which $\alpha_{\nu} = 0$ and also the coefficient of scattering
-$\beta_{\nu} = 0$ the medium is described as perfectly ``transparent'' or
-``diathermanous.'' But the properties of selective absorption
-and of diathermancy may for a given medium vary widely with
-the temperature. In general we shall assume a mean value for~$\alpha_{\nu}$.
-This implies that the absorption in a distance equal to a
-single wave length is very small, because the distance~$s$, while
-small, contains many wave lengths (\Sec{2}).
-%% -----File: 028.png---Folio 12-------
-
-\Section{13.} The foregoing considerations regarding the emission, the
-propagation, and the absorption of heat rays suffice for a mathematical
-treatment of the radiation phenomena. The calculation
-requires a knowledge of the value of the constants and the initial
-and boundary conditions, and yields a full account of the changes
-the radiation undergoes in a given time in one or more contiguous
-media of the kind stated, including the temperature changes
-caused by it. The actual calculation is usually very complicated.
-We shall, however, before entering upon the treatment of special
-cases discuss the general radiation phenomena from a different
-point of view, namely by fixing our attention not on a definite
-ray, but on a definite position in space.
-
-\Section{14.} Let $d\sigma$~be an arbitrarily chosen, infinitely small element of
-area in the interior of a medium through which radiation passes.
-At a given instant rays are passing through this element in many
-different directions. The energy radiated through it in an
-element of time~$dt$ in a definite direction is proportional to the area~$d\sigma$,
-the length of time~$dt$\Add{,} and to the cosine of the angle~$\theta$ made by
-the normal of~$d\sigma$ with the direction of the radiation. If we make
-$d\sigma$~sufficiently small, then, although this is only an approximation
-to the actual state of affairs, we can think of all points in~$d\sigma$ as
-being affected by the radiation in the same way. Then the
-energy radiated through~$d\sigma$ in a definite direction must be proportional
-to the solid angle in which $d\sigma$~intercepts that radiation
-and this solid angle is measured by $d\sigma \cos\theta$. It is readily seen
-that, when the direction of the element is varied relatively to the
-direction of the radiation, the energy radiated through it vanishes
-when
-\[
-\theta = \frac{\pi}{2}.
-\]
-
-Now in general a pencil of rays is propagated from every point
-of the element~$d\sigma$ in all directions, but with different intensities
-in different directions, and any two pencils emanating from two
-points of the element are identical save for differences of higher
-order. A single one of these pencils coming from a single point
-does not represent a finite quantity of energy, because a finite
-amount of energy is radiated only through a finite area. This
-holds also for the passage of rays through a so-called focus. For
-%% -----File: 029.png---Folio 13-------
-example, when sunlight passes through a converging lens and is
-concentrated in the focal plane of the lens, the solar rays do not
-converge to a single point, but each pencil of parallel rays forms
-a separate focus and all these foci together constitute a surface
-representing a small but finite image of the sun. A finite amount
-of energy does not pass through less than a finite portion of this
-surface.
-
-\Section{15.} Let us now consider quite generally the pencil, which is
-propagated from a point of the element~$d\sigma$ as vertex in all directions
-of space and on both sides of~$d\sigma$. A certain direction may
-be specified by the angle~$\theta$ (between $0$ and~$\pi$), as already used,
-and by an azimuth~$\phi$ (between $0$ and~$2\pi$). The intensity in this
-direction is the energy propagated in an infinitely thin cone limited
-by $\theta$ and $\theta + d\theta$ and $\phi$~and $\phi + d\phi$. The solid angle of this
-cone is
-\[
-d\Omega = \sin\theta ˇ d\theta ˇ d\phi.
-\Tag{(5)}
-\]
-Thus the energy radiated in time~$dt$ through the element of area~$d\sigma$
-in the direction of the cone~$d\Omega$ is:
-\[
-dt\, d\sigma\, \cos\theta\, d\Omega\, K = K \sin\theta \cos\theta\, d\theta\, d\phi\, d\sigma\, dt.
-\Tag{(6)}
-\]
-
-The finite quantity~$K$ we shall term the ``specific intensity''
-or the ``brightness,'' $d\Omega$~the ``solid angle'' of the pencil emanating
-from a point of the element~$d\sigma$ in the direction $(\theta, \phi)$. $K$~is a
-positive function of position, time, and the angles $\theta$~and~$\phi$. In
-general the specific intensities of radiation in different directions
-are entirely independent of one another. For example, on substituting
-$\pi - \theta$ for~$\theta$ and $\pi + \phi$ for~$\phi$ in the function~$K$, we obtain the
-specific intensity of radiation in the diametrically opposite
-direction, a quantity which in general is quite different from the
-preceding one.
-
-For the total radiation through the element of area~$d\sigma$ toward
-one side, say the one on which $\theta$~is an acute angle, we get, by
-integrating with respect to~$\phi$ from $0$ to~$2\pi$ and with respect to~$\theta$
-from $0$ to~$\dfrac{\pi}{2}$
-\[
-\int_{0}^{2\pi} d\phi \int_{0}^{\frac{\pi}{2}} d\theta\, K\sin\theta \cos\theta\, d\sigma\, dt.
-\]
-%% -----File: 030.png---Folio 14-------
-Should the radiation be uniform in all directions and hence $K$~be
-a constant, the total radiation on one side will be
-\[
-\pi K\, d\sigma\, dt.
-\Tag{(7)}
-\]
-
-\Section{16.} In speaking of the radiation in a definite direction
-$(\theta, \phi)$ one should always keep in mind that the energy radiated in a
-cone is not finite unless the angle of the cone is finite. No finite
-radiation of light or heat takes place in one definite direction only,
-or expressing it differently, in nature there is no such thing as
-absolutely parallel light or an absolutely plane wave front.
-From a pencil of rays called ``parallel'' a finite amount of energy of
-radiation can only be obtained if the rays or wave normals of the
-pencil diverge so as to form a finite though perhaps exceedingly
-narrow cone.
-
-\Section{17.} The specific intensity~$K$ of the whole energy radiated in
-a certain direction may be further divided into the intensities of
-the separate rays belonging to the different regions of the spectrum
-which travel independently of one another. Hence we
-consider the intensity of radiation within a certain range of frequencies,
-say from $\nu$ to~$\nu'$. If the interval $\nu' - \nu$ be taken sufficiently
-small and be denoted by~$d\nu$, the intensity of radiation
-within the interval is proportional to~$d\nu$. Such radiation is called
-homogeneous or monochromatic.
-
-A last characteristic property of a ray of definite direction,
-intensity, and color is its state of polarization. If we break up a
-ray, which is in any state of polarization whatsoever and which
-travels in a definite direction and has a definite frequency~$\nu$,
-into two plane polarized components, the sum of the intensities
-of the components will be just equal to the intensity of the ray
-as a whole, independently of the direction of the two planes,
-provided the two planes of polarization, which otherwise may be
-taken at random, are at right angles to each other. If their position
-be denoted by the azimuth~$\psi$ of one of the planes of vibration
-(plane of the electric vector), then the two components of the
-intensity may be written in the form
-\[
-\begin{alignedat}{2}
-&\ssfK_{\nu} \cos^{2} \psi &&+ \ssfK_{\nu}' \sin^{2} \psi \\
-\LeftText{and}
-&\ssfK_{\nu} \sin^{2} \psi &&+ \ssfK_{\nu}' \cos^2 \psi\Add{.}
-\end{alignedat}
-\Tag{(8)}
-\]
-Herein $\ssfK$~is independent of~$\psi$. These expressions we shall call
-%% -----File: 031.png---Folio 15-------
-the ``components of the specific intensity of radiation of frequency~$\nu$.''
-The sum is independent of~$\psi$ and is always equal to the
-intensity of the whole ray $\ssfK_{\nu} + \ssfK_{\nu}'$. At the same time $\ssfK_{\nu}$ and
-$\ssfK_{\nu}'$ represent respectively the largest and smallest values which
-either of the components may have, namely, when $\psi = 0$ and $\psi = \dfrac{\pi}{2}$.
-Hence we call these values the ``principal values of the intensities,''
-or the ``principal intensities,'' and the corresponding planes
-of vibration we call the ``principal planes of vibration'' of the
-ray. Of course both, in general, vary with the time. Thus we
-may write generally
-\[
-\ssfK = \int_{\DPtypo{c}{0}}^{\infty} d\nu\, (\ssfK_{\nu} + \ssfK_{\nu}')
-\Tag{(9)}
-\]
-where the positive quantities $\ssfK_{\nu}$ and $\ssfK_{\nu}'$, the two principal values
-of the specific intensity of the radiation (brightness) of frequency~$\nu$,
-depend not only on~$\nu$ but also on their position, the time,
-and on the angles $\theta$~and~$\phi$. By substitution in~\Eq{(6)} the energy
-radiated in the time~$dt$ through the element of area~$d\sigma$ in the direction
-of the conical element~$d\Omega$ assumes the value
-\[
-dt\, d\sigma \cos\theta\, d\Omega \int_{0}^{\infty} d\nu\, (\ssfK_{\nu} + \ssfK_{\nu}')
-\Tag{(10)}
-\]
-and for monochromatic plane polarized radiation of brightness~$\ssfK_{\nu}$:
-\[
-dt\, d\sigma \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu
- = dt\, d\sigma \sin\theta \cos\theta\, d\theta\, d\phi\, \ssfK_{\nu}\, d\nu.
-\Tag{(11)}
-\]
-For unpolarized rays $\ssfK_{\nu} = \ssfK_{\nu}'$, and hence
-\[
-K = 2 \int_{0}^{\infty} d\nu\, \ssfK_{\nu},
-\Tag{(12)}
-\]
-and the energy of a monochromatic ray of frequency~$\nu$ will be:
-\[
-2\, dt\, d\sigma \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu
- = 2\, dt\, d\sigma \sin\theta \cos\theta\, d\theta\, d\phi\, \ssfK_{\nu}\, d\nu.
-\Tag{(13)}
-\]
-When, moreover, the radiation is uniformly distributed in all
-directions, the total radiation through~$d\sigma$ toward one side may be
-found from \Eq{(7)}~and~\Eq{(12)}; it is
-\[
-2\pi\, d\sigma\, dt \int_{0}^{\infty} \ssfK_{\nu}\, d\nu.
-\Tag{(14)}
-\]
-%% -----File: 032.png---Folio 16-------
-
-\Section{18.} Since in nature $\ssfK_{\nu}$~can never be infinitely large, $K$~will not
-have a finite value unless $\ssfK_{\nu}$~differs from zero over a finite range
-of frequencies. Hence there exists in nature no absolutely
-homogeneous or monochromatic radiation of light or heat. A
-finite amount of radiation contains always a finite although possibly
-very narrow range of the spectrum. This implies a fundamental
-difference from the corresponding phenomena of acoustics,
-where a finite intensity of sound may correspond to a single
-definite frequency. This difference is, among other things, the
-cause of the fact that the second law of thermodynamics has an
-important bearing on light and heat rays, but not on sound waves.
-This will be further discussed later on.
-
-\Section{19.} From equation~\Eq{(9)} it is seen that the quantity~$\ssfK_{\nu}$, the
-intensity of radiation of frequency~$\nu$, and the quantity~$K$, the
-intensity of radiation of the whole spectrum, are of different
-dimensions. Further it is to be noticed that, on subdividing
-the spectrum according to wave lengths~$\lambda$, instead of frequencies~$\nu$,
-the intensity of radiation~$E_{\lambda}$ of the wave lengths~$\lambda$ corresponding
-to the frequency~$\nu$ is not obtained simply by replacing~$\nu$ in
-the expression for~$\ssfK_{\nu}$ by the corresponding value of~$\lambda$ deduced
-from
-\[
-\nu = \frac{q}{\lambda}
-\Tag{(15)}
-\]
-where $q$~is the velocity of propagation. For if $d\lambda$~and~$d\nu$ refer to
-the same interval of the spectrum, we have, not $E_{\lambda} = \ssfK_{\nu}$, but
-$E_{\lambda}\, d\lambda = \ssfK_{\nu}\, d\nu$. By differentiating~\Eq{(15)} and paying attention
-to the signs of corresponding values of $d\lambda$~and~$d\nu$ the equation
-\[
-d\nu = \frac{q\, d\lambda}{\lambda^{2}}
-\]
-is obtained. Hence we get by substitution:
-\[
-E_{\lambda} = \frac{q \ssfK_{\nu}}{\lambda^{2}}.
-\Tag{(16)}
-\]
-This relation shows among other things that in a certain spectrum
-the maxima of~$E_{\lambda}$ and~$\ssfK_{\nu}$ lie at different points of the spectrum.
-
-\Section{20.} When the principal intensities $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ of all monochromatic
-rays are given at all points of the medium and for all
-directions, the state of radiation is known in all respects and all
-%% -----File: 033.png---Folio 17-------
-questions regarding it may be answered. We shall show this by
-one or two applications to special cases. Let us first find the
-amount of energy which is radiated through any element of area~$d\sigma$
-toward any other element~$d\sigma'$. The distance~$r$ between the
-two elements may be thought of as large compared with the
-linear dimensions of the elements $d\sigma$ and~$d\sigma'$ but still so small
-that no appreciable amount of radiation is absorbed or scattered
-along it. This condition is, of course, superfluous for diathermanous
-media.
-
-From any definite point of~$d\sigma$ rays pass to all points of~$d\sigma'$.
-These rays form a cone whose vertex lies in~$d\sigma$ and whose solid
-angle is
-\[
-d\Omega = \frac{d\sigma' \cos (\DPchg{\nu'}{\Norm'}, r)}{r^{2}}
-\]
-%[** TN: \nu briefly denotes both a surface normal and frequency]
-where $\DPchg{\nu'}{\Norm'}$~denotes the normal of~$d\sigma'$ and the angle $(\DPchg{\nu'}{\Norm'}, r)$ is to be
-taken as an acute angle. This value of~$d\Omega$ is, neglecting small
-quantities of higher order, independent of the particular position
-of the vertex of the cone on~$d\sigma$.
-
-If we further denote the normal to~$d\sigma$ by~$\DPchg{\nu}{\Norm}$ the angle~$\theta$ of~\Eq{(14)}
-will be the angle $(\DPchg{\nu}{\Norm}, r)$ and hence from expression~\Eq{(6)} the energy of
-radiation required is found to be:
-\[
-K ˇ \frac{d\sigma\, d\sigma' \cos(\DPchg{\nu}{\Norm}, r) \DPchg{ˇ}{} \cos(\DPchg{\nu'}{\Norm'}, r)}{r^{2}}\DPchg{\,}{ˇ} dt.
-\Tag{(17)}
-\]
-For monochromatic plane polarized radiation of frequency~$\nu$ the
-energy will be, according to equation~\Eq{(11)},
-\[
-\ssfK_{\nu}\, d\nu ˇ \frac{d\sigma\, d\sigma' \cos(\DPchg{\nu}{\Norm}, r) \cos(\DPchg{\nu'}{\Norm'}, r)}{r^{2}} ˇ dt.
-\Tag{(18)}
-\]
-
-The relative size of the two elements $d\sigma$~and~$d\sigma'$ may have any
-value whatever. They may be assumed to be of the same or of a
-different order of magnitude, provided the condition remains
-satisfied that $r$~is large compared with the linear dimensions of
-each of them. If we choose $d\sigma$~small compared with~$d\sigma'$, the rays
-diverge from $d\sigma$ to~$d\sigma'$, whereas they converge from $d\sigma$ to~$d\sigma'$\DPtypo{,}{}
-if we choose~$d\sigma$ large compared with~$d\sigma'$.
-
-\Section{21.} Since every point of~$d\sigma$ is the vertex of a cone spreading
-out toward~$d\sigma'$, the whole pencil of rays here considered, which is
-%% -----File: 034.png---Folio 18-------
-defined by $d\sigma$~and~$d\sigma'$, consists of a double infinity of point pencils
-or of a fourfold infinity of rays which must all be considered
-equally for the energy radiation. Similarly the pencil of rays
-may be thought of as consisting of the cones which, emanating
-from all points of~$d\sigma$, converge in one point of~$d\sigma'$ respectively
-as a vertex. If we now imagine the whole pencil of rays to be
-cut by a plane at any arbitrary distance from the elements $d\sigma$
-and~$d\sigma'$ and lying either between them or outside, then the
-cross-sections of any two point pencils on this plane will not be
-identical, not even approximately. In general they will partly
-overlap and partly lie outside of each other, the amount of overlapping
-being different for different intersecting planes. Hence
-it follows that there is no definite cross-section of the pencil of
-rays so far as the uniformity of radiation is concerned. If, however,
-the intersecting plane coincides with either $d\sigma$ or~$d\sigma'$, then
-the pencil has a definite cross-section. Thus these two planes
-show an exceptional property. We shall call them the two
-``focal planes'' of the pencil.
-
-In the special case already mentioned above, namely, when one
-of the two focal planes is infinitely small compared with the other,
-the whole pencil of rays shows the character of a point pencil inasmuch
-as its form is approximately that of a cone having its vertex
-in that focal plane which is small compared with the other. In
-that case the ``cross-section'' of the whole pencil at a definite
-point has a definite meaning. Such a pencil of rays, which is
-similar to a cone, we shall call an elementary pencil, and the
-small focal plane we shall call the first focal plane of the elementary
-pencil. The radiation may be either converging toward the
-first focal plane or diverging from the first focal plane. All
-the pencils of rays passing through a medium may be considered
-as consisting of such elementary pencils, and hence we may base
-our future considerations on elementary pencils only, which is a
-great convenience, owing to their simple nature.
-
-As quantities necessary to define an elementary pencil with a
-given first focal plane~$d\sigma$, we may choose not the second focal
-plane~$d\sigma'$ but the magnitude of that solid angle~$d\Omega$ under which
-$d\sigma'$~is seen from~$d\sigma$. On the other hand, in the case of an arbitrary
-pencil, that is, when the two focal planes are of the same
-order of magnitude, the second focal plane in general cannot be
-%% -----File: 035.png---Folio 19-------
-replaced by the solid angle~$d\Omega$ without the pencil changing
-markedly in character. For if, instead of $d\sigma'$~being given, the
-magnitude and direction of~$d\Omega$, to be taken as constant for all
-points of~$d\sigma$, is given, then the rays emanating from~$d\sigma$ do not
-any longer form the original pencil, but rather an elementary
-pencil whose first focal plane is~$d\sigma$ and whose second focal plane
-lies at an infinite distance.
-
-\Section{22.} Since the energy radiation is propagated in the medium
-with a finite velocity~$q$, there must be in a finite space a finite
-amount of energy. We shall therefore speak of the ``space density
-of radiation,'' meaning thereby the ratio of the total quantity of
-energy of radiation contained in a volume-element to the magnitude
-of the latter. Let us now calculate the space density of
-radiation~$u$ at any arbitrary point of the medium. When we
-consider an infinitely small element of volume~$v$ at the point in
-question, having any shape whatsoever, we must allow for all
-rays passing through the volume-element~$v$. For this purpose
-we shall construct about any point~$O$ of~$v$ as center a sphere
-\Figure[2.5in]{1}
-of radius~$r$, $r$~being large compared
-with the linear dimensions of~$v$ but
-still so small that no appreciable
-absorption or scattering of the radiation
-takes place in the distance~$r$
-(\Fig{1}). Every ray which reaches~$v$
-must then come from some point
-on the surface of the sphere. If,
-then, we at first consider only all the
-rays that come from the points of an
-infinitely small element of area~$d\sigma$
-on the surface of the sphere, and
-reach~$v$, and then sum up for all elements of the spherical surface,
-we shall have accounted for all rays and not taken any one
-more than once.
-
-Let us then calculate first the amount of energy which is contributed
-to the energy contained in~$v$ by the radiation sent from
-such an element~$d\sigma$ to~$v$. We choose~$d\sigma$ so that its linear dimensions
-are small compared with those of~$v$ and consider the cone of
-rays which, starting at a point of~$d\sigma$, meets the volume~$v$. This
-cone consists of an infinite number of conical elements with the
-%% -----File: 036.png---Folio 20-------
-common vertex at~$P$, a point of~$d\sigma$, each cutting out of the volume~$v$
-a certain element of length, say~$s$. The solid angle of such a
-conical element is $\dfrac{f}{r^{2}}$ where $f$~denotes the area of cross-section
-normal to the axis of the cone at a distance~$r$ from the vertex.
-The time required for the radiation to pass through the distance~$s$
-is:
-\[
-\tau = \frac{s}{q}\Add{.}
-\]
-From expression~\Eq{(6)} we may find the energy radiated through a
-certain element of area. In the present case $d\Omega = \dfrac{f}{r^{2}}$ and $\theta = 0$;
-hence the energy is:
-\[
-\tau\, d\sigma\, \frac{f}{r^{2}} K = \frac{fs}{r^{2}q} ˇ K\, d\sigma.
-\Tag{(19)}
-\]
-This energy enters the conical element in~$v$ and spreads out into
-the volume~$fs$. Summing up over all conical elements that start
-from~$d\sigma$ and enter~$v$ we have
-\[
-\frac{K\, d\sigma}{r^{2}q} \sum fs = \frac{K\, d\sigma}{r^{2}q} v.
-\]
-This represents the entire energy of radiation contained in the
-volume~$v$, so far as it is caused by radiation through the element~$d\sigma$.
-In order to obtain the total energy of radiation contained
-in~$v$ we must integrate over all elements~$d\sigma$ contained in the surface
-of the sphere. Denoting by~$d\Omega$ the solid angle $\dfrac{d\sigma}{r^{2}}$ of a
-cone which has its center in~$O$ and intersects in~$d\sigma$ the surface of
-the sphere, we get for the whole energy:
-\[
-\frac{v}{q} \int K\, d\Omega.
-\]
-The volume density of radiation required is found from this by
-dividing by~$v$. It is
-\[
-u = \frac{1}{q} \int K\, d\Omega.
-\Tag{(20)}
-\]
-%% -----File: 037.png---Folio 21-------
-
-Since in this expression $r$~has disappeared, we can think of~$K$
-as the intensity of radiation at the point~$O$ itself. In integrating,
-it is to be noted that $K$~in general depends on the direction $(\theta, \phi)$.
-For radiation that is uniform in all directions $K$~is a constant and
-on integration we get:
-\[
-u = \frac{4\pi K}{q}\Add{.}
-\Tag{(21)}
-\]
-
-\Section{23.} A meaning similar to that of the volume density of the
-total radiation~$u$ is attached to the volume density of radiation
-of a definite frequency~$\ssfu_{\nu}$. Summing up for all parts of the spectrum
-we get:
-\[
-u = \int_{0}^{\infty} \ssfu_{\nu}\, d\nu.
-\Tag{(22)}
-\]
-
-Further by combining equations \Eq{(9)}~and~\Eq{(20)} we have:
-\[
-\ssfu_{\nu} = \frac{1}{q} \int (\ssfK_{\nu} + \ssfK_{\nu}')\, d\Omega,
-\Tag{(23)}
-\]
-and finally for unpolarized radiation uniformly distributed in all
-directions:
-\[
-\ssfu_{\nu} = \frac{8\pi \ssfK_{\nu}}{q}\Add{.}
-\Tag{(24)}
-\]
-%% -----File: 038.png---Folio 22-------
-
-\Chapter[Radiation at Thermodynamic Equilibrium]{II}{Radiation at Thermodynamic Equilibrium.
-Kirchhoff's Law. Black Radiation}
-
-\Section{24.} We shall now apply the laws enunciated in the last chapter
-to the special case of thermodynamic equilibrium, and hence
-we begin our consideration by stating a certain consequence of
-the second principle of thermodynamics: A system of bodies of
-arbitrary nature, shape, and position which is at rest and is surrounded
-by a rigid cover impermeable to heat will, no matter
-what its initial state may be, pass in the course of time into a
-permanent state, in which the temperature of all bodies of the
-system is the same. This is the state of thermodynamic equilibrium,
-in which the entropy of the system has the maximum value
-compatible with the total energy of the system as fixed by the
-initial conditions. This state being reached, no further increase
-in entropy is possible.
-
-In certain cases it may happen that, under the given conditions,
-the entropy can assume not only one but several maxima, of
-which one is the absolute one, the others having only a relative
-significance.\footnote
- {See, \eg, \Name{M.~Planck}, Vorlesungen über Thermodynamik, Leipzig, Veit and Comp., 1911
- (or English Translation, Longmans Green \&~Co.), Secs.\ 165 and~189, \textit{et~seq.}}
-In these cases every state corresponding to a maximum
-value of the entropy represents a state of thermodynamic
-equilibrium of the system. But only one of them, the one corresponding
-to the absolute maximum of entropy, represents the
-absolutely stable equilibrium. All the others are in a certain
-sense unstable, inasmuch as a suitable, however small, disturbance
-may produce in the system a permanent change in the
-equilibrium in the direction of the absolutely stable equilibrium.
-An example of this is offered by supersaturated steam enclosed in
-a rigid vessel or by any explosive substance. We shall also meet
-such unstable equilibria in the case of radiation phenomena
-(\Sec{52}).
-%% -----File: 039.png---Folio 23-------
-
-\Section{25.} We shall now, as in the previous chapter, assume that we
-are dealing with homogeneous isotropic media whose condition
-depends only on the temperature, and we shall inquire what laws
-the radiation phenomena in them must obey in order to be consistent
-with the deduction from the second principle mentioned
-in the preceding section. The means of answering this inquiry
-is supplied by the investigation of the state of thermodynamic
-equilibrium of one or more of such media, this investigation to be
-conducted by applying the conceptions and laws established in
-the last chapter.
-
-We shall begin with the simplest case, that of a single medium
-extending very far in all directions of space, and, like all systems
-we shall here consider, being surrounded by a rigid cover impermeable
-to heat. For the present we shall assume that the
-medium has finite coefficients of absorption, emission, and
-scattering.
-
-Let us consider, first, points of the medium that are far away
-from the surface. At such points the influence of the surface is,
-of course, vanishingly small and from the homogeneity and the
-isotropy of the medium it will follow that in a state of thermodynamic
-equilibrium the radiation of heat has everywhere and in all
-directions the same properties. Then $\ssfK_{\nu}$, the specific intensity of
-radiation of a plane polarized ray of frequency~$\nu$ (\Sec{17}), must be
-independent of the azimuth of the plane of polarization as well as
-of position and direction of the ray. Hence to each pencil of
-rays starting at an element of area~$d\sigma$ and diverging within
-a conical element~$d\Omega$ corresponds an exactly equal pencil of opposite
-direction converging within the same conical element toward
-the element of area.
-
-Now the condition of thermodynamic equilibrium requires
-that the temperature shall be everywhere the same and shall not
-vary in time. Therefore in any given arbitrary time just as
-much radiant heat must be absorbed as is emitted in each volume-element
-of the medium. For the heat of the body depends
-only on the heat radiation, since, on account of the uniformity in
-temperature, no conduction of heat takes place. This condition
-is not influenced by the phenomenon of scattering, because scattering
-refers only to a change in direction of the energy radiated,
-not to a creation or destruction of it. We shall, therefore, calculate
-%% -----File: 040.png---Folio 24-------
-the energy emitted and absorbed in the time~$dt$ by a
-volume-element~$v$.
-
-According to equation~\Eq{(2)} the energy emitted has the value
-\[
-dt\, v ˇ 8\pi \int_{0}^{\infty} \epsilon_{\nu}\, d\nu
-\]
-where $\epsilon_{\nu}$, the coefficient of emission of the medium, depends only
-on the frequency~$\nu$ and on the temperature in addition to the
-chemical nature of the medium.
-
-\Section{26.} For the calculation of the energy absorbed we shall employ
-the same reasoning as was illustrated by \Fig{1} (\Sec{22}) and
-shall retain the notation there used. The radiant energy
-absorbed by the volume-element~$v$ in the time~$dt$ is found by considering
-the intensities of all the rays passing through the element~$v$
-and taking that fraction of each of these rays which is absorbed
-in~$v$. Now, according to~\Eq{(19)}, the conical element that starts
-from~$d\sigma$ and cuts out of the volume~$v$ a part equal to~$fs$ has the
-intensity (energy radiated per unit time)
-\[
-d\sigma ˇ \frac{f}{r^{2}} ˇ K
-\]
-or, according to~\Eq{(12)}, by considering the different parts of the
-spectrum separately:
-\[
-2\, d\sigma\, \frac{f}{r^{2}} \int_{0}^{\infty} \ssfK_{\nu}\, d\nu.
-\]
-Hence the intensity of a monochromatic ray is:
-\[
-2\, d\sigma\, \frac{f}{r^{2}} \ssfK_{\nu}\, d\nu.
-\]
-The amount of energy of this ray absorbed in the distance~$s$ in
-the time~$dt$ is, according to~\Eq{(4)},
-\[
-dt\, \alpha_{\nu}s\, 2\, d\sigma\, \frac{f}{r^{2}} \ssfK_{\nu}\, d\nu.
-\]
-Hence the absorbed part of the energy of this small cone of rays,
-as found by integrating over all frequencies, is:
-\[
-dt\, 2\, d\sigma\, \frac{fs}{r^{2}} \int_{0}^{\infty} \alpha_{\nu} \ssfK_{\nu}\, d\nu.
-\]
-%% -----File: 041.png---Folio 25-------
-When this expression is summed up over all the different cross-sections~$f$
-of the conical elements starting at~$d\sigma$ and passing
-through~$v$, it is evident that $\sum fs = v$, and when we sum up over
-all elements~$d\sigma$ of the spherical surface of radius~$r$ we have
-\[
-\int \frac{d\sigma}{r^{2}} = 4\pi.
-\]
-Thus for the total radiant energy absorbed in the time~$d$t by the
-volume-element~$v$ the following expression is found:
-\[
-dt\, v\, 8\pi \int_{0}^{\infty} \alpha_{\nu} \ssfK_{\nu}\, d\nu.
-\Tag{(25)}
-\]
-By equating the emitted and absorbed energy we obtain:
-\[
-\int_{0}^{\infty} \epsilon_{\nu}\, d\nu
- = \int_{0}^{\infty} \alpha_{\nu} \ssfK_{\nu}\, d\nu.
-\]
-
-A similar relation may be obtained for the separate parts of the
-spectrum. For the energy emitted and the energy absorbed in the
-state of thermodynamic equilibrium are equal, not only for the
-entire radiation of the whole spectrum, but also for each monochromatic
-radiation. This is readily seen from the following. The
-magnitudes of $\epsilon_{\nu}$,~$\alpha_{\nu}$, and~$\ssfK_{\nu}$ are independent of position. Hence,
-if for any single color the absorbed were not equal to the emitted
-energy, there would be everywhere in the whole medium a continuous
-increase or decrease of the energy radiation of that
-particular color at the expense of the other colors. This would be
-contradictory to the condition that $\ssfK_{\nu}$~for each separate frequency
-does not change with the time. We have therefore for each
-frequency the relation:
-\begin{align*}
- \epsilon_{\nu} &= \alpha_{\nu} \ssfK_{\nu}, \textrm{ or}
-\Tag{(26)}\\
- \ssfK_{\nu} &= \frac{\epsilon_{\nu}}{\alpha_{\nu}},
-\Tag{(27)}
-\end{align*}
-%[** Semantic theorem]
-\ie: \emph{in the interior of a medium in a state of thermodynamic equilibrium
-the specific intensity of radiation of a certain frequency is
-equal to the coefficient of emission divided by the coefficient of absorption
-of the medium for this frequency}.
-%% -----File: 042.png---Folio 26-------
-
-\Section{27.} Since $\epsilon_{\nu}$~and~$\alpha_{\nu}$ depend only on the nature of the medium,
-the temperature, and the frequency~$\nu$, the intensity of radiation of
-a definite color in the state of thermodynamic equilibrium is
-completely defined by the nature of the medium and the temperature.
-An exceptional case is when $\alpha_{\nu} = 0$, that is, when the
-medium does not at all absorb the color in question. Since $\ssfK_{\nu}$~cannot
-become infinitely large, a first consequence of this is that
-in that case $\epsilon_{\nu} = 0$ also, that is, a medium does not emit any color
-which it does not absorb. A second consequence is that if $\epsilon_{\nu}$~and~$\alpha_{\nu}$
-both vanish, equation~\Eq{(26)} is satisfied by every value of~$\ssfK_{\nu}$.
-%[** Theorem]
-\emph{In a medium which is diathermanous for a certain color
-thermodynamic equilibrium can exist for any intensity of radiation
-whatever of that color.}
-
-This supplies an immediate illustration of the cases spoken of
-before (\Sec{24}), where, for a given value of the total energy of a
-system enclosed by a rigid cover impermeable to heat, several
-states of equilibrium can exist, corresponding to several relative
-maxima of the entropy. That is to say, since the intensity of
-radiation of the particular color in the state of thermodynamic
-equilibrium is quite independent of the temperature of a medium
-which is diathermanous for this color, the given total energy may
-be arbitrarily distributed between radiation of that color and the
-heat of the body, without making thermodynamic equilibrium
-impossible. Among all these distributions there is one particular
-one, corresponding to the absolute maximum of entropy, which
-represents absolutely stable equilibrium. This one, unlike all the
-others, which are in a certain sense unstable, has the property of
-not being appreciably affected by a small disturbance. Indeed
-we shall see later (\Sec{48}) that among the infinite number of
-values, which the quotient $\dfrac{\epsilon_{\nu}}{\alpha_{\nu}}$ can have, if numerator and denominator
-both vanish, there exists one particular one which depends
-in a definite way on the nature of the medium, the frequency~$\nu$,
-and the temperature. This distinct value of the fraction is
-accordingly called the stable intensity of radiation~$\ssfK_{\nu}$, in the medium,
-which at the temperature in question is diathermanous for
-rays of the frequency~$\nu$.
-
-Everything that has just been said of a medium which is diathermanous
-for a certain kind of rays holds true for an absolute
-%% -----File: 043.png---Folio 27-------
-vacuum, which is a medium diathermanous for rays of all kinds,
-the only difference being that one cannot speak of the heat and
-the temperature of an absolute vacuum in any definite sense.
-
-For the present we again shall put the special case of diathermancy
-aside and assume that all the media considered have a
-finite coefficient of absorption.
-
-\Section{28.} Let us now consider briefly the phenomenon of scattering
-at thermodynamic equilibrium. Every ray meeting the volume-element~$v$
-suffers there, apart from absorption, a certain weakening
-of its intensity because a certain fraction of its energy is
-diverted in different directions. The value of the total energy
-of scattered radiation received and diverted, in the time~$dt$ by
-the volume-element~$v$ in all directions, may be calculated from
-expression~\Eq{(3)} in exactly the same way as the value of the absorbed
-energy was calculated in \Sec{26}. Hence we get an expression
-similar to~\Eq{(25)}, namely,
-\[
-dt\, v\, 8\pi \int_{0}^{\infty} \beta_{\nu} \ssfK_{\nu}\, d\nu.
-\Tag{(28)}
-\]
-The question as to what becomes of this energy is readily answered.
-On account of the isotropy of the medium, the energy
-scattered in~$v$ and given by~\Eq{(28)} is radiated uniformly in all directions
-just as in the case of the energy entering~$v$. Hence that part
-of the scattered energy received in~$v$ which is radiated out in a
-cone of solid angle~$d\Omega$ is obtained by multiplying the last expression
-by~$\dfrac{d\Omega}{4\pi}$. This gives
-\[
-2\, dt\, v\, d\Omega \int_{0}^{\infty} \beta_{\nu} \ssfK_{\nu}\, d\nu,
-\]
-and, for monochromatic plane polarized radiation,
-\[
-dt\, v\, d\Omega\, \beta_{\nu} \ssfK_{\nu}\, d\nu.
-\Tag{(29)}
-\]
-
-Here it must be carefully kept in mind that this uniformity of
-radiation in all directions holds only for all rays striking the element~$v$
-taken together; a single ray, even in an isotropic medium,
-is scattered in different directions with different intensities and
-different directions of polarization. (See end of \Sec{8}.)
-%% -----File: 044.png---Folio 28-------
-
-It is thus found that, when thermodynamic equilibrium of radiation
-exists inside of the medium, the process of scattering produces,
-on the whole, no effect. The radiation falling on a volume-element
-from all sides and scattered from it in all directions behaves
-exactly as if it had passed directly through the volume-element
-without the least modification. Every ray loses by
-scattering just as much energy as it regains by the scattering of
-other rays.
-
-\WrapFigure[0.75in]{2}
-\Section{29.} We shall now consider from a different point of view the
-radiation phenomena in the interior of a very extended homogeneous
-isotropic medium which is in thermodynamic
-equilibrium. That is to say, we shall confine our
-attention, not to a definite volume-element, but to a
-definite pencil, and in fact to an elementary pencil
-(\Sec{21}). Let this pencil be specified by the infinitely
-small focal plane~$d\sigma$ at the point~$O$ (\Fig{2}), perpendicular
-to the axis of the pencil, and by the solid
-angle~$d\Omega$, and let the radiation take place toward the
-focal plane in the direction of the arrow. We shall
-consider exclusively rays which belong to this pencil.
-
-The energy of monochromatic plane polarized radiation
-of the pencil considered passing in unit time
-through~$d\sigma$ is represented, according to~\Eq{(11)}, since in
-this case $dt = 1$, $\theta = 0$, by
-\[
-d\sigma\, d\Omega\, \ssfK_{\nu}\, d\nu.
-\Tag{(30)}
-\]
-The same value holds for any other cross-section of the pencil.
-For first, $\ssfK_{\nu}\, d\nu$ has everywhere the same magnitude (\Sec{25}),
-and second, the product of any right section of the pencil and
-the solid angle at which the focal plane~$d\sigma$ is seen from this section
-has the constant value~$d\sigma\, d\Omega$, since the magnitude of the
-cross-section increases with the distance from the vertex~$O$ of
-the pencil in the proportion in which the solid angle decreases.
-Hence the radiation inside of the pencil takes place just as if the
-medium were perfectly diathermanous.
-
-On the other hand, the radiation is continuously modified along
-its path by the effect of emission, absorption, and scattering. We
-shall consider the magnitude of these effects separately.
-
-\Section{30.} Let a certain volume-element of the pencil be bounded by
-%% -----File: 045.png---Folio 29-------
-two cross-sections at distances equal to~$r_{0}$ (of arbitrary length)
-and $r_{0} + dr_{0}$ respectively from the vertex~$O$. The volume will be
-represented by $dr_{0} ˇ r_{0}^{2}\, d\Omega$. It emits in unit time toward the
-focal plane~$d\sigma$ at~$O$ a certain quantity~$E$ of energy of monochromatic
-plane polarized radiation. $E$~may be obtained from~\Eq{(1)}
-by putting
-\[
-dt = 1,\quad
-d\tau = dr_{0}\, r_{0}^{2}\, d\Omega,\quad
-d\Omega = \frac{d\sigma}{r_{0}^{2}}
-\]
-and omitting the numerical factor~$2$. We thus get
-\[
-%[** Inconstsistent dot]
-E = dr_{0} ˇ d\Omega\, d\sigma\, \epsilon_{\nu}\, d\nu.
-\Tag{(31)}
-\]
-
-Of the energy~$E$, however, only a fraction~$E_{0}$ reaches~$O$, since
-in every infinitesimal element of distance~$s$ which it traverses
-before reaching~$O$ the fraction $(\alpha_{\nu} + \beta_{\nu})s$ is lost by absorption and
-scattering. Let $E_{r}$~represent that part of~$E$ which reaches a
-cross-section at a distance~$r$ ($< r_{0}$) from~$O$. Then for a small
-distance $s = dr$ we have
-\[
-E_{r + dr} - E_{r} = E_{r}(\alpha_{\nu} + \beta_{\nu})\, dr,
-\]
-or,
-\[
-\frac{dE_{r}}{dr} = E_{r}(\alpha_{\nu} + \beta_{\nu}),
-\]
-and, by integration,
-\[
-E_{r} = E e^{(\alpha_{\nu} + \beta_{\nu})(r-r_{0})}
-\]
-since, for $r = r_{0}$, $E_{r} = E$ is given by equation~\Eq{(31)}. From this, by
-putting $r = 0$, the energy emitted by the volume-element at~$r_{0}$
-which reaches~$O$ is found to be
-\[
-E_{0} = E e^{-(\alpha_{\nu} + \beta_{\nu})r_{0}}
- = dr_{0}\, d\Omega\, d\sigma\, \epsilon_{\nu} e^{-(\alpha_{\nu} + \beta_{\nu})r_{0}}\, d\nu.
-\Tag{(32)}
-\]
-All volume-elements of the pencils combined produce by their
-emission an amount of energy reaching~$d\sigma$ equal to
-\[
-d\Omega\, d\sigma\, d\nu\, \epsilon_{\nu} \int_{0}^{\infty} dr_{0}\, e^{-(\alpha_{\nu} + \beta_{\nu})r_{0}}
- = d\Omega\, d\sigma\, \frac{\epsilon_{\nu}}{\alpha_{\nu} + \beta_{\nu}}\, d\nu.
-\Tag{(33)}
-\]
-
-\Section{31.} If the scattering did not affect the radiation, the total
-energy reaching~$d\sigma$ would necessarily consist of the quantities of
-energy emitted by the different volume-elements of the pencil,
-allowance being made, however, for the losses due to absorption
-%% -----File: 046.png---Folio 30-------
-on the way. For $\beta_{\nu} = 0$ expressions \Eq{(33)} and \Eq{(30)} are identical,
-as may be seen by comparison with~\Eq{(27)}. Generally, however,
-\Eq{(30)}~is larger than~\Eq{(33)} because the energy reaching~$d\sigma$ contains
-also some rays which were not at all emitted from elements inside
-of the pencil, but somewhere else, and have entered later on by
-scattering. In fact, the volume-elements of the pencil do not
-merely scatter outward the radiation which is being transmitted
-inside the pencil, but they also collect into the pencil rays coming
-from without. The radiation~$E'$ thus collected by the volume-element
-at~$r_{0}$ is found, by putting in~\Eq{(29)},
-\[
-dt = 1,\quad
-\nu = dr_{0}\, d\Omega\, r_{0}^{2},\quad
-d\Omega = \frac{d\sigma}{r_{0}^{2}},
-\]
-to be
-\[
-E' = dr_{0}\, d\Omega\, d\sigma\, \beta_{\nu} \ssfK_{\nu}\, d\nu.
-\]
-
-This energy is to be added to the energy~$E$ emitted by the volume-element,
-which we have calculated in~\Eq{(31)}. Thus for the
-total energy contributed to the pencil in the volume-element at~$r_{0}$
-we find:
-\[
-E + E' = dr_{0}\, d\Omega\, d\sigma\, (\epsilon_{\nu} + \beta_{\nu} \ssfK_{\nu})\, d\nu.
-\]
-The part of this reaching~$O$ is, similar to~\Eq{(32)}:
-\[
-dr_{0}\, d\Omega\, d\sigma\, (\epsilon_{\nu} + \beta_{\nu} \ssfK_{\nu})\, d\nu\, e^{-r_{0} (\alpha_{\nu} + \beta_{\nu})}\Add{.}
-\]
-Making due allowance for emission and collection of scattered
-rays entering on the way, as well as for losses by absorption and
-scattering, all volume-elements of the pencil combined give for
-the energy ultimately reaching~$d\sigma$
-\[
-d\Omega\, d\sigma\, (\epsilon_{\nu} + \beta_{\nu} \ssfK_{\nu})\, d\nu
- \int_{0}^{\infty} dr_{0}\, e^{-r_{0} (\alpha_{\nu} + \beta_{\nu})}
- = d\Omega\, d\sigma\, \frac{\epsilon_{\nu} + \beta_{\nu} \ssfK_{\nu}}{\alpha_{\nu} + \beta_{\nu}}\, d\nu,
-\]
-and this expression is really exactly equal to that given by~\Eq{(30)},
-as may be seen by comparison with~\Eq{(26)}.
-
-\Section{32.} The laws just derived for the state of radiation of a homogeneous
-isotropic medium when it is in thermodynamic equilibrium
-hold, so far as we have seen, only for parts of the medium
-which lie very far away from the surface, because for such parts
-only may the radiation be considered, by symmetry, as independent
-of position and direction. A simple consideration, however,
-%% -----File: 047.png---Folio 31-------
-shows that the value of~$\ssfK_{\nu}$, which was already calculated and given
-by~\Eq{(27)}, and which depends only on the temperature and the
-nature of the medium, gives the correct value of the intensity of
-radiation of the frequency considered for all directions up to
-points directly below the surface of the medium. For in the state
-of thermodynamic equilibrium every ray must have just the same
-intensity as the one travelling in an exactly opposite direction,
-since otherwise the radiation would cause a unidirectional transport
-of energy. Consider then any ray coming from the surface
-of the medium and directed inward; it must have the same
-intensity as the opposite ray, coming from the interior. A
-%[** Theorem]
-further immediate consequence of this is \emph{that the total state of
-radiation of the medium is the same on the surface as in the interior}.
-
-\Section{33.} While the radiation that starts from a surface element and
-is directed toward the interior of the medium is in every respect
-equal to that emanating from an equally large parallel element of
-area in the interior, it nevertheless has a different history. That
-is to say, since the surface of the medium was assumed to be
-impermeable to heat, it is produced only by reflection at the surface
-of radiation coming from the interior. So far as special
-details are concerned, this can happen in very different ways,
-depending on whether the surface is assumed to be smooth, \ie,
-in this case reflecting, or rough, \eg, white (\Sec{10}). In the first
-case there corresponds to each pencil which strikes the surface
-another perfectly definite pencil, symmetrically situated and
-having the same intensity, while in the second case every incident
-pencil is broken up into an infinite number of reflected pencils,
-each having a different direction, intensity, and polarization.
-While this is the case, nevertheless the rays that strike a surface-element
-from all different directions with the same intensity~$\ssfK_{\nu}$
-also produce, all taken together, a uniform radiation of the same
-intensity~$\ssfK_{\nu}$, directed toward the interior of the medium.
-
-\Section{34.} Hereafter there will not be the slightest difficulty in
-dispensing with the assumption made in \Sec{25} that the medium
-in question extends very far in all directions. For after thermodynamic
-equilibrium has been everywhere established in our medium,
-the equilibrium is, according to the results of the last
-paragraph, in no way disturbed, if we assume any number of
-rigid surfaces impermeable to heat and rough or smooth to be
-%% -----File: 048.png---Folio 32-------
-inserted in the medium. By means of these the whole system is
-divided into an arbitrary number of perfectly closed separate
-systems, each of which may be chosen as small as the general
-restrictions stated in \Sec{2} permit. It follows from this that
-the value of the specific intensity of radiation~$\ssfK_{\nu}$ given in~\Eq{(27)}
-remains valid for the thermodynamic equilibrium of a substance
-enclosed in a space as small as we please and of any shape whatever.
-
-\Section{35.} From the consideration of a system consisting of a single
-homogeneous isotropic substance we now pass on to the treatment
-of a system consisting of two different homogeneous isotropic
-substances contiguous to each other, the system being, as before,
-enclosed by a rigid cover impermeable to heat. We consider the
-state of radiation when thermodynamic equilibrium exists, at
-first, as before, with the assumption that the media are of considerable
-extent. Since the equilibrium is nowise disturbed, if we
-think of the surface separating the two media as being replaced
-for an instant by an area entirely impermeable to heat radiation,
-the laws of the last paragraphs must hold for each of the two
-substances separately. Let the specific intensity of radiation of
-frequency~$\nu$ polarized in an arbitrary plane be $\ssfK_{\nu}$~in the first substance
-(the upper one in \Fig{3}), and $\ssfK_{\nu}'$~in the second, and, in
-general, let all quantities referring to the second substance be
-indicated by the addition of an accent. Both of the quantities
-$\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ depend, according to equation~\Eq{(27)}, only on the temperature,
-the frequency~$\nu$, and the nature of the two substances,
-and these values of the intensities of radiation hold up to very
-small distances from the bounding surface of the substances, and
-hence are entirely independent of the properties of this surface.
-
-\Section{36.} We shall now suppose, to begin with, that the bounding
-surface of the media is smooth (\Sec{9}). Then every ray coming
-from the first medium and falling on the bounding surface is
-divided into two rays, the reflected and the transmitted ray.
-The directions of these two rays vary with the angle of incidence
-and the color of the incident ray; the intensity also varies with
-its polarization. Let us denote by~$\rho$ (coefficient of reflection) the
-fraction of the energy reflected, then the fraction transmitted is~$(1 - \rho)$,
-$\rho$~depending on the angle of incidence, the frequency, and
-the polarization of the incident ray. Similar remarks apply to
-%% -----File: 049.png---Folio 33-------
-$\rho'$~the coefficient of reflection of a ray coming from the second
-medium and falling on the bounding surface.
-
-Now according to~\Eq{(11)} we have for the monochromatic plane
-polarized radiation of frequency~$\nu$, emitted in time~$dt$ toward the
-first medium (in the direction of the feathered arrow upper left
-\Figure[4in]{3}
-hand in \Fig{3}), from an element~$d\sigma$ of the bounding surface and
-contained in the conical element~$d\Omega$,
-\[
-dt\, d\sigma\, \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu,
-\Tag{(34)}
-\]
-where
-\[
-d\Omega = \sin\theta\, d\theta\, d\phi.
-\Tag{(35)}
-\]
-This energy is supplied by the two rays which come from the first
-and the second medium and are respectively reflected from or
-transmitted by the element~$d\sigma$ in the corresponding direction
-(the unfeathered arrows). (Of the element~$d\sigma$ only the one point~$O$
-is indicated.) The first ray, according to the law of reflection,
-continues in the symmetrically situated conical element~$d\Omega$, the
-second in the conical element
-\[
-d\Omega' = \sin\theta'\, d\theta'\, d\phi'
-\Tag{(36)}
-\]
-where, according to the law of refraction,
-\[
-\phi' = \phi\quad \text{and}\quad
-\frac{\sin\theta}{\sin\theta'} = \frac{q}{q'}\Add{.}
-\Tag{(37)}
-\]
-%% -----File: 050.png---Folio 34-------
-
-If we now assume the radiation~\Eq{(34)} to be polarized either in
-the plane of incidence or at right angles thereto, the same will
-be true for the two radiations of which it consists, and the
-radiation coming from the first medium and reflected from~$d\sigma$
-contributes the part
-\[
-\rho\, dt\, d\sigma\, \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu
-\Tag{(38)}
-\]
-while the radiation coming from the second medium and transmitted
-through~$d\sigma$ contributes the part
-\[
-(1 - \rho')\, dt\, d\sigma\, \cos\theta'\, d\Omega'\, \ssfK_{\nu}'\, d\nu.
-\Tag{(39)}
-\]
-The quantities $dt$,~$d\sigma$,~$\nu$ and~$d\nu$ are written without the accent,
-because they have the same values in both media.
-
-By adding \Eq{(38)}~and~\Eq{(39)} and equating their sum to the expression~\Eq{(34)}
-we find
-\[
-\rho\, \cos\theta\, d\Omega\, \ssfK_{\nu} + (1 - \rho') \cos\theta'\, d\Omega'\, \ssfK_{\nu}'
- = \cos\theta\, d\Omega\, \ssfK_{\nu}.
-\]
-
-Now from~\Eq{(37)} we have
-\[
-\frac{\cos\theta\, d\theta}{q} = \frac{\cos\theta'\, d\theta'}{q'}
-\]
-and further by \Eq{(35)}~and~\Eq{(36)}
-\[
-d\Omega'\, \cos\theta' = \frac{d\Omega\, \cos\theta\, q'^{2}}{q^{2}}.
-\]
-Therefore we find
-\[
-\rho \ssfK_{\nu} + (1 - \rho') \frac{q'^{2}}{q^{2}} \ssfK_{\nu}' = \ssfK
-\]
-or
-\[
-\frac{\ssfK_{\nu}}{\ssfK_{\nu}'} ˇ \frac{q^{2}}{q'^{2}} = \frac{1 - \rho'}{1 - \rho}.
-\]
-
-\Section{37.} In the last equation the quantity on the left side is independent
-of the angle of incidence~$\theta$ and of the particular kind of
-polarization; hence the same must be true for the right side.
-Hence, whenever the value of this quantity is known for a single
-angle of incidence and any definite kind of polarization, this
-value will remain valid for all angles of incidence and all kinds
-of polarization. Now in the special case when the rays are
-polarized at right angles to the plane of incidence and strike the
-%% -----File: 051.png---Folio 35-------
-bounding surface at the angle of polarization, $\rho = 0$, and $\rho' = 0$.
-The expression on the right side of the last equation then becomes~$1$;
-hence it must always be~$1$ and we have the general relations:
-\[
-\rho = \rho'
-\Tag{(40)}
-\]
-and
-\[
-q^{2} \ssfK_{\nu} = q'^{2} \ssfK_{\nu}'\Add{.}
-\Tag{(41)}
-\]
-
-\Section{38.} The first of these two relations, which states that the
-coefficient of reflection of the bounding surface is the same on
-both sides, is a special case of a general law of reciprocity first
-stated by \Name{Helmholtz}.\footnote
- {\Name{H.~v.\ Helmholtz}, Handbuch d.\ physiologischen Optik~1. Lieferung, Leipzig, Leop.\ Voss,
- 1856, p.~169. See also \Name{Helmholtz}, Vorlesungen über die Theorie der Wärme herausgegeben
- von \Name{F.~Richarz}, Leipzig, J.~A. Barth, 1903, p.~161. The restrictions of the law of reciprocity
- made there do not bear on our problems, since we are concerned with temperature radiation
- only (\Sec{7}).}
-According to this law the loss of intensity
-which a ray of definite color and polarization suffers on its way
-through any media by reflection, refraction, absorption, and
-scattering is exactly equal to the loss suffered by a ray of the
-same intensity, color, and polarization pursuing an exactly
-opposite path. An immediate consequence of this law is that the
-radiation striking the bounding surface of any two media is
-always transmitted as well as reflected equally on both sides, for
-every color, direction, and polarization.
-
-\Section{39.} The second formula~\Eq{(41)} establishes a relation between the
-intensities of radiation in the two media, for it states that, when
-%[** Theorems]
-thermodynamic equilibrium exists, \emph{the specific intensities of radiation
-of a certain frequency in the two media are in the inverse
-ratio of the squares of the velocities of propagation or in the direct
-ratio of the squares of the indices of refraction}.\footnote
- {\Name{G.~Kirchhoff}, Gesammelte Abhandlungen, Leipzig, J.~A. Barth, 1882, p.~594.
- \Name{R.~Clausius}, Pogg.\ Ann.~\textbf{121}, p.~1, 1864.}
-
-By substituting for~$\ssfK_{\nu}$ its value from~\Eq{(27)} we obtain the following
-theorem: \emph{The quantity
-\[
-q^{2} \ssfK_{\nu} = q^{2} \frac{\epsilon_{\nu}}{\alpha_{\nu}}
-\Tag{(42)}
-\]
-does not depend on the nature of the substance, and is, therefore,
-a universal function of the temperature~$T$ and the frequency~$\nu$ alone.}
-
-The great importance of this law lies evidently in the fact that
-it states a property of radiation which is the same for all bodies
-%% -----File: 052.png---Folio 36-------
-in nature, and which need be known only for a single arbitrarily
-chosen body, in order to be stated quite generally for all bodies.
-We shall later on take advantage of the opportunity offered by
-this statement in order actually to calculate this universal function
-(\Sec{165}).
-
-\Section{40.} We now consider the other case, that in which the
-bounding surface of the two media is rough. This case is much
-more general than the one previously treated, inasmuch as the
-energy of a pencil directed from an element of the bounding surface
-into the first medium is no longer supplied by two definite
-pencils, but by an arbitrary number, which come from both
-media and strike the surface. Here the actual conditions may be
-very complicated according to the peculiarities of the bounding
-surface, which moreover may vary in any way from one element
-to another. However, according to \Sec{35}, the values of the
-specific intensities of radiation $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ remain always the
-same in all directions in both media, just as in the case of a smooth
-bounding surface. That this condition, necessary for thermodynamic
-equilibrium, is satisfied is readily seen from \Name{Helmholtz's}
-law of reciprocity, according to which, in the case of stationary
-radiation, for each ray striking the bounding surface and
-diffusely reflected from it on both sides, there is a corresponding
-ray at the same point, of the same intensity and opposite direction,
-produced by the inverse process at the same point on the
-bounding surface, namely by the gathering of diffusely incident
-rays into a definite direction, just as is the case in the interior of
-each of the two media.
-
-\Section{41.} We shall now further generalize the laws obtained.
-First, just as in \Sec{34}, the assumption made above, namely,
-that the two media extend to a great distance, may be abandoned
-since we may introduce an arbitrary number of bounding surfaces
-without disturbing the thermodynamic equilibrium. Thereby
-we are placed in a position enabling us to pass at once to the case
-of any number of substances of any size and shape. For when a
-system consisting of an arbitrary number of contiguous substances
-is in the state of thermodynamic equilibrium, the equilibrium is
-in no way disturbed, if we assume one or more of the surfaces of
-contact to be wholly or partly impermeable to heat. Thereby
-we can always reduce the case of any number of substances to
-%% -----File: 053.png---Folio 37-------
-that of two substances in an enclosure impermeable to heat, and,
-therefore, the law may be stated quite generally, that, when any
-arbitrary system is in the state of thermodynamic equilibrium,
-the specific intensity of radiation~$\ssfK_{\nu}$ is determined in each
-separate substance by the universal function~\Eq{(42)}.
-
-\Section{42.} We shall now consider a system in a state of thermodynamic
-equilibrium, contained within an enclosure impermeable
-to heat and consisting of $n$~emitting and absorbing adjacent bodies
-of any size and shape whatever. As in \Sec{36}, we again confine
-our attention to a monochromatic plane polarized pencil
-which proceeds from an element~$d\sigma$ of the bounding surface of the
-two media in the direction toward the first medium (\Fig{3},
-feathered arrow) within the conical element~$d\Omega$. Then, as in~\Eq{(34)},
-the energy supplied by the pencil in unit time is
-\[
-d\sigma\, \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu = I.
-\Tag{(43)}
-\]
-This energy of radiation~$I$ consists of a part coming from the first
-medium by regular or diffuse reflection at the bounding surface
-and of a second part coming through the bounding surface from
-the second medium. We shall, however, not stop at this mode of
-division, but shall further subdivide~$I$ according to that one of
-the $n$~media from which the separate parts of the radiation~$I$
-have been emitted. This point of view is distinctly different
-from the preceding, since, \eg, the rays transmitted from the
-second medium through the bounding surface into the pencil
-considered have not necessarily been emitted in the second
-medium, but may, according to circumstances, have traversed a
-long and very complicated path through different media and may
-have undergone therein the effect of refraction, reflection, scattering,
-and partial absorption any number of times. Similarly
-the rays of the pencil, which coming from the first medium are
-reflected at~$d\sigma$, were not necessarily all emitted in the first
-medium. It may even happen that a ray emitted from a certain
-medium, after passing on its way through other media, returns to
-the original one and is there either absorbed or emerges from this
-medium a second time.
-
-We shall now, considering all these possibilities, denote that
-part of~$I$ which has been emitted by volume-elements of the first
-medium by~$I_{1}$ no matter what paths the different constituents
-%% -----File: 054.png---Folio 38-------
-have pursued, that which has been emitted by volume-elements
-of the second medium by~$I_{2}$, etc. Then since every part of~$I$
-must have been emitted by an element of some body, the following
-equation must hold,
-\[
-I = I_{1} + I_{2} + I_{3} + \dots I_{n}.
-\Tag{(44)}
-\]
-
-\Section{43.} The most adequate method of acquiring more detailed
-information as to the origin and the paths of the different rays
-of which the radiations $I_{1}$,~$I_{2}$, $I_{3}$,~$\dots I_{n}$ consist, is to
-pursue the opposite course and to inquire into the future fate of
-that pencil, which travels exactly in the opposite direction to
-the pencil~$I$ and which therefore comes from the first medium in
-the cone~$d\Omega$ and falls on the surface element~$d\sigma$ of the second medium.
-For since every optical path may also be traversed in the
-opposite direction, we may obtain by this consideration all paths
-along which rays can pass into the pencil~$I$, however complicated
-they may otherwise be. Let $J$ represent the intensity of this
-inverse pencil, which is directed toward the bounding surface
-and is in the same state of polarization. Then, according to
-\Sec{40},
-\[
-J = I.
-\Tag{(45)}
-\]
-
-At the bounding surface~$d\sigma$ the rays of the pencil~$J$ are partly
-reflected and partly transmitted regularly or diffusely, and
-thereafter, travelling in both media, are partly absorbed, partly
-scattered, partly again reflected or transmitted to different
-media,~etc., according to the configuration of the system. But
-finally the whole pencil~$J$ after splitting into many separate rays
-will be completely absorbed in the $n$~media. Let us denote that
-part of~$J$ which is finally absorbed in the first medium by~$J_{1}$, that
-which is finally absorbed in the second medium by~$J_{2}$, etc., then
-we shall have
-\[
-J = J_{1} + J_{2} + J_{3} + \dots + J_{n}.
-\]
-
-Now the volume-elements of the $n$~media, in which the absorption
-of the rays of the pencil~$J$ takes place, are precisely the same
-as those in which takes place the emission of the rays constituting
-the pencil~$I$, the first one considered above. For, according to
-\Name{Helmholtz's} law of reciprocity, no appreciable radiation of the pencil~$J$
-can enter a volume-element which contributes no appreciable
-radiation to the pencil~$I$ and \textit{vice versa}.
-%% -----File: 055.png---Folio 39-------
-
-Let us further keep in mind that the absorption of each volume-element
-is, according to~\Eq{(42)}, proportional to its emission and that,
-according to \Name{Helmholtz's} law of reciprocity, the decrease which
-the energy of a ray suffers on any path is always equal to the decrease
-suffered by the energy of a ray pursuing the opposite path.
-It will then be clear that the volume-elements considered absorb
-the rays of the pencil~$J$ in just the same ratio as they contribute
-by their emission to the energy of the opposite pencil~$I$. Since,
-moreover, the sum~$I$ of the energies given off by emission by all
-volume-elements is equal to the sum~$J$ of the energies absorbed
-by all elements, the quantity of energy absorbed by each separate
-volume-element from the pencil~$J$ must be equal to the quantity
-of energy emitted by the same element into the pencil~$I$. In
-%[** Theorem]
-other words: \emph{the part of a pencil~$I$ which has been emitted from a
-certain volume of any medium is equal to the part of the pencil~$J$
-\($= I$\) oppositely directed, which is absorbed in the same volume}.
-
-Hence not only are the sums $I$~and~$J$ equal, but their constituents
-are also separately equal or
-\[
-J_{1} = I_{1},\quad
-J_{2} = I_{2},\ \dots\quad
-J_{n} = I_{n}.
-\Tag{(46)}
-\]
-
-\Section{44.} Following \Name{G.~Kirchhoff}\footnote
- {\Name{G.~Kirchhoff}, Gesammelte Abhandlungen, 1882, p.~574.}
-we call the quantity~$I_{2}$, \ie,~the
-intensity of the pencil emitted from the second medium into the
-first, the \emph{emissive power~$E$} of the second medium, while we call
-the ratio of $J_{2}$ to~$J$, \ie,~that fraction of a pencil incident on the
-second medium which is absorbed in this medium, the \emph{absorbing
-power~$A$} of the second medium. Therefore
-\[
-E = I_{2}\ (\leq I),\quad
-A = \frac{J_{2}}{J}\ (\leq 1).
-\Tag{(47)}
-\]
-
-The quantities $E$~and~$A$ depend (a)~on the nature of the two
-media, (b)~on the temperature, the frequency~$\nu$, and the direction
-and the polarization of the radiation considered, (c)~on the nature
-of the bounding surface and on the magnitude of the surface
-element~$d\sigma$ and that of the solid angle~$d\Omega$, (d)~on the geometrical
-extent and the shape of the total surface of the two media, (e)~on
-the nature and form of all other bodies of the system. For a ray
-may pass from the first into the second medium, be partly transmitted
-by the latter, and then, after reflection somewhere else,
-%% -----File: 056.png---Folio 40-------
-may return to the second medium and may be there entirely
-absorbed.
-
-With these assumptions, according to equations \Eq{(46)},~\Eq{(45)},
-and~\Eq{(43)}, \Name{Kirchhoff's} law holds,
-\[
-\frac{E}{A} = I = d\sigma\, \cos\theta\, d\Omega\, \ssfK_{\nu}\, d\nu,
-\Tag{(48)}
-\]
-%[** Theorem]
-\ie, \emph{the ratio of the emissive power to the absorbing power of any body
-is independent of the nature of the body}. For this ratio is equal to
-the intensity of the pencil passing through the \emph{first} medium,
-which, according to equation~\Eq{(27)}, does not depend on the second
-medium at all. The value of this ratio does, however, depend on
-the nature of the first medium, inasmuch as, according to~\Eq{(42)},
-it is not the quantity~$\ssfK_{\nu}$ but the quantity~$q^{2}\ssfK_{\nu}$, which is a universal
-function of the temperature and frequency. The proof of this
-%[** TN: "l. c." italicized in the original]
-law given by \Name{G.~Kirchhoff} \lc\ was later greatly simplified by
-\Name{E.~Pringsheim}.\footnote
- {\Name{E.~Pringsheim}, Verhandlungen der Deutschen Physikalischen Gesellschaft, \textbf{3}, p.~81, 1901.}
-
-\Section{45.} When in particular the second medium is a black body
-(\Sec{10}) it absorbs all the incident radiation. Hence in that case
-%[** Theorems]
-$J_{2} = J$, $A = 1$, and $E = \DPtypo{A}{I}$, \ie, \emph{the emissive power of a black body is
-independent of its nature. Its emissive power is larger than that
-of any other body at the same temperature and, in fact, is just equal to
-the intensity of radiation in the contiguous medium}.
-
-\Section{46.} We shall now add, without further proof, another general
-law of reciprocity, which is closely connected with that stated at
-the end of \Sec{43} and which may be stated thus: \emph{When any
-emitting and absorbing bodies are in the state of thermodynamic
-equilibrium, the part of the energy of definite color emitted by a body~$A$,
-which is absorbed by another body~$B$, is equal to the part of the
-energy of the same color emitted by~$B$ which is absorbed by~$A$.} Since
-a quantity of energy emitted causes a decrease of the heat of the
-body, and a quantity of energy absorbed an increase of the heat of
-the body, it is evident that, when thermodynamic equilibrium
-exists, any two bodies or elements of bodies selected at random
-exchange by radiation equal amounts of heat with each other.
-Here, of course, care must be taken to distinguish between the
-radiation emitted and the total radiation which reaches one body
-from the other.
-%% -----File: 057.png---Folio 41-------
-
-\Section{47.} The law holding for the quantity~\Eq{(42)} can be expressed in a
-different form, by introducing, by means of~\Eq{(24)}, the volume
-density~$\ssfu_{\nu}$ of monochromatic radiation instead of the intensity
-of radiation~$\ssfK_{\nu}$. We then obtain the law that, for radiation in
-a state of thermodynamic equilibrium, the quantity
-\[
-\ssfu_{\nu} q^{3}
-\Tag{(49)}
-\]
-is a function of the temperature~$T$ and the frequency~$\nu$, and is
-the same for all substances.\footnote
- {In this law it is assumed that the quantity~$q$ in~\Eq{(24)} is the same as in~\Eq{(37)}. This does
- not hold for strongly dispersing or absorbing substances. For the generalization applying
- to such cases see \Name{M.~Laue}, Annalen d.\ Physik\DPchg{,}{} \textbf{32}, p.~1085, 1910.}
-This law becomes clearer if we
-consider that the quantity
-\[
-\ssfu_{\nu}\, d\nu\, \frac{q^{3}}{\nu^{3}}
-\Tag{(50)}
-\]
-also is a universal function of $T$,~$\nu$, and~$\nu + d\nu$, and that the
-product $\ssfu_{\nu}\, d\nu$ is, according to~\Eq{(22)}, the volume density of the
-radiation whose frequency lies between $\nu$ and $\nu + d\nu$, while the
-quotient~$\dfrac{q}{\nu}$ represents the wave length of a ray of frequency~$\nu$ in
-the medium in question. The law then takes the following simple
-%[** Theorem]
-form: \emph{When any bodies whatever are in thermodynamic equilibrium,
-the energy of monochromatic radiation of a definite frequency,
-contained in a cubical element of side equal to the wave length, is
-the same for all bodies.}
-
-\Section{48.} We shall finally take up the case of diathermanous (\Sec{12})
-media, which has so far not been considered. In \Sec{27} we
-saw that, in a medium which is diathermanous for a given color
-and is surrounded by an enclosure impermeable to heat, there can
-be thermodynamic equilibrium for any intensity of radiation
-of this color. There must, however, among all possible intensities
-of radiation be a definite one, corresponding to the absolute
-maximum of the total entropy of the system, which designates
-the absolutely stable equilibrium of radiation. In fact, in equation~\Eq{(27)}
-the intensity of radiation~$\ssfK_{\nu}$ for $\alpha_{\nu} = 0$ and $\epsilon_{\nu} = 0$
-assumes the value~$\dfrac{0}{0}$, and hence cannot be calculated from this
-equation. But we see also that this indeterminateness is removed
-by equation~\Eq{(41)}, which states that in the case of thermodynamic
-%% -----File: 058.png---Folio 42-------
-equilibrium the product~$q^{2}\ssfK_{\nu}$ has the same value for all substances.
-From this we find immediately a definite value of~$\ssfK_{\nu}$
-which is thereby distinguished from all other values. Furthermore
-the physical significance of this value is immediately seen
-by considering the way in which that equation was obtained.
-It is that intensity of radiation which exists in a diathermanous
-medium, if it is in thermodynamic equilibrium when in contact
-with an arbitrary absorbing and emitting medium. The volume
-and the form of the second medium do not matter in the least,
-in particular the volume may be taken as small as we please.
-%[** Thm]
-Hence we can formulate the following law: \emph{Although generally
-speaking thermodynamic equilibrium can exist in a diathermanous
-medium for any intensity of radiation whatever, nevertheless there
-exists in every diathermanous medium for a definite frequency at a
-definite temperature an intensity of radiation defined by the universal
-function~\Eq{(42)}. This may be called the stable intensity, inasmuch
-as it will always be established, when the medium is exchanging
-stationary radiation with an arbitrary emitting and absorbing
-substance.}
-
-\Section{49.} According to the law stated in \Sec{45}, the intensity of a
-pencil, when a state of stable heat radiation exists in a diathermanous
-medium, is equal to the emissive power~$E$ of a black
-body in contact with the medium. On this fact is based the
-possibility of measuring the emissive power of a black body,
-although absolutely black bodies do not exist in nature.\footnote
- {\Name{W.~Wien} and \Name{O.~Lummer}, Wied.\ Annalen, \textbf{56}, p.~451, 1895.}
-A
-diathermanous cavity is enclosed by strongly emitting walls\footnote
- {The strength of the emission influences only the time required to establish stationary
- radiation, but not its character. It is essential, however, that the walls transmit no radiation
- to the exterior.}
-and the walls kept at a certain constant temperature~$T$. Then
-the radiation in the cavity, when thermodynamic equilibrium is
-established for every frequency~$\nu$, assumes the intensity corresponding
-to the velocity of propagation~$q$ in the diathermanous
-medium, according to the universal function~\Eq{(42)}. Then any
-element of area of the walls radiates into the cavity just as if the
-wall were a black body of temperature~$T$. The amount lacking
-in the intensity of the rays actually emitted by the walls as
-compared with the emission of a black body is supplied by rays
-%% -----File: 059.png---Folio 43-------
-which fall on the wall and are reflected there. Similarly every
-element of area of a wall receives the same radiation.
-
-In fact, the radiation~$I$ starting from an element of area of a
-wall consists of the radiation~$E$ emitted by the element of area and
-of the radiation reflected from the element of area from the
-incident radiation~$I$, \ie,~the radiation which is not absorbed
-$(1 - A)I$. We have, therefore, in agreement with \Name{Kirchhoff's}
-law~\Eq{(48)},
-\[
-I = E + (1 - A)I.
-\]
-
-If we now make a hole in one of the walls of a size~$d\sigma$, so small
-that the intensity of the radiation directed toward the hole is
-not changed thereby, then radiation passes through the hole to
-the exterior where we shall suppose there is the same diathermanous
-medium as within. This radiation has exactly the same
-properties as if $d\sigma$~were the surface of a black body, and this
-radiation may be measured for every color together with the
-temperature~$T$.
-
-\Section{50.} Thus far all the laws derived in the preceding sections for
-diathermanous media hold for a definite frequency, and it is to
-be kept in mind that a substance may be diathermanous for one
-color and adiathermanous for another. Hence the radiation of a
-medium completely enclosed by absolutely reflecting walls is,
-when thermodynamic equilibrium has been established for all
-colors for which the medium has a finite coefficient of absorption,
-always the stable radiation corresponding to the temperature
-of the medium such as is represented by the emission of a black
-body. Hence this is briefly called ``black'' radiation.\footnote
- {\Name{M.~Thiesen}, Verhandlungen d.\ Deutschen Physikal.\ Gesellschaft, \textbf{2}, p.~65, 1900.}
-On the
-other hand, the intensity of colors for which the medium is diathermanous
-is not necessarily the stable black radiation, unless
-the medium is in a state of stationary exchange of radiation with
-an absorbing substance.
-
-There is but one medium that is diathermanous for all kinds of
-rays, namely, the absolute vacuum, which to be sure cannot be
-produced in nature except approximately. However, most gases,
-\eg, the air of the atmosphere, have, at least if they are not too
-dense, to a sufficient approximation the optical properties of a
-vacuum with respect to waves of not too short length. So far as
-%% -----File: 060.png---Folio 44-------
-this is the case the velocity of propagation~$q$ may be taken as the
-same for all frequencies, namely,
-\[
-c = 3 × 10^{10}\, \frac{\cm}{\sec}\Add{.}
-\Tag{(51)}
-\]
-
-\Section{51.} Hence in a vacuum bounded by totally reflecting walls any
-state of radiation may persist. But as soon as an arbitrarily
-small quantity of matter is introduced into the vacuum, a stationary
-state of radiation is gradually established. In this the
-radiation of every color which is appreciably absorbed by the
-substance has the intensity~$\ssfK_{\nu}$ corresponding to the temperature
-of the substance and determined by the universal function~\Eq{(42)}
-for $q = c$, the intensity of radiation of the other colors remaining
-indeterminate. If the substance introduced is not diathermanous
-for any color, \eg, a piece of carbon however small, there
-exists at the stationary state of radiation in the whole vacuum for
-all colors the intensity~$\ssfK_{\nu}$ of black radiation corresponding to the
-temperature of the substance. The magnitude of~$\ssfK_{\nu}$ regarded as
-a function of~$\nu$ gives the spectral distribution of black radiation in
-a vacuum, or the so-called \emph{normal energy spectrum}, which depends
-on nothing but the temperature. In the normal spectrum,
-since it is the spectrum of emission of a black body, the intensity
-of radiation of every color is the largest which a body can emit at
-that temperature at all.
-
-\Section{52.} It is therefore possible to change a perfectly arbitrary
-radiation, which exists at the start in the evacuated cavity with
-perfectly reflecting walls under consideration, into black radiation
-by the introduction of a minute particle of carbon. The characteristic
-feature of this process is that the heat of the carbon particle
-may be just as small as we please, compared with the energy
-of radiation contained in the cavity of arbitrary magnitude.
-Hence, according to the principle of the conservation of energy,
-the total energy of radiation remains essentially constant during
-the change that takes place, because the changes in the heat of the
-carbon particle may be entirely neglected, even if its changes in
-temperature should be finite. Herein the carbon particle exerts
-only a releasing (auslösend) action. Thereafter the intensities
-of the pencils of different frequencies originally present and having
-different frequencies, directions, and different states of polarization
-%% -----File: 061.png---Folio 45-------
-change at the expense of one another, corresponding to
-the passage of the system from a less to a more stable state of
-radiation or from a state of smaller to a state of larger entropy.
-From a thermodynamic point of view this process is perfectly
-analogous, since the time necessary for the process is not essential,
-to the change produced by a minute spark in a quantity of oxy-hydrogen
-gas or by a small drop of liquid in a quantity of supersaturated
-vapor. In all these cases the magnitude of the disturbance
-is exceedingly small and cannot be compared with the
-magnitude of the energies undergoing the resultant changes, so
-that in applying the two principles of thermodynamics the cause
-of the disturbance of equilibrium, \viz, the carbon particle, the
-spark, or the drop, need not be considered. It is always a case of
-a system passing from a more or less unstable into a more stable
-state, wherein, according to the first principle of thermodynamics,
-the energy of the system remains constant, and, according to the
-second principle, the entropy of the system increases.
-%% -----File: 062.png---Folio 46-------
-% [Blank Page]
-%% -----File: 063.png---Folio 47-------
-
-\Part[Deductions From Electrodynamics]{II}{Deductions From Electrodynamics
-And Thermodynamics}
-%% -----File: 064.png---Folio 48-------
-%[Blank Page]
-%% -----File: 065.png---Folio 49-------
-
-\Chapter{I}{Maxwell's Radiation Pressure}
-
-\Section{53.} While in the preceding part the phenomena of radiation
-have been presented with the assumption of only well known
-elementary laws of optics summarized in \Sec{2}, which are common
-to all optical theories, we shall hereafter make use of the
-electromagnetic theory of light and shall begin by deducing a
-consequence characteristic of that theory. We shall, namely,
-calculate the magnitude of the mechanical force, which is exerted
-by a light or heat ray passing through a vacuum on striking a
-reflecting (\Sec{10}) surface assumed to be at rest.
-
-For this purpose we begin by stating Maxwell's general equations
-for an electromagnetic process in a vacuum. Let the vector~$\ssfE$
-denote the electric field-strength (intensity of the electric field)
-in electric units and the vector~$\ssfH$ the magnetic field-strength in
-magnetic units. Then the equations are, in the abbreviated
-notation of the vector calculus,
-\[
-\begin{aligned}
- \dot{\ssfE} &= c \curl\ssfH & \dot{\ssfH} &= -c \curl\ssfE\\
- \div \ssfE &= 0 & \div \ssfH &= 0\Add{.}
-\end{aligned}
-\Tag{(52)}
-\]
-Should the reader be unfamiliar with the symbols of this notation,
-he may readily deduce their meaning by working backward from
-the subsequent equations~\Eq{(53)}.
-
-\Section{54.} In order to pass to the case of a plane wave in any direction
-we assume that all the quantities that fix the state depend only
-on the time~$t$ and on one of the coordinates $x'$,~$y'$,~$z'$, of an orthogonal
-right-handed system of coordinates, say on~$x'$. Then the
-equations~\Eq{(52)} reduce to
-\[
-\begin{aligned}
-\frac{\dd\ssfE_{x'}}{\dd t} &= 0 &
-\frac{\dd\ssfH_{x'}}{\dd t} &= 0 \\[4pt]
-%
-\frac{\dd\ssfE_{y'}}{\dd t} &= -c \frac{\dd \ssfH_{z'}}{\dd x'} \qquad &
-\frac{\dd\ssfH_{y'}}{\dd t} &= \Neg c \frac{\dd \ssfE_{z'}}{\dd x'} \\[4pt]
-%% -----File: 066.png---Folio 50-------
-\frac{\dd\ssfE_{x'}}{\dd t} &= \Neg c\frac{\dd \ssfH_{y'}}{\dd x'} &
-\frac{\dd\ssfH_{z'}}{\dd t} &= -c\frac{\dd \ssfE_{y'}}{\dd x'} \\[4pt]
-\frac{\dd\ssfE_{x'}}{\dd x'} &= 0 &
-\frac{\dd\ssfH_{x'}}{\dd x'} &= 0\Add{.}
-\end{aligned}
-\Tag{(53)}
-\]
-Hence the most general expression for a plane wave passing
-through a vacuum in the direction of the positive $x'$-axis is
-\[
-\begin{aligned}
-\ssfE_{x'} &= 0 &
-\ssfH_{x'} &= 0 \\
-\ssfE_{y'} &= f\left(t - \frac{x'}{c}\right) &
-\ssfH_{y'} &= -g\left(t - \frac{x'}{c}\right) \\
-\ssfE_{z'} &= g\left(t - \frac{x'}{c}\right) &
-\ssfH_{z'} &= \Neg f\left(t - \frac{x'}{c}\right)
-\end{aligned}
-\Tag{(54)}
-\]
-where $f$~and~$g$ represent two arbitrary functions of the same
-argument.
-
-\Section{55.} Suppose now that this wave strikes a reflecting surface,
-\eg, the surface of an absolute conductor (metal) of infinitely
-\Figure[2.25in]{4}
-large conductivity. In such a
-conductor even an infinitely
-small electric field-strength produces
-a finite conduction current;
-hence the electric field-strength~$\ssfE$
-in it must be always
-and everywhere infinitely small.
-For simplicity we also suppose
-the conductor to be non-magnetizable,
-\ie, we assume the
-magnetic induction~$\ssfB$ in it to be
-equal to the magnetic field-strength~$\ssfH$,
-just as is the case
-in a vacuum.
-
-%[** F2: Comma-less coordinate system notation starts here]
-If we place the $x$-axis of a
-right-handed coordinate system
-$(xyz)$ along the normal of the surface
-directed toward the interior
-of the conductor, the $x$-axis is the normal of incidence. We
-place the $(x'y')$~plane in the plane of incidence and take this as
-the plane of the figure (\Fig{4}). Moreover, we can also, without
-%% -----File: 067.png---Folio 51-------
-any restriction of generality, place the $y$-axis in the plane of the
-figure, so that the $z$-axis coincides with the $z'$-axis (directed from
-the figure toward the observer). Let the common origin~$O$ of
-the two coordinate systems lie in the surface. If finally $\theta$~represents
-the angle of incidence, the coordinates with and without
-accent are related to each other by the following equations:
-\begin{align*}
- x &= x' \cos\theta - y' \sin\theta & x' &= \Neg x \cos\theta + y \sin\theta\\
- y &= x' \sin\theta + y' \cos\theta & y' &= -x \sin\theta + y \cos\theta\\
- z &= z' & z' &= z\Add{.}
-\end{align*}
-
-By the same transformation we may pass from the components
-of the electric or magnetic field-strength in the first coordinate
-system to their components in the second system. Performing
-this transformation the following values are obtained from~\Eq{(54)}
-for the components of the electric and magnetic field-strengths
-of the incident wave in the coordinate system without accent,
-\[
-\begin{aligned}
- \ssfE_{x} &= -\sin\theta ˇ f \qquad & \ssfH_{x} &= \Neg\sin\theta ˇ g \\
- \ssfE_{y} &= \Neg\cos\theta ˇ f & \ssfH_{y} &= -\cos\theta ˇ g \\
- \ssfE_{z} &= g & \ssfH_{z} &= f\Add{.}
-\end{aligned}
-\Tag{(55)}
-\]
-Herein the argument of the functions $f$~and~$g$ is
-\[
-t - \frac{x'}{c} = t - \frac{x \cos\theta + y \sin\theta}{c}\Add{.}
-\Tag{(56)}
-\]
-
-\Section{56.} In the surface of separation of the two media $x = 0$. According
-to the general electromagnetic boundary conditions the
-components of the field-strengths in the surface of separation,
-\ie, the four quantities $\ssfE_{y}$,~$\ssfE_{z}$, $\ssfH_{y}$,~$\ssfH_{z}$ must be equal to each
-other on the two sides of the surface of separation for this value
-of~$x$. In the conductor the electric field-strength~$\ssfE$ is infinitely
-small in accordance with the assumption made above. Hence
-$\ssfE_{y}$~and~$\ssfE_{z}$ must vanish also in the vacuum for $x = 0$. This condition
-cannot be satisfied unless we assume in the vacuum,
-besides the incident, also a reflected wave superposed on the former
-in such a way that the components of the electric field of the
-two waves in the $y$~and~$z$ direction just cancel at every instant
-and at every point in the surface of separation. By this assumption
-and the condition that the reflected wave is a plane wave
-returning into the interior of the vacuum, the other four components
-%% -----File: 068.png---Folio 52-------
-of the reflected wave are also completely determined. They
-are all functions of the single argument
-\[
-t - \frac{-x \cos\theta + y \sin\theta}{c}.
-\Tag{(57)}
-\]
-The actual calculation yields as components of the total electromagnetic
-field produced in the vacuum by the superposition of
-the two waves, the following expressions valid for points of the
-surface of separation $x = 0$,
-\[
-\begin{aligned}
- \ssfE_{x} &= -\sin\theta ˇ f - \sin\theta ˇ f = -2\sin\theta ˇ f\\
- \ssfE_{y} &= \Neg\cos\theta ˇ f - \cos\theta ˇ f = 0\\
- \ssfE_{z} &= g - g = 0 \\
- \ssfH_{x} &= \Neg\sin\theta ˇ g - \sin\theta ˇ g = 0\\
- \ssfH_{y} &= -\cos\theta ˇ g - \cos\theta ˇ g = -2\cos\theta ˇ g\\
- \ssfH_{z} &= f + f = 2f.
-\end{aligned}
-\Tag{(58)}
-\]
-In these equations the argument of the functions $f$~and~$g$ is, according
-to \Eq{(56)}~and~\Eq{(57)},
-\[
-t - \frac{y \sin\theta}{c}\Add{.}
-\]
-From these values the electric and magnetic field-strength within
-the conductor in the immediate neighborhood of the separating
-surface $x = 0$ is obtained:
-\[
-\begin{aligned}
- \ssfE_{x} &=0 \qquad & \ssfH_{x} &= 0\\
- \ssfE_{y} &=0 & \ssfH_{y} &= -2\cos\theta ˇ g\\
- \ssfE_{z} &=0 & \ssfH_{z} &= 2f
-\end{aligned}
-\Tag{(59)}
-\]
-where again the argument $t - \dfrac{y \sin\theta}{c}$ is to be substituted in the
-functions $f$~and~$g$. For the components of~$\ssfE$ all vanish in an absolute
-conductor and the components $\ssfH_{x}$,~$\ssfH_{y}$,~$\ssfH_{z}$ are all continuous
-at the separating surface, the two latter since they are tangential
-components of the field-strength, the former since it is the normal
-component of the magnetic induction~$\ssfB$ (\Sec{55}), which likewise
-remains continuous on passing through any surface of separation.
-
-On the other hand, the normal component of the electric field-strength~$\ssfE_{x}$
-is seen to be discontinuous; the discontinuity shows
-%% -----File: 069.png---Folio 53-------
-the existence of an electric charge on the surface, the surface
-density of which is given in magnitude and sign as follows:
-\[
-\frac{1}{4\pi} 2\sin\theta ˇ f = \frac{1}{2\pi} \sin\theta ˇ f.
-\Tag{(60)}
-\]
-In the interior of the conductor at a finite distance from the
-bounding surface, \ie, for $x > 0$, all six field components are infinitely
-small. Hence, on increasing~$x$, the values of $\ssfH_{y}$~and~$\ssfH_{z}$,
-which are finite for $x = 0$, approach the value~$0$ at an infinitely
-rapid rate.
-
-\Section{57.} A certain mechanical force is exerted on the substance of
-the conductor by the electromagnetic field considered. We shall
-calculate the component of this force normal to the surface. It
-is partly of electric, partly of magnetic, origin. Let us first consider
-the former,~$\ssfF_{e}$. Since the electric charge existing on the
-surface of the conductor is in an electric field, a mechanical force
-equal to the product of the charge and the field-strength is exerted
-on it. Since, however, the field-strength is discontinuous, having
-the value $-2\sin\theta f$ on the side of the vacuum and $0$~on the side
-of the conductor, from a well-known law of electrostatics the magnitude
-of the mechanical force~$\ssfF_{e}$ acting on an element of surface~$d\sigma$
-of the conductor is obtained by multiplying the electric charge
-of the element of area calculated in~\Eq{(60)} by the arithmetic mean
-of the electric field-strength on the two sides. Hence
-\[
-\ssfF_{e} = \frac{\sin\theta}{2\pi} f\, d\sigma\, (-\sin\theta f)
- = -\frac{\sin^{2}\theta}{2\pi} f^{2}\, d\sigma\Add{.}
-\]
-This force acts in the direction toward the vacuum and therefore
-exerts a tension.
-
-\Section{58.} We shall now calculate the mechanical force of magnetic
-origin~$\ssfF_{m}$. In the interior of the conducting substance there are
-certain conduction currents, whose intensity and direction are
-determined by the vector~$\ssfI$ of the current density
-\[
-\ssfI = \frac{c}{4\pi} \curl \ssfH.
-\Tag{(61)}
-\]
-A mechanical force acts on every element of space~$d\tau$ of the conductor
-through which a conduction current flows, and is given by
-the vector product
-\[
-\frac{d\tau}{c} \DPchg{[\ssfI \ssfH]}{[\ssfI × \ssfH]}\Add{.}
-\Tag{(62)}
-\]
-%% -----File: 070.png---Folio 54-------
-Hence the component of this force normal to the surface of the
-conductor $x = 0$ is equal to
-\[
-\frac{d\tau}{c} (\ssfI_{y} \ssfH_{z} - \ssfI_{z} \ssfH_{y}).
-\]
-On substituting the values of $\ssfI_{y}$~and~$\ssfI_{z}$ from~\Eq{(61)} we obtain
-\[
-\frac{d\tau}{4\pi}
- \left[\ssfH_{z} \left(\frac{\dd\ssfH_{x}}{\dd z} - \frac{\dd\ssfH_{z}}{\dd x}\right)
- - \ssfH_{y} \left(\frac{\dd\ssfH_{y}}{\dd x} - \frac{\dd\ssfH_{x}}{\dd y}\right)\right].
-\]
-In this expression the differential coefficients with respect to
-$y$~and~$z$ are negligibly small in comparison to those with respect to~$x$,
-according to the remark at the end of \Sec{56}; hence the expression
-reduces to
-\[
--\frac{d\tau}{4\pi}
- \left(\ssfH_{y} \frac{\dd\ssfH_{y}}{\dd x}
- + \ssfH_{z} \frac{\dd\ssfH_{z}}{\dd x}\right).
-\]
-Let us now consider a cylinder cut out of the conductor perpendicular
-to the surface with the cross-section~$d\sigma$, and extending
-from $x = 0$ to $x = \infty$. The entire mechanical force of magnetic
-origin acting on this cylinder in the direction of the $x$-axis, since
-$d\tau = d\sigma\, x$, is given by
-\[
-\ssfF_{m} = -\frac{d\sigma}{4\pi} \int_{0}^{\infty} dx
- \left(\ssfH_{y} \frac{\dd\ssfH_{y}}{\dd x}
- + \ssfH_{z} \frac{\dd\ssfH_{z}}{\dd x}\right).
-\]
-On integration, since $\ssfH$~vanishes for $x = \infty$, we obtain
-\[
-\ssfF_{m} = \frac{d\sigma}{8\pi} \left(\ssfH_{y}^{2} + \ssfH_{z}^{2}\right)_{x = 0}
-\]
-or by equation~\Eq{(59)}
-\[
-\ssfF_{m} = \frac{d\sigma}{2\pi} (\cos^{2} \theta ˇ g^{2} + f^{2}).
-\]
-
-By adding $\ssfF_{e}$~and~$\ssfF_{m}$ the total mechanical force acting on the
-cylinder in question in the direction of the $x$-axis is found to be
-\[
-\ssfF = \frac{d\sigma}{2\pi} \cos^{2} \theta (f^{2} + g^{2}).
-\Tag{(63)}
-\]
-This force exerts on the surface of the conductor a pressure, which
-acts in a direction normal to the surface toward the interior and is
-%% -----File: 071.png---Folio 55-------
-called ``\Name{Maxwell's} radiation pressure.'' The existence and the
-magnitude of the radiation pressure as predicted by the theory
-was first found by delicate measurements with the radiometer by
-\Name{P.~Lebedew.}\footnote
- {\Name{P. Lebedew}, Annalen d.\ Phys.\DPchg{,}{}\ \textbf{6}, p.~433, 1901. See also \Name{E.~F. Nichols} and \Name{G.~F. Hull},
- Annalen d.\ Phys.\DPchg{,}{}\ \textbf{12}, p.~225, 1903.}
-
-\Section{59.} We shall now establish a relation between the radiation
-pressure and the energy of radiation~$I\, dt$ falling on the surface
-element~$d\sigma$ of the conductor in a time element~$dt$. The latter
-from \Name{Poynting's} law of energy flow is
-\[
-I\, dt = \frac{c}{4\pi} (\ssfE_{y} \ssfH_{z} - \ssfE_{z} \ssfH_{y})\, d\sigma\, dt,
-\]
-hence from~\Eq{(55)}
-\[
-I\, dt = \frac{c}{4\pi} \cos\theta (f^{2} + g^{2})\, d\sigma\, dt.
-\]
-By comparison with~\Eq{(63)} we obtain
-\[
-\ssfF = \frac{2\cos\theta}{c} I.
-\Tag{(64)}
-\]
-
-From this we finally calculate the total pressure~$p$, \ie, that
-mechanical force, which an arbitrary radiation proceeding from
-the vacuum and totally reflected upon incidence on the conductor
-exerts in a normal direction on a unit surface of the conductor.
-The energy radiated in the conical element
-\[
-d\Omega = \sin\theta\, d\theta\, d\phi
-\]
-in the time~$dt$ on the element of area~$d\sigma$ is, according to~\Eq{(6)},
-\[
-I\, dt = K \cos\theta\, d\Omega\, d\sigma\, dt,
-\]
-where $K$~represents the specific intensity of the radiation in the
-direction~$d\Omega$ toward the reflector. On substituting this in~\Eq{(64)} and
-integrating over~$d\Omega$ we obtain for the total pressure of all pencils
-which fall on the surface and are reflected by it
-\[
-p = \frac{2}{c} \int K \cos^{2} \theta\, d\Omega,
-\Tag{(65)}
-\]
-the integration with respect to~$\phi$ extending from $0$ to~$2\pi$ and with
-respect to~$\theta$ from $0$ to~$\dfrac{\pi}{2}$.
-%% -----File: 072.png---Folio 56-------
-
-In case $K$~is independent of direction as in the case of black
-radiation, we obtain for the radiation pressure
-\[
-p = \frac{2K}{c} \int_{0}^{2\pi} d\phi \int_{0}^{\frac{\pi}{2}} d\theta \cos^{2} \theta \sin\theta
- = \frac{4\pi K}{3c}
-\]
-or, if we introduce instead of~$K$ the volume density of radiation~$u$
-from~\Eq{(21)}
-\[
-p = \frac{u}{3}.
-\Tag{(66)}
-\]
-
-This value of the radiation pressure holds only when the reflection
-of the radiation occurs at the surface of an absolute non-magnetizable
-conductor. Therefore we shall in the thermodynamic
-deductions of the next chapter make use of it only in such
-cases. Nevertheless it will be shown later on (\Sec{66}) that
-equation~\Eq{(66)} gives the pressure of uniform radiation against any
-totally reflecting surface, no matter whether it reflects uniformly
-or diffusely.
-
-\Section{60.} In view of the extraordinarily simple and close relation
-between the radiation pressure and the energy of radiation, the
-question might be raised whether this relation is really a special
-consequence of the electromagnetic theory, or whether it might
-not, perhaps, be founded on more general energetic or thermodynamic
-considerations. To decide this question we shall calculate
-the radiation pressure that would follow by Newtonian
-mechanics from \Name{Newton's} (emission) theory of light, a theory
-which, in itself, is quite consistent with the energy principle.
-According to it the energy radiated onto a surface by a light ray
-passing through a vacuum is equal to the kinetic energy of the
-light particles striking the surface, all moving with the constant
-velocity~$c$. The decrease in intensity of the energy radiation
-with the distance is then explained simply by the decrease of the
-volume density of the light particles.
-
-Let us denote by $n$ the number of the light particles contained
-in a unit volume and by $m$ the mass of a particle. Then for a
-beam of parallel light the number of particles impinging in unit
-time on the element~$d\sigma$ of a reflecting surface at the angle of
-incidence~$\theta$ is
-\[
-nc \cos\theta\, d\sigma.
-\Tag{(67)}
-\]
-%% -----File: 073.png---Folio 57-------
-
-Their kinetic energy is given according to Newtonian mechanics
-by
-\[
-I = nc \cos\theta\, d\sigma\, \frac{mc^{2}}{2} = nm \cos\theta \frac{c^{3}}{2}\, d\sigma.
-\Tag{(68)}
-\]
-Now, in order to determine the normal pressure of these particles
-on the surface, we may note that the normal component of the
-velocity $c \cos\theta$ of every particle is changed on reflection into a
-component of opposite direction. Hence the normal component
-of the momentum of every particle (impulse-coordinate) is
-changed through reflection by $-2mc \cos\theta$. Then the change
-in momentum for all particles considered will be, according to~\Eq{(67)},
-\[
--2nm \cos^{2} \theta\, c^{2}\, d\sigma.
-\Tag{(69)}
-\]
-
-Should the reflecting body be free to move in the direction of
-the normal of the reflecting surface and should there be no force
-acting on it except the impact of the light particles, it would be
-set into motion by the impacts. According to the law of action
-and reaction the ensuing motion would be such that the momentum
-acquired in a certain interval of time would be equal and
-opposite to the change in momentum of all the light particles
-reflected from it in the same time interval. But if we allow a
-separate constant force to act from outside on the reflector, there
-is to be added to the change in momenta of the light particles
-the impulse of the external force, \ie, the product of the force
-and the time interval in question.
-
-Therefore the reflector will remain continuously at rest, whenever
-the constant external force exerted on it is so chosen that its
-impulse for any time is just equal to the change in momentum
-of all the particles reflected from the reflector in the same time.
-Thus it follows that the force~$\ssfF$ itself which the particles exert
-by their impact on the surface element~$d\sigma$ is equal and opposite
-to the change of their momentum in unit time as expressed in~\Eq{(69)}
-\[
-\ssfF = 2nm \cos^{2} \theta\, c^{2}\, d\sigma
-\]
-and by making use of~\Eq{(68)},
-\[
-\ssfF = \frac{4\cos\theta}{c} I.
-\]
-
-On comparing this relation with equation~\Eq{(64)} in which all
-symbols have the same physical significance, it is seen that
-%% -----File: 074.png---Folio 58-------
-\Name{Newton's} radiation pressure is twice as large as \Name{Maxwell's} for the
-same energy radiation. A necessary consequence of this is that
-the magnitude of \Name{Maxwell's} radiation pressure cannot be deduced
-from general energetic considerations, but is a special feature of
-the electromagnetic theory and hence all deductions from \Name{Maxwell's}
-radiation pressure are to be regarded as consequences of the
-electromagnetic theory of light and all confirmations of them
-are confirmations of this special theory.
-%% -----File: 075.png---Folio 59-------
-
-\Chapter{II}{Stefan-Boltzmann Law of Radiation}
-
-\Section{61.} For the following we imagine a perfectly evacuated hollow
-cylinder with an absolutely tight-fitting piston free to move in a
-vertical direction with no friction. A part of the walls of the
-cylinder, say the rigid bottom, should consist of a black body,
-whose temperature~$T$ may be regulated arbitrarily from the outside.
-The rest of the walls including the inner surface of the piston
-may be assumed as totally reflecting. Then, if the piston
-remains stationary and the temperature,~$T$, constant, the radiation
-in the vacuum will, after a certain time, assume the character
-of black radiation (\Sec{50}) uniform in all directions. The
-specific intensity,~$K$, and the volume density,~$u$, depend only on
-the temperature,~$T$, and are independent of the volume,~$V$, of
-the vacuum and hence of the position of the piston.
-
-If now the piston is moved downward, the radiation is compressed
-into a smaller space; if it is moved upward the radiation
-expands into a larger space. At the same time the temperature
-of the black body forming the bottom may be arbitrarily changed
-by adding or removing heat from the outside. This always
-causes certain disturbances of the stationary state. If, however,
-the arbitrary changes in $V$~and~$T$ are made sufficiently slowly, the
-departure from the conditions of a stationary state may always be
-kept just as small as we please. Hence the state of radiation in
-the vacuum may, without appreciable error, be regarded as a
-state of thermodynamic equilibrium, just as is done in the thermodynamics
-of ordinary matter in the case of so-called infinitely
-slow processes, where, at any instant, the divergence from the
-state of equilibrium may be neglected, compared with the changes
-which the total system considered undergoes as a result of the
-entire process.
-
-If, \eg, we keep the temperature~$T$ of the black body forming
-the bottom constant, as can be done by a suitable connection
-%% -----File: 076.png---Folio 60-------
-between it and a heat reservoir of large capacity, then, on raising
-the piston, the black body will emit more than it absorbs, until
-the newly made space is filled with the same density of radiation
-as was the original one. \textit{Vice versa}, on lowering the piston the
-black body will absorb the superfluous radiation until the original
-radiation corresponding to the temperature~$T$ is again established.
-Similarly, on raising the temperature~$T$ of the black body, as
-can be done by heat conduction from a heat reservoir which is
-slightly warmer, the density of radiation in the vacuum will be
-correspondingly increased by a larger emission,~etc. To accelerate
-the establishment of radiation equilibrium the reflecting
-mantle of the hollow cylinder may be assumed white (\Sec{10}),
-since by diffuse reflection the predominant directions of radiation
-that may, perhaps, be produced by the direction of the motion
-of the piston, are more quickly neutralized. The reflecting
-surface of the piston, however, should be chosen for the present as
-a perfect metallic reflector, to make sure that the radiation pressure~\Eq{(66)}
-on the piston is \Name{Maxwell's}. Then, in order to produce
-mechanical equilibrium, the piston must be loaded by a weight
-equal to the product of the radiation pressure~$p$ and the cross-section
-of the piston. An exceedingly small difference of the
-loading weight will then produce a correspondingly slow motion
-of the piston in one or the other direction.
-
-Since the effects produced from the outside on the system in
-question, the cavity through which the radiation travels, during
-the processes we are considering, are partly of a mechanical
-nature (displacement of the loaded piston), partly of a thermal
-nature (heat conduction away from and toward the reservoir),
-they show a certain similarity to the processes usually considered
-in thermodynamics, with the difference that the system here
-considered is not a material system, \eg,~a gas, but a purely energetic
-one. If, however, the principles of thermodynamics hold
-quite generally in nature, as indeed we shall assume, then they
-must also hold for the system under consideration. That is to
-say, in the case of any change occurring in nature the energy of
-all systems taking part in the change must remain constant
-(first principle), and, moreover, the entropy of all systems taking
-part in the change must increase, or in the limiting case of reversible
-processes must remain constant (second principle).
-%% -----File: 077.png---Folio 61-------
-
-\Section{62.} Let us first establish the equation of the first principle for
-an infinitesimal change of the system in question. That the
-cavity enclosing the radiation has a certain energy we have
-already (\Sec{22}) deduced from the fact that the energy radiation
-is propagated with a finite velocity. We shall denote the energy
-by~$U$. Then we have
-\[
-U = Vu,
-\Tag{(70)}
-\]
-where $u$~the volume density of radiation depends only on the
-temperature \DPtypo{of~$T$}{$T$~of} the black body at the bottom.
-
-The work done by the system, when the volume~$V$ of the cavity
-increases by~$dV$ against the external forces of pressure (weight of
-the loaded piston), is~$p\, dV$, where $p$~represents \Name{Maxwell's} radiation
-pressure~\Eq{(66)}. This amount of mechanical energy is therefore
-gained by the surroundings of the system, since the weight is
-raised. The error made by using the radiation pressure on a
-stationary surface, whereas the reflecting surface moves during
-the volume change, is evidently negligible, since the motion may
-be thought of as taking place with an arbitrarily small velocity.
-
-If, moreover, $Q$~denotes the infinitesimal quantity of heat in
-mechanical units, which, owing to increased emission, passes
-from the black body at the bottom to the cavity containing the
-radiation, the bottom or the heat reservoir connected to it loses
-this heat~$Q$, and its internal energy is decreased by that amount.
-Hence, according to the first principle of thermodynamics, since
-the sum of the energy of radiation and the energy of the material
-bodies remains constant, we have
-\[
-dU + p\, dV - Q = 0.
-\Tag{(71)}
-\]
-
-According to the second principle of thermodynamics the cavity
-containing the radiation also has a definite entropy. For
-when the heat~$Q$ passes from the heat reservoir into the cavity,
-the entropy of the reservoir decreases, the change being
-\[
--\frac{Q}{T}\Add{.}
-\]
-
-Therefore, since no changes occur in the other bodies---inasmuch
-as the rigid absolutely reflecting piston with the weight on
-it does not change its internal condition with the motion---there
-%% -----File: 078.png---Folio 62-------
-must somewhere in nature occur a compensation of entropy having
-at least the value~$\dfrac{Q}{T}$, by which the above diminution is compensated,
-and this can be nowhere except in the entropy of the
-cavity containing the radiation. Let the entropy of the latter be
-denoted by~$S$.
-
-Now, since the processes described consist entirely of states
-of equilibrium, they are perfectly reversible and hence there is no
-increase in entropy. Then we have
-\[
-dS - \frac{Q}{T} = 0,
-\Tag{(72)}
-\]
-or from~\Eq{(71)}
-\[
-dS = \frac{dU + p\, dV}{T}\Add{.}
-\Tag{(73)}
-\]
-
-In this equation the quantities $U$,~$p$, $V$,~$S$ represent certain
-properties of the heat radiation, which are completely defined by
-the instantaneous state of the radiation. Therefore the quantity~$T$
-is also a certain property of the state of the radiation, \ie,~the
-black radiation in the cavity has a certain temperature~$T$ and
-this temperature is that of a body which is in heat equilibrium
-with the radiation.
-
-\Section{63.} We shall now deduce from the last equation a consequence
-which is based on the fact that the state of the system considered,
-and therefore also its entropy, is determined by the values of two
-independent variables. As the first variable we shall take~$V$, as
-the second either $T$,~$u$, or~$p$ may be chosen. Of these three quantities
-any two are determined by the third together with~$V$.
-We shall take the volume~$V$ and the temperature~$T$ as independent
-variables. Then by substituting from \Eq{(66)}~and~\Eq{(70)} in~\Eq{(73)}
-we have
-\[
-dS = \frac{V}{T}\, \frac{du}{dT}\, dT + \frac{4u}{3T}\, dV.
-\Tag{(74)}
-\]
-From this we obtain
-\[
-\left(\frac{\dd S}{\dd T}\right)_{V} = \frac{V}{T}\, \frac{du}{dT}\qquad
-\left(\frac{\dd S}{\dd V}\right)_{T} = \frac{4u}{3T}.
-\]
-%% -----File: 079.png---Folio 63-------
-On partial differentiation of these equations, the first with respect
-to~$V$, the second with respect to~$T$, we find
-\[
-\frac{\dd^{2} S}{\dd T\, \dd V}
- = \frac{1}{T}\, \frac{du}{dT}
- = \frac{4}{3T}\, \frac{du}{dT} - \frac{4u}{3T^{2}}
-\]
-or
-\[
-\frac{du}{dT} = \frac{4u}{T}
-\]
-and on integration
-\[
-u = aT^{4}
-\Tag{(75)}
-\]
-and from~\Eq{(21)} for the specific intensity of black radiation
-\[
-K = \frac{c}{4\pi} ˇ u = \frac{ac}{4\pi} T^{4}.
-\Tag{(76)}
-\]
-Moreover for the pressure of black radiation
-\[
-p = \frac{a}{3} T^{4},
-\Tag{(77)}
-\]
-and for the total radiant energy
-\[
-U = aT^{4} ˇ V.
-\Tag{(78)}
-\]
-This law, which states that the volume density and the specific
-intensity of black radiation are proportional to the fourth power
-of the absolute temperature, was first established by \Name{J.~Stefan}\footnote
- {\Name{J.~Stefan}, Wien.\ Berichte, \textbf{79}, p.~391, 1879.}
-on
-a basis of rather rough measurements. It was later deduced
-by \Name{L.~Boltzmann}\footnote
- {\Name{L.~Boltzmann}, Wied.\ Annalen, \textbf{22}, p.~291, 1884.}
-on a thermodynamic basis from \Name{Maxwell's}
-radiation pressure and has been more recently confirmed by
-\Name{O.~Lummer} and \Name{E.~Pringsheim}\footnote
- {\Name{O.~Lummer} und \Name{E.~Pringsheim}, Wied.\ Annalen, \textbf{63}, p.~395, 1897. Annalen d.\ Physik\DPchg{,}{} \textbf{3},
- p.~159, 1900.}
-by exact measurements between
-$100°$ and $1300° \Celsius$, the temperature being defined by the gas
-thermometer. In ranges of temperature and for requirements
-of precision for which the readings of the different gas thermometers
-no longer agree sufficiently or cannot be obtained at all, the
-\Name{Stefan-Boltzmann} law of radiation can be used for an absolute
-definition of temperature independent of all substances.
-
-\Section{64.} The numerical value of the constant~$a$ is obtained from
-measurements made by \Name{F.~Kurlbaum}.\footnote
- {\Name{F.~Kurlbaum}, Wied.\ Annalen, \textbf{65}, p.~759, 1898.}
-According to them, if
-%% -----File: 080.png---Folio 64-------
-we denote by~$S_{t}$ the total energy radiated in one second into air
-by a square centimeter of a black body at a temperature of $t° \Celsius$,
-the following equation holds
-\[
-S_{100} - S_{0}
- = 0.0731\, \frac{\watt}{\cm^{2}}
- = 7.31 × 10^{5}\, \frac{\erg}{\cm^{2}\, \sec}\Add{.}
-%[** TN: No equation number in the original]
-\Tag{(79)}
-\]
-Now, since the radiation in air is approximately identical with
-the radiation into a vacuum, we may according to \Eq{(7)}~and~\Eq{(76)}
-put
-\[
-S_{t} = \pi K = \frac{ac}{4} (273 + t)^{4}
-\]
-and from this
-\[
-S_{100} - S_{0} = \frac{ac}{4} (373^{4} - 273^{4}),
-\]
-therefore
-\[
-a = \frac{4 × 7.31 × 10^{5}}{3 × 10^{10} × (373^{4} - 273^{4})}
- = 7.061 × 10^{-15}\, \frac{\erg}{\cm^{3}\, \degree^{4}}\Add{.}
-\]
-
-Recently \Name{Kurlbaum} has increased the value measured by him
-by $2.5$~per cent.,\footnote
- {\Name{F. Kurlbaum}, Verhandlungen d.\ Deutsch.\ physikal.\ Gesellschaft, \textbf{14}, p.~580, 1912.}
-on account of the bolometer used being not
-perfectly black, whence it follows that $a = 7.24 ˇ 10^{-15}$.
-
-Meanwhile the radiation constant has been made the object
-of as accurate measurements as possible in various places. Thus
-it was measured by \Name{Féry}, \Name{Bauer} and \Name{Moulin}, \Name{Valentiner}, \Name{Féry} and
-\Name{Drecq}, \Name{Shakespear}, \Name{Gerlach}, with in some cases very divergent
-results, so that a mean value may hardly be formed.
-
-For later computations we shall use the most recent determination
-made in the physical laboratory of the University of Berlin\footnote
- {According to private information kindly furnished by my colleague \Name{H.~Rubens} (July,
- \Label{64}%[** TN: Page label]
- 1912). (These results have since been published. See \Name{W.~H. Westphal}, Verhandlungen d.\
- Deutsch.\ physikal.\ Gesellschaft, \textbf{14}, p.~987, 1912,~Tr.)}
-\[
-\frac{ac}{4}
- = \sigma
- = 5.46 ˇ 10^{-12}\, \frac{\watt}{\cm^{2}\, \degree^{4}}\Add{.}
-\]
-From this $a$~is found to be
-\[
-a = \frac{4 ˇ 5.46 ˇ 10^{-12} ˇ 10^{7}}{3 ˇ 10^{10}}
- = 7.28 ˇ 10^{-15}\, \frac{\erg}{\cm^{3}\, \degree^{4}}
-\]
-which agrees rather closely with \Name{Kurlbaum's} corrected value.
-%% -----File: 081.png---Folio 65-------
-
-\Section{65.} The magnitude of the entropy~$S$ of black radiation found
-by integration of the differential equation~\Eq{(73)} is
-\[
-S = \frac{4}{3} aT^{3}V.
-\Tag{(80)}
-\]
-
-In this equation the additive constant is determined by a choice
-that readily suggests itself, so that at the zero of the absolute
-scale of temperature, that is to say, when $u$~vanishes, $S$~shall
-become zero. From this the entropy of unit volume or the
-volume density of the entropy of black radiation is obtained,
-\[
-\frac{S}{V} = s = \frac{4}{3} aT^{3}.
-\Tag{(81)}
-\]
-
-\Section{66.} We shall now remove a restricting assumption made in
-order to enable us to apply the value of \Name{Maxwell's} radiation
-pressure, calculated in the preceding chapter. Up to now we
-have assumed the cylinder to be fixed and only the piston to be
-free to move. We shall now think of the whole of the vessel,
-consisting of the cylinder, the black bottom, and the piston, the
-latter attached to the walls in a definite height above the bottom,
-as being free to move in space. Then, according to the principle
-of action and reaction, the vessel as a whole must remain constantly
-at rest, since no external force acts on it. This is the
-conclusion to which we must necessarily come, even without,
-in this case, admitting \textit{a~priori} the validity of the principle of
-action and reaction. For if the vessel should begin to move,
-the kinetic energy of this motion could originate only at the expense
-of the heat of the body forming the bottom or the energy of
-radiation, as there exists in the system enclosed in a rigid cover
-no other available energy; and together with the decrease of
-energy the entropy of the body or the radiation would also decrease,
-an event which would contradict the second principle,
-since no other changes of entropy occur in nature. Hence the
-vessel as a whole is in a state of mechanical equilibrium. An
-immediate consequence of this is that the pressure of the radiation
-on the black bottom is just as large as the oppositely directed
-pressure of the radiation on the reflecting piston. Hence the
-pressure of black radiation is the same on a black as on a reflecting
-body of the same temperature and the same may be readily proven
-%% -----File: 082.png---Folio 66-------
-for any completely reflecting surface whatsoever, which we may
-assume to be at the bottom of the cylinder without in the least
-disturbing the stationary state of radiation. Hence we may also
-in all the foregoing considerations replace the reflecting metal
-by any completely reflecting or black body whatsoever, at the
-same temperature as the body forming the bottom, and it may
-be stated as a quite general law that the radiation pressure
-depends only on the properties of the radiation passing to and
-fro, not on the properties of the enclosing substance.
-
-\Section{67.} If, on raising the piston, the temperature of the black body
-forming the bottom is kept constant by a corresponding addition
-of heat from the heat reservoir, the process takes place isothermally.
-Then, along with the temperature~$T$ of the black body,
-the energy density~$u$, the radiation pressure~$p$, and the density of
-the entropy~$s$ also remain constant; hence the total energy of
-radiation increases from $U = uV$ to $U' = uV'$, the entropy from
-$S = sV$ to $S' = sV'$ and the heat supplied from the heat reservoir
-is obtained by integrating~\Eq{(72)} at constant~$T$,
-\[
-Q = T(S'- S) = Ts(V' - V)
-\]
-or, according to \Eq{(81)}~and~\Eq{(75)},
-\[
-Q = \frac{4}{3} aT^{4}(V' - V) = \frac{4}{3} (U' - U).
-\]
-
-Thus it is seen that the heat furnished from the outside exceeds
-the increase in energy of radiation $(U' - U)$ by $\frac{1}{3}(U' - U)$.
-This excess in the added heat is necessary to do the external work
-accompanying the increase in the volume of radiation.
-
-\Section{68.} Let us also consider a reversible adiabatic process. For
-this it is necessary not merely that the piston and the mantle but
-also that the bottom of the cylinder be assumed as completely
-reflecting, \eg,~as white. Then the heat furnished on compression
-or expansion of the volume of radiation is $Q = 0$ and the energy
-of radiation changes only by the value~$p\, dV$ of the external work.
-To insure, however, that in a finite adiabatic process the radiation
-shall be perfectly stable at every instant, \ie, shall have the character
-of black radiation, we may assume that inside the evacuated
-cavity there is a carbon particle of minute size. This particle,
-which may be assumed to possess an absorbing power differing
-%% -----File: 083.png---Folio 67-------
-from zero for all kinds of rays, serves merely to produce stable
-equilibrium of the radiation in the cavity (\Sec{51} \textit{et~seq.})\ and
-thereby to insure the reversibility of the process, while its heat
-contents may be taken as so small compared with the energy of
-radiation,~$U$, that the addition of heat required for an appreciable
-temperature change of the particle is perfectly negligible. Then
-at every instant in the process there exists absolutely stable
-equilibrium of radiation and the radiation has the temperature of
-the particle in the cavity. The volume, energy, and entropy of
-the particle may be entirely neglected.
-
-On a reversible adiabatic change, according to~\Eq{(72)}, the entropy~$S$
-of the system remains constant. Hence from~\Eq{(80)} we have as
-a condition for such a process
-\[
-T^{3}V = \const.,
-\]
-or, according to~\Eq{(77)},
-\[
-pV^{\efrac{4}{3}} = \const.,
-\]
-\ie, on an adiabatic compression the temperature and the pressure
-of the radiation increase in a manner that may be definitely
-stated. The energy of the radiation,~$U$, in such a case varies
-according to the law
-\[
-\frac{U}{T} = \frac{3}{4} S = \const.,
-\]
-\ie, it increases in proportion to the absolute temperature, although
-the volume becomes smaller.
-
-\Section{69.} Let us finally, as a further example, consider a simple case
-of an irreversible process. Let the cavity of volume~$V$, which is
-everywhere enclosed by absolutely reflecting walls, be uniformly
-filled with black radiation. Now let us make a small hole
-through any part of the walls, \eg,~by opening a stopcock, so
-that the radiation may escape into another completely evacuated
-space, which may also be surrounded by rigid, absolutely reflecting
-walls. The radiation will at first be of a very irregular character;
-after some time, however, it will assume a stationary condition
-and will fill both communicating spaces uniformly, its total
-volume being, say,~$V'$. The presence of a carbon particle will
-cause all conditions of black radiation to be satisfied in the new
-%% -----File: 084.png---Folio 68-------
-state. Then, since there is neither external work nor addition of
-heat from the outside, the energy of the new state is, according
-to the first principle, equal to that of the original one, or $U' = U$
-and hence from~\Eq{(78)}
-\begin{gather*}
-T'^{4} V' = T^{4} V \\
-\frac{T'}{T} = \sqrt[4]{\frac{V}{V'}}
-\end{gather*}
-which defines completely the new state of equilibrium. Since
-$V' > V$ the temperature of the radiation has been lowered by the
-process.
-
-According to the second principle of thermodynamics the
-entropy of the system must have increased, since no external
-changes have occurred; in fact we have from~\Eq{(80)}
-\[
-\frac{S'}{S} = \frac{T'^{3}V'}{T^{3}V} = \sqrt[4]{\frac{V'}{V}} > 1.
-\Tag{(82)}
-\]
-
-\Section{70.} If the process of irreversible adiabatic expansion of the
-radiation from the volume~$V$ to the volume~$V'$ takes place as
-just described with the single difference that there is no carbon
-particle present in the vacuum, after the stationary state of radiation
-is established, as will be the case after a certain time on
-account of the diffuse reflection from the walls of the cavity, the
-radiation in the new volume~$V'$ will not any longer have the
-character of black radiation, and hence no definite temperature.
-Nevertheless the radiation, like every system in a definite physical
-state, has a definite entropy, which, according to the second principle,
-is larger than the original~$S$, but not as large as the $S'$ given
-in~\Eq{(82)}. The calculation cannot be performed without the use
-of laws to be taken up later (see \Sec{103}). If a carbon particle
-is afterward introduced into the vacuum, absolutely stable
-equilibrium is established by a second irreversible process, and,
-the total energy as well as the total volume remaining constant,
-the radiation assumes the normal energy distribution of black
-radiation and the entropy increases to the maximum value~$S'$
-given by~\Eq{(82)}.
-%% -----File: 085.png---Folio 69-------
-
-\Chapter{III}{Wien's Displacement Law}
-
-\Section{71.} Though the manner in which the volume density~$u$ and the
-specific intensity~$K$ of black radiation depend on the temperature
-is determined by the \Name{Stefan-Boltzmann} law, this law is of comparatively
-little use in finding the volume density~$\ssfu_{\nu}$ corresponding
-to a definite frequency~$\nu$, and the specific intensity of radiation~$\ssfK_{\nu}$
-of monochromatic radiation, which are related to each other
-by equation~\Eq{(24)} and to $u$~and~$K$ by equations \Eq{(22)}~and~\Eq{(12)}.
-There remains as one of the principal problems of the theory of
-heat radiation the problem of determining the quantities $\ssfu_{\nu}$~and~$\ssfK_{\nu}$
-for black radiation in a vacuum and hence, according to~\Eq{(42)},
-in any medium whatever, as functions of $\nu$~and~$T$, or, in other
-words, to find the distribution of energy in the normal spectrum
-for any arbitrary temperature. An essential step in the solution
-of this problem is contained in the so-called ``displacement
-law'' stated by \Name{W.~Wien},\footnote
- {\Name{W.~Wien}, Sitzungsberichte d.\ Akad.\ d.\ Wissensch.\ Berlin, Febr.~9, 1893, p.~55. Wiedemann's
- Annal., \textbf{52}, p.~132, 1894. See also among others \Name{M.~Thiesen}, Verhandl.\ d.\ Deutsch.\
- phys.\ Gesellsch\DPtypo{}{.}, \textbf{2}, p.~65, 1900. \Name{H.~A. Lorentz}, Akad.\ d.\ Wissensch.\ Amsterdam, May~18,
- 1901, p.~607. \Name{M.~Abraham}, Annal.\ d.\ Physik\DPtypo{.}{}\ \textbf{14}, p.~236, 1904.}
-the importance of which lies in the
-fact that it reduces the functions $\ssfu_{\nu}$~and~$\ssfK_{\nu}$ of the two arguments
-$\nu$~and~$T$ to a function of a single argument.
-
-The starting point of \Name{Wien's} displacement law is the following
-theorem. If the black radiation contained in a perfectly evacuated
-cavity with absolutely reflecting walls is compressed or
-expanded adiabatically and infinitely slowly, as described above
-%[** Theorem]
-in \Sec{68}, \emph{the radiation always retains the character of black radiation,
-even without the presence of a carbon particle}. Hence the
-process takes place in an absolute vacuum just as was calculated
-in \Sec{68} and the introduction, as a precaution, of a carbon
-particle is shown to be superfluous. But this is true only in this
-special case, not at all in the case described in \Sec{70}.
-
-The truth of the proposition stated may be shown as follows:
-%% -----File: 086.png---Folio 70-------
-Let the completely evacuated hollow cylinder, which is at the
-start filled with black radiation, be compressed adiabatically
-and infinitely slowly to a finite fraction of the original volume.
-If, now, the compression being completed, the radiation were no
-longer black, there would be no stable thermodynamic equilibrium
-of the radiation (\Sec{51}). It would then be possible to
-produce a finite change at constant volume and constant total
-energy of radiation, namely, the change to the absolutely stable
-state of radiation, which would cause a finite increase of entropy.
-This change could be brought about by the introduction of a
-carbon particle, containing a negligible amount of heat as compared
-with the energy of radiation. This change, of course,
-refers only to the spectral density of radiation~$\ssfu_{\nu}$, whereas the
-total density of energy~$u$ remains constant. After this has been
-accomplished, we could, leaving the carbon particle in the space,
-allow the hollow cylinder to return adiabatically and infinitely
-slowly to its original volume and then remove the carbon particle.
-The system will then have passed through a cycle without any
-external changes remaining. For heat has been neither added
-nor removed, and the mechanical work done on compression has
-been regained on expansion, because the latter, like the radiation
-pressure, depends only on the total density~$u$ of the energy of radiation,
-not on its spectral distribution. Therefore, according to
-the first principle of thermodynamics, the total energy of radiation
-is at the end just the same as at the beginning, and hence
-also the temperature of the black radiation is again the same.
-The carbon particle and its changes do not enter into the calculation,
-for its energy and entropy are vanishingly small compared
-with the corresponding quantities of the system. The
-process has therefore been reversed in all details; it may be
-repeated any number of times without any permanent change
-occurring in nature. This contradicts the assumption, made
-above, that a finite increase in entropy occurs; for such a finite
-increase, once having taken place, cannot in any way be completely
-reversed. Therefore no finite increase in entropy can have
-been produced by the introduction of the carbon particle in the
-space of radiation, but the radiation was, before the introduction
-and always, in the state of stable equilibrium.
-
-\Section{72.} In order to bring out more clearly the essential part of
-%% -----File: 087.png---Folio 71-------
-this important proof, let us point out an analogous and more or
-less obvious consideration. Let a cavity containing originally
-a vapor in a state of saturation be compressed adiabatically and
-infinitely slowly.
-
-%[** Quoted; semantics?]
-``Then on an arbitrary adiabatic compression the vapor remains
-always just in the state of saturation. For let us suppose that it
-becomes supersaturated on compression. After the compression
-to an appreciable fraction of the original volume has taken place,
-condensation of a finite amount of vapor and thereby a change
-into a more stable state, and hence a finite increase of entropy of
-the system, would be produced at constant volume and constant
-total energy by the introduction of a minute drop of liquid, which
-has no appreciable mass or heat capacity. After this has been
-done, the volume could again be increased adiabatically and
-infinitely slowly until again all liquid is evaporated and thereby
-the process completely reversed, which contradicts the assumed
-increase of entropy.''
-
-Such a method of proof would be erroneous, because, by the
-process described, the change that originally took place is not
-at all completely reversed. For since the mechanical work
-expended on the compression of the supersaturated steam is not
-equal to the amount gained on expanding the saturated steam,
-there corresponds to a definite volume of the system when it is
-being compressed an amount of energy different from the one
-during expansion and therefore the volume at which all liquid is
-just vaporized cannot be equal to the original volume. The
-supposed analogy therefore breaks down and the statement made
-above in quotation marks is incorrect.
-
-\Section{73.} We shall now again suppose the reversible adiabatic process
-described in \Sec{68} to be carried out with the black radiation
-contained in the evacuated cavity with white walls and white
-bottom, by allowing the piston, which consists of absolutely
-reflecting metal, to move downward infinitely slowly, with the
-single difference that now there shall be no carbon particle in the
-cylinder. The process will, as we now know, take place exactly
-as there described, and, since no absorption or emission of radiation
-takes place, we can now give an account of the changes of
-color and intensity which the separate pencils of the system
-undergo. Such changes will of course occur only on reflection
-%% -----File: 088.png---Folio 72-------
-from the moving metallic reflector, not on reflection from the
-stationary walls and the stationary bottom of the cylinder.
-
-If the reflecting piston moves down with the constant, infinitely
-small, velocity~$v$, the monochromatic pencils striking it during
-the motion will suffer on reflection a change of color, intensity,
-and direction. Let us consider these different influences in order.\footnote
- {The complete solution of the problem of reflection of a pencil from a moving absolutely
- reflecting surface including the case of an arbitrarily large velocity of the surface may be
- found in the paper by \Name{M.~Abraham} quoted in \Sec{71}. See also the text-book by the same
- author. Electromagnetische Theorie der Strahlung, 1908 (Leipzig, B.~G. Teubner).}
-
-\Section{74.} To begin with, we consider the \emph{change of color} which a monochromatic
-ray suffers by reflection from the reflector, which is
-\Figure[2in]{5}
-moving with an infinitely small velocity.
-For this purpose we consider
-first the case of a ray which falls
-normally from below on the reflector
-and hence is reflected normally downward.
-Let the plane~$A$ (\Fig{5}) represent
-the position of the reflector at the
-time~$t$, the plane~$A'$ the position at
-the time $t + \delta t$, where the distance~$AA'$
-equals $v\, \delta t$, $v$~denoting the velocity
-of the reflector. Let us now suppose a stationary plane~$B$ to be
-placed parallel to~$A$ at a suitable distance and let us denote by~$\lambda$
-the wave length of the ray incident on the reflector and by~$\lambda'$
-the wave length of the ray reflected from it. Then at a time~$t$
-there are in the interval~$AB$ in the vacuum containing the radiation
-$\dfrac{AB}{\lambda}$ waves of the incident and $\dfrac{AB}{\lambda'}$ waves of the reflected ray,
-as can be seen, \eg, by thinking of the electric field-strength as
-being drawn at the different points of each of the two rays at
-the time~$t$ in the form of a sine curve. Reckoning both incident
-and reflected ray there are at the time~$t$
-\[
-AB\left(\frac{1}{\lambda} + \frac{1}{\lambda'}\right)
-\]
-waves in the interval between $A$~and~$B$. Since this is a large number,
-it is immaterial whether the number is an integer or not.
-%% -----File: 089.png---Folio 73-------
-
-Similarly at the time $t + \delta t$, when the reflector is at~$A'$, there are
-\[
-A'B\left(\frac{1}{\lambda} + \frac{1}{\lambda'}\right)
-\]
-waves in the interval between $A'$~and~$B$ all told.
-
-The latter number will be smaller than the former, since in the
-shorter distance~$A'B$ there is room for fewer waves of both kinds
-than in the longer distance~$AB$. The remaining waves must have
-been expelled in the time~$\delta t$ from the space between the stationary
-plane~$B$ and the moving reflector, and this must have taken place
-through the plane~$B$ downward; for in no other way could a
-wave disappear from the space considered.
-
-Now $\nu\, \delta t$~waves pass in the time~$\delta t$ through the stationary
-plane~$B$ in an upward direction and $\nu'\, \delta t$~waves in a downward
-direction; hence we have for the difference
-\[
-(\nu' - \nu)\, \delta t = (AB - A'B) \left(\frac{1}{\lambda} + \frac{1}{\lambda'}\right)
-\]
-or, since
-\[
-AB - A'B = v\, \delta t,
-\]
-and
-\begin{gather*}
-\lambda = \frac{c}{\nu}\quad
-\lambda' = \frac{c}{\nu'} \\
-\nu' = \frac{c + v}{c - v} \nu
-\end{gather*}
-or, since $v$~is infinitely small compared with~$c$,
-\[
-\nu' = \nu \left(1 + \frac{2v}{c}\right)\Add{.}
-\]
-
-\Section{75.} When the radiation does not fall on the reflector normally
-but at an acute angle of incidence~$\theta$, it is possible to pursue a very
-similar line of reasoning, with the difference that then~$A$, the
-point of intersection of a definite ray~$BA$ with the reflector at
-the time~$t$, has not the same position on the reflector as the point
-of intersection,~$A'$, of the same ray with the reflector at the time
-$t + \delta t$ (\Fig{6}). The number of waves which lie in the interval~$BA$
-at the time~$t$ is~$\dfrac{BA}{\lambda}$. Similarly, at the time~$t$ the number of
-waves in the interval~$AC$ representing the distance of the point~$A$
-%% -----File: 090.png---Folio 74-------
-from a wave plane~$CC'$, belonging to the reflected ray and
-stationary in the vacuum, is~$\dfrac{AC}{\lambda'}$.
-
-Hence there are, all told, at the time~$t$ in the interval~$BAC$
-\[
-\frac{BA}{\lambda} + \frac{AC}{\lambda'}
-\]
-waves of the ray under consideration. We may further note
-that the angle of reflection~$\theta'$ is not exactly equal to the angle
-\Figure[4in]{6}
-of incidence, but is a little smaller as can be shown by a simple
-geometric consideration based on \Name{Huyghens'} principle. The
-difference of $\theta$ and~$\theta'$, however, will be shown to be non-essential
-for our calculation. Moreover there are at the time $t + \delta t$, when
-the reflector passes through~$A'$,
-\[
-\frac{BA'}{\lambda} + \frac{A'C'}{\lambda'}
-\]
-waves in the distance~$BA'C'$. The latter number is smaller than
-the former and the difference must equal the total number of
-waves which are expelled in the time~$\delta t$ from the space which is
-bounded by the stationary planes $BB'$~and~$CC'$.
-
-Now $\nu\, \delta t$~waves enter into the space through the plane~$BB'$ in
-the time~$\delta t$ and $\nu'\, \delta t$~waves leave the space through the plane~$CC'$\Add{.}
-Hence we have
-\[
-(\nu' - \nu)\, \delta t
- = \left(\frac{BA}{\lambda} + \frac{AC}{\lambda'}\right)
- - \left(\frac{BA'}{\lambda} + \frac{A'C'}{\lambda'}\right)
-\]
-%% -----File: 091.png---Folio 75-------
-but
-\begin{gather*}
-BA - BA' = AA' = \frac{v\, \delta t}{\cos\theta} \\
-AC - A'C' = AA' \cos(\theta + \theta') \\
-\lambda = \frac{c}{\nu},\quad
-\lambda' = \frac{c}{\nu'}.
-\end{gather*}
-Hence
-\[
-\nu' = \frac{c \cos\theta + v}{c \cos\theta - v \cos(\theta + \theta')} \nu.
-\]
-
-This relation holds for any velocity~$v$ of the moving reflector.
-Now, since in our case $v$~is infinitely small compared with~$c$, we
-have the simpler expression
-\[%[** Size of parentheses]
-\nu' = \nu (1 + \frac{v}{c \cos\theta} [1 + \cos(\theta + \theta')])\Add{.}
-\]
-The difference between the two angles $\theta$~and~$\theta'$ is in any case of
-the order of magnitude~$\dfrac{v}{c}$; hence we may without appreciable
-error replace $\theta'$ by~$\theta$, thereby obtaining the following expression
-for the frequency of the reflected ray for oblique incidence
-\[
-\nu' = \nu \left(1 + \frac{2v \cos\theta}{c}\right)\Add{.}
-\Tag{(83)}
-\]
-
-\Section{76.} From the foregoing it is seen that the frequency of all rays
-which strike the moving reflector are increased on reflection, when
-the reflector moves toward the radiation, and decreased, when the
-reflector moves in the direction of the incident rays ($v < 0$).
-However, the total radiation of a definite frequency~$\nu$ striking the
-moving reflector is by no means reflected as monochromatic radiation
-but the change in color on reflection depends also essentially
-on the angle of incidence~$\theta$. Hence we may not speak of a certain
-spectral ``displacement'' of color except in the case of a single
-pencil of rays of definite direction, whereas in the case of the
-entire monochromatic radiation we must refer to a spectral
-``dispersion.'' The change in color is the largest for normal incidence
-and vanishes entirely for grazing incidence.
-
-\Section{77.} Secondly, let us calculate the \emph{change in energy}, which the
-%% -----File: 092.png---Folio 76-------
-moving reflector produces in the incident radiation, and let us
-consider from the outset the general case of oblique incidence.
-Let a monochromatic, infinitely thin, unpolarized pencil of rays\DPtypo{.}{,}
-which falls on a surface element of the reflector at the angle of
-incidence~$\theta$, transmit the energy~$I\, \delta t$ to the reflector in the time~$\delta t$.
-Then, ignoring vanishingly small quantities, the mechanical
-pressure of the pencil of rays normally to the reflector is, according
-to equation~\Eq{(64)},
-\[
-\ssfF = \frac{2 \cos\theta}{c} I,
-\]
-and to the same degree of approximation the work done from the
-outside on the incident radiation in the time~$\delta t$ is
-\[
-\ssfF v\, \delta t = \frac{2v \cos\theta}{c} I\, \delta t.
-\Tag{(84)}
-\]
-According to the principle of the conservation of energy this
-amount of work must reappear in the energy of the reflected radiation.
-Hence the reflected pencil has a larger intensity than the
-incident one. It produces, namely, in the time~$\delta t$ the energy\footnote
- {It is clear that the change in intensity of the reflected radiation caused by the motion of
- the reflector can also be derived from purely electrodynamical considerations, since electrodynamics
- are consistent with the energy principle. This method is somewhat lengthy,
- but it affords a deeper insight into the details of the phenomenon of reflection.}
-\[
-I\, \delta t + \ssfF v\, \delta t
- = I\left(1 + \frac{2v \cos\theta}{c}\right) \delta t
- = I'\, \delta t.
-\Tag{(85)}
-\]
-Hence we may summarize as follows: By the reflection of a
-monochromatic unpolarized pencil, incident at an angle~$\theta$ on a
-reflector moving toward the radiation with the infinitely small
-velocity~$v$, the radiant energy~$I\, \delta t$, whose frequencies extend from
-$\nu$ to $\nu + d\nu$, is in the time~$\delta t$ changed into the radiant energy
-$I'\, \delta t$ with the interval of frequency $(\nu', \nu' + d\nu')$, where $I'$~is given
-by~\Eq{(85)}, $\nu'$~by~\Eq{(83)}, and accordingly $d\nu'$, the spectral breadth of
-the reflected pencil, by
-\[
-d\nu' = d\nu \left(1 + \frac{2v \cos\theta}{c}\right)\Add{.}
-\Tag{(86)}
-\]
-A comparison of these values shows that
-\[
-\frac{I'}{I} = \frac{\nu'}{\nu} = \frac{d\nu'}{d\nu}\Add{.}
-\Tag{(87)}
-\]
-%% -----File: 093.png---Folio 77-------
-The absolute value of the radiant energy which has disappeared
-in this change is, from equation~\Eq{(13)},
-\[
-I\, \delta t = 2\ssfK_{\nu}\, d\sigma \cos\theta\, d\Omega\, d\nu\, \delta t,
-\Tag{(88)}
-\]
-and hence the absolute value of the radiant energy which has
-been formed is, according to~\Eq{(85)},
-\[
-I'\, \delta t = 2\ssfK_{\nu}\, d\sigma \cos\theta\, d\Omega\, d\nu \left(1 + \frac{2v \cos\theta}{c}\right) \delta t.
-\Tag{(89)}
-\]
-
-Strictly speaking these last two expressions would require an
-infinitely small correction, since the quantity~$I$ from equation~\Eq{(88)}
-represents the energy radiation on a stationary element of area~$d\sigma$,
-while, in reality, the incident radiation is slightly increased
-by the motion of~$d\sigma$ toward the incident pencil. \Erratum{The additional
-terms resulting therefrom may, however, be omitted here without
-appreciable error.}{The corresponding additional terms may, however, be omitted
-here without appreciable error, since the correction caused by
-them would consist merely of the addition to the energy change
-here calculated of a comparatively infinitesimal energy change of
-the same kind with an external work that is infinitesimal of the
-second order.}
-
-\Section{78.} As regards finally the \emph{changes in direction}, which are imparted
-to the incident ray by reflection from the moving reflector,
-we need not calculate them at all at this stage. For if the motion
-of the reflector takes place sufficiently slowly, all irregularities
-in the direction of the radiation are at once equalized by further
-reflection from the walls of the vessel. We may, indeed, think of
-the whole process as being accomplished in a very large number of
-short intervals, in such a way that the piston, after it has moved
-a very small distance with very small velocity, is kept at rest for
-a while, namely, until all irregularities produced in the directions
-of the radiation have disappeared as the result of the reflection
-from the white walls of the hollow cylinder. If this procedure
-be carried on sufficiently long, the compression of the radiation
-may be continued to an arbitrarily small fraction of the original
-volume, and while this is being done, the radiation may be always
-regarded as uniform in all directions. This continuous process
-of equalization refers, of course, only to difference in the direction
-of the radiation; for changes in the color or intensity of the
-radiation of however small size, having once occurred, can
-evidently never be equalized by reflection from totally reflecting
-stationary walls but continue to exist forever.
-
-\Section{79.} With the aid of the theorems established we are now in a
-position to calculate the change of the density of radiation for
-%% -----File: 094.png---Folio 78-------
-every frequency for the case of infinitely slow adiabatic compression
-of the perfectly evacuated hollow cylinder, which is filled
-with uniform radiation. For this purpose we consider the radiation
-at the time~$t$ in a definite infinitely small interval of frequencies,
-from $\nu$ to $\nu + d\nu$, and inquire into the change which
-the total energy of radiation contained in this definite constant
-interval suffers in the time~$\delta t$.
-
-At the time~$t$ this radiant energy is, according to \Sec{23}, $V\ssfu\, d\nu$;
-at the time $t + \delta t$ it is $\bigl(V\ssfu + \delta (V\ssfu)\bigr)\, d\nu$, hence the change to be
-calculated is
-\[
-\delta (V\ssfu)\, d\nu.
-\Tag{(90)}
-\]
-In this the density of monochromatic radiation~$\ssfu$ is to be regarded
-as a function of the mutually independent variables $\nu$~and~$t$, the
-differentials of which are distinguished by the symbols $d$~and~$\delta$.
-
-The change of the energy of monochromatic radiation is produced
-only by the reflection from the moving reflector, that is
-to say, firstly by certain rays, which at the time~$t$ belong to the
-interval $(\nu, d\nu)$, leaving this interval on account of the change in
-color suffered by reflection, and secondly by certain rays, which at
-the time~$t$ do not belong to the interval $(\nu, d\nu)$, coming into this
-interval on account of the change in color suffered on reflection.
-Let us calculate these influences in order. The calculation is
-greatly simplified by taking the width of this interval~$d\nu$ so small
-that
-\[
-d\nu \text{ is small compared with } \frac{v}{c} \nu,
-\Tag{(91)}
-\]
-a condition which can always be satisfied, since $d\nu$~and~$v$ are
-mutually independent.
-
-\Section{80.} The rays which at the time~$t$ belong to the interval $(\nu, d\nu)$
-and leave this interval in the time~$\delta t$ on account of reflection from
-the moving reflector, are simply those rays which strike the
-moving reflector in the time~$\delta t$. For the change in color which
-such a ray undergoes is, from \Eq{(83)}~and~\Eq{(91)}, large compared with~$d\nu$,
-the width of the whole interval. Hence we need only calculate
-the energy, which in the time~$\delta t$ is transmitted to the reflector
-by the rays in the interval $(\nu, d\nu)$.
-
-For an elementary pencil, which falls on the element~$d\sigma$ of the
-%% -----File: 095.png---Folio 79-------
-reflecting surface at the angle of incidence~$\theta$, this energy is,
-according to \Eq{(88)}~and~\Eq{(5)},
-\[
-I\, \delta t = 2\ssfK_{\nu}\, d\sigma \cos\theta\, d\Omega\, d\nu\, \delta t
- = 2\ssfK_{\nu}\, d\sigma \sin\theta \cos\theta\, d\theta\, d\phi\, d\nu\, \delta t.
-\]
-Hence we obtain for the total monochromatic radiation, which
-falls on the whole surface~$F$ of the reflector, by integration with
-respect to~$\phi$ from $0$ to~$2\pi$, with respect to~$\theta$ from $0$ to~$\dfrac{\pi}{2}$, and with
-respect to~$d\sigma$ from $0$ to~$F$,
-\[
-2\pi F \ssfK_{\nu}\, d\nu\, \delta t.
-\Tag{(92)}
-\]
-Thus this radiant energy leaves, in the time~$\delta t$, the interval of
-frequencies $(\nu, d\nu)$ considered.
-
-\Section{81.} In calculating the radiant energy which enters the interval
-$(\nu, d\nu)$ in the time~$\delta t$ on account of reflection from the moving
-reflector, the rays falling on the reflector at different angles of
-incidence must be considered separately. Since in the case of a
-positive~$v$, the frequency is increased by the reflection, the rays
-which must be considered have, at the time~$t$, the frequency
-$\nu_{1} < \nu$. If we now consider at the time~$t$ a monochromatic pencil
-of frequency $(\nu_{1}, d\nu_{1})$, falling on the reflector at an angle of incidence~$\theta$,
-a necessary and sufficient condition for its entrance, by
-reflection, into the interval $(\nu, d\nu)$ is
-\[
-\nu = \nu_{1} \left(1 + \frac{2v \cos\theta}{c}\right) \quad\text{and}\quad
-d\nu = d\nu_{1} \left(1 + \frac{2v \cos\theta}{c}\right)\Add{.}
-\]
-These relations are obtained by substituting $\nu_{1}$~and $\nu$ respectively
-in the equations \Eq{(83)}~and~\Eq{(86)} in place of the frequencies before
-and after reflection $\nu$~and~$\nu'$.
-
-The energy which this pencil carries into the interval $(\nu_{1}, d\nu)$
-in the time~$\delta t$ is obtained from~\Eq{(89)}, likewise by substituting $\nu_{1}$
-for~$\nu$. It is
-\[
-2\ssfK_{\nu_{1}}\, d\sigma \cos\theta\, d\Omega\, d\nu_{1} \left(1 + \frac{2v \cos\theta}{c}\right) \delta t
- = 2\ssfK_{\nu_{1}}\, d\sigma \cos\theta\, d\Omega\, d\nu\, \delta t.
-\]
-Now we have
-\[
-\ssfK_{\nu_{1}} = \ssfK_{\nu} + (\nu_{1} - \nu) \frac{\dd \ssfK}{\dd \nu} + \dots
-\]
-where we shall assume $\dfrac{\dd \ssfK}{\dd \nu}$ to be finite.
-%% -----File: 096.png---Folio 80-------
-
-Hence, neglecting small quantities of higher order,
-\[
-\ssfK_{\nu_{1}} = \ssfK_{\nu} - \frac{2\nu v \cos\theta}{c}\, \frac{\dd \ssfK}{\dd \nu}\Add{.}
-\]
-Thus the energy required becomes
-\[
-2\, d\sigma \left(\ssfK_{\nu} - \frac{2\nu v \cos\theta}{c}\, \frac{\dd \ssfK}{\dd \nu}\right) \sin\theta \cos\theta\, d\theta\, d\phi\, d\nu\, \delta t,
-\]
-and, integrating this expression as above, with respect to $d\sigma$,~$\phi$,
-and~$\theta$, the total radiant energy which enters into the interval
-$(\nu, d\nu)$ in the time~$\delta t$ becomes
-\[
-2\pi F \left(\ssfK_{\nu} - \frac{4}{3}\, \frac{\nu v}{c}\, \frac{\dd \ssfK}{\dd \nu}\right) d\nu\, \delta t.
-\Tag{(93)}
-\]
-
-\Section{82.} The difference of the two expressions \Eq{(93)}~and~\Eq{(92)} is equal
-to the whole change~\Eq{(90)}, hence
-\[
--\frac{8\pi}{3} F \frac{\nu v}{c}\, \frac{\dd \ssfK}{\dd \nu}\, \delta t = \delta (V\ssfu),
-\]
-or, according to~\Eq{(24)},
-\[
--\frac{1}{3} F\nu v \frac{\dd \ssfu}{\dd \nu}\, \delta t = \delta (V\ssfu),
-\]
-or, finally, since $Fv\, \delta t$ is equal to the decrease of the volume~$V$,
-\[
-\frac{1}{3} \nu \frac{\dd \ssfu}{\dd \nu}\, \delta V
- = \delta (V\ssfu)
- = \ssfu\, \delta V + V\, \delta \ssfu,
-\Tag{(94)}
-\]
-whence it follows that
-\[
-\delta \ssfu = \left(\frac{\nu}{3}\, \frac{\dd \ssfu}{\dd \nu} - \ssfu\right) \frac{\delta V}{V}.
-\Tag{(95)}
-\]
-This equation gives the change of the energy density of any
-definite frequency~$\nu$, which occurs on an infinitely slow adiabatic
-compression of the radiation. It holds, moreover, not only for
-black radiation, but also for \emph{radiation originally of a perfectly
-arbitrary distribution of energy}, as is shown by the method of
-derivation.
-
-Since the changes taking place in the state of the radiation in
-the time~$\delta t$ are proportional to the infinitely small velocity~$v$ and
-are reversed on changing the sign of the latter, this equation
-holds for any sign of~$\delta V$; \emph{hence the process is reversible}.
-%% -----File: 097.png---Folio 81-------
-
-\Section{83.} Before passing on to the general integration of equation~\Eq{(95)}
-let us examine it in the manner which most easily suggests
-itself. According to the energy principle, the change in the
-radiant energy
-\[
-U = Vu = V \int_{0}^{\infty} \ssfu\, d\nu,
-\]
-occurring on adiabatic compression, must be equal to the external
-work done against the radiation pressure
-\[
--p\, \delta V = -\frac{u}{3}\, \delta V
- = -\frac{\delta V}{3} \int_{0}^{\infty} \ssfu\, d\nu.
-\Tag{(96)}
-\]
-Now from~\Eq{(94)} the change in the total energy is found to be
-\[
-\delta U = \int_{0}^{\infty} d\nu\, \delta(V\ssfu)
- = \frac{\delta V}{3} \int_{0}^{\infty} \nu \frac{\dd \ssfu}{\dd \nu}\, d\nu,
-\]
-or, by partial integration,
-\[
-\delta U = \frac{\delta V}{3} \bigl(
- [\nu \ssfu]\Strut_{0}^{\infty} - \int_{0}^{\infty} \ssfu\, d\nu
-\bigr),
-\]
-and this expression is, in fact, identical with~\Eq{(96)}, since the product~$\nu\ssfu$
-vanishes for $\nu = 0$ as well as for $\nu = \infty$. The latter might
-at first seem doubtful; but it is easily seen that, if $\nu\ssfu$ for $\nu = \infty$
-had a value different from zero, the integral of~$\ssfu$ with respect to~$\nu$
-taken from $0$ to~$\infty$ could not have a finite value, which, however,
-certainly is the case.
-
-\Section{84.} We have already emphasized (\Sec{79}) that $\ssfu$~must be
-regarded as a function of two independent variables, of which we
-have taken as the first the frequency~$\nu$ and as the second the time~$t$.
-Since, now, in equation~\Eq{(95)} the time~$t$ does not explicitly
-appear, it is more appropriate to introduce the volume~$V$, which
-depends only on~$t$, as the second variable instead of $t$~itself. Then
-equation~\Eq{(95)} may be written as a partial differential equation as
-follows:
-\[
-V \frac{\dd \ssfu}{\dd V} = \frac{\nu}{3}\, \frac{\dd \ssfu}{\dd \nu} - \ssfu.
-\Tag{(97)}
-\]
-From this equation, if, for a definite value of~$V$, $\ssfu$~is known as a
-function of~$\nu$, it may be calculated for all other values of~$V$ as a
-%% -----File: 098.png---Folio 82-------
-function of~$\nu$. The general integral of this differential equation,
-as may be readily seen by substitution, is
-\[
-\ssfu = \frac{1}{V} \phi(\nu^{3} V),
-\Tag{(98)}
-\]
-where $\phi$~denotes an arbitrary function of the single argument
-$\nu^{3} V$. Instead of this we may, on substituting $\nu^{3} V \phi(\nu^{3} V)$ for
-$\phi(\nu^{3} V)$, write
-\[
-\ssfu = \nu^{3} \phi(\nu^{3} V).
-\Tag{(99)}
-\]
-Either of the last two equations is the general expression of
-\Name{Wien's} displacement law.
-
-If for a definitely given volume~$V$ the spectral distribution of
-energy is known (\ie, $\ssfu$~as a function of~$\nu$), it is possible to deduce
-therefrom the dependence of the function~$\phi$ on its argument, and
-thence the distribution of energy for any other volume~$V'$, into
-which the radiation filling the hollow cylinder may be brought by
-a reversible adiabatic process.
-
-\Section{84a.} The characteristic feature of this new distribution of
-energy may be stated as follows: If we denote all quantities
-referring to the new state by the addition of an accent, we have
-the following equation in addition to~\Eq{(99)}
-\[
-\ssfu' = \nu'^{3} \phi(\nu'^{3} V').
-\]
-Therefore, if we put
-\[
-\nu'^{3} V' = \nu^{3} V,
-\Tag{(99a)}
-\]
-we shall also have
-\[
-\frac{\ssfu'}{\nu'^{3}} = \frac{\ssfu}{\nu^{3}} \quad\text{and}\quad
-\ssfu' V' = \ssfu V,
-\Tag{(99b)}
-\]
-\ie, if we coordinate with every frequency~$\nu$ in the original state
-that frequency~$\nu'$ which is to~$\nu$ in the inverse ratio of the cube
-roots of the respective volumes, the corresponding energy
-densities $\ssfu'$~and~$\ssfu$ will be in the inverse ratio of the volumes.
-
-The meaning of these relations will be more clearly seen, if we
-write
-\[
-\frac{V'}{\lambda'^{3}} = \frac{V}{\lambda^{3}}\Add{.}
-\]
-This is the number of the cubes of the wave lengths, which
-correspond to the frequency~$\nu$ and are contained in the volume
-%% -----File: 099.png---Folio 83-------
-of the radiation. Moreover $\ssfu\, d\nu\, V = \ssfU\, d\nu$ denotes the radiant
-energy lying between the frequencies $\nu$ and $\nu + d\nu$, which is contained
-in the volume~$V$. Now since, according to~\Eq{(99a)},
-\[
-\sqrt[3]{V'}\, d\nu' = \sqrt[3]{V}\, d\nu \quad\text{or}\quad
-\frac{d\nu'}{\nu'} = \frac{d\nu}{\nu}
-\Tag{(99c)}
-\]
-we have, taking account of~\Eq{(99b)},
-\[
-\ssfU' \frac{d\nu'}{\nu'} = \ssfU \frac{d\nu}{\nu}\Add{.}
-\]
-These results may be summarized thus: On an infinitely slow
-reversible adiabatic change in volume of radiation contained in
-a cavity and uniform in all directions, the frequencies change in
-such a way that the number of cubes of wave lengths of every
-frequency contained in the total volume remains unchanged, and
-the radiant energy of every infinitely small spectral interval
-changes in proportion to the frequency.
-\Erratum{}{These laws hold for any original distribution of energy whatever;
-hence, \eg, an originally monochromatic radiation remains
-monochromatic during the process described, its color changing
-in the way stated.}
-
-\Section{85.} Returning now to the discussion of \Sec{73} we introduce
-the assumption that at first the spectral distribution of energy is
-the normal one, corresponding to black radiation. Then, according
-to the law there proven, the radiation retains this property
-without change during a reversible adiabatic change of volume
-and the laws derived in \Sec{68} hold for the process. The radiation
-then possesses in every state a definite temperature~$T$, which
-depends on the volume~$V$ according to the equation derived in
-that paragraph,
-\[
-T^{3}V = \const. = T'^{3}V'.
-\Tag{(100)}
-\]
-Hence we may now write equation~\Eq{(99)} as follows:
-\[
-\ssfu = \nu^{3} \phi\left(\frac{\nu^{3}}{T^{3}}\right)
-\]
-or
-\[
-\ssfu = \nu^{3} \phi \left(\frac{T}{\nu}\right)\Add{.}
-\]
-Therefore, if for a single temperature the spectral distribution
-of black radiation, \ie, $\ssfu$~as a function of~$\nu$, is known, the dependence
-of the function~$\phi$ on its argument, and hence the spectral
-distribution for any other temperature, may be deduced
-therefrom.
-%% -----File: 100.png---Folio 84-------
-
-If we also take into account the law proved in \Sec{47}, that,
-for the black radiation of a definite temperature, the product~$\ssfu q^{3}$
-has for all media the same value, we may also write
-\[
-\ssfu = \frac{\nu^{3}}{c^{3}} F\left(\frac{T}{\nu}\right)
-\Tag{(101)}
-\]
-where now the function~$F$ no longer contains the velocity of
-propagation.
-
-\Section{86.} For the total radiation density in space of the black radiation
-in the vacuum we find
-\[
-u = \int_{0}^{\infty} \ssfu\, d\nu
- = \frac{1}{c^{3}} \int_{0}^{\infty} \nu^{3} F\left(\frac{T}{\nu}\right) d\nu,
-\Tag{(102)}
-\]
-or, on introducing $\dfrac{T}{\nu} = x$ as the variable of integration instead
-of~$\nu$,
-\[
-u = \frac{T^{4}}{c^{3}} \int_{0}^{\infty} \frac{F(x)}{x^{5}}\, dx.
-\Tag{(103)}
-\]
-If we let the absolute constant
-\[
-\frac{1}{c^{3}} \int_{0}^{\infty} \frac{F(x)}{x^{5}}\, dx = a
-\Tag{(104)}
-\]
-the equation reduces to the form of the \Name{Stefan-Boltzmann} law of
-radiation expressed in equation~\Eq{(75)}.
-
-\Section{87.} If we combine equation~\Eq{(100)} with equation~\Eq{(99a)} we
-obtain
-\[
-\frac{\nu'}{T'} = \frac{\nu}{T}\Add{.}
-\Tag{(105)}
-\]
-
-Hence the laws derived at the end of \Sec{84a} assume the following
-form: On infinitely slow reversible adiabatic change in
-volume of black radiation contained in a cavity, the temperature~$T$
-varies in the inverse ratio of the cube root of the volume~$V$,
-the frequencies~$\nu$ vary in proportion to the temperature, and
-the radiant energy~$\ssfU\, d\nu$ of an infinitely small spectral interval
-varies in the same ratio. Hence the total radiant energy~$U$ as
-the sum of the energies of all spectral intervals varies also in
-proportion to the temperature, a statement which agrees with the
-%% -----File: 101.png---Folio 85-------
-conclusion arrived at already at the end of \Sec{68}, while the
-space density of radiation, $u = \dfrac{U}{V}$, varies in proportion to the
-fourth power of the temperature, in agreement with the \Name{Stefan-Boltzmann}
-law.
-
-\Section{88.} \Name{Wien's} displacement law may also in the case of black
-radiation be stated for the specific intensity of radiation~$\ssfK_{\nu}$ of
-a plane polarized monochromatic ray. In this form it reads
-according to~\Eq{(24)}
-\[
-\ssfK_{\nu} = \frac{\nu^{3}}{\DPtypo{c}{c^{2}}} F\left(\frac{T}{\nu}\right)\Add{.}
-\Tag{(106)}
-\]
-If, as is usually done in experimental physics, the radiation intensity
-is referred to wave lengths~$\lambda$ instead of frequencies~$\nu$, according
-to~\Eq{(16)}, namely
-\[
-\ssfE_{\lambda} = \frac{c \ssfK_{\nu}}{\lambda^{2}}\Add{,}
-\]
-equation~\Eq{(106)} takes the following form:
-\[
-E_{\lambda} = \frac{c^{2}}{\lambda^{5}} F\left(\frac{\lambda T}{c}\right).
-\Tag{(107)}
-\]
-This form of \Name{Wien's} displacement law has usually been the starting-point
-for an experimental test, the result of which has in all
-cases been a fairly accurate verification of the law.\footnote
- {\Eg, \Name{F.~Paschen}, Sitzungsber.\ d.\ Akad.\ d.\ Wissensch.\ Berlin, pp.\ 405 and 959, 1899.
- \Name{O.~Lummer} und \Name{E.~Pringsheim}, Verhandlungen d.\ Deutschen physikalischen Gesellschaft~\textbf{1},
- pp.\ 23 and 215, 1899. Annal.\ d.\ Physik~\textbf{6}, p.~192, 1901.}
-
-\Section{89.} Since $E_{\lambda}$~vanishes for $\lambda = 0$ as well as for $\lambda = \infty$, $E_{\lambda}$~must
-have a maximum with respect to~$\lambda$, which is found from the
-equation
-\[
-\frac{dE_{\lambda}}{d\lambda} = 0
- = -\frac{5}{\lambda^{6}} F\left(\frac{\lambda T}{c}\right)
- + \frac{1}{\lambda^{5}}\, \frac{T}{c} \dot{F}\left(\frac{\lambda T}{c}\right)
-\]
-where $\dot{F}$~denotes the differential coefficient of~$F$ with respect to
-its argument. Or
-\[
-\frac{\lambda T}{c} \dot{F}\left(\frac{\lambda T}{c}\right)
- - 5F\left(\frac{\lambda T}{c}\right) = 0.
-\Tag{(108)}
-\]
-This equation furnishes a definite value for the argument $\dfrac{\lambda T}{c}$, so
-%% -----File: 102.png---Folio 86-------
-that for the wave length~$\lambda_{m}$ corresponding to the maximum of the
-radiation intensity~$E_{\lambda}$ the relation holds
-\[
-\lambda_{m}T = b.
-\Tag{(109)}
-\]
-With increasing temperature the maximum of radiation is
-therefore displaced in the direction of the shorter wave lengths.
-
-The numerical value of the constant~$b$ as determined by
-\Name{Lummer} and \Name{Pringsheim}\footnote
- {\Name{O.~Lummer} und \Name{E.~Pringsheim}, \lc}
-is
-\[
-b = 0.294\, \cm\DPchg{.}{}\, \degree.
-\Tag{(110)}
-\]
-
-\Name{Paschen}\footnote
- {\Name{F.~Paschen}, Annal.\ d.\ Physik\DPchg{,}{} \textbf{6}, p.~657, 1901.}
-has found a slightly smaller value, about $0.292$.
-
-We may emphasize again at this point that, according to
-\Sec{19}, the maximum of~$E_{\lambda}$ does not by any means occur at the
-same point in the spectrum as the maximum of~$\ssfK_{\nu}$ and that hence
-the significance of the constant~$b$ is essentially dependent on the
-fact that the intensity of monochromatic radiation is referred to
-wave lengths, not to frequencies.
-
-\Section{90.} The value also of the maximum of~$E_{\lambda}$ is found from~\Eq{(107)}
-by putting $\lambda = \lambda_{m}$. Allowing for~\Eq{(109)} we obtain
-\[
-E_{\max} = \const.\; T^{5},
-\Tag{(111)}
-\]
-\ie, the value of the maximum of radiation in the spectrum of the
-black radiation is proportional to the fifth power of the absolute
-temperature.
-
-Should we measure the intensity of monochromatic radiation
-not by~$E_{\lambda}$ but by~$\ssfK_{\nu}$, we would obtain for the value of the radiation
-maximum a quite different law, namely,
-\[
-\ssfK_{max} = \const.\;T^{3}.
-\Tag{(112)}
-\]
-%% -----File: 103.png---Folio 87-------
-
-\Chapter[Spectral Distribution of Energy]
-{IV}{Radiation of Any Arbitrary Spectral Distribution
-of Energy. Entropy and Temperature
-of Monochromatic Radiation}
-
-\Section{91.} We have so far applied \Name{Wien's} displacement law only to
-the case of black radiation; it has, however, a much more general
-importance. For equation~\Eq{(95)}, as has already been stated, gives,
-for any original spectral distribution of the energy radiation contained
-in the evacuated cavity and radiated uniformly in all directions,
-the change of this energy distribution accompanying a
-reversible adiabatic change of the total volume. Every state of
-radiation brought about by such a process is perfectly stationary
-and can continue infinitely long, subject, however, to the condition
-that no trace of an emitting or absorbing substance exists
-in the radiation space. For otherwise, according to \Sec{51}, the
-distribution of energy would, in the course of time, change
-through the releasing action of the substance irreversibly, \ie,
-with an increase of the total entropy, into the stable distribution
-\DPtypo{correponding}{corresponding} to black radiation.
-
-The difference of this general case from the special one dealt
-with in the preceding chapter is that we can no longer, as in the
-case of black radiation, speak of a definite temperature of the
-radiation. Nevertheless, since the second principle of thermodynamics
-is supposed to hold quite generally, the radiation, like
-every physical system which is in a definite state, has a definite
-entropy, $S = Vs$. This entropy consists of the entropies of the
-monochromatic radiations, and, since the separate kinds of rays
-are independent of one another, may be obtained by addition.
-Hence
-\[
-s = \int_{0}^{\infty} \ssfs\, d\nu,\qquad
-S = V \int_{0}^{\infty} \ssfs\, d\nu,
-\Tag{(113)}
-\]
-where $\ssfs\, d\nu$ denotes the entropy of the radiation of frequencies
-between $\nu$ and $\nu + d\nu$ contained in unit volume. $\ssfs$~is a definite
-%% -----File: 104.png---Folio 88-------
-function of the two independent variables $\nu$~and~$\ssfu$ and in the
-following will always be treated as such.
-
-\Section{92.} If the analytical expression of the function~$\ssfs$ were known,
-the law of energy distribution in the normal spectrum could
-immediately be deduced from it; for the normal spectral distribution
-of energy or that of black radiation is distinguished from
-all others by the fact that it has the maximum of the entropy of
-radiation~$S$.
-
-Suppose then we take $\ssfs$ to be a known function of $\nu$~and~$\ssfu$.
-Then as a condition for black radiation we have
-\[
-\delta S = 0,
-\Tag{(114)}
-\]
-for any variations of energy distribution, which are possible
-with a constant total volume~$V$ and constant total energy of
-radiation~$U$. Let the variation of energy distribution be characterized
-by making an infinitely small change~$\delta \ssfu$ in the energy~$\ssfu$
-of every separate definite frequency~$\nu$. Then we have as fixed
-conditions
-\[
-\delta V = 0 \quad\text{and}\quad
-\int_{0}^{\infty} \delta \ssfu\, d\nu = 0.
-\Tag{(115)}
-\]
-The changes $d$~and~$\delta$ are of course quite independent of each
-other.
-
-Now since $\delta V = 0$, we have from \Eq{(114)}~and~\Eq{(113)}
-\[
-\int_{0}^{\infty} \delta \ssfs\, d\nu = 0,
-\]
-or, since $\nu$~remains unvaried
-\[
-\int_{0}^{\infty} \frac{\dd \ssfs}{\dd \ssfu}\, \delta \ssfu\, d\nu = 0,
-\]
-and, by allowing for~\Eq{(115)}, the validity of this equation for all
-values of $\delta \ssfu$ whatever requires that
-\[
-\frac{\dd \ssfs}{\dd \ssfu} = \const.
-\Tag{(116)}
-\]
-for all different frequencies. This equation states the law of
-energy distribution in the case of black radiation.
-
-\Section{93.} The constant of equation~\Eq{(116)} bears a simple relation to
-the temperature of black radiation. For if the black radiation,
-%% -----File: 105.png---Folio 89-------
-by conduction into it of a certain amount of heat at constant volume~$V$,
-undergoes an infinitely small change in energy~$\delta U$, then,
-according to~\Eq{(73)}, its change in entropy is
-\[
-\delta S = \frac{\delta U}{T}.
-\]
-However, from \Eq{(113)}~and~\Eq{(116)},
-\[
-\delta S = V \int_{0}^{\infty} \frac{\dd \ssfs}{\dd \ssfu}\, \delta \ssfu\, d\nu
- = \frac{\dd \ssfs}{\dd \ssfu} V \int_{0}^{\infty} \delta\ssfu\, d\nu
- = \frac{\dd\ssfs}{\dd\ssfu}\, \delta U
-\]
-hence
-\[
-\frac{\dd\ssfs}{\dd\ssfu} = \frac{1}{T}
-\Tag{(117)}
-\]
-and the above quantity, which was found to be the same for all
-frequencies in the case of black radiation, is shown to be the reciprocal
-of the temperature of black radiation.
-
-Through this law the concept of temperature gains significance
-also for radiation of a quite arbitrary distribution of
-%[** Thm]
-energy. For since $\ssfs$~depends only on $\ssfu$~and~$\nu$, \emph{monochromatic
-radiation, which is uniform in all directions and has a definite
-energy density~$\ssfu$, has also a definite temperature given by~\Eq{(117)},
-and, among all conceivable distributions of energy, the normal one
-is characterized by the fact that the radiations of all frequencies
-have the same temperature}.
-
-Any change in the energy distribution consists of a passage of
-energy from one monochromatic radiation into another, and, if
-the temperature of the first radiation is higher, the energy
-transformation causes an increase of the total entropy and is
-hence possible in nature without compensation; on the other hand,
-if the temperature of the second radiation is higher, the total
-entropy decreases and therefore the change is impossible in nature,
-unless compensation occurs simultaneously, just as is the case
-with the transfer of heat between two bodies of different temperatures.
-
-\Section{94.} Let us now investigate \Name{Wien's} displacement law with regard
-to the dependence of the quantity~$\ssfs$ on the variables $\ssfu$~and~$\nu$.
-%% -----File: 106.png---Folio 90-------
-From equation~\Eq{(101)} it follows, on solving for~$T$ and substituting
-the value given in~\Eq{(117)}, that
-\[
-\frac{1}{T}
- = \frac{1}{\nu} F\left(\frac{c^{3} \ssfu}{\nu^{3}}\right)
- = \frac{\dd \ssfs}{\dd \ssfu}
-\Tag{(118)}
-\]
-where again $F$~represents a function of a single argument and the
-constants do not contain the velocity of propagation~$c$. On
-integration with respect to the argument we obtain
-\[
-\ssfs = \frac{\nu^{2}}{c^{3}} F_{1}\left(\frac{c^{3} \ssfu}{\nu^{3}}\right)
-\Tag{(119)}
-\]
-the notation remaining the same. In this form \Name{Wien's} displacement
-law has a significance for every separate monochromatic
-radiation and hence also for radiations of any arbitrary energy
-distribution.
-
-\Section{95.} According to the second principle of thermodynamics, the
-total entropy of radiation of quite arbitrary distribution of
-energy must remain constant on adiabatic reversible compression.
-We are now able to give a direct proof of this proposition on the
-basis of equation~\Eq{(119)}. For such a process, according to
-equation~\Eq{(113)}, the relation holds:
-\begin{align*}
-\delta S &= \int_{0}^{\infty} d\nu\, (V\, \delta \ssfs + \ssfs\, \delta V) \\
- &= \int_{0}^{\infty} d\nu\, (V \frac{\dd \ssfs}{\dd \ssfu}\, \delta \ssfu + \ssfs\, \delta V).
-\Tag{(120)}
-\end{align*}
-Here, as everywhere, $\ssfs$~should be regarded as a function of $\ssfu$~and~$\nu$,
-and $\delta \nu = 0$.
-
-Now for a reversible adiabatic change of state the relation~\Eq{(95)}
-holds. Let us take from the latter the value of~$\delta \ssfu$ and substitute.
-Then we have
-\[
-\delta S = \delta V \int_{0}^{\infty} d\nu \left\{\frac{\dd \ssfs}{\dd \ssfu} \left(\frac{\nu\, d\ssfu}{3\, d\nu} - \ssfu\right) + \ssfs\right\}.
-\]
-In this equation the differential coefficient of~$\ssfu$ with respect to~$\nu$
-refers to the spectral distribution of energy originally assigned
-arbitrarily and is therefore, in contrast to the partial differential
-coefficients, denoted by the letter~$d$.
-%% -----File: 107.png---Folio 91-------
-
-Now the complete differential is:
-\[
-\frac{d\ssfs}{d\nu} = \frac{\dd \ssfs}{\dd \ssfu}\, \frac{d\ssfu}{d\nu} + \frac{\dd \ssfs}{\dd \nu}\Add{.}
-\]
-Hence by substitution:
-\[
-\delta S = \delta V \int_{0}^{\infty} d\nu \left\{\frac{\nu}{3} \left(\frac{d\ssfs}{d\nu} -\frac{\dd \ssfs}{\dd \nu}\right) - \ssfu \frac{\dd \ssfs}{\dd \ssfu} + \ssfs\right\}.
-\Tag{(121)}
-\]
-But from equation~\Eq{(119)} we obtain by differentiation
-\[
-\frac{\dd \ssfs}{\dd \ssfu} = \frac{1}{\nu} \dot{F}\left(\frac{c^3\ssfu}{\nu^3}\right)
-\quad\text{and}\quad
-\frac{\dd \ssfs}{\dd \nu}
- = \frac{2\nu}{c^{3}} F\left(\frac{c^{3} \ssfu}{\nu^{3}}\right)
- - \frac{3\ssfu}{\nu^{2}} \dot{F}\left(\frac{c^{3} \ssfu}{\nu^{3}}\right)\Add{.}
-\Tag{(122)}
-\]
-Hence
-\[
-\DPchg{\frac{\nu\, \dd \ssfs}{\dd \nu}}{\nu \frac{\dd \ssfs}{\dd \nu}}
- = 2\ssfs - 3\ssfu \frac{\dd \ssfs}{\dd \ssfu}\Add{.}
-\Tag{(123)}
-\]
-On substituting this in~\Eq{(121)}, we obtain
-\[
-\delta S = \delta V \int_{0}^{\infty} d\nu \left(\frac{\nu}{3}\, \frac{d\ssfs}{d\nu} + \frac{1}{3} \ssfs\right)
-\Tag{(124)}
-\]
-or,
-\[
-\delta S = \frac{\delta V}{3} [\nu\ssfs]\Strut_{0}^{\infty} = 0,
-\]
-as it should be. That the product~$\nu\ssfs$ vanishes also for $\nu = \infty$
-may be shown just as was done in \Sec{83} for the product~$\nu\ssfu$.
-
-\Section{96.} By means of equations \Eq{(118)}~and~\Eq{(119)} it is possible to give
-to the laws of reversible adiabatic compression a form in which
-their meaning is more clearly seen and which is the generalization
-of the laws stated in \Sec{87} for black radiation and a supplement
-to them. It is, namely, possible to derive~\Eq{(105)} again from \Eq{(118)}~and~\Eq{(99b)}.
-Hence the laws deduced in \Sec{87} for the change of
-frequency and temperature of the monochromatic radiation
-energy remain valid for a radiation of an originally quite arbitrary
-distribution of energy. The only difference as compared with
-the black radiation consists in the fact that now every frequency
-has its own distinct temperature.
-
-Moreover it follows from \Eq{(119)}~and~\Eq{(99b)} that
-\[
-\frac{\ssfs'}{\nu'^{2}} = \frac{\ssfs}{\nu^{2}}\Add{.}
-\Tag{(125)}
-\]
-%% -----File: 108.png---Folio 92-------
-Now $\ssfs\, d\nu\, V = \ssfS\, d\nu$ denotes the radiation entropy between the
-frequencies $\nu$ and $\nu + d\nu$ contained in the volume~$V$. Hence on
-account of \Eq{(125)},~\Eq{(99a)}, and~\Eq{(99c)}
-\[
-\ssfS'\, d\nu' = \ssfS\, d\nu,
-\Tag{(126)}
-\]
-\ie, the radiation entropy of an infinitely small spectral interval
-remains constant. This is another statement of the fact that the
-total entropy of radiation, taken as the sum of the entropies of all
-monochromatic radiations contained therein, remains constant.
-
-\Section{97.} We may go one step further, and, from the entropy~$\ssfs$
-and the temperature~$T$ of an unpolarized monochromatic radiation
-which is uniform in all directions, draw a certain conclusion
-regarding the entropy and temperature of a single, plane polarized,
-monochromatic pencil. That every separate pencil also has
-a certain entropy follows by the second principle of thermodynamics
-from the phenomenon of emission. For since, by the
-act of emission, heat is changed into radiant heat, the entropy
-of the emitting body decreases during emission, and, along with
-this decrease, there must be, according to the principle of increase
-of the total entropy, an increase in a different form of entropy as
-a compensation. This can only be due to the energy of the
-emitted radiation. Hence every separate, plane polarized, monochromatic
-pencil has its definite entropy, which can depend only
-on its energy and frequency and which is propagated and
-spreads into space with it. We thus gain the idea of entropy
-radiation, which is measured, as in the analogous case of energy
-radiation, by the amount of entropy which passes in unit time
-through unit area in a definite direction. Hence statements,
-exactly similar to those made in \Sec{14} regarding energy radiation,
-will hold for the radiation of entropy, inasmuch as every
-pencil possesses and conveys, not only its energy, but also its
-entropy. Referring the reader to the discussions of \Sec{14},
-we shall, for the present, merely enumerate the most important
-laws for future use.
-
-\Section{98.} In a space filled with any radiation whatever the entropy
-radiated in the time~$dt$ through an element of area~$d\sigma$ in the
-direction of the conical element~$d\Omega$ is given by an expression of
-the form
-\[
-dt\, d\sigma \cos\theta\, d\Omega\, L = L \sin\theta \cos\theta\, d\theta\, d\phi\, d\sigma\, dt.
-\Tag{(127)}
-\]
-%% -----File: 109.png---Folio 93-------
-The positive quantity~$L$ we shall call the ``specific intensity of
-entropy radiation'' at the position of the element of area~$d\sigma$
-in the direction of the solid angle~$d\Omega$. $L$~is, in general, a function
-of position, time, and direction.
-
-The total radiation of entropy through the element of area~$d\sigma$
-toward one side, say the one where $\theta$~is an acute angle, is obtained
-by integration with respect to~$\phi$ from $0$ to~$2\pi$ and with
-respect to~$\theta$ from $0$ to~$\dfrac{\pi}{2}$. It is
-\[
-d\sigma\, dt \int_{0}^{2\pi} d\phi \int_{0}^{\frac{\pi}{2}} d\theta\, L \sin\theta \cos\theta.
-\]
-When the radiation is uniform in all directions, and hence $L$~constant,
-the entropy radiation through~$d\sigma$ toward one side is
-\[
-\pi L\, d\sigma\, dt.
-\Tag{(128)}
-\]
-
-The specific intensity~$L$ of the entropy radiation in every direction
-consists further of the intensities of the separate rays belonging
-to the different regions of the spectrum, which are propagated
-independently of one another. Finally for a ray of definite color
-and intensity the nature of its polarization is characteristic.
-When a monochromatic ray of frequency~$\nu$ consists of two
-mutually independent\footnote
- {``Independent'' in the sense of ``\DPchg{noncoherent}{non-coherent}.'' If, \eg, a ray with the principal intensities
- $\ssfK$~and~$\ssfK'$ is elliptically polarized, its entropy is not equal to~$\ssfL + \ssfL'$, but equal to the
- entropy of a plane polarized ray of intensity $\ssfK + \ssfK'$. For an elliptically polarized ray may
- be transformed at once into a plane polarized one, \eg, by total reflection. For the entropy
- of a ray with coherent components see below \Sec{104}, \textit{et~seq.}}
-components, polarized at right angles to
-each other, with the principal intensities of energy radiation
-(\Sec{17}) $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$, the specific intensity of entropy radiation
-is of the form
-\[
-L = \int_{0}^{\infty} d\nu\, (\ssfL_{\nu} + \ssfL_{\nu}').
-\Tag{(129)}
-\]
-
-The positive quantities $L_{\nu}$~and~$L_{\nu}'$ in this expression, the
-principal intensities of entropy radiation of frequency~$\nu$, are
-determined by the values of $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$. By substitution in~\Eq{(127)},
-this gives for the entropy which is radiated in the time
-%% -----File: 110.png---Folio 94-------
-%[** tie]
-$dt$ through the element of area~$d\sigma$ in the direction of the conical
-element~$d\Omega$ the expression
-\[
-dt\, d\sigma \cos\theta\, d\Omega \int_{0}^{\infty} d\nu\, (\ssfL_{\nu} + \ssfL_{\nu}'),
-\]
-and, for monochromatic plane polarized radiation,
-\[
-dt\, d\sigma \cos\theta\, d\Omega\, \ssfL_{\nu}\, d\nu
- = \ssfL_{\nu}\, d\nu\, \sin\theta \cos\theta\, d\theta\, d\phi\, d\sigma\, dt.
-\Tag{(130)}
-\]
-For unpolarized rays $\ssfL_{\nu} = \ssfL_{\nu}'$ and \Eq{(129)}~becomes
-\[
-L = 2 \int_{0}^{\infty} \ssfL_{\nu}\, d\nu.
-\]
-For radiation which is uniform in all directions the total entropy
-radiation toward one side is, according to~\Eq{(128)},
-\[
-2\pi\, d\sigma\, dt \int_{0}^{\infty} \ssfL_{\nu}\, d\nu.
-\]
-
-\Section{99.} From the intensity of the propagated entropy radiation
-the expression for the \emph{space density} of the radiant entropy may also
-be obtained, just as the space density of the radiant energy
-follows from the intensity of the propagated radiant energy.
-(Compare \Sec{22}.) In fact, in analogy with equation~\Eq{(20)}, the
-space density,~$s$, of the entropy of radiation at any point in a
-vacuum is
-\[
-s = \frac{1}{c} \int L\, d\Omega,
-\Tag{(131)}
-\]
-where the integration is to be extended over the conical elements
-which spread out from the point in question in all directions.
-$L$~is constant for uniform radiation and we obtain
-\[
-s = \frac{4\pi L}{c}\Add{.}
-\Tag{(132)}
-\]
-By spectral resolution of the quantity~$L$, according to equation~\Eq{(129)},
-we obtain from~\Eq{(131)} also the space density of the monochromatic
-radiation entropy:
-\[
-\ssfs = \frac{1}{c} \int (\ssfL + \ssfL')\, d\Omega,
-\]
-and for unpolarized radiation, which is uniform in all directions
-\[
-\ssfs = \frac{8\pi \ssfL}{c}\Add{.}
-\Tag{(133)}
-\]
-%% -----File: 111.png---Folio 95-------
-
-\Section{100.} As to how the entropy radiation~$\ssfL$ depends on the energy
-radiation~$\ssfK$ \Name{Wien's} displacement law in the form of~\Eq{(119)} affords
-immediate information. It follows, namely, from it, considering
-\Eq{(133)}~and~\Eq{(24)}, that
-\[
-\ssfL = \frac{\nu^{2}}{c^{2}} F\left(\frac{c^{2}\ssfK}{\nu^{3}}\right)
-\Tag{(134)}
-\]
-and, moreover, on taking into account~\Eq{(118)},
-\[
-\frac{\dd \ssfL}{\dd \ssfK} = \frac{\dd \ssfs}{\dd \ssfu} = \frac{1}{T}\Add{.}
-\Tag{(135)}
-\]
-Hence also
-\[
-T = \nu F_{1}\left(\frac{c^{2} \ssfK}{\nu^{3}}\right)
-\Tag{(136)}
-\]
-or
-\[
-\ssfK = \frac{\nu^{3}}{c^{2}} F_{2}\left(\frac{T}{\nu}\right)\Add{.}
-\Tag{(137)}
-\]
-
-It is true that these relations, like the equations \Eq{(118)} and~\Eq{(119)},
-were originally derived for radiation which is unpolarized
-and uniform in all directions. They hold, however, generally in
-the case of any radiation whatever for each separate monochromatic
-plane polarized ray. For, since the separate rays behave
-and are propagated quite independently of one another, the intensity,~$\ssfL$,
-of the entropy radiation of a ray can depend only on the
-intensity of the energy radiation,~$\ssfK$, of the same ray. Hence
-every separate monochromatic ray has not only its energy but
-also its entropy defined by~\Eq{(134)} and its temperature defined by~\Eq{(136)}.
-
-\Section{101.} The extension of the conception of temperature to a
-single monochromatic ray, just discussed, implies that at the
-same point in a medium, through which any rays whatever pass,
-there exist in general an infinite number of temperatures, since
-every ray passing through the point has its separate temperature,
-and, moreover, even the rays of different color traveling in the
-same direction show temperatures that differ according to the
-spectral distribution of energy. In addition to all these temperatures
-there is finally the temperature of the medium itself, which
-at the outset is entirely independent of the temperature of the
-radiation. This complicated method of consideration lies in the
-%% -----File: 112.png---Folio 96-------
-nature of the case and corresponds to the complexity of the
-physical processes in a medium through which radiation travels
-in such a way. It is only in the case of stable thermodynamic
-equilibrium that there is but one temperature, which then is
-common to the medium itself and to all rays of whatever color
-crossing it in different directions.
-
-In practical physics also the necessity of separating the conception
-of radiation temperature from that of body temperature
-has made itself felt to a continually increasing degree. Thus it
-has for some time past been found advantageous to speak, not
-only of the real temperature of the sun, but also of an ``apparent''
-or ``effective'' temperature of the sun, \ie, that temperature
-which the sun would need to have in order to send to the earth
-the heat radiation actually observed, if it radiated like a black
-body. Now the apparent temperature of the sun is obviously
-nothing but the actual temperature of the solar rays,\footnote
- {On the average, since the solar rays of different color do not have exactly the same
- temperature.}
-depending
-entirely on the nature of the rays, and hence a property of the
-rays and not a property of the sun itself. Therefore it would be,
-not only more convenient, but also more correct, to apply this
-notation directly, instead of speaking of a fictitious temperature
-of the sun, which can be made to have a meaning only by the
-introduction of an assumption that does not hold in reality.
-
-Measurements of the brightness of monochromatic light have
-recently led \Name{L.~Holborn} and \Name{F.~Kurlbaum}\footnote
- {\Name{L.~Holborn} und \Name{F.~Kurlbaum}, Annal.\ d.\ Physik\DPtypo{.}{}\DPchg{,}{} \textbf{10}, p.~229, 1903.}
-to the introduction of
-the concept of ``black'' temperature of a radiating surface. The
-black temperature of a radiating surface is measured by the
-brightness of the rays which it emits. It is in general a separate
-one for each ray of definite color, direction, and polarization,
-which the surface emits, and, in fact, merely represents the
-temperature of such a ray. It is, according to equation~\Eq{(136)},
-determined by its brightness (specific intensity),~$\ssfK$, and its
-frequency,~$\nu$, without any reference to its origin and previous
-states. The definite numerical form of this equation will be
-given below in \Sec{166}. Since a black body has the maximum
-emissive power, the temperature of an emitted ray can never be
-higher than that of the emitting body.
-%% -----File: 113.png---Folio 97-------
-
-\Section{102.} Let us make one more simple application of the laws just
-found to the special case of black radiation. For this, according
-to~\Eq{(81)}, the total space density of entropy is
-\[
-s = \frac{4}{3} a^{3} T.
-\Tag{(138)}
-\]
-Hence, according to~\Eq{(132)}, the specific intensity of the total
-entropy radiation in any direction is
-\[
-L = \frac{c}{3\pi} a T^{3},
-\Tag{(139)}
-\]
-and the total entropy radiation through an element of area~$d\sigma$
-toward one side is, according to~\Eq{(128)},
-\[
-\frac{c}{3} aT^{3}\, d\sigma\, dt.
-\Tag{(140)}
-\]
-As a special example we shall now apply the two principles of
-thermodynamics to the case in which the surface of a black body
-of temperature~$T$ and of infinitely large heat capacity is struck
-by black radiation of temperature~$T'$ coming from all directions.
-Then, according to \Eq{(7)}~and~\Eq{(76)}, the black body emits per unit
-area and unit time the energy
-\[
-\pi K = \frac{ac}{4} T^{4},
-\]
-and, according to~\Eq{(140)}, the entropy
-\[
-\frac{ac}{3} T^{3}.
-\]
-On the other hand, it absorbs the energy
-\[
-\frac{ac}{4} T'^{4}
-\]
-and the entropy
-\[
-\frac{ac}{3} T'^{3}.
-\]
-Hence, according to the first principle, the total heat added to the
-body, positive or negative according as $T'$~is larger or smaller
-than~$T$, is
-\[
-Q = \frac{ac}{4} T'^{4} - \frac{ac}{4} T^{4} = \frac{ac}{4} (T'^{4} - T^{4}),
-\]
-%% -----File: 114.png---Folio 98-------
-and, according to the second principle, the change of the entire
-entropy is positive or zero. Now the entropy of the body changes
-by~$\dfrac{Q}{T}$, the entropy of the radiation in the vacuum by
-\[
-\frac{ac}{3} (T^{3} - T'^{3}).
-\]
-Hence the change per unit time and unit area of the entire entropy
-of the system considered is
-\[
-\frac{ac}{4}\, \frac{T'^{4} - T^{4}}{T} + \frac{ac}{3} (T^{3} - T'^{3}) \geq 0.
-\]
-In fact this relation is satisfied for all values of $T$~and~$T'$. The
-minimum value of the expression on the left side is zero; this value
-is reached when $T = T'$. In that case the process is reversible.
-If, however, $T$~differs from~$T'$, we have an appreciable increase
-of entropy; hence the process is irreversible. In particular we
-find that if $T = 0$ the increase in entropy is~$\infty$, \ie,~the absorption
-of heat radiation by a black body of vanishingly small temperature
-is accompanied by an infinite increase in entropy and
-cannot therefore be reversed by any finite compensation. On the
-other hand for $T' = 0$, the increase in entropy is only equal to~$\dfrac{ac}{12} T^{3}$,
-\ie,~the emission of a black body of temperature~$T$ without
-simultaneous absorption of heat radiation is irreversible without
-compensation, but can be reversed by a compensation of at least
-the stated finite amount. For example, if we let the rays emitted
-by the body fall back on it, say by suitable reflection, the body,
-while again absorbing these rays, will necessarily be at the same
-time emitting new rays, and this is the compensation required by
-the second principle.
-
-Generally we may say: Emission without simultaneous absorption
-is irreversible, while the opposite process, absorption without
-emission, is impossible in nature.
-
-\Section{103.} A further example of the application of the two principles
-of thermodynamics is afforded by the irreversible expansion of
-originally black radiation of volume~$V$ and temperature~$T$ to
-the larger volume~$V'$ as considered above in \Sec{70}, but in the
-absence of any absorbing or emitting substance whatever. Then
-%% -----File: 115.png---Folio 99-------
-not only the total energy but also the energy of every separate
-frequency~$\nu$ remains constant; hence, when on account of diffuse
-reflection from the walls the radiation has again become uniform
-in all directions, $\ssfu_{\nu}V = \ssfu_{\nu}'V'$; moreover by this relation, according
-to~\Eq{(118)}, the temperature~$T_{\nu}$' of the monochromatic radiation of
-frequency~$\nu$ in the final state is determined. The actual calculation,
-however, can be performed only with the help of equation~\Eq{(275)}
-(see below). The total entropy of radiation, \ie,~the sum
-of the entropies of the radiations of all frequencies,
-\[
-V'\int_{0}^{\infty} \ssfs_{\nu}'\, d\nu,
-\]
-must, according to the second principle, be larger in the final state
-than in the original state. Since $T_{\nu}'$~has different values for the
-different frequencies~$\nu$, the final radiation is no longer black.
-Hence, on subsequent introduction of a carbon particle into the
-cavity, a finite change of the distribution of energy is obtained,
-and simultaneously the entropy increases further to the value~$S'$
-calculated in~\Eq{(82)}.
-
-\Section{104.} In \Sec{98} we have found the intensity of entropy radiation
-of a definite frequency in a definite direction by adding the
-entropy radiations of the two independent components $\ssfK$~and~$\ssfK'$,
-polarized at right angles to each other, or
-\[
-\ssfL(\ssfK) + \ssfL(\ssfK'),
-\Tag{(141)}
-\]
-where $\ssfL$~denotes the function of~$\ssfK$ given in equation~\Eq{(134)}.
-This method of procedure is based on the general law that the
-entropy of two mutually independent physical systems is equal
-to the sum of the entropies of the separate systems.
-
-If, however, the two components of a ray, polarized at right
-angles to each other, are not independent of each other, this
-method of procedure no longer remains correct. This may be
-seen, \eg, on resolving the radiation intensity, not with reference
-to the two principal planes of polarization with the principal
-intensities $\ssfK$~and~$\ssfK'$, but with reference to any other two planes
-at right angles to each other, where, according to equation~\Eq{(8)},
-the intensities of the two components assume the following
-values
-\[
-\begin{alignedat}{3}
-&\ssfK \cos^{2} \psi &&+ \ssfK' \sin^{2} \psi &&= \ssfK'' \\
-&\ssfK \sin^{2} \psi &&+ \ssfK' \cos^{2} \psi &&= \ssfK'''.
-\end{alignedat}
-\Tag{(142)}
-\]
-%% -----File: 116.png---Folio 100-------
-In that case, of course, the entropy radiation is not equal to
-$\ssfL(\ssfK'') + \ssfL(\ssfK''')$.
-
-Thus, while the energy radiation is always obtained by the
-summation of any two components which are polarized at right
-angles to each other, no matter according to which azimuth the
-resolution is performed, since always
-\[
-\ssfK'' + \ssfK''' = \ssfK + \ssfK',
-\Tag{(143)}
-\]
-a corresponding equation does not hold in general for the entropy
-radiation. The cause of this is that the two components, the
-intensities of which we have denoted by $\ssfK''$~and~$\ssfK'''$, are, unlike
-$\ssfK$~and~$\ssfK'$, not independent or \DPchg{noncoherent}{non-coherent} in the optic sense.
-In such a case
-\[
-\ssfL(\ssfK'') + \ssfL(\ssfK''') > \ssfL(\ssfK) + \ssfL(\ssfK'),
-\Tag{(144)}
-\]
-as is shown by the following consideration.
-
-Since in the state of thermodynamic equilibrium all rays of
-the same frequency have the same intensity of radiation, the
-intensities of radiation of any two plane polarized rays will tend
-to become equal, \ie, the passage of energy between them will
-be accompanied by an increase of entropy, when it takes place
-in the direction from the ray of greater intensity toward that of
-smaller intensity. Now the left side of the inequality~\Eq{(144)}
-represents the entropy radiation of two \DPchg{noncoherent}{non-coherent} plane polarized
-rays with the intensities $\ssfK''$ and $\ssfK'''$, and the right side the
-entropy radiation of two \DPchg{noncoherent}{non-coherent} plane polarized rays with the
-intensities $\ssfK$ and~$\ssfK'$. But, according to~\Eq{(142)}, the values of $\ssfK''$
-and $\ssfK'''$ lie between $\ssfK$ and~$\ssfK'$; therefore the inequality~\Eq{(144)}
-holds.
-
-At the same time it is apparent that the error committed, when
-the entropy of two coherent rays is calculated as if they were
-\DPchg{noncoherent}{non-coherent}, is always in such a sense that the entropy found is
-too large. The radiations $\ssfK''$ and $\ssfK'''$ are called ``partially
-coherent,'' since they have some terms in common. In the
-special case when one of the two principal intensities $\ssfK$ and $\ssfK'$
-vanishes entirely, the radiations $\ssfK''$ and $\ssfK'''$ are said to be
-``completely coherent,'' since in that case the expression for one
-radiation may be completely reduced to that for the other. The
-entropy of two completely coherent plane polarized rays is equal
-%% -----File: 117.png---Folio 101-------
-to the entropy of a single plane polarized ray, the energy of which
-is equal to the sum of the two separate energies.
-
-\Section{105.} Let us for future use solve also the more general problem
-of calculating the entropy radiation of a ray consisting of an
-arbitrary number of plane polarized \DPchg{noncoherent}{non-coherent} components
-$\ssfK_{1}$,~$\ssfK_{2}$, $\ssfK_{3},~\dots$, the planes of vibration (planes of
-the electric vector) of which are given by the azimuths $\psi_{1}$,~$\psi_{2}$,
-$\psi_{3},~\dots$. This problem amounts to finding the principal
-intensities $\ssfK_{0}$ and $\ssfK_{0}'$ of the whole ray; for the ray behaves in
-every physical respect as if it consisted of the \DPchg{noncoherent}{non-coherent} components
-$\ssfK_{0}$ and~$\ssfK_{0}'$. For this purpose we begin by establishing
-the value~$\ssfK_{\psi}$ of the component of the ray for an azimuth~$\psi$
-taken arbitrarily. Denoting by~$f$ the electric vector of the ray
-in the direction~$\psi$, we obtain this value~$\ssfK_{\psi}$ from the equation
-\[
-f = f_{1} \cos(\psi_{1} - \psi)
- + f_{2} \cos(\psi_{2} - \psi)
- + f_{3} \cos(\psi_{3} - \psi) + \dots,
-\]
-where the terms on the right side denote the projections of the
-vectors of the separate components in the direction~$\psi$, by squaring
-and averaging and taking into account the fact that $f_{1}$,~$f_{2}$, $f_{3},~\dots$
-are \DPchg{noncoherent}{non-coherent}
-\[
-\begin{aligned}
-\ssfK_{\psi}
- &= \ssfK_{1} \cos^{2}(\psi_{1} - \psi) + \ssfK_{2} \cos^{2}(\psi_{2} - \psi) + \dots \\
-\LeftText[\qquad]{or}
-\ssfK_{\psi} &= A\cos^{2}\psi + B\sin^{2}\psi + C\sin\psi \cos\psi \\
-\LeftText[\qquad]{where}
-A &= \ssfK_{1} \cos^{2}\psi_{1} + \ssfK_{2} \cos^{2}\psi_{2} + \dots \\
-B &= \ssfK_{1} \sin^{2}\psi_{1} + \ssfK_{2} \sin^{2}\psi_{2} + \dots \\
-C &= 2(\ssfK_{1} \sin\psi_{1} \cos\psi_{1} + \ssfK_{2} \sin\psi_{2} \cos\psi_{2} + \dots).
-\end{aligned}
-\Tag{(145)}
-\]
-
-The principal intensities $\ssfK_{0}$ and $\ssfK_{0}'$ of the ray follow from this
-expression as the maximum and the minimum value of~$\ssfK_{\psi}$
-according to the equation
-\[
-\frac{d\ssfK_{\psi}}{d\psi} = 0 \quad\text{or,}\quad
-\tan 2\psi = \frac{C}{A - B}\Add{.}
-\]
-Hence it follows that the principal intensities are
-\[
-\left.
-\begin{aligned}
-&\ssfK_{0} \\
-&\ssfK_{0}'
-\end{aligned}
-\right\} = \tfrac{1}{2}(A + B ą \sqrt{(A - B)^{2} + C^{2}}),
-\Tag{(146)}
-\]
-or, by taking~\Eq{(145)} into account,
-\begin{multline*}
-\left.
-\begin{aligned}
-&\ssfK_{0} \\
-&\ssfK_{0}'
-\end{aligned}
-\right\} = \frac{1}{2}\Biggl(\ssfK_{1} + \ssfK_{2} + \dots \\
- ą \sqrt{
- \begin{aligned}
- (\ssfK_{1} \cos 2\psi_{1} &+ \ssfK_{2} \cos 2\psi_{2} + \dots)^{2} \\
- &\quad + (\ssfK_{1} \sin 2\psi_{1} + \ssfK_{2} \sin 2\psi_{2} + \dots)^{2}
- \end{aligned}
- }\DPtypo{.}{}\,\Biggr)\DPtypo{}{.}
-\Tag{(147)}
-\end{multline*}
-%% -----File: 118.png---Folio 102-------
-Then the entropy radiation required becomes:
-\[
-\ssfL(\ssfK_{0}) + \ssfL(\ssfK_{0}').
-\Tag{(148)}
-\]
-
-\Section{106.} When two ray components $\ssfK$ and~$\ssfK'$, polarized at right
-angles to each other, are \DPchg{noncoherent}{non-coherent}, $\ssfK$~and $\ssfK'$ are also the principal
-intensities, and the entropy radiation is given by~\Eq{(141)}.
-The converse proposition, however, does not hold in general, that
-is to say, the two components of a ray polarized at right angles to
-each other, which correspond to the principal intensities $\ssfK$ and~$\ssfK'$,
-are not necessarily \DPchg{noncoherent}{non-coherent}, and hence the entropy radiation
-is not always given by~\Eq{(141)}.
-
-This is true, \eg, in the case of elliptically polarized light.
-There the radiations $\ssfK$ and $\ssfK'$ are completely coherent and their
-entropy is equal to~$\ssfL(\ssfK + \ssfK')$. This is caused by the fact that
-it is possible to give the two ray components an arbitrary displacement
-of phase in a reversible manner, say by total reflection.
-Thereby it is possible to change elliptically polarized light to
-plane polarized light and \textit{vice versa}.
-
-The entropy of completely or partially coherent rays has been
-investigated most thoroughly by \Name{M.~Laue}.\footnote
- {\Name{M.~Laue}, Annalen d.\ Phys.\DPchg{,}{}\ \textbf{23}, p.~1, 1907.}
-For the significance
-of optical coherence for thermodynamic probability see the next
-part, \Sec{119}.
-%% -----File: 119.png---Folio 103-------
-
-\Chapter[Stationary Field of Radiation]
-{V}{Electrodynamical Processes in a Stationary
-Field of Radiation}
-
-\Section{107.} We shall now consider from the standpoint of pure electrodynamics
-the processes that take place in a vacuum, which
-is bounded on all sides by reflecting walls and through which
-heat radiation passes uniformly in all directions, and shall then
-inquire into the relations between the electrodynamical and the
-thermodynamic quantities.
-
-The electrodynamical state of the field of radiation is determined
-at every instant by the values of the electric field-strength~$\ssfE$
-and the magnetic field-strength~$\ssfH$ at every point in the field,
-and the changes in time of these two vectors are completely
-determined by \Name{Maxwell's} field equations~\Eq{(52)}, which we have
-already used in \Sec{53}, together with the boundary conditions,
-which hold at the reflecting walls. In the present case, however,
-we have to deal with a solution of these equations of much greater
-complexity than that expressed by~\Eq{(54)}, which corresponds to a
-plane wave. For a plane wave, even though it be periodic with
-a wave length lying within the optical or thermal spectrum, can
-never be interpreted as heat radiation. For, according to \Sec{16},
-a finite intensity~$K$ of heat radiation requires a finite solid angle
-of the rays and, according to \Sec{18}, a spectral interval of finite
-width. But an absolutely plane, absolutely periodic wave has a
-zero solid angle and a zero spectral width. Hence in the case of
-a plane periodic wave there can be no question of either entropy
-or temperature of the radiation.
-
-\Section{108.} Let us proceed in a perfectly general way to consider the
-components of the field-strengths $\ssfE$~and~$\ssfH$ as functions of the
-time at a definite point, which we may think of as the origin of
-the coordinate system. Of these components, which are produced
-by all rays passing through the origin, there are six; we
-select one of them, say~$\ssfE_{z}$, for closer consideration. However
-%% -----File: 120.png---Folio 104-------
-complicated it may be, it may under all circumstances be written
-as a \Name{Fourier's} series for a limited time interval, say from $t = 0$
-to $t = \ssfT$; thus
-\[
-\ssfE_{z} = \sum_{n=1}^{n=\infty} C_{n} \cos\left(\frac{2\pi nt}{\ssfT} - \theta_{n}\right)
-\Tag{(149)}
-\]
-where the summation is to extend over all positive integers~$n$,
-while the constants $C_{n}$ (positive) and~$\theta_{n}$ may vary arbitrarily
-from term to term. The time interval~$\ssfT$, the fundamental
-period of the \Name{Fourier's} series, we shall choose so large that all
-times~$t$ which we shall consider hereafter are included in this
-time interval, so that $0 < t < \ssfT$. Then we may regard $\ssfE_{z}$ as
-identical in all respects with the \Name{Fourier's} series, \ie,~we may
-regard $\ssfE_{z}$ as consisting of ``partial vibrations,'' which are strictly
-periodic and of frequencies given by
-\[
-\nu = \frac{n}{\ssfT}\Add{.}
-\]
-
-Since, according to \Sec{3}, the time differential~$dt$ required for
-the definition of the intensity of a heat ray is necessarily large
-compared with the periods of vibration of all colors contained
-in the ray, a single time differential~$dt$ contains a large number of
-vibrations, \ie, the product~$\nu\, dt$ is a large number. Then it
-follows \textit{a~fortiori} that $\nu t$ and, still more,
-\[%[** Attn]
-\nu \ssfT = n \text{ is enormously large}
-\Tag{(150)}
-\]
-for all values of~$\nu$ entering into consideration. From this we
-must conclude that all amplitudes~$C_{n}$ with a moderately large
-value for the ordinal number~$n$ do not appear at all in the
-\Name{Fourier's} series, that is to say, they are negligibly small.
-
-\Section{109.} Though we have no detailed special information about
-the function~$\ssfE_{z}$, nevertheless its relation to the radiation of heat
-affords some important information as to a few of its general
-properties. Firstly, for the space density of radiation in a vacuum
-we have, according to Maxwell's theory,
-\[
-u = \frac{1}{8\pi} (\bar{\ssfE_{x}^{2}} + \bar{\ssfE_{y}^{2}} + \bar{\ssfE_{z}^{2}}
- + \bar{\ssfH_{x}^{2}} + \bar{\ssfH_{y}^{2}} + \bar{\ssfH_{z}^{2}}).
-\]
-Now the radiation is uniform in all directions and in the stationary
-%% -----File: 121.png---Folio 105-------
-state, hence the six mean values named are all equal to one
-another, and it follows that
-\[
-u = \frac{3}{4\pi} \bar{\ssfE_{z}^{2}}.
-\Tag{(151)}
-\]
-Let us substitute in this equation the value of~$\ssfE_{z}$ as given by~\Eq{(149)}.
-Squaring the latter and integrating term by term through a
-time interval, from $0$ to~$t$, assumed large in comparison with all
-periods of vibration~$\dfrac{1}{\nu}$ but otherwise arbitrary, and then dividing
-by~$t$, we obtain, since the radiation is perfectly stationary,
-\[
-u = \frac{3}{8\pi} \sum C_{n}^{2}.
-\Tag{(152)}
-\]
-
-From this relation we may at once draw an important conclusion
-as to the nature of~$\ssfE_{z}$ as a function of time. Namely,
-since the \Name{Fourier's} series~\Eq{(149)} consists, as we have seen, of a
-great many terms, the squares,~$C_{n}^{2}$, of the separate amplitudes
-of vibration the sum of which gives the space density of radiation,
-must have exceedingly small values. Moreover in the integral of
-the square of the \Name{Fourier's} series the terms which depend on the
-time~$t$ and contain the products of any two different amplitudes
-all cancel; hence the amplitudes~$C_{n}$ and the phase-constants~$\theta_{n}$
-must vary from one ordinal number to another in a quite irregular
-manner. We may express this fact by saying that the separate
-partial vibrations of the series are very small and in a ``chaotic''\footnote
- {Compare footnote to \PageRef{page}{116} (Tr.).}
-state.
-
-For the specific intensity of the radiation travelling in any
-direction whatever we obtain from~\Eq{(21)}
-\[
-K = \frac{cu}{4\pi} = \frac{3c}{32\pi^{2}} \sum C_{n}^{2}.
-\Tag{(153)}
-\]
-
-\Section{110.} Let us now perform the spectral resolution of the last two
-equations. To begin with we have from~\Eq{(22)}:
-\[
-u = \int_{0}^{\infty} \ssfu_{\nu}\, d\nu = \frac{3}{8\pi} \sum_{1}^{\infty} C_{n}^{2}.
-\Tag{(154)}
-\]
-On the right side of the equation the sum~$\sum$ consists of separate
-%% -----File: 122.png---Folio 106-------
-terms, every one of which corresponds to a separate ordinal
-number $n$ and to a simple periodic partial vibration. Strictly
-speaking this sum does not represent a continuous sequence of
-frequencies $\nu$, since $n$~is an integral number. But $n$~is, according
-to~\Eq{(150)}, so enormously large for all frequencies which need be
-considered that the frequencies~$\nu$ corresponding to the successive
-values of~$n$ lie very close together. Hence the interval~$d\nu$,
-though infinitesimal compared with~$\nu$, still contains a large
-number of partial vibrations, say~$n'$, where
-\[
-d\nu = \frac{n'}{\ssfT}\Add{.}
-\Tag{(155)}
-\]
-If now in~\Eq{(154)} we equate, instead of the total energy densities,
-the energy densities corresponding to the interval $d\nu$ only,
-which are independent of those of the other spectral regions, we
-obtain
-\[
-\ssfu_{\nu}\, d\nu = \frac{3}{8\pi} \sum_{n}^{n + n'} C_{n}^{2},
-\]
-or, according to~\Eq{(155)},
-\[
-\ssfu_{\nu}
- = \frac{3\ssfT}{8\pi} ˇ \frac{1}{n'} \sum_{n}^{n + n'} C_{n}^{2}
- = \frac{3\ssfT}{8\pi} ˇ \bar{C_{n}^{2}},
-\Tag{(156)}
-\]
-where we denote by~$\bar{C_{n}^{2}}$ the average value of~$C_{n}^{2}$ in the interval
-from $n$ to~$n + n'$. The existence of such an average value, the
-magnitude of which is independent of~$n$, provided $n'$~be taken
-small compared with~$n$, is, of course, not self-evident at the
-outset, but is due to a special property of the function~$\ssfE_{z}$ which is
-peculiar to stationary heat radiation. On the other hand, since
-many terms contribute to the mean value, nothing can be said
-either about the magnitude of a separate term~$C_{n}^{2}$, or about the
-connection of two consecutive terms, but they are to be regarded
-as perfectly independent of each other.
-
-In a very similar manner, by making use of~\Eq{(24)}, we find for
-the specific intensity of a monochromatic plane polarized ray,
-travelling in any direction whatever,
-\[
-\ssfK_{\nu} = \frac{3c\ssfT}{64\pi^{2}} \bar{C_{n}^{2}}.
-\Tag{(157)}
-\]
-%% -----File: 123.png---Folio 107-------
-
-From this it is apparent, among other things, that, according
-to the electromagnetic theory of radiation, a monochromatic
-light or heat ray is represented, not by a simple periodic wave, but
-by a superposition of a large number of simple periodic waves,
-the mean value of which constitutes the intensity of the ray. In
-accord with this is the fact, known from optics, that two rays of
-the same color and intensity but of different origin never interfere
-with each other, as they would, of necessity, if every ray were a
-simple periodic one.
-
-Finally we shall also perform the spectral resolution of the mean
-value of~$\ssfE_{z}^{2}$, by writing
-\[
-\ssfE_{z}^{2} = J = \int_{0}^{\infty} \ssfJ_{\nu}\, d\nu\Add{.}
-\Tag{(158)}
-\]
-
-Then by comparison with \Eq{(151)},~\Eq{(154)}, and~\Eq{(156)} we find
-\[
-\ssfJ_{\nu} = \frac{4\pi}{3} \ssfu_{\nu} = \frac{\ssfT}{2} \bar{C_{n}^{2}}\Add{.}
-\Tag{(159)}
-\]
-According to~\Eq{(157)}, $\ssfJ_{\nu}$~is related to~$\ssfK_{\nu}$, the specific intensity of
-radiation of a plane polarized ray, as follows:
-\[
-\ssfK_{\nu} = \frac{3c}{32\pi^{2}} \ssfJ_{\nu}.
-\Tag{(160)}
-\]
-
-\Section{111.} Black radiation is frequently said to consist of a large
-number of regular periodic vibrations. This method of expression
-is perfectly justified, inasmuch as it refers to the resolution
-of the total vibration in a \Name{Fourier's} series, according to equation~\Eq{(149)},
-and often is exceedingly well adapted for convenience and
-clearness of discussion. It should, however, not mislead us into
-believing that such a ``regularity'' is caused by a special physical
-property of the elementary processes of vibration. For the
-resolvability into a Fourier's series is mathematically self-evident
-and hence, in a physical sense, tells us nothing new. In fact, it
-is even always possible to regard a vibration which is damped
-to an arbitrary extent as consisting of a sum of regular periodic
-partial vibrations with constant amplitudes and constant phases.
-On the contrary, it may just as correctly be said that in all nature
-there is no process more complicated than the vibrations of black
-%% -----File: 124.png---Folio 108-------
-radiation. In particular, these vibrations do not depend in any
-characteristic manner on the special processes that take place
-in the centers of emission of the rays, say on the period or the
-damping of the emitting particles; for the normal spectrum is
-distinguished from all other spectra by the very fact that all
-individual differences caused by the special nature of the emitting
-substances are perfectly equalized and effaced. Therefore to
-attempt to draw conclusions concerning the special properties
-of the particles emitting the rays from the elementary vibrations
-in the rays of the normal spectrum would be a hopeless
-undertaking.
-
-In fact, black radiation may just as well be regarded as consisting,
-not of regular periodic vibrations, but of absolutely
-irregular separate impulses. The special regularities, which we
-observe in monochromatic light resolved spectrally, are caused
-merely by the special properties of the spectral apparatus used,
-\eg, the dispersing prism (natural periods of the molecules), or
-the diffraction grating (width of the slits). Hence it is also incorrect
-to find a characteristic difference between light rays and
-Roentgen\DPnote{**F2: No italics; check other instances?} rays (the latter assumed as an electromagnetic process
-in a vacuum) in the circumstance that in the former the vibrations
-take place with greater regularity. Roentgen rays may,
-under certain conditions, possess more selective properties than
-light rays. The resolvability into a \Name{Fourier's} series of partial
-vibrations with constant amplitudes and constant phases exists
-for both kinds of rays in precisely the same manner. What
-especially distinguishes light vibrations from Roentgen vibrations
-is the much smaller frequency of the partial vibrations of the
-former. To this is due the possibility of their spectral resolution,
-and probably also the far greater regularity of the changes of the
-radiation intensity in every region of the spectrum in the course of
-time, which, however, is not caused by a special property of the
-elementary processes of vibration, but merely by the constancy
-of the mean values.
-
-\Section{112.} The elementary processes of radiation exhibit regularities
-only when the vibrations are restricted to a narrow spectral region,
-that is to say in the case of spectroscopically resolved light, and
-especially in the case of the natural spectral lines. If, \eg, the
-amplitudes~$C_{n}$ of the \Name{Fourier's} series~\Eq{(149)} differ from zero only
-%% -----File: 125.png---Folio 109-------
-between the ordinal numbers $n = n_{0}$ and $n = n_{1}$, where $\dfrac{n_{1} - n_{0}}{n_{0}}$
-is small, we may write
-\[
-\ssfE_{z} = C_{0} \cos\left(\frac{2\pi n_{0}t}{\ssfT} - \theta_{0}\right),
-\Tag{(161)}
-\]
-where
-\begin{align*}
-C_{0} \cos\theta_{0} &= \sum_{n_{0}}^{n_{1}} C_{n} \cos\left(\frac{2\pi(n - n_{0})t}{\ssfT}-\theta_{n}\right)\\
-C_{0} \sin\theta_{0} &=-\sum_{n_{0}}^{n_{1}} C_{n} \sin\left(\frac{2\pi(n - n_{0})t}{\ssfT}-\theta_{n}\right)
-\end{align*}
-and $\ssfE_{z}$~may be regarded as a single approximately periodic vibration
-of frequency $\nu_{0} = \dfrac{n_{0}}{\ssfT}$ with an amplitude~$C_{0}$ and a phase-constant~$\theta_{0}$
-which vary slowly and irregularly.
-
-The smaller the spectral region, and accordingly the smaller
-$\dfrac{n_{1} - n_{0}}{n_{0}}$, the slower are the fluctuations (``Schwankungen'') of
-$C_{0}$~and~$\theta_{0}$, and the more regular is the resulting vibration and also
-the larger is the difference of path for which radiation can interfere
-with itself. If a spectral line were absolutely sharp, the
-radiation would have the property of being capable of interfering
-with itself for differences of path of any size whatever. This
-case, however, according to \Sec{18}, is an ideal abstraction, never
-occurring in reality.
-%% -----File: 126.png---Folio 110-------
-% [Blank Page]
-%% -----File: 127.png---Folio 111-------
-
-\Part{III}{Entropy and Probability}
-%% -----File: 128.png---Folio 112-------
-% [Blank Page]
-%% -----File: 129.png---Folio 113-------
-
-\Chapter[Fundamental Definitions and Laws]
-{I}{Fundamental Definitions and Laws.
-Hypothesis of Quanta}
-
-\Section{113.} Since a wholly new element, entirely unrelated to the
-fundamental principles of electrodynamics, enters into the range
-of investigation with the introduction of probability considerations
-into the electrodynamic theory of heat radiation, the question
-arises at the outset, whether such considerations are justifiable
-and necessary. At first sight we might, in fact, be inclined
-to think that in a purely electrodynamical theory there would be
-no room at all for probability calculations. For since, as is well
-known, the electrodynamic equations of the field together with
-the initial and boundary conditions determine uniquely the way
-in which an electrodynamical process takes place, in the course
-of time, considerations which lie outside of the equations of the
-field would seem, theoretically speaking, to be uncalled for and in
-any case dispensable. For either they lead to the same results
-as the fundamental equations of electrodynamics and then they
-are superfluous, or they lead to different results and in this case
-they are wrong.
-
-In spite of this apparently unavoidable dilemma, there is a
-flaw in the reasoning. For on closer consideration it is seen
-that what is understood in electrodynamics by ``initial and
-boundary'' conditions, as well as by the ``way in which a process
-takes place in the course of time,'' is entirely different from what
-is denoted by the same words in thermodynamics. In order to
-make this evident, let us consider the case of radiation \textit{in~vacuo},
-uniform in all directions, which was treated in the last chapter.
-
-From the standpoint of thermodynamics the state of radiation
-is completely determined, when the intensity of monochromatic
-radiation~$\ssfK_{\nu}$ is given for all frequencies~$\nu$. The electrodynamical
-observer, however, has gained very little by this single statement;
-because for him a knowledge of the state requires that every one
-%% -----File: 130.png---Folio 114-------
-of the six components of the electric and magnetic field-strength
-be given at all points of the space; and, while from the thermodynamic
-point of view the question as to the way in which the
-process takes place in time is settled by the constancy of the
-intensity of radiation~$\ssfK_{\nu}$, from the electrodynamical point of
-view it would be necessary to know the six components of the
-field at every point as functions of the time, and hence the amplitudes~$C_{n}$
-and the phase-constants~$\theta_{n}$ of all the several partial
-vibrations contained in the radiation would have to be calculated.
-This, however, is a problem whose solution is quite impossible,
-for the data obtainable from the measurements are by no
-means sufficient. The thermodynamically measurable quantities,
-looked at from the electrodynamical standpoint, represent
-only certain mean values, as we saw in the special case of
-stationary radiation in the last chapter.
-
-We might now think that, since in thermodynamic measurements
-we are always concerned with mean values only, we need
-consider nothing beyond these mean values, and, therefore, need
-not take any account of the particular values at all. This method
-is, however, impracticable, because frequently and that too just
-in the most important cases, namely, in the cases of the processes
-of emission and absorption, we have to deal with mean values
-which cannot be calculated unambiguously by electrodynamical
-methods from the measured mean values. For example, the
-mean value of~$C_{n}$ cannot be calculated from the mean value of~$C_{n}^{2}$,
-if no special information as to the particular values of~$C_{n}$ is
-available.
-
-Thus we see that the electrodynamical state is not by any
-means determined by the thermodynamic data and that in cases
-where, according to the laws of thermodynamics and according
-to all experience, an unambiguous result is to be expected, a purely
-electrodynamical theory fails entirely, since it admits not one
-definite result, but an infinite number of different results.
-
-\Section{114.} Before entering on a further discussion of this fact and
-of the difficulty to which it leads in the electrodynamical theory
-of heat radiation, it may be pointed out that exactly the same case
-and the same difficulty are met with in the mechanical theory of
-heat, especially in the kinetic theory of gases. For when, for
-example, in the case of a gas flowing out of an opening at the time
-%% -----File: 131.png---Folio 115-------
-$t = 0$, the velocity, the density, and the temperature are given
-at every point, and the boundary conditions are completely
-known, we should expect, according to all experience, that these
-data would suffice for a unique determination of the way in which
-the process takes place in time. This, however, from a purely
-mechanical point of view is not the case at all; for the positions
-and velocities of all the separate molecules are not at all given
-by the visible velocity, density, and temperature of the gas, and
-they would have to be known exactly, if the way in which the
-process takes place in time had to be completely calculated from
-the equations of motion. In fact, it is easy to show that, with
-given initial values of the visible velocity, density, and temperature,
-an infinite number of entirely different processes is mechanically
-possible, some of which are in direct contradiction to the
-principles of thermodynamics, especially the second principle.
-
-\Section{115.} From these considerations we see that, if we wish to calculate
-the way in which a thermodynamic process takes place
-in time, such a formulation of initial and boundary conditions
-as is perfectly sufficient for a unique determination of the process
-in thermodynamics, does not suffice for the mechanical theory of
-heat or for the electrodynamical theory of heat radiation. On
-the contrary, from the standpoint of pure mechanics or electrodynamics
-the solutions of the problem are infinite in number.
-Hence, unless we wish to renounce entirely the possibility of
-representing the thermodynamic processes mechanically or electrodynamically,
-there remains only one way out of the difficulty,
-namely, to supplement the initial and boundary conditions by
-special hypotheses of such a nature that the mechanical or
-electrodynamical equations will lead to an unambiguous result
-in agreement with experience. As to how such an hypothesis
-is to be formulated, no hint can naturally be obtained from the
-principles of mechanics or electrodynamics, for they leave the
-question entirely open. Just on that account any mechanical or
-electrodynamical hypothesis containing some further specialization
-of the given initial and boundary conditions, which cannot
-be tested by direct measurement, is admissible \textit{a~priori}. What
-hypothesis is to be preferred can be decided only by testing the
-results to which it leads in the light of the thermodynamic principles
-based on experience.
-%% -----File: 132.png---Folio 116-------
-
-\Section{116.} Although, according to the statement just made, a decisive
-test of the different admissible hypotheses can be made only
-\textit{a~posteriori}, it is nevertheless worth while noticing that it is possible
-to obtain \textit{a~priori}, without relying in any way on thermodynamics,
-a definite hint as to the nature of an admissible hypothesis.
-Let us again consider a flowing gas as an illustration (\Sec{114}).
-The mechanical state of all the separate gas molecules is not at
-all completely defined by the thermodynamic state of the gas,
-as has previously been pointed out. If, however, we consider all
-conceivable positions and velocities of the separate gas molecules,
-consistent with the given values of the visible velocity, density,
-and temperature, and calculate for every combination of them the
-mechanical process, assuming some simple law for the impact
-of two molecules, we shall arrive at processes, the vast majority
-of which agree completely in the mean values, though perhaps
-not in all details. Those cases, on the other hand, which show
-appreciable deviations, are vanishingly few, and only occur
-when certain very special and far-reaching conditions between the
-coordinates and velocity-components of the molecules are
-satisfied. Hence, if the assumption be made that such special
-conditions do not exist, however different the mechanical details
-may be in other respects, a form of flow of gas will be found,
-which may be called quite definite with respect to all measurable
-mean values---and they are the only ones which can be tested
-experimentally---although it will not, of course, be quite definite
-in all details. And the remarkable feature of this is that it is
-just the motion obtained in this manner that satisfies the postulates
-of the second principle of thermodynamics.
-
-\Section{117.} From these considerations it is evident that the hypotheses
-whose introduction was proven above to be necessary completely
-answer their purpose, if they state nothing more than that
-exceptional cases, corresponding to special conditions which exist
-between the separate quantities determining the state and which
-cannot be tested directly, do not occur in nature. In mechanics
-this is done by the hypothesis\footnote
- {\Name{L.~Boltzmann}, Vorlesungen über Gastheorie \textbf{1}, p.~21, 1896. Wiener Sitzungsberichte
- \textbf{78}, Juni, 1878, at the end. Compare also \Name{S.~H. Burbury}, Nature, \textbf{51}, p.~78, 1894.}
-that the heat motion is a ``molecular
-chaos'';\footnote
- {Hereafter \Name{Boltzmann's} ``Unordnung'' will be rendered by chaos, ``ungeordnet'' by
- \Label{116}% [** TN: Page label]
- chaotic (Tr.).}
-in electrodynamics the same thing is accomplished
-%% -----File: 133.png---Folio 117-------
-by the hypothesis of ``natural radiation,'' which states that
-there exist between the numerous different partial vibrations~\Eq{(149)}
-of a ray no other relations than those caused by the measurable
-mean values (compare below, \Sec{148}). If, for brevity, we
-denote any condition or process for which such an hypothesis
-%[** Thm]
-holds as an ``elemental chaos,'' the principle, \emph{that in nature any
-state or any process containing numerous elements not in themselves
-measurable is an elemental chaos}, furnishes the necessary condition
-for a unique determination of the measurable processes in mechanics
-as well as in electrodynamics and also for the validity of the
-second principle of thermodynamics. This must also serve as a
-mechanical or electrodynamical explanation of the conception of
-entropy, which is characteristic of the second law and of the
-closely allied concept of temperature.\footnote
- {To avoid misunderstanding I must emphasize that the question, whether the hypothesis
- of elemental chaos is really everywhere satisfied in nature, is not touched upon by the preceding
- considerations. I intended only to show at this point that, wherever this hypothesis
- does not hold, the natural processes, if viewed from the thermodynamic (macroscopic) point
- of view, do not take place unambiguously.}
-It also follows from this
-that the significance of entropy and temperature is, according to
-their nature, connected with the condition of an elemental
-chaos. The terms entropy and temperature do not apply to a
-purely periodic, perfectly plane wave, since all the quantities in
-such a wave are in themselves measurable, and hence cannot be
-an elemental chaos any more than a single rigid atom in motion
-can. The necessary condition for the hypothesis of an elemental
-chaos and with it for the existence of entropy and temperature
-can consist only in the irregular simultaneous effect of
-very many partial vibrations of different periods, which are
-propagated in the different directions in space independent
-of one another, or in the irregular flight of a multitude of
-atoms.
-
-\Section{118.} But what mechanical or electrodynamical quantity
-represents the entropy of a state? It is evident that this quantity
-depends in some way on the ``probability'' of the state.
-For since an elemental chaos and the absence of a record of any
-individual element forms an essential feature of entropy, the
-tendency to neutralize any existing temperature differences,
-which is connected with an increase of entropy, can mean nothing
-for the mechanical or electrodynamical observer but that uniform
-%% -----File: 134.png---Folio 118-------
-distribution of elements in a chaotic state is more probable than
-any other distribution.
-
-Now since the concept of entropy as well as the second principle
-of thermodynamics are of universal application, and since
-on the other hand the laws of probability have no less universal
-validity, it is to be expected that the connection between entropy
-and probability should be very close. Hence we make the
-following proposition the foundation of our further discussion:
-%[** Thm]
-\emph{The entropy of a physical system in a definite state depends solely
-on the probability of this state.} The fertility of this law will be
-seen later in several cases. We shall not, however, attempt to
-give a strict general proof of it at this point. In fact, such an
-attempt evidently would have no meaning at this point. For,
-so long as the ``probability'' of a state is not numerically defined,
-the correctness of the proposition cannot be quantitatively
-tested. One might, in fact, suspect at first sight that on this
-account the proposition has no definite physical meaning. It
-may, however, be shown by a simple deduction that it is possible
-by means of this fundamental proposition to determine quite
-generally the way in which entropy depends on probability,
-without any further discussion of the probability of a state.
-
-\Section{119.} For let $S$ be the entropy, $W$~the probability of a physical
-system in a definite state; then the \DPtypo{propositon}{proposition}
-states that
-\[
-S = f(W)
-\Tag{(162)}
-\]
-where $f(W)$~represents a universal function of the argument~$W$.
-In whatever way $W$~may be defined, it can be safely inferred from
-the mathematical concept of probability that the probability of
-a system which consists of two entirely independent\footnote
- {It is well known that the condition that the two systems be independent of each other is
- essential for the validity of the expression~\Eq{(163)}. That it is also a necessary condition for the
- additive combination of the entropy was proven first by \Name{M.~Laue} in the case of optically
- coherent rays. Annalen d.\ Physik\DPchg{,}{} \textbf{20}, p.~365, 1906.}
-systems
-is equal to the product of the probabilities of these two systems
-separately. If we think, \eg, of the first system as any body
-whatever on the earth and of the second system as a cavity containing
-radiation on Sirius, then the probability that the terrestrial
-body be in a certain state~$1$ and that simultaneously the
-radiation in the cavity in a definite state~$2$ is
-\[
-W = W_{1}W_{2},
-\Tag{(163)}
-\]
-%% -----File: 135.png---Folio 119-------
-where $W_{1}$~and~$W_{2}$ are the probabilities that the systems involved
-are in the states in question.
-
-If now $S_{1}$~and~$S_{2}$ are the entropies of the separate systems in
-the two states, then, according to~\Eq{(162)}, we have
-\[
-S_{1} = f(W_{1})\qquad
-S_{2} = f(W_{2}).
-\]
-But, according to the second principle of thermodynamics, the
-total entropy of the two systems, which are independent (see
-\DPchg{footnote to preceding page}{preceding footnote}) of each other, is $S = S_{1} + S_{2}$ and hence
-from \Eq{(162)}~and~\Eq{(163)}
-\[
-f(W_{1}W_{2}) = f(W_{1}) + f(W_{2}).
-\]
-
-From this functional equation $f$~can be determined. For on
-differentiating both sides with respect to~$W_{1}$, $W_{2}$~remaining constant,
-we obtain
-\[
-W_{2} \dot{f}(W_{1}W_{2}) = \dot{f}(W_{1}).
-\]
-On further differentiating with respect to~$W_{2}$, $W_{1}$~now remaining
-constant, we get
-\[
-\dot{f}(W_{1}W_{2}) + W_{1}W_{2} \ddot{f}(W_{1}W_{2}) = 0
-\]
-or
-\[
-\dot{f}(W) + W \ddot{f}(W) = 0.
-\]
-The general integral of this differential equation of the second
-order is
-\[
-f(W) = k \log W + \const.
-\]
-Hence from~\Eq{(162)} we get
-\[
-S = k \log W + \const.,
-%[** TN: No equation number in the original]
-\Tag{(164)}
-\]
-an equation which determines the general way in which the entropy
-depends on the probability. The universal constant of
-integration~$k$ is the same for a terrestrial as for a cosmic system,
-and its value, having been determined for the former, will remain
-valid for the latter. The second additive constant of integration
-may, without any restriction as regards generality, be included
-as a constant multiplier in the quantity~$W$, which here has not yet
-been completely defined, so that the equation reduces to
-\[
-S = k \log W.
-\]
-
-\Section{120.} The logarithmic connection between entropy and probability
-was first stated by \Name{L.~Boltzmann}\footnote
- {\Name{L.~Boltzmann}, Vorlesungen über Gastheorie, \textbf{1}, \Sec{6}.}
-in his kinetic theory of
-%% -----File: 136.png---Folio 120-------
-gases. Nevertheless our equation~\Eq{(164)} differs in its meaning
-from the corresponding one of Boltzmann in two essential points.
-
-Firstly, \Name{Boltzmann's} equation lacks the factor~$k$, which is due
-to the fact that \Name{Boltzmann} always used gram-molecules, not the
-molecules themselves, in his calculations. Secondly, and this is
-of greater consequence, \Name{Boltzmann} leaves an additive constant
-undetermined in the entropy~$S$ as is done in the whole of classical
-thermodynamics, and accordingly there is a constant factor of
-proportionality, which remains undetermined in the value of the
-probability~$W$.
-
-In contrast with this we assign a definite absolute value to the
-entropy~$S$. This is a step of fundamental importance, which
-can be justified only by its consequences. As we shall see later,
-this step leads necessarily to the ``hypothesis of quanta'' and
-moreover it also leads, as regards radiant heat, to a definite law
-of distribution of energy of black radiation, and, as regards heat
-energy of bodies, to \Name{Nernst's} heat theorem.
-
-From \Eq{(164)} it follows that with the entropy~$S$ the probability~$W$
-is, of course, also determined in the absolute sense. We shall
-designate the quantity~$W$ thus defined as the ``thermodynamic
-probability,'' in contrast to the ``mathematical probability,'' to
-which it is proportional but not equal. For, while the mathematical
-probability is a proper fraction, the thermodynamic
-probability is, as we shall see, always an integer.
-
-\Section{121.} The relation~\Eq{(164)} contains a general method for calculating
-the entropy~$S$ by probability considerations. This,
-however, is of no practical value, unless the thermodynamic
-probability~$W$ of a system in a given state can be expressed
-numerically. The problem of finding the most general and most
-precise definition of this quantity is among the most important
-problems in the mechanical or electrodynamical theory of heat.
-It makes it necessary to discuss more fully what we mean by the
-``state'' of a physical system.
-
-By the state of a physical system at a certain time we mean the
-aggregate of all those mutually independent quantities, which
-determine uniquely the way in which the processes in the system
-take place in the course of time for given boundary conditions.
-Hence a knowledge of the state is precisely equivalent to a knowledge
-of the ``initial conditions.'' If we now take into account
-%% -----File: 137.png---Folio 121-------
-the considerations stated above in \Sec{113}, it is evident that we
-must distinguish in the theoretical treatment two entirely different
-kinds of states, which we may denote as ``microscopic'' and
-``macroscopic'' states. The microscopic state is the state as
-described by a mechanical or electrodynamical observer; it contains
-the separate values of all coordinates, velocities, and field-strengths.
-The microscopic processes, according to the laws of
-mechanics and electrodynamics, take place in a perfectly unambiguous
-way; for them entropy and the second principle of thermodynamics
-have no significance. The macroscopic state,
-however, is the state as observed by a thermodynamic observer;
-any macroscopic state contains a large number of microscopic
-ones, which it unites in a mean value. Macroscopic processes
-take place in an unambiguous way in the sense of the second
-principle, when, and only when, the hypothesis of the elemental
-chaos (\Sec{117}) is satisfied.
-
-\Section{122.} If now the calculation of the probability~$W$ of a state is
-in question, it is evident that the state is to be thought of in the
-macroscopic sense. The first and most important question is
-now: How is a macroscopic state defined? An answer to it will
-dispose of the main features of the whole problem.
-
-For the sake of simplicity, let us first consider a special case,
-that of a very large number,~$N$, of simple similar molecules. Let
-the problem be solely the distribution of these molecules in space
-within a given volume,~$V$, irrespective of their velocities, and further
-the definition of a certain macroscopic distribution in space.
-The latter cannot consist of a statement of the coordinates of all
-the separate molecules, for that would be a definite microscopic
-distribution. We must, on the contrary, leave the positions of
-the molecules undetermined to a certain extent, and that can be
-done only by thinking of the whole volume~$V$ as being divided
-into a number of small but finite \emph{space elements},~$G$, each containing
-a specified number of molecules. By any such statement a
-definite macroscopic distribution in space is defined. The manner
-in which the molecules are distributed within every separate
-space element is immaterial, for here the hypothesis of elemental
-chaos (\Sec{117}) provides a supplement, which insures the unambiguity
-of the macroscopic state, in spite of the microscopic
-indefiniteness. If we distinguish the space elements in order by
-%% -----File: 138.png---Folio 122-------
-the numbers $1$,~$2$, $3,~\dots$ and, for any particular macroscopic
-distribution in space, denote the number of the molecules
-lying in the separate space elements by $N_{1}$,~$N_{2}$, $N_{3}~\dots$,
-then to every definite system of values $N_{1}$,~$N_{2}$, $N_{3}~\dots$,
-there corresponds a definite macroscopic distribution in space.
-We have of course always:
-\[
-N_{1} + N_{2} + N_{3} + \dots = N
-\Tag{(165)}
-\]
-or if
-\begin{gather*}
-\frac{N_{1}}{N} = w_{1}\Add{,}\quad
-\frac{N_{2}}{N} = w_{2},\ \dots
-\Tag{(166)} \\
-w_{1} + w_{2} + w_{3} + \dots = 1.
-\Tag{(167)}
-\end{gather*}
-The quantity~$w_{i}$ may be called the density of distribution of the
-molecules, or the mathematical probability that any molecule
-selected at random lies in the $i$th~space element.
-
-If we now had, \eg, only $10$~molecules and $7$~space elements, a
-definite space distribution would be represented by the values:
-\[
-N_{1} = 1,\ N_{2} = 2,\ N_{3} = 0,\ N_{4} = 0,\ N_{5} = 1,\ N_{6} = 4,\ N_{7} = 2,
-\Tag{(168)}
-\]
-which state that in the seven space elements there lie respectively
-$1$,~$2$, $0$, $0$, $1$, $4$, $2$~molecules.
-
-\Section{123.} The definition of a macroscopic distribution in space may
-now be followed immediately by that of its thermodynamic
-probability~$W$. The latter is founded on the consideration that
-a certain distribution in space may be realized in many different
-ways, namely, by many different individual coordinations or
-``complexions,'' according as a certain molecule considered will
-happen to lie in one or the other space element. For, with a
-given distribution of space, it is of consequence only how many, not
-which, molecules lie in every space element.
-
-The number of all complexions which are possible with a given
-distribution in space we equate to the thermodynamic probability~$W$
-of the space distribution.
-
-In order to form a definite conception of a certain complexion,
-we can give the molecules numbers, write these numbers in
-order from $1$ to~$N$, and place below the number of every molecule
-the number of that space element to which the molecule in question
-belongs in that particular complexion. Thus the following
-%% -----File: 139.png---Folio 123-------
-table represents one particular complexion, selected at random,
-for the distribution in the preceding illustration
-\[
-\begin{array}{r*{9}{>{\quad}r}}
-1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\
-6 & 1 & 7 & 5 & 6 & 2 & 2 & 6 & 6 & 7
-\end{array}
-\Tag{(169)}
-\]
-By this the fact is exhibited that the
-
-%[** Attn]
-Molecule~$2$ lies in space element~$1$.
-
-Molecules $6$~and~$7$ lie in space element~$2$.
-
-Molecule~$4$ lies in space element~$5$.
-
-Molecules $1$,~$5$,~$8$, and~$9$ lie in space element~$6$.
-
-Molecules $3$~and~$10$ lie in space element~$7$.
-
-As becomes evident on comparison with~\Eq{(168)}, this complexion
-does, in fact, correspond in every respect to the space
-distribution given above, and in a similar manner it is easy to
-exhibit many other complexions, which also belong to the same
-space distribution. The number of all possible complexions
-required is now easily found by inspecting the lower of the two
-lines of figures in~\Eq{(169)}. For, since the number of the molecules
-is given, this line of figures contains a definite number of places.
-Since, moreover, the distribution in space is also given, the number
-of times that every figure (\ie,~every space element) appears
-in the line is equal to the number of molecules which lie in that
-particular space element. But every change in the table gives
-a new particular coordination between molecules and space
-elements and hence a new complexion. Hence the number of
-the possible complexions, or the thermodynamic probability,~$W$,
-of the given space distribution, is equal to the number of ``permutations
-with repetition'' possible under the given conditions.
-In the simple numerical example chosen, we get for~$W$, according
-to a well-known formula, the expression
-\[
-\frac{10!}{1!\, 2!\, 0!\, 0!\, 1!\, 4!\, 2!} = 37,800.
-\]
-
-The form of this expression is so chosen that it may be applied
-easily to the general case. The numerator is equal to factorial~$N$,
-$N$~being the total number of molecules considered, and the
-denominator is equal to the product of the factorials of the numbers,
-$N_{1}$,~$N_{2}$, $N_{3},~\dots$ of the molecules, which lie in every
-separate space element and which, in the general case, must be
-%% -----File: 140.png---Folio 124-------
-thought of as large numbers. Hence we obtain for the required
-probability of the given space distribution
-\[
-W = \frac{N!}{N_{1}!\, N_{2}!\, N_{3}!\, \dots}\Add{.}
-\Tag{(170)}
-\]
-
-Since all the $N$'s are large numbers, we may apply to their
-factorials Stirling's formula, which for a large number may be
-abridged\footnote
- {Abridged in the sense that factors which in the logarithmic expression~\Eq{(173)} would give
- \Label{124}% [** TN: Page label]
- rise to small additive terms have been omitted at the outset. A brief derivation of equation~\Eq{(173)}
- may be found on \PageRef{p.}{218} (Tr.).}
-to\footnote
- {See for example \Name{E.~Czuber}, Wahrscheinlichkeitsrechnung (Leipzig, B.~G. Teubner)
- p.~22, 1903; \Name{H.~Poincaré}, Calcul des Probabilités (Paris, Gauthier-Villars), p.~85, 1912.}
-\[
-n! = \left(\frac{n}{e}\right)^{n}\Add{.}
-\Tag{(171)}
-\]
-Hence, by taking account of~\Eq{(165)}, we obtain
-\[
-W = \left(\frac{N}{N_{1}}\right)^{N_{1}}
- \left(\frac{N}{N_{2}}\right)^{N_{2}}
- \left(\frac{N}{N_{3}}\right)^{N_{3}} \dots\Add{.}
-\Tag{(172)}
-\]
-
-\Section{124.} Exactly the same method as in the case of the space distribution
-just considered may be used for the definition of a
-macroscopic state and of the thermodynamic probability in the
-general case, where not only the coordinates but also the velocities,
-the electric moments,~etc., of the molecules are to be dealt
-with. Every thermodynamic state of a system of $N$~molecules
-is, in the macroscopic sense, defined by the statement of the
-number of molecules, $N_{1}$,~$N_{2}$, $N_{3},~\dots$, which are contained
-in the region elements $1$,~$2$, $3,~\dots$ of the ``state
-space.'' This state space, however, is not the ordinary three-dimensional
-space, but an ideal space of as many dimensions as
-there are variables for every molecule. In other respects the
-definition and the calculation of the thermodynamic probability~$W$
-are exactly the same as above and the entropy of the state is
-accordingly found from~\Eq{(164)}, taking \Eq{(166)} also into account, to
-be
-\[
-S = -kN \sum w_{1} \log w_{1},%[** F2: Probably w_{i}]
-\Tag{(173)}
-\]
-where the sum $\sum$ is to be taken over all region elements. It is
-obvious from this expression that the entropy is in every case a
-\emph{positive quantity}.
-
-\Section{125.} By the preceding developments the calculation of the
-%% -----File: 141.png---Folio 125-------
-entropy of a system of $N$~molecules in a given thermodynamic
-state is, in general, reduced to the single problem of finding the
-magnitude~$G$ of the region elements in the state space. That
-such a definite finite quantity really exists is a characteristic
-feature of the theory we are developing, as contrasted with that
-due to \Name{Boltzmann}, and forms the content of the so-called \emph{hypothesis
-of quanta}. As is readily seen, this is an immediate consequence
-of the proposition of \Sec{120} that the entropy~$S$ has an
-absolute, not merely a relative, value; for this, according to~\Eq{(164)},
-necessitates also an absolute value for the magnitude of the thermodynamic
-probability~$W$, which, in turn, according to \Sec{123},
-is dependent on the number of complexions, and hence also
-on the number and size of the region elements which are used.
-Since all different complexions contribute uniformly to the value
-of the probability~$W$, the region elements of the state space
-represent also \emph{regions of equal probability}. If this were not so,
-the complexions would not be all equally probable.
-
-However, not only the magnitude, but also the shape and position
-of the region elements must be perfectly definite. For since,
-in general, the distribution density~$w$ is apt to vary appreciably
-from one region element to another, a change in the shape of a
-region element, the magnitude remaining unchanged, would, in
-general, lead to a change in the value of~$w$ and hence to a change
-in~$S$. We shall see that only in special cases, namely, when the
-distribution densities~$w$ are very small, may the absolute magnitude
-of the region elements become physically unimportant, inasmuch
-as it enters into the entropy only through an additive constant.
-This happens, \eg, at high temperatures, large volumes,
-slow vibrations (state of an ideal gas, \Sec{132}, \Name{Rayleigh's} radiation
-law, \DPtypo{Sec.~195}{\Sec{159}}). Hence it is permissible for such limiting
-cases to assume, without appreciable error, that $G$~is infinitely
-small in the macroscopic sense, as has hitherto been the practice
-in statistical mechanics. As soon, however, as the distribution
-densities~$w$ assume appreciable values, the classical statistical
-mechanics fail.
-
-\Section{126.} If now the problem be to determine the magnitude~$G$
-of the region elements of equal probability, the laws of the classical
-statistical mechanics afford a certain hint, since in certain
-limiting cases they lead to correct results.
-%% -----File: 142.png---Folio 126-------
-
-Let $\phi_{1}$,~$\phi_{2}$, $\phi_{3},~\dots$ be the ``generalized coordinates,''
-$\psi_{1}$,~$\psi_{2}$, $\psi_{3},~\dots$ the corresponding ``impulse coordinates''
-or ``moments,'' which determine the microscopic state of a certain
-molecule; then the state space contains as many dimensions
-as there are coordinates~$\phi$ and moments~$\psi$ for every molecule.
-Now the region element of probability, according to classical
-statistical mechanics, is identical with the infinitely small element
-of the state space (in the macroscopic sense)\footnote
- {Compare, for example, \Name{L.~Boltzmann}, Gastheorie, \textbf{2}, p.~62 \textit{et~seq.}, 1898, or \Name{J.~W.~Gibbs},
- Elementary principles in statistical mechanics, Chapter~I, 1902.}
-\[
-d\phi_{1}\, d\phi_{2}\, d\phi_{3} \dots d\psi_{1}\, d\psi_{2}\, d\psi_{3} \dots\Add{.}
-\Tag{(174)}
-\]
-
-According to the hypothesis of quanta, on the other hand,
-every region element of probability has a definite finite magnitude
-\[
-G = \int d\phi_{1}\, d\phi_{2}\, d\phi_{3} \dots d\psi_{1}\, d\psi_{2}\, d\psi_{3} \dots
-\Tag{(175)}
-\]
-whose value is the same for all different region elements and, moreover,
-depends on the nature of the system of molecules considered.
-The shape and position of the separate region elements are determined
-by the limits of the integral and must be determined anew
-in every separate case.
-%% -----File: 143.png---Folio 127-------
-
-\Chapter{II}{Ideal Monatomic Gases}
-
-\Section{127.} In the preceding chapter it was proven that the introduction
-of probability considerations into the mechanical and
-electrodynamical theory of heat is justifiable and necessary, and
-from the general connection between entropy~$S$ and probability~$W$,
-as expressed in equation~\Eq{(164)}, a method was derived for calculating
-the entropy of a physical system in a given state. Before
-we apply this method to the determination of the entropy of
-radiant heat we shall in this chapter make use of it for calculating
-the entropy of an ideal monatomic gas in an arbitrarily given
-state. The essential parts of this calculation are already contained
-in the investigations of \Name{L.~Boltzmann}\footnote
- {\Name{L.~Boltzmann}, Sitzungsber.\ d.\ Akad.\ d.\ Wissensch.\ zu Wien~(II) \textbf{76}, p.~373, 1877. Compare
- also Gastheorie, \textbf{1}, p.~38, 1896.}
-on the mechanical
-theory of heat; it will, however, be advisable to discuss this
-simple case in full, firstly to enable us to compare more readily
-the method of calculation and physical significance of mechanical
-entropy with that of radiation entropy, and secondly, what is
-more important, to set forth clearly the differences as compared
-with \Name{Boltzmann's} treatment, that is, to discuss the meaning of
-the universal constant~$k$ and of the finite region elements~$G$. For
-this purpose the treatment of a special case is sufficient.
-
-\Section{128.} Let us then take $N$~similar monatomic gas molecules in
-an arbitrarily given thermodynamic state and try to find the
-corresponding entropy. The state space is six-dimensional,
-with the three coordinates $x$,~$y$,~$z$, and the three corresponding
-moments $m\xi$,~$m\eta$,~$m\zeta$, of a molecule, where we denote the mass
-by~$m$ and velocity components by $\xi$,~$\eta$,~$\zeta$. Hence these quantities
-are to be substituted for the $\phi$~and~$\psi$ in \Sec{126}. We thus obtain
-for the size of a region element~$G$ the sextuple integral
-\[
-G = m^{3} \int d\sigma,
-\Tag{(176)}
-\]
-where, for brevity
-\[
-dx\, dy\, dz\, d\xi\, d\eta\, d\zeta = d\sigma\Add{.}
-\Tag{(177)}
-\]
-%% -----File: 144.png---Folio 128-------
-
-If the region elements are known, then, since the macroscopic
-state of the system of molecules was assumed as known, the
-numbers $N_{1}$,~$N_{2}$, $N_{3},~\dots$ of the molecules which lie in
-the separate region elements are also known, and hence the distribution
-densities $w_{1}$,~$w_{2}$, $w_{3},~\dots$ \Eq{(166)} are given and the
-entropy of the state follows at once from~\Eq{(173)}.
-
-\Section{129.} The theoretical determination of~$G$ is a problem as difficult
-as it is important. Hence we shall at this point restrict ourselves
-from the very outset to the special case in which the distribution
-density varies but slightly from one region element to the next---the
-characteristic feature of the state of an ideal gas. Then the
-summation over all region elements may be replaced by the integral
-over the whole state space. Thus we have from \Eq{(176)}~and~\Eq{(167)}
-\[
-%[** Again, w_{i}?]
-\sum w_{1}
- = \sum w_{1} \frac{m^{3}}{G} \int d\sigma
- = \frac{m^{3}}{G} \int w\, d\sigma = 1,
-\Tag{(178)}
-\]
-in which $w$~is no longer thought of as a discontinuous function
-of the ordinal number,~$i$, of the region element, where $i = 1$,
-$2$, $3,~\dots~n$, but as a continuous function of the variables,
-$x$,~$y$,~$z$, $\xi$,~$\eta$,~$\zeta$, of the state space. Since the whole state region
-contains very many region elements, it follows, according to~\Eq{(167)}
-and from the fact that the distribution density~$w$ changes
-slowly, that $w$ has everywhere a small value.
-
-Similarly we find for the entropy of the gas from~\Eq{(173)}:
-\[
-S = -kN \sum w_{1} \log w_{1} = -kN \frac{m^{3}}{G} \int w \log w\, d\sigma.
-\Tag{(179)}
-\]
-Of course the whole energy~$E$ of the gas is also determined by the
-distribution densities~$w$. If $w$~is sufficiently small in every
-region element, the molecules contained in any one region
-element are, on the average, so far apart that their energy depends
-only on the velocities. Hence:
-\begin{align*}
-E = &\sum N_{1}\, \frac{1}{2} m(\xi_{1}^{2} + \eta_{1}^{2} + \zeta_{1}^{2}) + E_{0}\\
- = N&\sum w_{1}\, \frac{1}{2} m(\xi_{1}^{2} + \eta_{1}^{2} + \zeta_{1}^{2}) + E_{0},
-\Tag{(180)}
-\end{align*}
-where $\xi_{1}\, \eta_{1}\, \zeta_{1}$ denotes any velocity lying within the region element~$1$
-and $E_{0}$~denotes the internal energy of the stationary molecules,
-%% -----File: 145.png---Folio 129-------
-which is assumed constant. In place of the latter expression we
-may write, again according to~\Eq{(176)},
-\[
-E = \frac{m^{4}N}{2G} \int (\xi^{2} + \eta^{2} + \zeta^{2}) w\, d\sigma + E_{0}.
-\Tag{(181)}
-\]
-
-\Section{130.} Let us consider the state of thermodynamic equilibrium.
-According to the second principle of thermodynamics this state
-is distinguished from all others by the fact that, for a given volume~$V$
-and a given energy~$E$ of the gas, the entropy~$S$ is a maximum.
-Let us then regard the volume
-\[
-V = \iiint dx\, dy\, dz
-\Tag{(182)}
-\]
-and the energy~$E$ of the gas as given. The condition for equilibrium
-is $\delta S = 0$, or, according to~\Eq{(179)},
-\[
-\sum (\log w_{1} + 1)\, \delta w_{1} = 0,
-\]
-and this holds for any variations of the distribution densities
-whatever, provided that, according to \Eq{(167)}~and~\Eq{(180)}, they
-satisfy the conditions
-\[
-\sum \delta w_{1} = 0 \quad\text{and}\quad
-\sum (\xi_{1}^{2} + \eta_{1}^{2} + \zeta_{1}^{2})\, \delta w_{1} = 0.
-\]
-This gives us as the necessary and sufficient condition for thermodynamic
-equilibrium for every separate distribution density~$w$:
-\[
-\log w + \beta(\xi^{2} + \eta^{2} + \zeta^{2}) + \const. = 0
-\]
-or
-\[
-w = \alpha e^{-\beta(\xi^{2} + \eta^{2} + \zeta^{2})},
-\Tag{(183)}
-\]
-where $\alpha$~and~$\beta$ are constants. Hence in the state of equilibrium
-the distribution of the molecules in space is independent of
-$x$,~$y$,~$z$, that is, macroscopically uniform, and the distribution of
-velocities is the well-known one of \Name{Maxwell}.
-
-\Section{131.} The values of the constants $\alpha$~and~$\beta$ may be found from
-those of $V$~and~$E$. For, on substituting the value of~$w$ just
-found in~\Eq{(178)} and taking account of \Eq{(177)}~and~\Eq{(182)}, we get
-\[%[** Attn: notation for integrating over space]
-\frac{G}{m^{3}}
- = \alpha V \iiint_{-\infty}^{\infty} e^{-\beta(\xi^{2} + \eta^{2} + \zeta^{2})}\, d\xi\, d\eta\, d\zeta
- = \alpha V \left(\frac{\pi}{\beta}\right)^{\efrac{3}{2}},
-\]
-%% -----File: 146.png---Folio 130-------
-and on substituting $w$ in~\Eq{(181)} we get
-\[
-E = E_{0} + \frac{\alpha m^{4} NV}{2G} \iiint_{-\infty}^{\infty} (\xi^{2} + \eta^{2} + \zeta^{2}) e^{-\beta(\xi^{2} + \eta^{2} + \zeta^{2})}\, d\xi\, d\eta\, d\zeta,
-\]
-or
-\[
-E = E_{0} + \frac{3\alpha m^{4} NV}{4G}\, \frac{1}{\beta} \left(\frac{\pi}{\beta}\right)^{\efrac{3}{2}}.
-\]
-Solving for $\alpha$~and~$\beta$ we have
-\begin{align*}
-\alpha &= \frac{G}{V} \left(\frac{3N}{4\pi m(E - E_{0})}\right)^{\efrac{3}{2}}
-\Tag{(184)} \\
-\beta &= \frac{3}{4}\, \frac{Nm}{E - E_{0}}.
-\Tag{(185)}
-\end{align*}
-From this finally we find, as an expression for the entropy~$S$ of
-the gas in the state of equilibrium with given values of $N$,~$V$,
-and~$E$,
-\[
-S = kN \log \left\{\frac{V}{G} \left(\frac{4\pi em(E - E_{0})}{3N}\right)^{\efrac{3}{2}}\right\}.
-\Tag{(186)}
-\]
-
-\Section{132.} This determination of the entropy of an ideal monatomic
-gas is based solely on the general connection between entropy and
-probability as expressed in equation~\Eq{(164)}; in particular, we have
-at no stage of our calculation made use of any special law of the
-theory of gases. It is, therefore, of importance to see how the
-entire thermodynamic behavior of a monatomic gas, especially
-the equation of state and the values of the specific heats, may be
-deduced from the expression found for the entropy directly by
-means of the principles of thermodynamics. From the general
-thermodynamic equation defining the entropy, namely,
-\[
-dS = \frac{dE + p\, dV}{T},
-\Tag{(187)}
-\]
-the partial differential coefficients of~$S$ with respect to $E$~and~$V$
-are found to be
-\[
-\left(\frac{\dd S}{\dd E}\right)_{V} = \frac{1}{T}, \quad
-\left(\frac{\dd S}{\dd V}\right)_{E} = \frac{p}{T}\Add{.}
-\]
-%% -----File: 147.png---Folio 131-------
-Hence, by using~\Eq{(186)}, we get for our gas
-\[
-\left(\frac{\dd S}{\dd E}\right)_{V} = \frac{3}{2}\, \frac{kN}{E - E_{0}} = \frac{1}{T}
-\Tag{(188)}
-\]
-and
-\[
-\left(\frac{\dd S}{\dd V}\right)_{E} = \frac{kN}{V} = \frac{p}{T}.
-\Tag{(189)}
-\]
-The second of these equations
-\[
-p = \frac{kNT}{V}
-\Tag{(190)}
-\]
-contains the laws of \Name{Boyle}, \Name{Gay Lussac}, and \Name{Avogadro}, the last
-named because the pressure depends only on the number~$N$, not
-on the nature of the molecules. If we write it in the customary
-form:
-\[
-p = \frac{RnT}{V},
-\Tag{(191)}
-\]
-where $n$~denotes the number of gram molecules or mols of the gas,
-referred to $O_{2} = 32\, \DPchg{g}{\gr.}$, and $R$~represents the absolute gas constant
-\[
-R = 831 × 10^{5}\, \frac{\erg}{\degree},
-\Tag{(192)}
-\]
-we obtain by comparison
-\[
-k = \frac{Rn}{N}.
-\Tag{(193)}
-\]
-If we now call the ratio of the number of mols to the number of
-molecules~$\omega$, or, what is the same thing, the ratio of the mass of a
-molecule to that of a mol, $\omega = \dfrac{n}{N}$, we shall have
-\[
-k = \omega R.
-\Tag{(194)}
-\]
-From this the universal constant~$k$ may be calculated, when $\omega$~is
-given, and \textit{vice versa}. According to~\Eq{(190)} this constant~$k$ is
-nothing but the absolute gas constant, if it is referred to molecules
-instead of mols.
-
-From equation~\Eq{(188)}
-\[
-E - E_{0} = \tfrac{3}{2} kNT.
-\Tag{(195)}
-\]
-%% -----File: 148.png---Folio 132-------
-Now, since the energy of an ideal gas is also given by
-\[
-E = Anc_{v}T + E_{0}
-\Tag{(196)}
-\]
-where $c_{v}$~is the heat capacity of a mol at constant volume in
-calories and $A$~is the mechanical equivalent of heat:
-\[
-A = 419 × 10^{5}\, \frac{\erg}{\cal}
-\Tag{(197)}
-\]
-it follows that
-\[
-c_{v} = \frac{3}{2}\, \frac{kN}{An}
-\]
-and further, by taking account of~\Eq{(193)}
-\[
-c_{v} = \frac{3}{2}\, \frac{R}{A}
- = \frac{3}{2}\DPchg{\,}{ˇ} \frac{831 × 10^{5}}{419 × 10^{5}} = 3.0
-\Tag{(198)}
-\]
-as an expression for the heat capacity per mol of any monatomic
-gas at constant volume in calories.\footnote
- {Compare \Name{F.~Richarz}, Wiedemann's Annal., \textbf{67}, p.~705, 1899.}
-
-For the heat capacity per mol at constant pressure,~$c_{p}$, we
-have as a consequence of the first principle of thermodynamics:
-\[
-c_{p} - c_{v} = \frac{R}{A}
-\]
-and hence by~\Eq{(198)}
-\[
-c_{p} = \frac{5}{2}\, \frac{R}{A},\qquad
-\frac{c_{p}}{c_{v}} = \frac{5}{3},
-\Tag{(199)}
-\]
-as is known to be the case for monatomic gases. It follows from~\Eq{(195)}
-that the kinetic energy~$L$ of the gas molecules is equal to
-\[
-L = E - E_{0} = \DPchg{\frac{3}{2}}{\tfrac{3}{2}} NkT\Add{.}
-\Tag{(200)}
-\]
-
-\Section{133.} The preceding relations, obtained simply by identifying
-the mechanical expression of the entropy~\Eq{(186)} with its thermodynamic
-expression~\Eq{(187)}, show the usefulness of the theory
-developed. In them an additive constant in the expression for
-the entropy is immaterial and hence the size~$G$ of the region element
-of probability does not matter. The hypothesis of quanta,
-however, goes further, since it fixes the absolute value of the
-entropy and thus leads to the same conclusion as the heat theorem
-%% -----File: 149.png---Folio 133-------
-of \Name{Nernst}. According to this theorem the ``characteristic function''
-of an ideal gas\footnote
- {\Eg, \Name{M.~Planck}, Vorlesungen über Thermodynamik, Leipzig, Veit und Comp., 1911,
- Sec.~287, equation~267.}
-is in our notation
-\[
-\Phi = S - \frac{E + pV}{T} = n\left(Ac_{p} \log T - R \log p + a - \frac{b}{T}\right),
-\]
-where $a$~denotes \Name{Nernst's} chemical constant, and $b$~the energy
-constant.
-
-On the other hand, the preceding formulć \Eq{(186)},~\Eq{(188)}, and~\Eq{(189)}
-give for the same function~$\Phi$ the following expression:
-\[
-\Phi = N\left(\frac{5}{2} k \log T - k \log p + a'\right) - \frac{E_{0}}{T}
-\]
-where for brevity $a'$~is put for:
-\[
-a' = k \log \left\{\frac{kN}{eG} (2\pi mk)^{\efrac{3}{2}}\right\}\Add{.}
-\]
-
-From a comparison of the two expressions for~$\Phi$ it is seen, by
-taking account of \Eq{(199)}~and~\Eq{(193)}, that they agree completely,
-provided
-\[
-% [** TN: Set on two lines in the original]
-a = \frac{N}{n} a'
- = R \log \left\{\frac{Nk^{\efrac{5}{2}}}{eG} (2\pi m)^{\efrac{3}{2}}\right\},
-\qquad
-b = \frac{E_{0}}{n}\Add{.}
-\Tag{(201)}
-\]
-This expresses the relation between the chemical constant~$a$ of
-the gas and the region element~$G$ of the probability.\footnote
- {Compare also \Name{O.~Sackur}, Annal.\ d.\ Physik\DPchg{,}{} \textbf{36}, p.~958, 1911, Nernst-Festschrift, p.~405,
- 1912, and \Name{H.~Tetrode}, Annal.\ d.\ Physik\DPchg{,}{} \textbf{38}, p.~434, 1912.}
-
-It is seen that $G$~is proportional to the total number,~$N$, of the
-molecules. Hence, if we put $G = Ng$, we see that $g$, the molecular
-region element, depends only on the chemical nature of the gas.
-
-Obviously the quantity~$g$ must be closely connected with the
-law, so far unknown, according to which the molecules act microscopically
-on one another. Whether the value of~$g$ varies with
-the nature of the molecules or whether it is the same for all
-kinds of molecules, may be left undecided for the present.
-%% -----File: 150.png---Folio 134-------
-
-If $g$~were known, \Name{Nernst's} chemical constant,~$a$, of the gas
-could be calculated from~\Eq{(201)} and the theory could thus be
-tested. For the present the reverse only is feasible, namely, to
-calculate $g$ from~$a$. For it is known that $a$~may be measured
-directly by the tension of the saturated vapor, which at sufficiently
-low temperatures satisfies the simple equation\footnote
- {\Name{M.~Planck}, \lc, Sec.~288, equation~271.}
-\[
-\log p = \frac{5}{2} \log T - \frac{Ar_{0}}{RT} + \frac{a}{R}
-\Tag{(202)}
-\]
-(where $r_{0}$ is the heat of vaporization of a mol at~$0°$ in calories).
-When $a$~has been found by measurement, the size~$g$ of the molecular
-region element is found from~\Eq{(201)} to be
-\[
-g = (2\pi m)^{\efrac{3}{2}} k^{\efrac{5}{2}} e^{-\efrac{a}{R} - 1}\Add{.}
-\Tag{(203)}
-\]
-Let us consider the dimensions of~$g$.
-
-According to~\Eq{(176)} $g$~is of the dimensions $[\erg^{3}\, \sec^{3}]$. The
-same follows from the present equation, when we consider that the
-dimension of the chemical constant~$a$ is not, as might at first be
-thought, that of~$R$, but, according to~\Eq{(202)}, that of $R \log \dfrac{p}{T^{\efrac{5}{2}}}$\Add{.}
-
-\Section{134.} To this we may at once add another quantitative relation.
-All the preceding calculations rest on the assumption that
-the distribution density~$w$ and hence also the constant~$\alpha$ in~\Eq{(183)}
-are small (\Sec{129}). Hence, if we take the value of~$\alpha$
-from~\Eq{(184)} and take account of \Eq{(188)},~\Eq{(189)} and~\Eq{(201)}, it follows
-that
-\[
-\frac{p}{T^{\efrac{5}{2}}} e^{-\efrac{a}{R} - 1} \quad\text{must be small.}
-\]
-When this relation is not satisfied, the gas cannot be in the ideal
-state. For the saturated vapor it follows then from~\Eq{(202)} that
-$e^{-\efrac{Ar_{0}}{RT}}$ is small. In order, then, that a saturated vapor may be
-assumed to be in the state of an ideal gas, the temperature~$T$
-must certainly be less than $\dfrac{A}{R}r_{0}$ or~$\dfrac{r_{0}}{2}$. Such a restriction is unknown
-to the classical thermodynamics.
-%% -----File: 151.png---Folio 135-------
-
-\Chapter{III}{Ideal Linear Oscillators}
-
-\Section{135.} The main problem of the theory of heat radiation is to
-determine the energy distribution in the normal spectrum of
-black radiation, or, what amounts to the same thing, to find the
-function which has been left undetermined in the general expression
-of \Name{Wien's} displacement law~\Eq{(119)}, the function which connects
-the entropy of a certain radiation with its energy. The
-purpose of this chapter is to develop some preliminary theorems
-leading to this solution. Now since, as we have seen in \Sec{48},
-the normal energy distribution in a diathermanous medium cannot
-be established unless the medium exchanges radiation with
-an emitting and absorbing substance, it will be necessary for the
-treatment of this problem to consider more closely the processes
-which cause the creation and the destruction of heat rays, that is,
-the processes of emission and absorption. In view of the complexity
-of these processes and the difficulty of acquiring knowledge of
-any definite details regarding them, it would indeed be quite
-hopeless to expect to gain any certain results in this way, if it
-were not possible to use as a reliable guide in this obscure region
-the law of \Name{Kirchhoff} derived in \Sec{51}. This law states that a
-vacuum completely enclosed by reflecting walls, in which any
-emitting and absorbing bodies are scattered in any arrangement
-whatever, assumes in the course of time the stationary state of
-black radiation, which is completely determined by one parameter
-only, namely, the temperature, and in particular does not
-depend on the number, the nature, and the arrangement of the
-material bodies present. Hence, for the investigation of the
-properties of the state of black radiation the nature of the bodies
-which are assumed to be in the vacuum is perfectly immaterial.
-In fact, it does not even matter whether such bodies really exist
-somewhere in nature, provided their existence and their properties
-are consistent with the laws of thermodynamics and electrodynamics.
-%% -----File: 152.png---Folio 136-------
-If, for any special arbitrary assumption regarding the
-nature and arrangement of emitting and absorbing systems, we
-can find a state of radiation in the surrounding vacuum which is
-distinguished by absolute stability, this state can be no other
-than that of black radiation.
-
-Since, according to this law, we are free to choose any system
-whatever, we now select from all possible emitting and absorbing
-systems the simplest conceivable one, namely, one consisting
-of a large number~$N$ of similar stationary oscillators, each consisting
-of two poles, charged with equal quantities of electricity of
-opposite sign, which may move relatively to each other on a fixed
-straight line, the axis of the oscillator.
-
-It is true that it would be more general and in closer accord with
-the conditions in nature to assume the vibrations to be those of an
-oscillator consisting of two poles, each of which has three degrees
-of freedom of motion instead of one, \ie, to assume the vibrations
-as taking place in space instead of in a straight line only. Nevertheless
-we may, according to the fundamental principle stated
-above, restrict ourselves from the beginning to the treatment of
-one single component, without fear of any essential loss of
-generality of the conclusions we have in view.
-
-It might, however, be questioned as a matter of principle,
-whether it is really permissible to think of the centers of mass
-of the oscillators as stationary, since, according to the kinetic
-theory of gases, all material particles which are contained in
-substances of finite temperature and free to move possess a certain
-finite mean kinetic energy of translatory motion. This
-objection, however, may also be removed by the consideration
-that the velocity is not fixed by the kinetic energy alone. We
-need only think of an oscillator as being loaded, say at its positive
-pole, with a comparatively large inert mass, which is perfectly
-neutral electrodynamically, in order to decrease its velocity for a
-given kinetic energy below any preassigned value whatever. Of
-course this consideration remains valid also, if, as is now frequently
-done, all inertia is reduced to electrodynamic action. For this
-action is at any rate of a kind quite different from the one to be
-considered in the following, and hence cannot influence it.
-
-Let the state of such an oscillator be completely determined
-by its moment~$f(t)$, that is, by the product of the electric charge
-%% -----File: 153.png---Folio 137-------
-of the pole situated on the positive side of the axis and the pole
-distance, and by the derivative of~$f$ with respect to the time or
-\[
-\frac{df(t)}{dt} = \dot{f}(t).
-\Tag{(204)}
-\]
-Let the energy of the oscillator be of the following simple form:
-\[
-U = \tfrac{1}{2} Kf^{2} + \tfrac{1}{2} L \dot{f}^{2},
-\Tag{(205)}
-\]
-where $K$~and~$L$ denote positive constants, which depend on the
-nature of the oscillator in some way that need not be discussed
-at this point.
-
-If during its vibration an oscillator neither absorbed nor
-emitted any energy, its energy of vibration,~$U$, would remain
-constant, and we would have:
-\[
-dU = Kf\, df + L \dot{f}\, d\dot{f} = 0,
-\Tag{(205a)}
-\]
-or, on account of~\Eq{(204)},
-\[
-Kf(t) + L \ddot{f}(t) = 0.
-\Tag{(206)}
-\]
-The general solution of this differential equation is found to be a
-purely periodical vibration:
-\[
-f = C \cos(2\pi \nu t - \theta)
-\Tag{(207)}
-\]
-where $C$~and~$\theta$ denote the integration constants and $\nu$~the number
-of vibrations per unit time:
-\[
-\nu = \frac{1}{2\pi} \sqrt{\frac{K}{L}}\Add{.}
-\Tag{(208)}
-\]
-
-\Section{136.} If now the assumed system of oscillators is in a space
-traversed by heat rays, the energy of vibration,~$U$, of an oscillator
-will not in general remain constant, but will be always changing
-by absorption and emission of energy. Without, for the present,
-considering in detail the laws to which these processes are subject,
-let us consider any one arbitrarily given thermodynamic state
-of the oscillators and calculate its entropy, irrespective of the
-surrounding field of radiation. In doing this we proceed entirely
-according to the principle advanced in the two preceding chapters,
-allowing, however, at every stage for the conditions caused by
-the peculiarities of the case in question.
-
-The first question is: What determines the thermodynamic
-state of the system considered? For this purpose, according to
-%% -----File: 154.png---Folio 138-------
-\Sec{124}, the numbers $N_{1}$,~$N_{2}$, $N_{3},~\dots$ of the oscillators,
-which lie in the region elements $1$,~$2$, $3,~\dots$ of the ``state
-space'' must be given. The state space of an oscillator contains
-those coordinates which determine the microscopic state of an
-oscillator. In the case in question these are only two in number,
-namely, the moment~$f$ and the rate at which it varies,~$\dot{f}$, or instead
-of the latter the quantity
-\[
-\psi = L \dot{f},
-\Tag{(209)}
-\]
-which is of the dimensions of an impulse. The region element
-of the state plane is, according to the hypothesis of quanta
-(\Sec{126}), the double integral
-\[
-\iint df\, d\psi = h.
-\Tag{(210)}
-\]
-The quantity~$h$ is the same for all region elements. \textit{A~priori},
-it might, however, depend also on the nature of the system considered,
-for example, on the frequency of the oscillators. The
-following simple consideration, however, leads to the assumption
-that $h$~is a universal constant. We know from the generalized
-displacement law of \Name{Wien} (equation~\DPtypo{119}{\Eq{(119)}}) that in the universal
-function, which gives the entropy radiation as dependent on the
-energy radiation, there must appear a universal constant of the
-dimension $\dfrac{c^{3} \ssfu}{\nu^{3}}$ and this is of the dimension of a quantity of action\footnote
- {The quantity from which the principle of \emph{least action} takes its name. (Tr.)}
-($\erg\, \sec.$). Now, according to~\Eq{(210)}, the quantity~$h$ has precisely
-this dimension, on which account we may denote it as ``element
-of action'' or ``quantity element of action.'' Hence, unless a
-second constant also enters, $h$~cannot depend on any other physical
-quantities.
-
-\Section{137.} The principal difference, compared with the calculations
-for an ideal gas in the preceding chapter, lies in the fact that we
-do not now assume the distribution densities $w_{1}$,~$w_{2}$, $w_{3}~\dots$
-of the oscillators among the separate region elements to vary but
-little from region to region as was assumed in \Sec{129}. Accordingly
-the $w$'s are not small, but finite proper fractions, and the
-summation over the region elements cannot be written as an
-integration.
-%% -----File: 155.png---Folio 139-------
-
-In the first place, as regards the shape of the region elements,
-the fact that in the case of undisturbed vibrations of an oscillator
-the phase is always changing, whereas the amplitude remains
-constant, leads to the conclusion that, for the macroscopic state
-of the oscillators, the amplitudes only, not the phases, must be
-considered, or in other words the region elements in the $f\psi$~plane
-are bounded by the curves $C = \const.$, that is, by ellipses, since
-from \Eq{(207)}~and~\Eq{(209)}
-\[
-\left(\frac{f}{C}\right)^{2} + \left(\frac{\psi}{2\pi\nu LC}\right)^{2} = 1.
-\Tag{(211)}
-\]
-
-The semi-axes of such an ellipse are:
-\[
-a = C \quad\text{and}\quad
-b = 2\pi\nu LC.
-\Tag{(212)}
-\]
-Accordingly the region elements $1$,~$2$, $3,~\dots$ $n~\dots$\DPnote{** regularize comas?}
-are the concentric, similar, and similarly situated elliptic rings,
-which are determined by the increasing values of~$C$:
-\[
-0,\ C_{1},\ C_{2},\ C_{3},\ \dots\ C_{n-1},\ C_{n}\ \dots.
-\Tag{(213)}
-\]
-{\Loosen The $n$th~region element is that which is bounded by the ellipses
-$C = C_{n-1}$ and $C = C_{n}$. The first region element is the full
-ellipse~$C_{1}$. All these rings have the same area~$h$, which is found
-by subtracting the area of the full ellipse~$C_{n-1}$ from that of the
-full ellipse~$C_{n}$; hence}
-\[
-h = (a_{n}b_{n} - a_{n-1}b_{n-1})\pi
-\]
-or, according to~\Eq{(212)},
-\[
-h = (C_{n}^{2} - C_{n-1}^{2}) 2\pi^{2}\nu L,
-\]
-where $n = 1$, $2$, $3,~\dots$.
-
-From the additional fact that $C_{0} = 0$, it follows that:
-\[
-C_{n}^{2} = \frac{nh}{2\pi^{2}\nu L}.
-\Tag{(214)}
-\]
-Thus the semi-axes of the bounding ellipses are in the ratio of
-the square roots of the integral numbers.
-
-\Section{138.} The thermodynamic state of the system of oscillators
-is fixed by the fact that the values of the distribution densities
-$w_{1}$,~$w_{2}$, $w_{3},~\dots$ of the oscillators among the separate
-region elements are given. \emph{Within} a region element the distribution
-of the oscillators is according to the law of elemental
-chaos (\Sec{122}), \ie,~it is approximately \emph{uniform}.
-%% -----File: 156.png---Folio 140-------
-
-These data suffice for calculating the entropy~$S$ as well as the
-energy~$E$ of the system in the given state, the former quantity
-directly from~\Eq{(173)}, the latter by the aid of~\Eq{(205)}. It must be
-kept in mind in the calculation that, since the energy varies
-appreciably within a region element, the energy~$E_{n}$ of all those
-oscillators which lie in the $n$th~region element is to be found by an
-integration. Then the whole energy~$E$ of the system is:
-\[
-E = E_{1} + E_{2} + \dots E_{n} + \dots.
-\Tag{(215)}
-\]
-$E_{n}$~may be calculated with the help of the law that within every
-region element the oscillators are uniformly distributed. If the
-$n$th~region element contains, all told, $N_{n}$~oscillators, there are per
-unit area $\dfrac{N_{n}}{h}$~oscillators and hence $\dfrac{N_{n}}{h}\, df ˇ d\psi$ per element of area.
-Hence we have:
-\[
-E_{n} = \frac{N_{n}}{h} \iint U\, df\, d\psi.
-\]
-In performing the integration, instead of $f$~and~$\psi$ we take $C$~and~$\phi$,
-as new variables, and since according to~\Eq{(211)},
-\[
-f = C \cos\phi\qquad \psi = 2\pi\nu LC \sin\phi
-\Tag{(216)}
-\]
-we get:
-\[
-E_{n} = 2\pi \nu L \frac{N_{n}}{h} \iint U C\, dC\, d\phi
-\]
-to be integrated with respect to~$\phi$ from $0$ to~$2\pi$ and with respect
-to~$C$ from $C_{n-1}$ to~$C_{n}$. If we substitute from \Eq{(205)},~\Eq{(209)}
-and~\Eq{(216)}
-\[
-U = \tfrac{1}{2} KC^{2},
-\Tag{(217)}
-\]
-we obtain by integration
-\[
-E_{n} = \frac{\pi^{2}}{2} \nu LK \frac{N_{n}}{h} (C_{n}^{4} - C_{n-1}^{4})
-\]
-and from \Eq{(214)}~and~\Eq{(208)}:
-\[
-E_{n} = N_{n}(n - \tfrac{1}{2})h\nu = Nw_{n}(n - \tfrac{1}{2})h\nu,
-%[** TN: No equation number in the original]
-\Tag{(218)}
-\]
-{\Loosen that is, the mean energy of an oscillator in the $n$th~region element
-is $(n - \frac{1}{2})h\nu$. This is exactly the arithmetic mean of the energies
-$(n - 1)h\nu$ and~$nh\nu$ which correspond to the two ellipses $C = C_{n-1}$
-and $C = C_{n}$ bounding the region, as may be seen from~\Eq{(217)}, if
-the values of $C_{n-1}$~and~$C_{n}$ are therein substituted from~\Eq{(214)}.}
-%% -----File: 157.png---Folio 141-------
-
-The total energy~$E$ is, according to~\Eq{(215)},
-\[
-E = Nh\nu \sum_{n=1}^{n=\infty} (n - \tfrac{1}{2}) w_{n}.
-\Tag{(219)}
-\]
-
-\Section{139.} Let us now consider the state of thermodynamic equilibrium
-of the oscillators. According to the second principle of
-thermodynamics, the entropy~$S$ is in that case a maximum for a
-given energy~$E$. Hence we assume $E$ in~\Eq{(219)} as given. Then
-from~\Eq{(179)} we have for the state of equilibrium:
-\[
-\delta S = 0 = \sum_{1}^{\infty} (\log w_{n} + 1)\, \delta w_{n},
-\]
-where according to \Eq{(167)}~and~\Eq{(219)}
-\[
-\sum_{1}^{\infty} \delta w_{n} = 0 \quad\text{and}\quad
-\sum_{1}^{\infty} (n - \tfrac{1}{2})\, \delta w_{n} = 0\Add{.}
-\]
-From these relations we find:
-\[
-\log w_{n} + \beta n + \const. = 0
-\]
-or
-\[
-w_{n} = \alpha \gamma^{n}.
-\Tag{(220)}
-\]
-The values of the constants $\alpha$~and~$\gamma$ follow from equations \Eq{(167)}
-and~\Eq{(219)}:
-\[
-\alpha = \frac{2Nh\nu}{2E - Nh\nu}\qquad
-\gamma = \frac{2E - Nh\nu}{2E + Nh\nu}.
-\Tag{(221)}
-\]
-Since $w_{n}$~is essentially positive it follows that equilibrium is not
-possible in the system of oscillators considered unless the total
-energy~$E$ has a greater value than~$\dfrac{Nh\nu}{2}$, that is unless the mean
-energy of the oscillators is at least~$\dfrac{h\nu}{2}$. This, according to~\Eq{(218)},
-is the mean energy of the oscillators lying in the first
-region element. In fact, in this extreme case all $N$ oscillators
-lie in the first region element, the region of smallest energy;
-within this element they are arranged uniformly.
-
-The entropy~$S$ of the system, which is in thermodynamic
-equilibrium, is found by combining \Eq{(173)} with \Eq{(220)}~and~\Eq{(221)}
-\begin{multline*}
-S = kN\biggl\{\left(\frac{E}{Nh\nu} + \frac{1}{2}\right) \log\left(\frac{E}{Nh\nu} + \frac{1}{2}\right) \\
- - \left(\frac{E}{Nh\nu} - \frac{1}{2}\right) \log\left(\frac{E}{Nh\nu} - \frac{1}{2}\right)\biggr\}\Add{.}
-\Tag{(222)}
-\end{multline*}
-%% -----File: 158.png---Folio 142-------
-
-\Section{140.} The connection between energy and entropy just obtained
-allows furthermore a certain conclusion as regards the temperature.
-For from the equation of the second principle of thermodynamics,
-$dS = \dfrac{dE}{T}$ and from differentiation of~\Eq{(222)} with respect
-to~$E$ it follows that
-\[
-E = N \frac{h\nu}{2}\, \frac{1 + e^{-\efrac{h\nu}{kT}}}{1 - e^{-\efrac{h\nu}{kT}}}
- = Nh\nu \left(\frac{1}{2} + \frac{1}{e^{\efrac{h\nu}{kT}} - 1}\right)\Add{.}
-%[** TN: No equation number in the original]
-\Tag{(223)}
-\]
-Hence, for the zero point of the absolute temperature~$E$ becomes,
-not~$0$, but~$N\dfrac{h\nu}{2}$. This is the extreme case discussed in the preceding
-paragraph, which just allows thermodynamic equilibrium
-to exist. That the oscillators are said to perform vibrations even
-at the temperature zero, the mean energy of which is as large as~$\dfrac{h\nu}{2}$
-and hence may become quite large for rapid vibrations, may
-at first sight seem strange. It seems to me, however, that certain
-facts point to the existence, inside the atoms, of vibrations
-independent of the temperature and supplied with appreciable
-energy, which need only a small suitable excitation to become
-evident externally. For example, the velocity, sometimes very
-large, of secondary cathode rays produced by Roentgen rays,
-and that of electrons liberated by photoelectric effect are independent
-of the temperature of the metal and of the intensity of
-the exciting radiation. Moreover the radioactive energies are
-also independent of the temperature. It is also well known that
-the close connection between the inertia of matter and its energy
-as postulated by the relativity principle leads to the assumption
-of very appreciable quantities of intra-atomic energy even at the
-zero of absolute temperature.
-
-For the extreme case, $T = \infty$, we find from~\Eq{(223)} that
-\[
-E = NkT,
-\Tag{(224)}
-\]
-\ie, the energy is proportional to the temperature and independent
-of the size of the quantum of action,~$h$, and of the nature of
-the oscillators. It is of interest to compare this value of the
-energy of vibration~$E$ of the system of oscillators, which holds at
-high temperatures, with the kinetic energy~$L$ of the molecular
-%% -----File: 159.png---Folio 143-------
-motion of an ideal monatomic gas at the same temperature as
-calculated in~\Eq{(200)}. From the comparison it follows that
-\[
-E = \tfrac{2}{3} L\Add{.}
-\Tag{(225)}
-\]
-This simple relation is caused by the fact that for high temperatures
-the contents of the hypothesis of quanta coincide with
-those of the classical statistical mechanics. Then the absolute
-magnitude of the region element, $G$~or $h$ respectively, becomes
-physically unimportant (compare \Sec{125}) and we have the
-simple law of equipartition of the energy among all variables in
-question (see below \Sec{169}). The factor~$\frac{2}{3}$ in equation~\Eq{(225)}
-is due to the fact that the kinetic energy of a moving molecule
-depends on three variables ($\xi$,~$\eta$,~$\zeta$,) and the energy of a vibrating
-oscillator on only two ($f$,~$\psi$).
-
-The heat capacity of the system of oscillators in question is,
-from~\Eq{(223)},
-\[
-\frac{dE}{dT} = Nk\left(\frac{h\nu}{kT}\right)^{2} \frac{e^{\efrac{h\nu}{kT}}}{(e^{\efrac{h\nu}{kT}} - 1)^{2}}\Add{.}
-\Tag{(226)}
-\]
-It vanishes for $T = 0$ and becomes equal to~$Nk$ for $T = \infty$.
-\Name{A.~Einstein}\footnote
- {\Name{A.~Einstein}, Ann.\ d.\ Phys.\ \textbf{22}, p.~180, 1907. Compare also \Name{M.~Born} und \Name{Th.\ von~Kárman},
- Phys.\ Zeitschr.\ \textbf{13}, p.~297, 1912.}
-has made an important application of this equation
-to the heat capacity of solid bodies, but a closer discussion of
-this would be beyond the scope of the investigations to be made
-in this book.
-
-For the constants $\alpha$~and~$\gamma$ in the expression~\Eq{(220)} for the distribution
-density~$w$ we find from~\Eq{(221)}:
-\[
-\alpha = e^{\efrac{h\nu}{kT}} - 1\qquad
-\gamma = e^{-\efrac{h\nu}{kT}}
-\Tag{(227)}
-\]
-and finally for the entropy~$S$ of our system as a function of temperature:
-\[
-S = kN \left\{\frac{\dfrac{h\nu}{kT}}{e^{\efrac{h\nu}{kT}-1}} - \log\left(1 - e^{-\efrac{h\nu}{kT}}\right)\right\}\Add{.}
-\Tag{(228)}
-\]
-%% -----File: 160.png---Folio 144-------
-
-\Chapter[Direct Calculation of the Entropy]
-{IV}{Direct Calculation of the Entropy in The
-Case of Thermodynamic Equilibrium}
-
-\Section{141.} In the calculation of the entropy of an ideal gas and of a
-system of resonators, as carried out in the preceding chapters, we
-proceeded in both cases, by first determining the entropy for an
-arbitrarily given state, then introducing the special condition of
-thermodynamic equilibrium, \ie, of the maximum of entropy,
-and then deducing for this special case an expression for the
-entropy.
-
-If the problem is only the determination of the entropy in the
-case of thermodynamic equilibrium, this method is a roundabout
-one, inasmuch as it requires a number of calculations, namely,
-the determination of the separate distribution densities $w_{1}$,~$w_{2}$,
-$w_{3},~\dots$ which do not enter separately into the final
-result. It is therefore useful to have a method which leads
-directly to the expression for the \emph{entropy} of a system in the state
-of thermodynamic equilibrium, without requiring any consideration
-of the \emph{state} of thermodynamic equilibrium. This method
-is based on an important general property of the thermodynamic
-probability of a state of equilibrium.
-
-We know that there exists between the entropy~$S$ and the thermodynamic
-probability~$W$ in any state whatever the general
-relation~\Eq{(164)}. In the state of thermodynamic equilibrium both
-quantities have maximum values; hence, if we denote the maximum
-values by a suitable index:
-\[
-S_{m} = k \log W_{m}.
-\Tag{(229)}
-\]
-It follows from the two equations that:
-\[
-\frac{W_{m}}{W} = e^{\efrac{S_{m} - S}{k}}\Add{.}
-\]
-Now, when the deviation from thermodynamic equilibrium is at
-all appreciable, $\dfrac{S_{m} - S}{k}$ is certainly a very large number. Accordingly
-%% -----File: 161.png---Folio 145-------
-$W_{m}$~is not only large but of a very high order large, compared
-with~$W$, that is to say: The thermodynamic probability
-of the state of equilibrium is enormously large compared with the
-thermodynamic probability of all states which, in the course of
-time, change into the state of equilibrium.
-
-This proposition leads to the possibility of calculating~$W_{m}$
-with an accuracy quite sufficient for the determination of~$S_{m}$,
-without the necessity of introducing the special condition of
-equilibrium. According to \Sec{123}, \textit{et~seq.}, $W_{m}$~is equal to the
-number of all different complexions possible in the state of thermodynamic
-equilibrium. This number is so enormously large compared
-with the number of complexions of all states deviating from
-equilibrium that we commit no appreciable error if we think of
-the number of complexions of all states, which as time goes on
-change into the state of equilibrium, \ie,~all states which are at
-all possible under the given external conditions, as being included
-in this number. The total number of all possible complexions
-may be calculated much more readily and directly than the
-number of complexions referring to the state of equilibrium only.
-
-\Section{142.} We shall now use the method just formulated to calculate
-the entropy, in the state of equilibrium, of the system of ideal
-linear oscillators considered in the last chapter, when the total
-energy~$E$ is given. The notation remains the same as above.
-
-We put then $W_{m}$ equal to the number of complexions of all
-stages which are at all possible with the given energy~$E$ of the
-system. Then according to~\Eq{(219)} we have the condition:
-\[
-E = h\nu \sum_{n=1}^{\infty} (n - \tfrac{1}{2}) N_{n}.
-\Tag{(230)}
-\]
-Whereas we have so far been dealing with the number of complexions
-with given~$N_{n}$, now the~$N_{n}$ are also to be varied in all ways
-consistent with the condition~\Eq{(230)}.
-
-The total number of all complexions is obtained in a simple
-way by the following consideration. We write, according to~\Eq{(165)},
-the condition~\Eq{(230)} in the following form:
-\[
-\frac{E}{h\nu} - \frac{N}{2} = \sum_{n=1}^{\infty} (n - 1) N_{n}
-\]
-%% -----File: 162.png---Folio 146-------
-or
-\[
-%[** TN: Set on two lines in the original]
-0 ˇ N_{1} + 1 ˇ N_{2} + 2 ˇ N_{3} + \dots + (n - 1)N_{n} + \dots
- = \frac{E}{h\nu} - \frac{N}{2} = P.
-\Tag{(231)}
-\]
-$P$~is a given large positive number, which may, without
-restricting the generality, be taken as an integer.
-
-According to \Sec{123} a complexion is a definite assignment of
-every individual oscillator to a definite region element $1$,~$2$,
-$3,~\dots$ of the state plane $(f, \psi)$. Hence we may characterize
-a certain complexion by thinking of the $N$~oscillators as
-being numbered from $1$ to~$N$ and, when an oscillator is assigned
-to the $n$th~region element, writing down the number of the
-oscillator $(n - 1)$~times. If in any complexion an oscillator is
-assigned to the first region element its number is not put down at
-all. Thus every complexion gives a certain row of figures, and
-\textit{vice versa} to every row of figures there corresponds a certain complexion.
-The position of the figures in the row is immaterial.
-
-What makes this form of representation useful is the fact that
-according to~\Eq{(231)} the number of figures in such a row is always
-equal to~$P$. Hence we have ``combinations with repetitions of
-$N$~elements taken $P$~at a time,'' whose total number is
-\[%[** Attn alignment of factors]
-\frac{N(N + 1)(N + 2) \dots (N + P - 1)}
- {\PadTo{N}{1} \PadTo{(N+1)}{2} \PadTo{(N+2)}{3} \dots \PadTo{(N + P - 1)}{P}}
- = \frac{(N + P - 1)!}{(N - 1)!\, P!}\Add{.}
-\Tag{(232)}
-\]
-If for example we had $N = 3$ and $P = 4$ all possible complexions
-would be represented by the rows of figures:
-\[
-\begin{array}{c*{2}{>{\qquad}c}}
-1111 & 1133 & 2222 \\
-1112 & 1222 & 2223 \\
-1113 & 1223 & 2233 \\
-1122 & 1233 & 2333 \\
-1123 & 1333 & 3333
-\end{array}
-\]
-
-The first row denotes that complexion in which the first oscillator
-lies in the $5$th~region element and the two others in the first.
-The number of complexions in this case is~$15$, in agreement with
-the formula.
-
-\Section{143.} For the entropy~$S$ of the system of oscillators which is
-%% -----File: 163.png---Folio 147-------
-in the state of thermodynamic equilibrium we thus obtain from
-equation~\Eq{(229)} since $N$~and~$P$ are large numbers:
-\[
-S = k \log\frac{(N + P)!}{N!\, P!}
-\]
-and by making use of Stirling's formula~\Eq{(171)}\footnote
- {Compare footnote to \PageRef{page}{124}. See also \PageRef{page}{218}.}
-\[
-S = kN \left\{\left(\frac{P}{N} + 1\right) \log\left(\frac{P}{N} + 1\right) - \frac{P}{N} \log \frac{P}{N}\right\}.
-\]
-If we now replace $P$ by~$E$ from~\Eq{(231)} we find for the entropy
-exactly the same value as given by~\Eq{(222)} and thus we have
-demonstrated in a special case both the admissibility and the
-practical usefulness of the method employed.\footnote
- {A complete mathematical discussion of the subject of this chapter has been given by
- \Name{H.~A. Lorentz}. Compare, \eg, Nature, \textbf{92}, p.~305, Nov.~6, 1913. (Tr.)}
-%% -----File: 164.png---Folio 148-------
-%[Blank Page]
-%% -----File: 165.png---Folio 149-------
-
-\Part[A System of Oscillators]{IV}{A System of Oscillators in a Stationary
-Field of Radiation}
-%% -----File: 166.png---Folio 150-------
-%[Blank Page]
-%% -----File: 167.png---Folio 151-------
-
-\Chapter[The Elementary Dynamical Law]
-{I}{The Elementary Dynamical Law for The
-Vibrations of an Ideal Oscillator.
-Hypothesis of Emission of Quanta}
-
-\Section{144.} All that precedes has been by way of preparation. Before
-taking the final step, which will lead to the law of distribution of
-energy in the spectrum of black radiation, let us briefly put
-together the essentials of the problem still to be solved. As we
-have already seen in \Sec{93}, the whole problem amounts to the
-determination of the temperature corresponding to a monochromatic
-radiation of given intensity. For among all conceivable
-distributions of energy the normal one, that is, the one
-peculiar to black radiation, is characterized by the fact that in it
-the rays of all frequencies have the same temperature. But the
-temperature of a radiation cannot be determined unless it be
-brought into thermodynamic equilibrium with a system of molecules
-or oscillators, the temperature of which is known from other
-sources. For if we did not consider any emitting and absorbing
-matter there would be no possibility of defining the entropy and
-temperature of the radiation, and the simple propagation of free
-radiation would be a reversible process, in which the entropy and
-temperature of the separate pencils would not undergo any
-change. (Compare below \Sec{166}.)
-
-Now we have deduced in the preceding section all the characteristic
-properties of the thermodynamic equilibrium of a system
-of ideal oscillators. Hence, if we succeed in indicating a state of
-radiation which is in thermodynamic equilibrium with the system
-of oscillators, the temperature of the radiation can be no other
-than that of the oscillators, and therewith the problem is solved.
-
-\Section{145.} Accordingly we now return to the considerations of \Sec{135}
-and assume a system of ideal linear oscillators in a stationary
-field of radiation. In order to make progress along the line
-proposed, it is necessary to know the elementary dynamical law,
-%% -----File: 168.png---Folio 152-------
-according to which the mutual action between an oscillator and the
-incident radiation takes place, and it is moreover easy to see that
-this law cannot be the same as the one which the classical electrodynamical
-theory postulates for the vibrations of a linear Hertzian
-oscillator. For, according to this law, all the oscillators, when
-placed in a stationary field of radiation, would, since their
-properties are exactly similar, assume the same energy of vibration,
-if we disregard certain irregular variations, which, however,
-will be smaller, the smaller we assume the damping constant of
-the oscillators, that is, the more pronounced their natural vibration
-is. This, however, is in direct contradiction to the
-definite discrete values of the distribution densities $w_{1}$,~$w_{2}$,
-$w_{3},~\dots$ which we have found in \Sec{139} for the stationary
-state of the system of oscillators. The latter allows us to conclude
-with certainty that in the dynamical law to be established the
-quantity element of action~$h$ must play a characteristic part.
-Of what nature this will be cannot be predicted \textit{a~priori}; this much,
-however, is certain, that the only type of dynamical law admissible
-is one that will give for the stationary state of the oscillators
-exactly the distribution densities~$w$ calculated previously. It is in
-this problem that the question of the dynamical significance of the
-quantum of action~$h$ stands for the first time in the foreground,
-a question the answer to which was unnecessary for the calculations
-of the preceding sections, and this is the principal reason
-why in our treatment the preceding section was taken up first.
-
-\Section{146.} In establishing the dynamical law, it will be rational to
-proceed in such a way as to make the deviation from the laws of
-classical electrodynamics, which was recognized as necessary, as
-slight as possible. Hence, as regards the influence of the field of
-radiation on an oscillator, we follow that theory closely. If the
-oscillator vibrates under the influence of any external electromagnetic
-field whatever, its energy~$U$ will not in general remain
-constant, but the energy equation~\Eq{(205a)} must be extended to
-include the work which the external electromagnetic field does on
-the oscillator, and, if the axis of the electric doublet coincides with
-the $z$-axis, this work is expressed by the term $\ssfE_{z}\, df = \ssfE_{z} \dot{f}\, dt$.
-Here $\ssfE_{z}$~denotes the $z$~component of the external electric field-strength
-at the position of the oscillator, that is, that electric
-field-strength which would exist at the position of the oscillator,
-%% -----File: 169.png---Folio 153-------
-if the latter were not there at all. The other components of
-the external field have no influence on the vibrations of the
-oscillator.
-
-Hence the complete energy equation reads:
-\[
-Kf\, df + L \dot{f}\, d\dot{f} = \ssfE_{z}\, df
-\]
-or:
-\[
-Kf + L \ddot{f} = \ssfE_{z},
-\Tag{(233)}
-\]
-and the energy absorbed by the oscillator during the time element~$dt$
-is:
-\[
-\ssfE_{z} \dot{f}\, dt\Add{.}
-\Tag{(234)}
-\]
-
-\Section{147.} While the oscillator is absorbing it must also be emitting,
-for otherwise a stationary state would be impossible. Now, since
-in the law of absorption just assumed the hypothesis of quanta
-has as yet found no room, it follows that it must come into play
-in some way or other in the emission of the oscillator, and this is
-provided for by the introduction of the hypothesis of emission of
-quanta. That is to say, we shall assume that the emission does
-not take place continuously, as does the absorption, but that it
-occurs only at certain definite times, suddenly, in pulses, and in
-particular we assume that an oscillator can emit energy only at
-the moment when its energy of vibration,~$U$, is an integral multiple~$n$
-of the quantum of energy, $\epsilon = h\nu$. Whether it then really
-emits or whether its energy of vibration increases further by
-absorption will be regarded as a matter of chance. This will not
-be regarded as implying that there is no causality for emission;
-but the processes which cause the emission will be assumed to be
-of such a concealed nature that for the present their laws cannot
-be obtained by any but statistical methods. Such an assumption
-is not at all foreign to physics; it is, \eg, made in the atomistic
-theory of chemical reactions and the disintegration theory of
-radioactive substances.
-
-It will be assumed, however, that if emission does take place,
-the entire energy of vibration,~$U$, is emitted, so that the vibration
-of the oscillator decreases to zero and then increases again by
-further absorption of radiant energy.
-
-It now remains to fix the law which gives the probability that
-an oscillator will or will not emit at an instant when its energy has
-reached an integral multiple of~$\epsilon$. For it is evident that the statistical
-state of equilibrium, established in the system of oscillators
-%% -----File: 170.png---Folio 154-------
-by the assumed alternations of absorption and emission
-will depend on this law; and evidently the mean energy~$U$ of the
-oscillators will be larger, the larger the probability that in such a
-critical state no emission takes place. On the other hand, since
-the mean energy~$U$ will be larger, the larger the intensity of the
-field of radiation surrounding the oscillators, we shall state the
-%[** Thm]
-law of emission as follows: \emph{The ratio of the probability that no
-emission takes place to the probability that emission does take place
-is proportional to the intensity~$\ssfI$} of the vibration which excites the
-oscillator and which was defined in equation~\Eq{(158)}. The value
-of the constant of proportionality we shall determine later on by
-the application of the theory to the special case in which the
-energy of vibration is very large. For in this case, as we know,
-the familiar formulć of the classical dynamics hold for any period
-of the oscillator whatever, since the quantity element of action~$h$
-may then, without any appreciable error, be regarded as infinitely
-small.
-
-These statements define completely the way in which the
-radiation processes considered take place, as time goes on, and
-the properties of the stationary state. We shall now, in the
-first place, consider in the second chapter the absorption, and,
-then, in the third chapter the emission and the stationary distribution
-of energy, and, lastly, in the fourth chapter we shall
-compare the stationary state of the system of oscillators thus
-found with the thermodynamic state of equilibrium which was
-derived directly from the hypothesis of quanta in the preceding
-part. If we find them to agree, the hypothesis of emission of
-quanta may be regarded as admissible.
-
-It is true that we shall not thereby prove that this hypothesis
-represents the only possible or even the most adequate expression
-of the elementary dynamical law of the vibrations of the oscillators.
-On the contrary I think it very probable that it may be
-greatly improved as regards form and contents. There is, however,
-no method of testing its admissibility except by the investigation
-of its consequences, and as long as no contradiction in
-itself or with experiment is discovered in it, and as long as no
-more adequate hypothesis can be advanced to replace it, it may
-justly claim a certain importance.
-%% -----File: 171.png---Folio 155-------
-
-\Chapter{II}{Absorbed Energy}
-
-\Section{148.} Let us consider an oscillator which has just completed an
-emission and which has, accordingly, lost all its energy of vibration.
-If we reckon the time~$t$ from this instant then for $t = 0$ we
-have $f = 0$ and $\DPchg{df/dt}{\dfrac{df}{dt}} = 0$, and the vibration takes place according
-to equation~\Eq{(233)}. Let us write~$\ssfE_{z}$ as in~\Eq{(149)} in the form of a
-Fourier's series:
-\[
-\ssfE_{z} = \sum_{n=1}^{n=\infty}
- \left[A_{n} \cos\frac{2\pi nt}{\ssfT}
- + B_{n} \sin\frac{2\pi nt}{\ssfT}\right],
-\Tag{(235)}
-\]
-where $\ssfT$~may be chosen very large, so that for all times~$t$ considered
-$t < \ssfT$. Since we assume the radiation to be stationary,
-the constant coefficients $A_{n}$~and~$B_{n}$ depend on the ordinal numbers~$n$
-in a wholly irregular way, according to the hypothesis of
-natural radiation (\Sec{117}). The partial vibration with the
-ordinal number~$n$ has the frequency~$\nu$, where
-\[
-\omega = 2\pi \nu = \frac{2\pi n}{\ssfT},
-\Tag{(236)}
-\]
-while for the frequency~$\nu_{0}$ of the natural period of the oscillator
-\[
-\omega_{0} = 2\pi \nu_{0} = \sqrt{\frac{K}{L}}.
-\]
-
-Taking the initial condition into account, we now obtain as
-the solution of the differential equation~\Eq{(233)} the expression
-\[
-f = \sum_{1}^{\infty}
- [a_{n} (\cos \omega t - \cos \omega_{0} t)
- + b_{n} (\sin \omega t - \frac{\omega}{\omega_{0}} \sin \omega_{0}t)],
-\Tag{(237)}
-\]
-where
-\[
-a_{n} = \frac{A_{n}}{L(\omega_{0}^{2} - \omega^{2})},\qquad
-b_{n} = \frac{B_{n}}{L(\omega_{0}^{2} - \omega^{2})}.
-\Tag{(238)}
-\]
-%% -----File: 172.png---Folio 156-------
-This represents the vibration of the oscillator up to the instant
-when the next emission occurs.
-
-The coefficients $a_{n}$~and~$b_{n}$ attain their largest values when $\omega$~is
-nearly equal to~$\omega_{0}$. (The case $\omega = \omega_{0}$ may be excluded by
-assuming at the outset that $\nu_{0} \ssfT$~is not an integer.)
-
-\Section{149.} Let us now calculate the total energy which is absorbed
-by the oscillator in the time from $t = 0$ to $t = \tau$, where
-\[
-\omega_{0} \tau \text{ is large}.
-\Tag{(239)}
-\]
-According to equation~\Eq{(234)}, it is given by the integral
-\[
-\int_{0}^{\tau} \ssfE_{z} \frac{df}{dt}\, dt,
-\Tag{(240)}
-\]
-the value of which may be obtained from the known expression
-for~$\ssfE_{z}$ \Eq{(235)} and from
-\[
-\frac{df}{dt} = \sum_{1}^{\infty}
- [a_{n}(-\omega \sin \omega t + \omega_{0} \sin \omega_{0} t)
- + b_{n}( \omega \cos \omega t - \omega \cos \omega_{0} t)].
-\Tag{(241)}
-\]
-By multiplying out, substituting for $a_{n}$~and~$b_{n}$ their values from~\Eq{(238)},
-and leaving off all terms resulting from the multiplication
-of two constants $A_{n}$ and~$B_{n}$, this gives for the absorbed energy
-the following value:
-\begin{multline*}
-%[** TN: Moved + from end of first line to beginning of second line]
-\frac{1}{L} \int_{0}^{\tau} dt \sum_{1}^{\infty}\biggl[
- \frac{A_{n}^{2}}{\omega_{0}^{2} - \omega^{2}} \cos \omega t (-\omega \sin \omega t + \omega_{0} \sin \omega_{0} t) \\
- + \frac{B_{n}^{2}}{\omega_{0}^{2} - \omega^{2}} \sin \omega t ( \omega \cos \omega t - \omega \cos \omega_{0} t)\biggr].
-\Tag{(241a)}
-\end{multline*}
-In this expression the integration with respect to~$t$ may be performed
-term by term. Substituting the limits $\tau$ and~$0$ it gives
-\begin{align*}
- &\frac{1}{L} \sum_{1}^{\infty} \frac{A_{n}^{2}}{\omega_{0}^{2} - \omega^{2}} \left[-\frac{\sin^{2} \omega \tau}{2} + \omega_{0} \left(\frac{\sin^{2}\dfrac{\omega_{0} + \omega}{2} \tau}{\omega_{0} + \omega} + \frac{\sin^{2}\dfrac{\omega_{0} - \omega}{2} \tau}{\omega_{0} - \omega}\right)\right]\\
-%
-+ &\frac{1}{L} \sum_{1}^{\infty} \frac{B_{n}^{2}}{\omega_{0}^{2} - \omega^{2}} \left[\Neg \frac{\sin^{2} \omega \tau}{2} - \omega \left(\frac{\sin^{2}\dfrac{\omega_{0} + \omega}{2} \tau}{\omega_{0} + \omega} - \frac{\sin^{2}\dfrac{\omega_{0} - \omega}{2} \tau}{\omega_{0} - \omega}\right)\right].
-\end{align*}
-%% -----File: 173.png---Folio 157-------
-In order to separate the terms of different order of magnitude, this
-expression is to be transformed in such a way that the difference
-$\omega_{0} - \omega$ will appear in all terms of the sum. This gives
-\begin{multline*}
-% [** TN: Reformatted slightly from the original]
-\frac{1}{L} \sum_{1}^{\infty} \frac{A_{n}^{2}}{\omega_{0}^{2} - \omega^{2}}
-\biggl[\frac{\omega_{0} - \omega}{2(\omega_{0} + \omega)} \sin^{2} \omega \tau
- + \frac{\omega_{0}}{\omega_{0} + \omega} \sin \frac{\omega_{0} - \omega}{2} \tau ˇ \sin \frac{\omega_{0} + 3\omega}{2} \tau \\
- \shoveright{+ \frac{\omega_{0}}{\omega_{0} - \omega} \sin^{2} \frac{\omega_{0} - \omega}{2} \tau \biggr]\phantom{.}} \\
-%
-+ \frac{1}{L} \sum_{1}^{\infty} \frac{B_{n}^{2}}{\omega_{0}^{2} - \omega^{2}}
-\biggl[\frac{\omega_{0} - \omega}{2(\omega_{0} + \omega)} \sin^{2} \omega \tau
- - \frac{\omega}{\omega_{0} + \omega} \sin \frac{\omega_{0} - \omega}{2} \tau ˇ \sin \frac{\omega_{0} + 3\omega}{2} \tau \\
- + \frac{\omega}{\omega_{0} - \omega} \sin^{2} \frac{\omega_{0} - \omega}{2} \tau \biggr].
-\end{multline*}
-The summation with respect to the ordinal numbers~$n$ of the
-Fourier's series may now be performed. Since the fundamental
-period~$\ssfT$ of the series is extremely large, there corresponds to
-the difference of two consecutive ordinal numbers, $\Delta n = 1$ only
-a very small difference of the corresponding values of $\omega$,~$d\omega$,
-namely, according to~\Eq{(236)},
-\[
-\Delta n = 1 = \ssfT\, d\nu = \frac{\ssfT\, d\omega}{2\pi},
-\Tag{(242)}
-\]
-and the summation with respect to~$n$ becomes an integration with
-respect to~$\omega$.
-
-The last summation with respect to~$A_{n}$ may be rearranged as
-the sum of three series, whose orders of magnitude we shall first
-compare. So long as only the order is under discussion we may
-disregard the variability of the~$A_{n}^{2}$ and need only compare the
-three integrals
-\begin{gather*}
-\int_{0}^{\infty} d\omega\, \frac{\sin^{2} \omega\tau}{2(\omega_{0} + \omega)^{2}} = J_{1}, \\
-\int_{0}^{\infty} d\omega\, \frac{\omega_{0}}{(\omega_{0} + \omega)^{2} (\omega_{0} - \omega)}
- \sin\frac{\omega_{0} - \omega}{2} \tau ˇ \sin\frac{\omega_{0} + 3\omega}{2} \tau = J_{2}, \\
-\intertext{and}
-\int_{0}^{\infty} d\omega\, \frac{\omega_{0}}{(\omega_{0} + \omega)(\omega_{0} - \omega)^{2}}
- \sin^{2} \frac{\omega_{0} - \omega}{2} \tau = J_{3}.
-\end{gather*}
-%% -----File: 174.png---Folio 158-------
-The evaluation of these integrals is greatly simplified by the fact
-that, according to~\Eq{(239)}, $\omega_{0}\tau$~and therefore also~$\omega\tau$ are large numbers,
-at least for all values of~$\omega$ which have to be considered.
-Hence it is possible to replace the expression $\sin^{2} \omega \tau$ in the integral~$J_{1}$
-by its mean value~$\frac{1}{2}$ and thus we obtain:
-\[
-J_{1} = \frac{1}{4 \omega_{0}}\Add{.}
-\]
-It is readily seen that, on account of the last factor, we obtain
-\[
-J_{2} = 0
-\]
-for the second integral.
-
-In order finally to calculate the third integral~$J_{3}$ we shall lay
-off in the series of values of~$\omega$ on both sides of~$\omega_{0}$ an interval
-extending from $\omega_{1}$ ($< \omega_{0}$) to $\omega_{2}$ ($> \omega_{0}$) such that
-\[
-\frac{\omega_{0} - \omega_{1}}{\omega_{0}} \quad\text{and}\quad
-\frac{\omega_{2} - \omega_{0}}{\omega_{0}} \quad\text{are small,}
-\Tag{(243)}
-\]
-and simultaneously
-\[
-(\omega_{0} - \omega_{1}) \tau \quad\text{and}\quad
-(\omega_{2} - \omega_{0}) \tau \quad\text{are large.}
-\Tag{(244)}
-\]
-This can always be done, since $\omega_{0}\tau$~is large. If we now break up
-the integral~$J_{3}$ into three parts, as follows:
-\[
-J_{3} = \int_{0}^{\infty}
- = \int_{0}^{\omega_{1}}
- + \int_{\omega_{1}}^{\omega_{2}}
- + \int_{\omega_{2}}^{\infty},
-\]
-it is seen that in the first and third partial integral the expression
-$\sin^{2} \dfrac{\omega_{0} - \omega}{2} \tau$ may, because of the condition~\Eq{(244)}, be replaced by its
-mean value~$\frac{1}{2}$. Then the two partial integrals become:
-\[
-\int_{0}^{\omega_{1}} \frac{\omega_{0}\, d\omega}{2(\omega_{0} + \omega)(\omega_{0} - \omega)^{2}}
-\quad\text{and}\quad
-\int_{\omega_{2}}^{\infty} \frac{\omega_{0}\, d\omega}{2(\omega_{0} + \omega)(\omega_{0} - \omega)^{2}}.
-\Tag{(245)}
-\]
-These are certainly smaller than the integrals:
-\[
-\int_{0}^{\omega_{1}} \frac{d\omega}{2(\omega_{0} - \omega)^{2}}
-\quad\text{and}\quad
-\int_{\omega_{2}}^{\infty} \frac{d\omega}{2(\omega_{0} - \omega)^{2}}
-\]
-%% -----File: 175.png---Folio 159-------
-which have the values
-\[
-\frac{1}{2}\, \frac{\omega_{1}}{\omega_{0} (\omega_{0} - \omega_{1})}
-\quad\text{and}\quad
-\frac{1}{2(\omega_{2} - \omega_{0})}
-\Tag{(246)}
-\]
-respectively. We must now consider the middle one of the three
-partial integrals:
-\[
-\int_{\omega_{1}}^{\omega_{2}} d\omega\, \frac{\omega_{0}}{(\omega_{0} + \omega)(\omega_{0} - \omega)^{2}} ˇ \sin^{2} \frac{\omega_{0} - \omega}{2} \tau.
-\]
-Because of condition~\Eq{(243)} we may write instead of this:
-\[
-\int_{\omega_{1}}^{\omega_{2}} d\omega ˇ \frac{\sin^{2} \dfrac{\omega_{0} - \omega}{2} \tau}{2(\omega_{0} - \omega)^{2}}
-\]
-and by introducing the variable of integration~$x$, where
-\[
-x = \frac{\omega - \omega_{0}}{2} \tau
-\]
-and taking account of condition~\Eq{(244)} for the limits of the integral,
-we get:
-\[
-\frac{\tau}{4} \int_{-\infty}^{+\infty} \frac{\sin^{2} x\, dx}{x^{2}}
- = \frac{\tau}{4}\, \pi.
-\]
-This expression is of a higher order of magnitude than the expressions~\Eq{(246)}
-and hence of still higher order than the partial integrals~\Eq{(245)}
-and the integrals $J_{1}$~and~$J_{2}$ given above. Thus for
-our calculation only those values of~$\omega$ will contribute an appreciable
-part which lie in the interval between $\omega_{1}$~and~$\omega_{2}$, and hence
-we may, because of~\Eq{(243)}, replace the separate coefficients $A_{n}^{2}$~and~$B_{n}^{2}$
-in the expression for the total absorbed energy by their mean
-values $A_{0}^{2}$~and~$B_{0}^{2}$ in the neighborhood of~$\omega_{0}$ and thus, by taking
-account of~\Eq{(242)}, we shall finally obtain for the total value of the
-energy absorbed by the oscillator in the time~$\tau$:
-\[
-\frac{1}{L}\, \frac{\tau}{8} (A_{0}^{2} + B_{0}^{2}) \ssfT\Add{.}
-\Tag{(247)}
-\]
-If we now, as in \Eq{(158)}, define $\ssfI$, the ``intensity of the vibration
-%% -----File: 176.png---Folio 160-------
-exciting the oscillator,'' by spectral resolution of the mean value
-of the square of the exciting field-strength~$\ssfE_{z}$:
-\[
-\bar{\ssfE_{z}^{2}} = \int_{0}^{\infty} \ssfI_{\nu}\, d\nu
-\Tag{(248)}
-\]
-we obtain from \Eq{(235)}~and~\Eq{(242)}:
-\[
-\bar{\ssfE_{z}^{2}}
- = \tfrac{1}{2} \sum_{1}^{\infty} (A_{n}^{2} + B_{n}^{2})
- = \tfrac{1}{2} \int_{0}^{\infty} (A_{n}^{2} + B_{n}^{2}) \ssfT\, d\nu,
-\]
-and by comparison with~\Eq{(248)}:
-\[
-\ssfI = \tfrac{1}{2} (A_{0}^{2} + B_{0}^{2}) \ssfT.
-\]
-Accordingly from~\Eq{(247)} the energy absorbed in the time~$\tau$ becomes:
-\[
-\frac{\ssfI}{4L} \tau,
-\]
-%[** Thm]
-that is, \emph{in the time between two successive emissions, the energy~$U$
-of the oscillator increases uniformly with the time}, according to the
-law
-\[
-\frac{dU}{dt} = \frac{\ssfI}{4L} = a.
-\Tag{(249)}
-\]
-Hence the energy absorbed by all $N$ oscillators in the time~$dt$ is:
-\[
-\frac{N \ssfI}{4L}\, dt = Na\, dt.
-\Tag{(250)}
-\]
-%% -----File: 177.png---Folio 161-------
-
-\Chapter{III}{Emitted Energy. Stationary State}
-
-\Section{150.} Whereas the absorption of radiation by an oscillator
-takes place in a perfectly continuous way, so that the energy of
-the oscillator increases continuously and at a constant rate, for
-its emission we have, in accordance with \Sec{147}, the following
-law: The oscillator emits in irregular intervals, subject to the
-laws of chance; it emits, however, only at a moment when its
-energy of vibration is just equal to an integral multiple~$n$ of the
-elementary quantum $\epsilon = h\nu$, and then it always emits its whole
-energy of vibration~$n\epsilon$.
-
-We may represent the whole process by the following figure in
-which the abscissć represent the time~$t$ and the ordinates the
-energy
-\[
-U = n\epsilon + \rho,\quad (\rho < \epsilon)
-\Tag{(251)}
-\]
-\Figure[3.5in]{7}
-of a definite oscillator under consideration. The oblique parallel
-lines indicate the continuous increase of energy at a constant
-rate\DPtypo{.}{}
-\[
-\frac{dU}{dt} = \frac{d\rho}{dt} = a,
-\Tag{(252)}
-\]
-%% -----File: 178.png---Folio 162-------
-which is, according to~\Eq{(249)}, caused by absorption at a constant
-rate. Whenever this straight line intersects one of the parallels
-to the axis of abscissć $U = \epsilon$, $U = 2\epsilon,~\dots$ emission may
-possibly take place, in which case the curve drops down to zero
-at that point and immediately begins to rise again.
-
-\Section{151.} Let us now calculate the most important properties of
-the state of statistical equilibrium thus produced. Of the $N$~oscillators
-situated in the field of radiation the number of those
-whose energy at the time~$t$ lies in the interval between $U = n\epsilon + \rho$
-and $U + dU = n\epsilon + \rho + d\rho$ may be represented by
-\[
-NR_{n, \rho}\, d\rho,
-\Tag{(253)}
-\]
-where $R$~depends in a definite way on the integer~$n$ and the quantity~$\rho$
-which varies continuously between $0$ and~$\epsilon$.
-
-After a time $dt = \dfrac{d\rho}{a}$ all the oscillators will have their energy increased
-by~$d\rho$ and hence they will all now lie outside of the energy
-interval considered. On the other hand, during the same time~$dt$,
-all oscillators whose energy at the time~$t$ was between $n\epsilon + \rho - d\rho$
-and $n\epsilon + \rho$ will have entered that interval. The number of all
-these oscillators is, according to the notation used above,
-\[
-NR_{n, \rho - d\rho}\, d\rho.
-\Tag{(254)}
-\]
-Hence this expression gives the number of oscillators which are
-at the time $t + dt$ in the interval considered.
-
-Now, since we assume our system to be in a state of statistical
-equilibrium, the distribution of energy is independent of the time
-and hence the expressions \Eq{(253)}~and~\Eq{(254)} are equal,~\ie,
-\[
-R_{n, \rho - d\rho} = R_{n, \rho} = R_{n}.
-\Tag{(255)}
-\]
-Thus $R_{n}$ does not depend on $\rho$.
-
-This consideration must, however, be modified for the special
-case in which $\rho = 0$. For, in that case, of the oscillators,
-$N = R_{n-1}\, d\rho$ in number, whose energy at the time~$t$ was between
-$n\epsilon$ and $n\epsilon - d\rho$, during the time $dt = \dfrac{d\rho}{a}$ some enter into the energy
-interval (from $U = n\epsilon$ to $U + dU = n\epsilon + d\rho$) considered; but all of
-them do not necessarily enter, for an oscillator may possibly emit
-all its energy on passing through the value $U = n\epsilon$. If the probability
-%% -----File: 179.png---Folio 163-------
-that emission takes place be denoted by~$\eta$ ($<1$) the number
-of oscillators which pass through the critical value without
-emitting will be
-\[
-NR_{n-1}(1 - \eta)\, d\rho,
-\Tag{(256)}
-\]
-and by equating \Eq{(256)}~and~\Eq{(253)} it follows that
-\[
-R_{n} = R_{n-1}(1 - \eta),
-\]
-and hence, by successive reduction,
-\[
-R_{n} = R_{0}(1 - \eta)^{n}.
-\Tag{(257)}
-\]
-To calculate~$R_{0}$ we repeat the above process for the special case
-when $n = 0$ and $\rho = 0$. In this case the energy interval in question
-extends from $U = 0$ to $dU = d\rho$. Into this interval enter in the
-time $dt = \dfrac{d\rho}{a}$ all the oscillators which perform an emission during
-this time, namely, those whose energy at the time~$t$ was between
-$\epsilon - d\rho$ and~$\epsilon$, $2\epsilon - d\rho$ and~$2\epsilon$, $3\epsilon - d\rho$ and~$3\epsilon~\dots$.
-The numbers of these oscillators are respectively
-\[
-NR_{0}\, d\rho,\quad
-NR_{1}\, d\rho,\quad
-NR_{2}\, d\rho,
-\]
-hence their sum multiplied by~$\eta$ gives the desired number of
-emitting oscillators, namely,
-\[
-N\eta(R_{0} + R_{1} + R_{2} + \dots)\, d\rho,
-\Tag{(258)}
-\]
-and this number is equal to that of the oscillators in the energy
-interval between $0$ and~$d\rho$ at the time~$t + dt$, which is~$NR_{0}\, d\rho$.
-Hence it follows that
-\[
-R_{0} = \eta (R_{0} + R_{1} + R_{2} + \dots).
-\Tag{(259)}
-\]
-
-Now, according to~\Eq{(253)}, the whole number of all the oscillators
-is obtained by integrating with respect to~$\rho$ from $0$ to~$\epsilon$, and
-summing up with respect to~$n$ from $0$ to~$\infty$. Thus
-\[
-N = N \sum_{n=0}^{n=\infty} \int_{0}^{\epsilon} R_{n, \rho}\, d\rho
- = N \sum R_{n}\epsilon
-\Tag{(260)}
-\]
-and
-\[
-\sum R_{n} = \frac{1}{\epsilon}.
-\Tag{(261)}
-\]
-Hence we get from \Eq{(257)}~and~\Eq{(259)}
-\[
-R_{0} = \frac{\eta}{\epsilon},\quad
-R_{n} = \frac{\eta}{\epsilon} (1 - \eta)^{n}.
-\Tag{(262)}
-\]
-%% -----File: 180.png---Folio 164-------
-
-\Section{152.} The total energy emitted in the time element $dt = \dfrac{d\rho}{a}$
-is found from~\Eq{(258)} by considering that every emitting oscillator
-expends all its energy of vibration and is
-\begin{align*}
- &N \eta\, d\rho\, (R_{0} + 2R_{1} + 3R_{2} + \dots) \epsilon\\
-= &N \eta\, d\rho\, \eta(1 + 2(1 - \eta) + 3(1 - \eta)^{2} + \dots)\\
-= &N\, d\rho = Na\, dt.
-\end{align*}
-It is therefore equal to the energy absorbed in the same time by
-all oscillators~\Eq{(250)}, as is necessary, since the state is one of
-statistical equilibrium.
-
-Let us now consider the \emph{mean energy}~$\bar{U}$ of an oscillator. It is
-evidently given by the following relation, which is derived in the
-same way as~\Eq{(260)}:
-\[
-N \bar{U} = N \sum_{0}^{\infty} \int_{0}^{\epsilon} (n\epsilon + \rho) R_{n}\, d\rho.
-\Tag{(263)}
-\]
-From this it follows by means of~\Eq{(262)}, that
-\[
-\bar{U}
- = \left(\frac{1}{\eta} - \frac{1}{2}\right) \epsilon
- = \left(\frac{1}{\eta} - \frac{1}{2}\right) h\nu\Add{.}
-\Tag{(264)}
-\]
-Since $\eta < 1$, $\bar{U}$~lies between $\dfrac{h\nu}{2}$ and~$\infty$. Indeed, it is immediately
-evident that $\bar{U}$~can never become less than $\dfrac{h\nu}{2}$ since the energy
-of \emph{every} oscillator, however small it may be, will assume the value
-$\epsilon = h\nu$ within a time limit, which can be definitely stated.
-
-\Section{153.} The probability constant~$\eta$ contained in the formulć for
-the stationary state is determined by the law of emission enunciated
-in \Sec{147}. According to this, the ratio of the probability
-that no emission takes place to the probability that emission does
-take place is proportional to the intensity~$\ssfI$ of the vibration
-exciting the oscillator, and hence
-\[
-\frac{1 - \eta}{\eta} = p \ssfI
-\Tag{(265)}
-\]
-where the constant of proportionality is to be determined in
-%% -----File: 181.png---Folio 165-------
-such a way that for very large energies of vibration the familiar
-formulć of classical dynamics shall hold.
-
-Now, according to~\Eq{(264)}, $\eta$~becomes small for large values of~$\bar{U}$
-and for this special case the equations \Eq{(264)} and~\Eq{(265)} give
-\[
-\bar{U} = ph\nu \ssfI,
-\]
-and the energy emitted or absorbed respectively in the time~$dt$ by
-all $N$ oscillators becomes, according to~\Eq{(250)},
-\[
-\frac{N \ssfI}{4L}\, dt = \frac{N \bar{U}}{4Lph\nu}\, dt.
-\Tag{(266)}
-\]
-
-On the other hand, \Name{H.~Hertz} has already calculated from
-\Name{Maxwell's} theory the energy emitted by a linear oscillator
-vibrating periodically. For the energy emitted in the time of
-one-half of one vibration he gives the expression\footnote
- {\Name{H.~Hertz}, Wied.\ Ann.\ \textbf{36}, p.~12, 1889.}
-\[
-\frac{\pi^{4} E^{2} l^{2}}{3\lambda^{3}}
-\]
-where $\lambda$~denotes half the wave length, and the product~$El$ (the $C$
-of our notation) denotes the amplitude of the moment~$f$
-(\Sec{135}) of the vibrations. This gives for the energy emitted
-in the time of a whole vibration
-\[
-\frac{16\pi^{4} C^{2}}{3\lambda^{3}}
-\]
-where $\lambda$~denotes the whole wave length, and for the energy
-emitted by $N$~similar oscillators in the time~$dt$
-\[
-N \frac{16\pi^{4} C^{2} \nu^{4}}{3c^{3}}\, dt
-\]
-since $\lambda = \dfrac{c}{\nu}$. On introducing into this expression the energy~$U$ of
-an oscillator from \Eq{(205)},~\Eq{(207)}, and~\Eq{(208)}, namely
-\[
-U = 2\pi^{2} \nu^{2} LC^{2},
-\]
-we have for the energy emitted by the system of oscillators
-\[
-N \frac{8\pi^{2} \nu^{2} U}{3c^{3}L}\, dt
-\Tag{(267)}
-\]
-%% -----File: 182.png---Folio 166-------
-and by equating the expressions \Eq{(266)}~and~\Eq{(267)} we find for the
-factor of proportionality~$p$
-\[
-p = \frac{3c^{3}}{32\pi^{2} h\nu^{3}}.
-\Tag{(268)}
-\]
-
-\Section{154.} By the determination of~$p$ the question regarding the
-properties of the state of statistical equilibrium between the
-system of the oscillators and the vibration exciting them receives
-a general answer. For from~\Eq{(265)} we get
-\[
-\eta = \frac{1}{1 + p \ssfI}
-\]
-and further from~\Eq{(262)}
-\[
-R_{n} = \frac{1}{\epsilon}\, \frac{(p \ssfI)^{n}}{(1 + p \ssfI)^{n+1}}.
-\Tag{(269)}
-\]
-Hence in the state of stationary equilibrium the number of
-oscillators whose energy lies between $nh\nu$ and $(n + 1)h\nu$ is, from
-equation~\Eq{(253)},
-\[
-N \int_{0}^{\epsilon} R_{n}\, d\rho
- = NR_{n}\epsilon
- = N \frac{(p \ssfI)^{n}}{(1 + p \ssfI)^{n+1}}
-\Tag{(270)}
-\]
-where $n = 0$, $1$, $2$, $3,~\dots$.
-%% -----File: 183.png---Folio 167-------
-
-\Chapter[Law of the Normal Distribution Of Energy]
-{IV}{The Law of the Normal Distribution Of
-Energy. Elementary Quanta Of
-Matter and Electricity}
-
-\Section{155.} In the preceding chapter we have made ourselves familiar
-with all the details of a system of oscillators exposed to uniform
-radiation. We may now develop the idea put forth at the end of
-\Sec{144}. That is to say, we may identify the stationary state
-of the oscillators just found with the state of maximum entropy
-of the system of oscillators which was derived directly from the
-hypothesis of quanta in the preceding part, and we may then
-equate the temperature of the radiation to the temperature of
-the oscillators. It is, in fact, possible to obtain perfect agreement
-of the two states by a suitable coordination of their corresponding
-quantities.
-
-According to \Sec{139}, the ``distribution density''~$w$ of the
-oscillators in the state of statistical equilibrium changes abruptly
-from one region element to another, while, according to \Sec{138},
-the distribution within a single region element is uniform. The
-region elements of the state plane~$(f\psi)$ are bounded by concentric
-similar and similarly situated ellipses which correspond to those
-values of the energy~$U$ of an oscillator which are integral multiples
-of~$h\nu$. We have found exactly the same thing for the stationary
-state of the oscillators when they are exposed to uniform radiation,
-and the distribution density~$w_{n}$ in the $n$th~region element
-may be found from~\Eq{(270)}, if we remember that the $n$th~region
-element contains the energies between $(n - 1)h \nu$ and~$nh \nu$. Hence:
-\[
-w_{n} = \frac{(p \ssfI)^{n-1}}{(1 + p \ssfI)^{n}}
- = \frac{1}{p \ssfI} \left(\frac{p \ssfI}{1 + p \ssfI}\right)^{n}.
-\Tag{(271)}
-\]
-This is in perfect agreement with the previous value~\Eq{(220)} of~$w_{n}$
-if we put
-\[
-\alpha = \frac{1}{p \ssfI} \quad\text{and}\quad
-\gamma = \frac{p \ssfI}{1 + p \ssfI},
-\]
-%% -----File: 184.png---Folio 168-------
-and each of these two equations leads, according to~\Eq{(221)}, to the
-following relation between the intensity of the exciting vibration~$\ssfI$
-and the total energy~$E$ of the $N$~oscillators:
-\[
-p \ssfI = \frac{E}{Nh \nu} - \frac{1}{2}.
-\Tag{(272)}
-\]
-
-\Section{156.} If we finally introduce the temperature~$T$ from~\Eq{(223)}, we
-get from the last equation, by taking account of the value~\Eq{(268)} of
-the factor of proportionality~$p$,
-\[
-\ssfI = \frac{32\pi^{2} h\nu^{3}}{3c^{3}}\,
- \frac{1}{e^{\efrac{h\nu}{kT}} - 1}\Add{.}
-\Tag{(273)}
-\]
-Moreover the specific intensity~$\ssfK$ of a monochromatic plane
-polarized ray of frequency~$\nu$ is, according to equation~\Eq{(160)},
-\[
-\ssfK = \frac{h\nu^{3}}{c^{2}}\, \frac{1}{e^{\efrac{h\nu}{kT}} - 1}
-\Tag{(274)}
-\]
-and the space density of energy of uniform monochromatic unpolarized
-radiation of frequency~$\nu$ is, from~\Eq{(159)},
-\[
-\ssfu = \frac{8\pi h\nu^{3}}{c^{3}}\, \frac{1}{e^{\efrac{h\nu}{kT}} - 1}\Add{.}
-\Tag{(275)}
-\]
-Since, among all the forms of radiation of differing constitutions,
-black radiation is distinguished by the fact that all monochromatic
-rays contained in it have the same temperature (\Sec{93})
- these equations also give the law of distribution of energy in
-the normal spectrum, \ie, in the emission spectrum of a body
-which is black with respect to the vacuum.
-
-If we refer the specific intensity of a monochromatic ray not to
-the frequency~$\nu$ but, as is usually done in experimental physics,
-to the wave length~$\lambda$, by making use of \Eq{(15)}~and~\Eq{(16)} we obtain the
-expression
-\[%[** F2: Check meaning of subscripted "c"s]
-E_{\lambda}
- = \frac{c^{2} h}{\lambda^{5}}\, \frac{1}{e^{\efrac{ch}{k\lambda T}} - 1}
- = \frac{c_{1}}{\lambda^{5}}\,
- \frac{1}{e^{\efrac{c_{2}}{\lambda T}} - 1}\Add{.}
-\Tag{(276)}
-\]
-This is the specific intensity of a monochromatic plane polarized
-ray of the wave length~$\lambda$ which is emitted from a black body at the
-temperature~$T$ into a vacuum in a direction perpendicular to the
-%% -----File: 185.png---Folio 169-------
-surface. The corresponding space density of unpolarized radiation
-is obtained by multiplying~$E_{\lambda}$ by~$\dfrac{8\pi}{c}$.
-
-Experimental tests have so far confirmed equation~\Eq{(276)}.\footnote
- {See among others \Name{H.~Rubens} und \Name{F.~Kurlbaum}, Sitz.\ Ber.\ d.\ Akad.\ d.\ Wiss.\ zu Berlin
- vom 25.~Okt., 1900, p.~929. Ann.\ d.\ Phys.\ \textbf{4}, p.~649, 1901. \Name{F.~Paschen}, Ann.\ d.\ Phys.\ \textbf{4},
- p.~277, 1901. \Name{O.~Lummer} und \Name{E.~Pringsheim}, Ann.\ d.\ Phys.\ \textbf{6}, p.~210, 1901. Tätigkeitsbericht
- der Phys.-Techn.\ Reichsanstalt vom J.~1911, Zeitschr.\ f.\ Instrumentenkunde, 1912,
- April, p.~134~ff.}
-According to the most recent measurements made in the Physikalisch-technische
-Reichsanstalt\footnote
- {According to private information kindly furnished by the president, \Name{Mr.~Warburg}.}
-the value of the second radiation
-constant~$c_{2}$ is approximately
-\[
-c_{2} = \frac{ch}{k} = 1.436\, \cm\, \degree.
-%[** TN: No equation number in the original]
-\Tag{(277)}
-\]
-
-More detailed information regarding the history of the equation
-of radiation is to be found in the original papers and in the
-first edition of this book. At this point it may merely be added
-that equation~\Eq{(276)} was not simply extrapolated from radiation
-measurements, but was originally found in a search after a
-connection between the entropy and the energy of an oscillator
-vibrating in a field, a connection which would be as simple as
-possible and consistent with known measurements.
-
-\Section{157.} The entropy of a ray is, of course, also determined
-by its temperature. In fact, by combining equations \Eq{(138)}
-and~\Eq{(274)} we readily obtain as an expression for the entropy
-radiation~$\ssfL$ of a monochromatic plane polarized ray of the
-specific intensity of radiation~$\ssfK$ and the frequency~$\nu$,
-\[
-\ssfL = \frac{k\nu^{2}}{c^{2}} \left\{
- \left(1 + \frac{c^{2}\ssfK}{h\nu^{3}}\right) \log\left(1 + \frac{c^{2}\ssfK}{h\nu^{3}}\right)
- - \frac{c^{2}\ssfK}{h\nu^{3}} \log\frac{c^{2}\ssfK}{h\nu^{3}}
-\right\}
-\Tag{(278)}
-\]
-which is a more definite statement of equation~\Eq{(134)} for \Name{Wien's}
-displacement law.
-
-Moreover it follows from~\Eq{(135)}, by taking account of~\Eq{(273)},
-that the space density of the entropy~$\ssfs$ of uniform monochromatic
-unpolarized radiation as a function of the space density of
-energy~$\ssfu$ is
-\[
-\Squeeze[0.95]{$\displaystyle
-\ssfs = \frac{8\pi k\nu^{2}}{c^{3}} \biggl\{
- \biggl(1 + \frac{c^{3}\ssfu}{8\pi h\nu^{3}}\biggr) \log\biggl(1 + \frac{c^{3}\ssfu}{8\pi h\nu^{3}}\biggr)
- - \frac{c^{3}\ssfu}{8\pi h\nu^{3}} \log\frac{c^{3}\ssfu}{8\pi h\nu^{3}}\biggr\}\Add{.}$}
-\Tag{(279)}
-\]
-This is a more definite statement of equation~\Eq{(119)}.
-%% -----File: 186.png---Folio 170-------
-
-%[** Size of ()]
-\Section{158.} For \emph{small} values of~$\lambda T$ (\ie,~small compared with the
-constant $\dfrac{ch}{k}$) equation~\Eq{(276)} becomes
-\[
-E_{\lambda} = \frac{c^{2}h}{\lambda^{5}} e^{-\efrac{ch}{k\lambda T}}
-\Tag{(280)}
-\]
-an equation which expresses \Name{Wien's}\footnote
- {\Name{W.~Wien}, Wied.\ Ann.\ \textbf{58}, p.~662, 1896.}
-law of energy distribution.
-
-The specific intensity of radiation~$\ssfK$ then becomes, according to~\Eq{(274)},
-\[
-\ssfK = \frac{h\nu^{3}}{c^{2}} e^{-\efrac{h\nu}{kT}}
-\Tag{(281)}
-\]
-and the space density of energy~$\ssfu$ is, from~\Eq{(275)},
-\[
-\ssfu = \frac{8\pi h\nu^{3}}{c^{3}} e^{-\efrac{h\nu}{kT}}\Add{.}
-\Tag{(282)}
-\]
-
-\Section{159.} On the other hand, for \emph{large} values of~$\lambda T$ \Eq{(276)}~becomes
-\[
-E_{\lambda} = \frac{ckT}{\lambda^{4}}\Add{,}
-\Tag{(283)}
-\]
-a relation which was established first by \Name{Lord Rayleigh}\footnote
- {\Name{Lord Rayleigh}, Phil.\ Mag.\ \textbf{49}, p.~539, 1900.}
-and which
-we may, therefore, call ``\Name{Rayleigh's} law of radiation.''
-
-We then find for the specific intensity of radiation~$\ssfK$ from~\Eq{(274)}
-\[
-\ssfK = \frac{k\nu^{2}T}{c^{2}}
-\Tag{(284)}
-\]
-and from~\Eq{(275)} for the space density of monochromatic radiation
-we get
-\[
-\ssfu = \frac{8\pi k\nu^{2}T}{c^{3}}\Add{.}
-\Tag{(285)}
-\]
-
-\Name{Rayleigh's} law of radiation is of very great theoretical interest,
-since it represents that distribution of energy which is obtained
-for radiation in statistical equilibrium with material molecules
-by means of the classical dynamics, and without introducing the
-hypothesis of quanta.\footnote
- {\Name{J.~H. Jeans}, Phil.\ Mag.\ Febr.,~1909, p.~229, \Name{H.~A. Lorentz}, Nuovo Cimento~V, vol.~\textbf{16},
- 1908.}
-This may also be seen from the fact
-that for a vanishingly small value of the quantity element of
-action,~$h$, the general formula~\Eq{(276)} degenerates into \Name{Rayleigh's}
-formula~\Eq{(283)}. See also below, \Sec{168} \textit{et~seq.}
-%% -----File: 187.png---Folio 171-------
-
-\Section{160.} For the total space density,~$u$, of black radiation at any
-temperature~$T$ we obtain, from~\Eq{(275)},
-\[
-u = \int_{0}^{\infty} \ssfu\, d\nu
- = \frac{8\pi h}{c^{3}} \int_{0}^{\infty} \frac{\nu^{3}\, d\nu}{e^{\efrac{h\nu}{kT}} - 1}
-\]
-or
-\[
-u = \frac{8\pi h}{c^{3}} \int_{0}^{\infty} \left(
- e^{-\efrac{h\nu}{kT}}
- + e^{-\efrac{2h\nu}{kT}}
- + e^{-\efrac{3h\nu}{kT}} + \dots\right) \nu^{3}\, d\nu
-\]
-and, integrating term by term,
-\[
-u = \frac{48\pi h}{c^{3}}\left(\frac{kT}{h}\right)^{4}\alpha
-\Tag{(286)}
-\]
-where $\alpha$~is an abbreviation for
-\[
-\alpha = 1 + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \frac{1}{4^{4}} + \dots = 1.0823.
-\Tag{(287)}
-\]
-
-This relation expresses the \Name{Stefan-Boltzmann} law~\Eq{(75)} and it
-also tells us that the constant of this law is given by
-\[
-a = \frac{48\pi\alpha k^{4}}{c^{3}h^{3}}.
-\Tag{(288)}
-\]
-
-\Section{161.} For that wave length~$\lambda_{m}$ the maximum of the
-intensity of radiation corresponds in the spectrum of black radiation,
-we find from~\Eq{(276)}
-\[
-\left(\frac{dE_{\lambda}}{d\lambda}\right)_{\lambda = \lambda_{m}} = 0.
-\]
-On performing the differentiation and putting as an abbreviation
-\[
-\frac{ch}{k\lambda_{m}T} = \beta,
-\]
-we get
-\[
-e^{-\beta} + \frac{\beta}{5} - 1 = 0.
-\]
-The root of this transcendental equation is
-\[
-\beta = 4.9651,
-\Tag{(289)}
-\]
-and accordingly $\lambda_{m}T = \dfrac{ch}{\beta k}$, and this is a constant, as demanded
-%% -----File: 188.png---Folio 172-------
-by \Name{Wien's} displacement law. By comparison with~\Eq{(109)} we
-find the meaning of the constant~$b$, namely,
-\[
-b = \frac{ch}{\beta k},
-\Tag{(290)}
-\]
-and, from~\Eq{(277)},
-\[
-b = \frac{c_{2}}{\beta}
- = \frac{1.436}{4.9651}
- = 0.289\, \cm \DPchg{ˇ}{}\, \degree,
-\Tag{(291)}
-\]
-while \Name{Lummer} and \Name{Pringsheim} found by measurements $0.294$ and
-\Name{Paschen} $0.292.$
-
-\Section{162.} By means of the measured values\footnote
- {Here as well as later on the value given above~\Eq{(79)} has been replaced by $a =
- 7.39 ˇ 10^{-15}$, obtained from $\sigma = \DPchg{ac/4}{\dfrac{ac}{4}} = 5.54 ˇ 10^{-5}$. This is the final result of the newest measurements
- made by \Name{W.~Westphal}, according to information kindly furnished by him and
- \Name{Mr.~H. Rubens}.\DPnote{** "Mr." not ital in orig} (Nov.,~1912). [Compare \PageRef{p.}{64}, footnote. Tr.]}
-of $a$~and~$c_{2}$ the universal
-constants $h$~and~$k$ may be readily calculated. For it follows from
-equations \Eq{(277)}~and~\Eq{(288)} that
-\[
-h = \frac{ac_{2}^{4}}{48\pi \alpha c}\qquad
-k = \frac{ac_{2}^{3}}{48\pi \alpha}\Add{.}
-\Tag{(292)}
-\]
-Substituting the values of the constants $a$,~$c_{2}$, $\alpha$,~$c$, we get
-\[
-h = 6.415 ˇ 10^{-27}\, \erg\, \sec.,\qquad
-k = 1.34 ˇ 10^{-16}\, \frac{\erg}{\degree}\Add{.}
-\Tag{(293)}
-\]
-
-\Section{163.} To ascertain the full physical significance of the quantity
-element of action,~$h$, much further research work will be required.
-On the other hand, the value obtained for~$k$ enables us readily
-to state numerically in the \CGS\ system the general connection
-between the entropy~$S$ and the thermodynamic probability~$W$
-as expressed by the universal equation~\Eq{(164)}. The general
-expression for the entropy of a physical system is
-\[
-S = 1.34 ˇ 10^{-16} \log W\, \frac{\erg}{\degree}\Add{.}
-\Tag{(294)}
-\]
-
-\emph{This equation may be regarded as the most general definition of
-entropy.} Herein the thermodynamic probability~$W$ is an integral
-number, which is completely defined by the macroscopic state of
-the system. Applying the result expressed in~\Eq{(293)} to the kinetic
-%% -----File: 189.png---Folio 173-------
-theory of gases, we obtain from equation~\Eq{(194)} for the ratio of the
-mass of a molecule to that of a mol,
-\[
-\omega = \frac{k}{R} = \frac{1.34 × 10^{-16}}{831 × 10^{5}} = 1.61 × 10^{-24},
-\Tag{(295)}
-\]
-that is to say, there are in one mol
-\[
-\frac{1}{\omega} = 6.20 × 10^{23}
-\]
-molecules, where the mol of oxygen, $O_{2}$, is always assumed as
-$32\, \gr$. Hence, for example, the absolute mass of a hydrogen
-atom ($\frac{1}{2}H_{2} = 1.008$) equals $1.62 × 10^{-24}\, \gr$. With these numerical
-values the number of molecules contained in $1\, \cm\DPchg{.}{}^{3}$ of an
-ideal gas at $0° \Celsius$ and $1$~atmosphere pressure becomes
-\[
-N = \frac{76 ˇ 13.6 ˇ 981}{831 ˇ 10^{5} ˇ 273 \omega} = 2.77 ˇ 10^{19}.
-\Tag{(296)}
-\]
-The mean kinetic energy of translatory motion of a molecule
-at the absolute temperature $T = 1$ is, in the absolute \CGS\
-system, according to~\Eq{(200)},
-\[
-\DPchg{\frac{3}{2}}{\tfrac{3}{2}} k = 2.01 ˇ 10^{-16}\Add{.}
-\Tag{(297)}
-\]
-In general the mean kinetic energy of translatory motion of a
-molecule is expressed by the product of this number and the
-absolute temperature~$T$.
-
-The elementary quantity of electricity or the free charge of a
-monovalent ion or electron is, in electrostatic units,
-\[
-e = \omega ˇ 9654 ˇ 3 ˇ 10^{10} = 4.67 ˇ 10^{-10}.
-\Tag{(298)}
-\]
-Since absolute accuracy is claimed for the formulć here employed,
-the degree of approximation to which these numbers
-represent the corresponding physical constants depends only on
-the accuracy of the measurements of the two radiation constants
-$a$~and~$c_{2}$.
-
-\Section[164.]{164. Natural Units.}---All the systems of units which have
-hitherto been employed, including the so-called absolute \CGS\
-system, owe their origin to the coincidence of accidental circumstances,
-%% -----File: 190.png---Folio 174-------
-inasmuch as the choice of the units lying at the base of
-every system has been made, not according to general points of
-view which would necessarily retain their importance for all
-places and all times, but essentially with reference to the special
-needs of our terrestrial civilization.
-
-Thus the units of length and time were derived from the present
-dimensions and motion of our planet, and the units of mass
-and temperature from the density and the most important
-temperature points of water, as being the liquid which plays the
-most important part on the surface of the earth, under a pressure
-which corresponds to the mean properties of the atmosphere
-surrounding us. It would be no less arbitrary if, let us say, the
-invariable wave length of Na-light were taken as unit of length.
-For, again, the particular choice of Na from among the many
-chemical elements could be justified only, perhaps, by its common
-occurrence on the earth, or by its double line, which is in
-the range of our vision, but is by no means the only one of its
-kind. Hence it is quite conceivable that at some other time,
-under changed external conditions, every one of the systems of
-units which have so far been adopted for use might lose, in part
-or wholly, its original natural significance.
-
-In contrast with this it might be of interest to note that, with
-the aid of the two constants $h$~and~$k$ which appear in the universal
-law of radiation, we have the means of establishing units of length,
-mass, time, and temperature, which are independent of special
-bodies or substances, which necessarily retain their significance
-for all times and for all environments, terrestrial and human or
-otherwise, and which may, therefore, be described as ``natural
-units.''
-
-The means of determining the four units of length, mass, time,
-and temperature, are given by the two constants $h$~and~$k$ mentioned,
-together with the magnitude of the velocity of propagation
-of light in a vacuum,~$c$, and that of the constant of gravitation,~$f$.
-Referred to centimeter, gram, second, and degrees
-Centigrade, the numerical values of these four constants are as
-follows:
-\begin{align*}
-h &= 6.415 ˇ 10^{-27}\, \frac{\DPchg{g}{\gr}\, \cm^{2}}{\sec} \displaybreak[0] \\
-%% -----File: 191.png---Folio 175-------
-k &= 1.34 ˇ 10^{-16}\, \frac{\DPchg{g}{\gr}\, \cm^{2}}{\sec^{2}\, \degree} \displaybreak[0] \\
-c &= 3 ˇ 10^{10}\, \frac{\cm}{\sec} \displaybreak[0] \\
-f &= 6.685 ˇ 10^{-8}\, \frac{\cm^{3}}{\DPchg{g}{\gr}\, \sec^{2}}\Add{\;.}\footnotemark
-\end{align*}
-\footnotetext{\Name{F.~Richarz} and \Name{O.~Krigar-Menzel}, Wied.\ Ann.\ \textbf{66}, p.~190, 1898.}%
-If we now choose the natural units so that in the new system of
-measurement each of the four preceding constants assumes the
-value~$1$, we obtain, as unit of length, the quantity
-\[
-\sqrt\frac{fh}{c^{3}} = 3.99 ˇ 10^{-33}\, \cm,
-\]
-as unit of mass
-\[
-\sqrt\frac{ch}{f} = 5.37 ˇ 10^{-5}\, \DPchg{g}{\gr},
-\]
-as unit of time
-\[
-\sqrt\frac{fh}{c^{5}} = 1.33 ˇ 10^{-43}\, \sec,
-\]
-as unit of temperature
-\[
-\frac{1}{k} \sqrt\frac{c^{5} h}{f} = 3.60 ˇ 10^{32}\, \degree.
-\]
-These quantities retain their natural significance as long as
-the law of gravitation and that of the propagation of light in
-a vacuum and the two principles of thermodynamics remain
-valid; they therefore must be found always the same, when
-measured by the most widely differing intelligences according to
-the most widely differing methods.
-
-\Section{165.} The relations between the intensity of radiation and the
-temperature expressed in \Sec{156} hold for radiation in a pure
-vacuum. If the radiation is in a medium of refractive index~$n$,
-the way in which the intensity of radiation depends on the
-frequency and the temperature is given by the proposition of
-\Sec{39}, namely, the product of the specific intensity of radiation~$\ssfK_{\nu}$
-and the square of the velocity of propagation of the radiation
-%% -----File: 192.png---Folio 176-------
-has the same value for all substances. The form of this universal
-function~\Eq{(42)} follows directly from~\Eq{(274)}
-\[
-\ssfK q^{2}
- = \frac{\epsilon_{\nu}}{\alpha_{\nu}} q^{2}
- = \frac{h\nu^{3}}{e^{\efrac{h\nu}{kT}} - 1}\Add{.}
-\Tag{(299)}
-\]
-Now, since the refractive index~$n$ is inversely proportional to the
-velocity of propagation, equation~\Eq{(274)} is, in the case of a medium
-with the index of refraction~$n$, replaced by the more general relation
-\[
-\ssfK_{\nu} = \frac{h\nu^{3} n^{2}}{c^{2}}\, \frac{1}{e^{\efrac{h\nu}{kT}} - 1}
-\Tag{(300)}
-\]
-and, similarly, in place of~\Eq{(275)} we have the more general relation
-\[
-\ssfu = \frac{8\pi h\nu^{3} n^{3}}{c_{3}}\, \frac{1}{e^{\efrac{h\nu}{kT}} - 1}\Add{.}
-\Tag{(301)}
-\]
-These expressions hold, of course, also for the emission of a body
-which is black with respect to a medium with an index of refraction~$n$.
-
-\Section{166.} We shall now use the laws of radiation we have obtained
-to calculate the temperature of a monochromatic unpolarized
-radiation of given intensity in the following case. Let the light
-pass normally through a small area (slit) and let it fall on an
-arbitrary system of diathermanous media separated by spherical
-surfaces, the centers of which lie on the same line, the axis of
-the system. Such radiation consists of homocentric pencils and
-hence forms behind every refracting surface a real or virtual
-image of the emitting surface, the image being likewise normal
-to the axis. To begin with, we assume the last as well as the first
-medium to be a pure vacuum. Then, for the determination of
-the temperature of the radiation according to equation~\Eq{(274)},
-we need calculate only the specific intensity of radiation~$\ssfK_{\nu}$ in
-the last medium, and this is given by the total intensity of the
-monochromatic radiation~$I_{\nu}$, the size of the area of the image~$F$,
-and the solid angle~$\Omega$ of the cone of rays passing through a point
-of the image. For the specific intensity of radiation~$\ssfK_{\nu}$ is,
-according to~\Eq{(13)}, determined by the fact that an amount
-\[
-2\ssfK_{\nu}\, d\sigma\, d\Omega\, d\nu\, dt
-\]
-%% -----File: 193.png---Folio 177-------
-of energy of unpolarized light corresponding to the interval of
-frequencies from $\nu$ to $\nu + d\nu$ is, in the time~$dt$, radiated in a normal
-direction through an element of area~$d\sigma$ within the conical element~$d\Omega$.
-If now $d\sigma$~denotes an element of the area of the surface
-image in the last medium, then the total monochromatic radiation
-falling on the image has the intensity
-\[
-I_{\nu} = 2\ssfK_{\nu} \int d\sigma \int d\Omega.
-\]
-$I_{\nu}$~is of the dimensions of energy, since the product $d\nu\, dt$ is a mere
-number. The first integral is the whole area,~$F$, of the image,
-the second is the solid angle,~$\Omega$, of the cone of rays passing
-through a point of the surface of the image. Hence we get
-\[
-I_{\nu} = 2\ssfK_{\nu} F\Omega,
-\Tag{(302)}
-\]
-and, by making use of~\Eq{(274)}, for the temperature of the radiation
-\[
-T = \frac{h\nu}{k}
- ˇ \frac{1}{\log\left(\dfrac{2h\nu^{3} F\Omega}{c^{2}I_{\nu}} + 1 \right)}\Add{.}
-\Tag{(303)}
-\]
-If the diathermanous medium considered is not a vacuum but
-has an index of refraction~$n$, \Eq{(274)}~is replaced by the more general
-relation~\Eq{(300)}, and, instead of the last equation, we obtain
-\[
-T = \frac{h\nu}{k}\,
- \frac{1}{\log\left(\dfrac{2h\nu^{3} F\Omega n^{2}}{c^{2}I_{\nu}} + 1 \right)}
-\Tag{(304)}
-\]
-or, on substituting the numerical values of $c$,~$h$, and~$k$,
-\[
-T = \frac{0.479 ˇ 10^{-10}\nu}{\log\left(\dfrac{1.43 ˇ 10^{-47}\nu^{3} F\Omega n^{2}}{I_{\nu}} + 1 \right)} \text{ degree Centigrade}.
-\]
-In this formula the natural logarithm is to be taken, and $I_{\nu}$~is
-to be expressed in ergs, $\nu$~in ``reciprocal seconds,'' \ie, $(\text{seconds})^{-1}$,
-$F$~in square centimeters. In the case of visible rays the second
-term,~$1$, in the denominator may usually be omitted.
-
-The temperature thus calculated is retained by the radiation
-considered, so long as it is propagated without any disturbing
-%% -----File: 194.png---Folio 178-------
-influence in the diathermanous medium, however great the distance
-to which it is propagated or the space in which it spreads.
-For, while at larger distances an ever decreasing amount of energy
-is radiated through an element of area of given size, this is contained
-in a cone of rays starting from the element, the angle of
-the cone continually decreasing in such a way that the value of~$\ssfK$
-remains entirely unchanged. Hence the free expansion of radiation
-is a perfectly reversible process. (Compare above, \Sec{144}.)
-It may actually be reversed by the aid of a suitable concave mirror
-or a converging lens.
-
-Let us next consider the temperature of the radiation in the
-other media, which lie between the separate refracting or reflecting
-spherical surfaces. In every one of these media the radiation
-has a definite temperature, which is given by the last formula
-when referred to the real or virtual image formed by the radiation
-in that medium.
-
-The frequency~$\nu$ of the monochromatic radiation is, of course,
-the same in all media; moreover, according to the laws of geometrical
-optics, the product $n^{2}F\Omega$ is the same for all media. Hence,
-if, in addition, the total intensity of radiation~$I_{\nu}$ remains constant
-on refraction (or reflection), $T$~also remains constant, or in other
-words: The temperature of a homocentric pencil is not changed
-by regular refraction or reflection, unless a loss in energy of
-radiation occurs. Any weakening, however, of the total intensity~$I_{\nu}$
-by a subdivision of the radiation, whether into two or
-into many different directions, as in the case of diffuse reflection,
-leads to a lowering of the temperature of the pencil. In fact, a
-certain loss of energy by refraction or reflection does occur, in
-general, on a refraction or reflection, and hence also a lowering of
-the temperature takes place. In these cases a fundamental
-difference appears, depending on whether the radiation is weakened
-merely by free expansion or by subdivision or absorption.
-In the first case the temperature remains constant, in the second
-it decreases.\footnote
- {Nevertheless regular refraction and reflection are not irreversible processes; for the
- refracted and the reflected rays are coherent and the entropy of two coherent rays is not
- equal to the sum of the entropies of the separate rays. (Compare above, \Sec{104}.) On
- the other hand, diffraction is an irreversible process. \Name{M.~Laue}, Ann.\ d.\ Phys.\ \textbf{31}, p.~547,
- 1910.}
-
-\Section{167.} The laws of emission of a black body having been determined,
-%% -----File: 195.png---Folio 179-------
-it is possible to calculate, with the aid of \Name{Kirchhoff's} law~\Eq{(48)},
-the emissive power~$E$ of any body whatever, when its
-absorbing power~$A$ or its reflecting power $1 - A$ is known. In the
-case of metals this calculation becomes especially simple for long
-waves, since \Name{E.~Hagen} and \Name{H.~Rubens}\footnote
- {\Name{E.~Hagen} und \Name{H.~Rubens}, Ann.\ d.\ \DPtypo{Phs.y}{Phys.}\ \textbf{11}, p.~873, 1903.}
-have shown experimentally
-that the reflecting power and, in fact, the entire optical behavior
-of the metals in the spectral region mentioned is represented by
-the simple equations of \Name{Maxwell} for an electromagnetic field with
-homogeneous conductors and hence depends only on the specific
-conductivity for steady electric currents. Accordingly, it is
-possible to express completely the emissive power of a metal for
-long waves by its electric conductivity combined with the formul{\ae}
-for black radiation.\footnote
- {\Name{E.~Aschkinass}, Ann.\ d.\ Phys.\ \textbf{17}, p.~960, 1905.}
-
-\Section{168.} There is, however, also a method, applicable to the case
-of long waves, for the direct theoretical determination of the electric
-conductivity and, with it, of the absorbing power,~$A$, as well
-as the emissive power,~$E$, of metals. This is based on the ideas
-of the electron theory, as they have been developed for the thermal
-and electrical processes in metals by \Name{E.~Riecke}\footnote
- {\Name{E.~Riecke}, Wied.\ Ann.\ \textbf{66}, p.~353, 1898.}
-and especially
-by \Name{P.~Drude}.\footnote
- {\Name{P.~Drude}, Ann.\ d.\ Phys.\ \textbf{1}, p.~566, 1900.}
-According to these, all such processes are based on
-the rapid irregular motions of the negative electrons, which fly
-back and forth between the positively charged molecules of matter
-(here of the metal) and rebound on impact with them as well
-as with one another, like gas molecules when they strike a rigid
-obstacle or one another. The velocity of the heat motions of the
-material molecules may be neglected compared with that of the
-electrons, since in the stationary state the mean kinetic energy of
-motion of a material molecule is equal to that of an electron, and
-since the mass of a material molecule is more than a thousand
-times as large as that of an electron. Now, if there is an electric
-field in the interior of the metal, the oppositely charged particles
-are driven in opposite directions with average velocities depending
-on the mean free path, among other factors, and this explains
-the conductivity of the metal for the electric current. On the
-other hand, the emissive power of the metal for the radiant heat
-follows from the calculation of the impacts of the electrons. For,
-%% -----File: 196.png---Folio 180-------
-so long as an electron flies with constant speed in a constant
-direction, its kinetic energy remains constant and there is no
-radiation of energy; but, whenever it suffers by impact a change
-of its velocity components, a certain amount of energy, which
-may be calculated from electrodynamics and which may always
-be represented in the form of a \Name{Fourier's} series, is radiated into the
-surrounding space, just as we think of \emph{Roentgen rays}\DPnote{cf. Roentgen non-italic elsewhere} as being
-caused by the impact on the anticathode of the electrons ejected
-from the cathode. From the standpoint of the hypothesis of
-quanta this calculation cannot, for the present, be carried out
-without ambiguity except under the assumption that, during the
-time of a partial vibration of the \Name{Fourier} series, a large number of
-impacts of electrons occurs, \ie, for comparatively long waves,
-for then the fundamental law of impact does not essentially
-matter.
-
-Now this method may evidently be used to derive the laws of
-black radiation in a new way, entirely independent of that previously
-employed. For if the emissive power,~$E$, of the metal,
-thus calculated, is divided by the absorbing power,~$A$, of the same
-metal, determined by means of its electric conductivity, then,
-according to \Name{Kirchhoff's} law~\Eq{(48)}, the result must be the emissive
-power of a black body, irrespective of the special substance used
-in the determination. In this manner \Name{H.~A. Lorentz}\footnote
- {\Name{H.~A. Lorentz}, Proc.\ Kon.\ Akad.\ v.\ Wet.\ Amsterdam, 1903, p.~666.}
-has, in a
-profound investigation, derived the law of radiation of a black
-body and has obtained a result the contents of which agree exactly
-with equation~\Eq{(283)}, and where also the constant~$k$ is related to
-the gas constant~$R$ by equation~\Eq{(193)}. It is true that this method
-of establishing the laws of radiation is, as already said, restricted
-to the range of long waves, but it affords a deeper and very important
-insight into the mechanism of the motions of the electrons
-and the radiation phenomena in metals caused by them. At the
-same time the point of view described above in \Sec{111}, according
-to which the normal spectrum may be regarded as consisting of a
-large number of quite irregular processes as elements, is expressly
-confirmed.
-
-\Section{169.} A further interesting confirmation of the law of radiation
-of black bodies for long waves and of the connection of the
-radiation constant~$k$ with the absolute mass of the material
-%% -----File: 197.png---Folio 181-------
-molecules was found by \Name{J.~H. Jeans}\footnote
- {\Name{J.~H. Jeans}, Phil.\ Mag.\ \textbf{10}, p.~91, 1905.}
-by a method previously
-used by \Name{Lord Rayleigh},\footnote
- {\Name{Lord Rayleigh}, Nature \textbf{72}, p.~54 and p.~243, 1905.}
-which differs essentially from the
-one pursued here, in the fact that it entirely avoids making
-use of any special mutual action between matter (molecules,
-oscillators) and the ether and considers essentially only the
-processes in the vacuum through which the radiation passes.
-The starting point for this method of treatment is given by the
-following proposition of statistical mechanics. (Compare above,
-\Sec{140}.) When irreversible processes take place in a system,
-which satisfies \Name{Hamilton's} equations of motion, and whose state
-is determined by a large number of independent variables and
-whose total energy is found by addition of different parts depending
-on the squares of the variables of state, they do so, on the
-average, in such a sense that the partial energies corresponding
-to the separate independent variables of state tend to equality,
-so that finally, on reaching statistical equilibrium, their mean
-values have become equal. From this proposition the stationary
-distribution of energy in such a system may be found, when the
-independent variables which determine the state are known.
-
-Let us now imagine a perfect vacuum, cubical in form, of
-edge~$l$, and with metallically reflecting sides. If we take the
-origin of coordinates at one corner of the cube and let the axes of
-coordinates coincide with the adjoining edges, an electromagnetic
-process which may occur in this cavity is represented by the
-following system of equations:
-\[
-\begin{aligned}
-\ssfE_{x} &= \cos\frac{\ssfa \pi x}{l} \sin\frac{\ssfb \pi y}{l} \sin\frac{\ssfc \pi z}{l} (e_{1} \cos 2\pi \nu t + e_{1}' \sin 2\pi \nu t), \\
-\ssfE_{y} &= \sin\frac{\ssfa \pi x}{l} \cos\frac{\ssfb \pi y}{l} \sin\frac{\ssfc \pi z}{l} (e_{2} \cos 2\pi \nu t + e_{2}' \sin 2\pi \nu t), \\
-\ssfE_{z} &= \sin\frac{\ssfa \pi x}{l} \sin\frac{\ssfb \pi y}{l} \cos\frac{\ssfc \pi z}{l} (e_{3} \cos 2\pi \nu t + e_{3}' \sin 2\pi \nu t), \\
-\ssfH_{x} &= \sin\frac{\ssfa \pi x}{l} \cos\frac{\ssfb \pi y}{l} \cos\frac{\ssfc \pi z}{l} (h_{1} \sin 2\pi \nu t - h_{1}' \cos 2\pi \nu t), \\
-%% -----File: 198.png---Folio 182-------
-\ssfH_{y} &= \cos\frac{\ssfa \pi x}{l} \sin\frac{\ssfb \pi y}{l} \cos\frac{\ssfc \pi z}{l} (h_{2} \sin 2\pi \nu t - h_{2}' \cos 2\pi \nu t), \\
-\ssfH_{z} &= \cos\frac{\ssfa \pi x}{l} \cos\frac{\ssfb \pi y}{l} \sin\frac{\ssfc \pi z}{l} (h_{3} \sin 2\pi \nu t - h_{3}' \cos 2\pi \nu t),
-\end{aligned}
-%[** TN: No equation number in the original]
-\Tag{(305)}
-\]
-where $\ssfa$,~$\ssfb$,~$\ssfc$ represent any three positive integral numbers.
-The boundary conditions in these expressions are satisfied by the
-fact that for the six bounding surfaces $x = 0$, $x = l$, $y = 0$, $y = l$,
-$z = 0$, $z = l$ the tangential components of the electric field-strength~$\ssfE$
-vanish. Maxwell's equations of the field~\Eq{(52)} are also satisfied,
-as may be seen on substitution, provided there exist certain conditions
-between the constants which may be stated in a single
-proposition as follows: Let $a$~be a certain positive constant, then
-there exist between the nine quantities written in the following
-square:
-\[
-\begin{array}{c*{2}{>{\qquad}c}}
-\dfrac{\ssfa c}{2l \nu} & \dfrac{\ssfb c}{2l \nu} & \dfrac{\ssfc c}{2l \nu} \\
-\Strut[28pt]\dfrac{h_{1}}{a} & \dfrac{h_{2}}{a} & \dfrac{h_{3}}{a} \\
-\Strut[28pt]\dfrac{e_{1}}{a} & \dfrac{e_{2}}{a} & \dfrac{e_{3}}{a}
-\end{array}
-\]
-all the relations which are satisfied by the nine so-called ``direction
-cosines'' of two orthogonal right-handed coordinate systems,
-\ie,~the cosines of the angles of any two axes of the systems.
-
-Hence the sum of the squares of the terms of any horizontal
-or vertical row equals~$1$, for example,
-\[
-\begin{gathered}
-\frac{c^{2}}{4l^{2}} \nu^{2} (\ssfa^{2} + \ssfb^{2} + \ssfc^{2}) = 1 \\[4pt]
-h_{1}^{2} + h_{2}^{2} + h_{3}^{2} = a^{2} = e_{1}^{2} + e_{2}^{2} + e_{3}^{2}.
-\end{gathered}
-\Tag{(306)}
-\]
-Moreover the sum of the products of corresponding terms in any
-two parallel rows is equal to zero, for example,
-\[
-\begin{alignedat}{3}
-\ssfa e_{1} &+ \ssfb e_{2} &&+ \ssfc e_{3} &&= 0 \\
-\ssfa h_{1} &+ \ssfb h_{2} &&+ \ssfc h_{3} &&= 0.
-\end{alignedat}
-\Tag{(307)}
-\]
-%% -----File: 199.png---Folio 183-------
-Moreover there are relations of the following form:
-\[
-\frac{h_{1}}{a}
- = \frac{e_{2}}{a} ˇ \frac{\ssfc c}{2l \nu}
- - \frac{e_{3}}{a}\, \frac{\ssfb c}{2l \nu}, %[** Inconsistent dots]
-\]
-and hence
-\[
-h_{1} = \frac{c}{2l \nu} (\ssfc e_{2} - \ssfb e_{3}), \text{ etc.}
-\Tag{(308)}
-\]
-If the integral numbers $\ssfa$,~$\ssfb$,~$\ssfc$ are given, then the frequency~$\nu$ is
-immediately determined by means of~\Eq{(306)}. Then among the
-six quantities $e_{1}$,~$e_{2}$,~$e_{3}$, $h_{1}$,~$h_{2}$,~$h_{3}$, only two may be chosen arbitrarily,
-the others then being uniquely determined by them by
-linear homogeneous relations. If, for example, we assume $e_{1}$
-and $e_{2}$ arbitrarily, $e_{3}$~follows from~\Eq{(307)} and the values of $h_{1}$,~$h_{2}$,~$h_{3}$
-are then found by relations of the form~\Eq{(308)}. Between the
-quantities with accent $e_{1}'$,~$e_{2}'$,~$e_{3}'$, $h_{1}'$,~$h_{2}'$,~$h_{3}'$ there exist exactly
-the same relations as between those without accent, of which
-they are entirely independent. Hence two also of them, say
-$h_{1}'$ and~$h_{2}'$, may be chosen arbitrarily so that in the equations
-given above for given values of $\ssfa$,~$\ssfb$,~$\ssfc$ four constants remain
-undetermined. If we now form, for all values of $\ssfa$~$\ssfb$~$\ssfc$\DPnote{** commas?} whatever,
-expressions of the type~\Eq{(305)} and add the corresponding field
-components, we again obtain a solution for \Name{Maxwell's} equations
-of the field and the boundary conditions, which, however, is now
-so general that it is capable of representing any electromagnetic
-process possible in the hollow cube considered. For it is always
-possible to dispose of the constants $e_{1}$,~$e_{2}$, $h_{1}'$,~$h_{2}'$ which have
-remained undetermined in the separate particular solutions in
-such a way that the process may be adapted to any initial state
-($t = 0$) whatever.
-
-If now, as we have assumed so far, the cavity is entirely void
-of matter, the process of radiation with a given initial state is
-uniquely determined in all its details. It consists of a set of
-stationary vibrations, every one of which is represented by one
-of the particular solutions considered, and which take place
-entirely independent of one another. Hence in this case there
-can be no question of irreversibility and hence also none of any
-tendency to equality of the partial energies corresponding to the
-separate partial vibrations. As soon, however, as we assume the
-%% -----File: 200.png---Folio 184-------
-presence in the cavity of only the slightest trace of matter which
-can influence the electrodynamic vibrations, \eg,~a few gas
-molecules, which emit or absorb radiation, the process becomes
-chaotic and a passage from less to more probable states will take
-place, though perhaps slowly. Without considering any further
-details of the electromagnetic constitution of the molecules, we
-may from the law of statistical mechanics quoted above draw
-the conclusion that, among all possible processes, that one in
-which the energy is distributed uniformly among all the independent
-variables of the state has the stationary character.
-
-From this let us determine these independent variables. In
-the first place there are the velocity components of the gas molecules.
-In the stationary state to every one of the three mutually
-independent velocity components of a molecule there corresponds
-on the average the energy $\frac{1}{3}\bar{L}$ where $\bar{L}$~represents the mean energy
-of a molecule and is given by~\Eq{(200)}. Hence the partial energy,
-which on the average corresponds to any one of the independent
-variables of the electromagnetic system, is just as large.
-
-Now, according to the above discussion, the \DPchg{electro-magnetic}{electromagnetic}
-state of the whole cavity for every stationary vibration corresponding
-to any one system of values of the numbers $\ssfa$~$\ssfb$~$\ssfc$\DPnote{** commas?} is
-determined, at any instant, by four mutually independent quantities.
-Hence for the radiation processes the number of independent
-variables of state is four times as large as the number
-of the possible systems of values of the positive integers $\ssfa$, $\ssfb$,~$\ssfc$.
-
-We shall now calculate the number of the possible systems of
-values $\ssfa$, $\ssfb$,~$\ssfc$, which correspond to the vibrations within a certain
-small range of the spectrum, say between the frequencies $\nu$ and
-$\nu + d \nu$. According to~\Eq{(306)}, these systems of values satisfy the
-inequalities
-\[
-\left(\frac{2l \nu}{c}\right)^{2}
- < \ssfa^{2} + \ssfb^{2} + \ssfc^{2}
- < \left(\frac{2l (\nu + d \nu)}{c}\right)^{2},
-\Tag{(309)}
-\]
-where not only $\dfrac{2l \nu}{c}$ but also $\dfrac{2l\, d\nu}{c}$ is to be thought of as a large
-number. If we now represent every system of values of $\ssfa$, $\ssfb$,~$\ssfc$
-graphically by a point, taking $\ssfa$,~$\ssfb$,~$\ssfc$ as coordinates in an orthogonal
-coordinate system, the points thus obtained occupy one
-octant of the space of infinite extent, and condition~\Eq{(309)} is
-%% -----File: 201.png---Folio 185-------
-equivalent to requiring that the distance of any one of these
-points from the origin of the coordinates shall lie between $\dfrac{2l\nu}{c}$
-and $\dfrac{2l (\nu + d\nu)}{c}$. Hence the required number is equal to the
-number of points which lie between the two spherical surface-octants
-corresponding to the radii $\dfrac{2l \nu}{c}$ and $\dfrac{2l (\nu + d\nu)}{c}$. Now since
-to every point there corresponds a cube of volume~$1$ and \textit{vice
-versa}, that number is simply equal to the space between the two
-spheres mentioned, and hence equal to
-\[
-\frac{1}{8}\, 4\pi \left(\frac{2l \nu}{c}\right)^{2} \frac{2l\, d\nu}{c},
-\]
-and the number of the independent variables of state is four times
-as large or
-\[
-\frac{16\pi l^{3} \nu^{2}\, d\nu}{c^{3}}\Add{.}
-\]
-
-Since, moreover, the partial energy~$\dfrac{\bar{L}}{3}$ corresponds on the average
-to every independent variable of state in the state of equilibrium,
-the total energy falling in the interval from $\nu$ to $\nu + d\nu$
-becomes
-\[
-\frac{16\pi l^{3} \nu^{2}\, d\nu}{3c^{3}} \bar{L}.
-\]
-Since the volume of the cavity is~$l^{3}$, this gives for the space
-density of the energy of frequency~$\nu$
-\[
-\ssfu\, d\nu = \frac{16\pi \nu^{2}\, d\nu}{3c^{3}} \bar{L},
-\]
-and, by substitution of the value of $\bar{L} = \dfrac{L}{N}$ from~\Eq{(200)},
-\[
-\ssfu = \frac{8\pi \nu^{2} kT}{c^{3}},
-\Tag{(310)}
-\]
-which is in perfect agreement with \Name{Rayleigh's} formula~\Eq{(285)}.
-
-If the law of the equipartition of energy held true in all
-%% -----File: 202.png---Folio 186-------
-cases, \Name{Rayleigh's} law of radiation would, in consequence, hold for
-all wave lengths and temperatures. But since this possibility
-is excluded by the measurements at hand, the only possible
-conclusion is that the law of the equipartition of energy and,
-with it, the system of \Name{Hamilton's} equations of motion does not
-possess the general importance attributed to it in classical dynamics.
-Therein lies the strongest proof of the necessity of a fundamental
-modification of the latter.
-%% -----File: 203.png---Folio 187-------
-
-\Part{V}{Irreversible Radiation Processes}
-%% -----File: 204.png---Folio 188-------
-%[Blank Page]
-%% -----File: 205.png---Folio 189-------
-
-\Chapter{I}{Fields of Radiation in General}
-
-\Section{170.} According to the theory developed in the preceding section,
-the nature of heat radiation within an isotropic medium,
-when the state is one of stable thermodynamic equilibrium, may
-be regarded as known in every respect. The intensity of the
-radiation, uniform in all directions, depends for all wave lengths
-only on the temperature and the velocity of propagation, according
-to equation~\Eq{(300)}, which applies to black radiation in any
-medium whatever. But there remains another problem to be
-solved by the theory. It is still necessary to explain how and by
-what processes the radiation which is originally present in the
-medium and which may be assigned in any way whatever,
-passes gradually, when the medium is bounded by walls impermeable
-to heat, into the stable state of black radiation, corresponding
-to the maximum of entropy, just as a gas which is
-enclosed in a rigid vessel and in which there are originally currents
-and temperature differences assigned in any way whatever
-gradually passes into the state of rest and of uniform distribution
-of temperature.
-
-To this much more difficult question only a partial answer can,
-at present, be given. In the first place, it is evident from the
-extensive discussion in the first chapter of the third part that,
-since irreversible processes are to be dealt with, the principles of
-pure electrodynamics alone will not suffice. For the second principle
-of thermodynamics or the principle of increase of entropy is
-foreign to the contents of pure electrodynamics as well as of pure
-mechanics. This is most immediately shown by the fact that the
-fundamental equations of mechanics as well as those of electrodynamics
-allow the direct reversal of every process as regards
-time, which contradicts the principle of increase of entropy.
-Of course all kinds of friction and of electric conduction of currents
-%% -----File: 206.png---Folio 190-------
-must be assumed to be excluded; for these processes, since
-they are always connected with the production of heat, do not
-belong to mechanics or electrodynamics proper.
-
-This assumption being made, the time~$t$ occurs in the fundamental
-equations of mechanics only in the components of
-acceleration; that is, in the form of the square of its differential.
-Hence, if instead of~$t$ the quantity~$-t$ is introduced as time variable
-in the equations of motion, they retain their form without change,
-and hence it follows that if in any motion of a system of material
-points whatever the velocity components of all points are suddenly
-reversed at any instant, the process must take place
-in the reverse direction. For the electrodynamic processes in
-a homogeneous non-conducting medium a similar statement
-holds. If in \Name{Maxwell's} equations of the electrodynamic field
-$-t$~is written everywhere instead of~$t$, and if, moreover, the sign of
-the magnetic field-strength~$\ssfH$ is reversed, the equations remain
-unchanged, as can be readily seen, and hence it follows that if in
-any electrodynamic process whatever the magnetic field-strength
-is everywhere suddenly reversed at a certain instant, while the
-electric field-strength keeps its value, the whole process must take
-place in the opposite sense.
-
-If we now consider any radiation processes whatever, taking
-place in a perfect vacuum enclosed by reflecting walls, it is found
-that, since they are completely determined by the principles of
-classical electrodynamics, there can be in their case no question of
-irreversibility of any kind. This is seen most clearly by considering
-the perfectly general formulć~\Eq{(305)}, which hold for a
-cubical cavity and which evidently have a periodic, \ie, reversible
-character. Accordingly we have frequently (Sec.\ \SecNo{144}~and~\SecNo{166})
-pointed out that the simple propagation of free radiation
-represents a reversible process. An irreversible element is
-introduced by the addition of emitting and absorbing substance.
-
-\Section{171.} Let us now try to define for the general case the state of
-radiation in the thermodynamic-macroscopic sense as we did
-above in \Sec{107}, \textit{et~seq.}, for a stationary radiation. Every one
-of the three components of the electric field-strength, \eg,~$\ssfE_{z}$ may,
-for the long time interval from $t = 0$ to $t = \ssfT$, be represented at
-every point, \eg, at the origin of coordinates, by a \Name{Fourier's}
-%% -----File: 207.png---Folio 191-------
-integral, which in the present case is somewhat more convenient
-than the \Name{Fourier's} series~\Eq{(149)}:
-\[
-\ssfE_{z} = \int_{0}^{\infty} d\nu\, C_{\nu} \cos (2\pi \nu t - \theta_{\nu}),
-\Tag{(311)}
-\]
-where $C_{\nu}$ (positive) and $\theta_{\nu}$ denote certain functions of the positive
-variable of integration~$\nu$. The values of these functions are
-not wholly determined by the behavior of~$\ssfE_{z}$ in the time interval
-mentioned, but depend also on the manner in which $\ssfE_{z}$~varies
-as a function of the time beyond both ends of that interval.
-Hence the quantities $C_{\nu}$~and~$\theta_{\nu}$ possess separately no definite
-physical significance, and it would be quite incorrect to think
-of the vibration~$\ssfE_{z}$ as, say, a continuous spectrum of periodic
-vibrations with the constant amplitudes~$C_{\nu}$. This may, by the
-way, be seen at once from the fact that the character of the vibration~$\ssfE_{z}$
-may vary with the time in any way whatever. How the
-spectral resolution of the vibration~$\ssfE_{z}$ is to be performed and to
-what results it leads will be shown below (\Sec{174}).
-
-\Section{172.} We shall, as heretofore~\Eq{(158)}, define~$J$, the ``intensity of
-the exciting vibration,''\footnote
- {Not to be confused with the ``field intensity'' (field-strength)~$\ssfE_{z}$ of the exciting vibration.}
-as a function of the time to be the mean
-value of~$\ssfE_{z}^{2}$ in the time interval from $t$ to $t + \tau$, where $\tau$~is taken
-as large compared with the time~$\DPchg{1/\nu}{\dfrac{1}{\nu}}$, which is the duration of one
-of the periodic partial vibrations contained in the radiation, but
-as small as possible compared with the time~$\ssfT$. In this statement
-there is a certain indefiniteness, from which results the fact that
-$J$~will, in general, depend not only on~$t$ but also on~$\tau$. If this is
-the case one cannot speak of the intensity of the exciting vibration
-at all. For it is an essential feature of the conception of the
-intensity of a vibration that its value should change but unappreciably
-within the time required for a single vibration. (Compare
-above, \Sec{3}.) Hence we shall consider in future only those
-processes for which, under the conditions mentioned, there exists
-a mean value of~$\ssfE_{z}^{2}$ depending only on~$t$. We are then obliged
-to assume that the quantities~$C_{\nu}$ in~\Eq{(311)} are negligible for all
-values of~$\nu$ which are of the same order of magnitude as~$\dfrac{1}{\tau}$ or
-smaller,~\ie,
-\[
-\nu\tau \text{ is large.}
-\Tag{(312)}
-\]
-%% -----File: 208.png---Folio 192-------
-
-In order to calculate~$J$ we now form from~\Eq{(311)} the value of~$\ssfE_{z}^{2}$
-and determine the mean value~$\bar{\ssfE_{z}^{2}}$ of this quantity by integrating
-with respect to~$t$ from $t$ to $t + \tau$, then dividing by~$\tau$ and
-passing to the limit by decreasing $\tau$ sufficiently. Thus we get
-\[
-\ssfE_{z}^{2} = \int_{0}^{\infty} \int_{0}^{\infty} d\nu'\, d\nu\, C_{\nu'} C_{\nu} \cos(2\pi \nu' t - \theta_{\nu'}) \cos(2\pi \nu t - \theta_{\nu}).
-\]
-If we now exchange the values of $\nu$~and~$\nu'$, the function under
-the sign of integration does not change; hence we assume
-\[
-\nu' > \nu
-\]
-and write:
-\[
-\ssfE_{z}^{2} = 2 \iint d\nu'\, d\nu\, C_{\nu'} C_{\nu} \cos(2\pi \nu' t - \theta_{\nu'}) \cos(2\pi \nu t - \theta_{\nu}),
-\]
-or
-\begin{multline*}
-\ssfE_{z}^{2} = \iint d\nu'\, d\nu\, C_{\nu'} C_{\nu} \{\cos[2\pi(\nu' - \nu)t - \theta_{\nu'} + \theta_{\nu}] \\
- + \cos[2\pi(\nu' + \nu)t - \theta_{\DPtypo{v'}{\nu'}} - \theta_{\nu}]\}.
-\end{multline*}
-And hence
-\begin{multline*}
-J = \bar{\ssfE_{z}^{2}}
- = \frac{1}{\tau} \int_{t}^{t+\tau} \ssfE_{z}^{2}\, dt \\
- = \iint d\nu'\, d\nu\, C_{\nu'} C_{\nu} \biggl\{
- \frac{\sin \pi(\nu' - \nu)\tau ˇ \cos[\pi(\nu' - \nu)(2t + \tau) - \theta_{\nu'} + \theta_{\nu}]}{\pi(\nu' - \nu) \tau} \\
- + \frac{\sin \pi(\nu' + \nu)\tau ˇ \cos[\pi(\nu' + \nu)(2t + \tau) - \theta_{\nu'} - \theta_{\nu}]}{\pi(\nu' + \nu) \tau}\biggr\}.
-\end{multline*}
-
-If we now let $\tau$~become smaller and smaller, since $\nu\tau$~remains
-large, the denominator $(\nu' + \nu)\tau$ of the second fraction remains
-large under all circumstances, while that of the first fraction
-$(\nu' - \nu)\tau$ may decrease with decreasing value of~$\tau$ to less than any
-finite value. Hence for sufficiently small values of $(\nu' - \nu)$ the integral
-reduces to
-\[
-\iint d\nu'\, d\nu\, C_{\nu'} C_{\nu} \cos[2\pi(\nu' - \nu)t - \theta_{\nu'} + \theta_{\nu}]
-\]
-which is in fact independent of~$\tau$. The remaining terms of the
-double integral, which correspond to larger values of $\nu' - \nu$, \ie,
-to more rapid changes with the time, depend in general on~$\tau$ and
-%% -----File: 209.png---Folio 193-------
-therefore must vanish, if the intensity~$J$ is not to depend on~$\tau$.
-Hence in our case on introducing as a second variable of integration
-instead of~$\nu$
-\[
-\mu = \nu' - \nu\ (> 0)
-\]
-we have
-\[
-J = \iint d\mu\, d\nu\, C_{\nu+\mu} C_{\nu} \cos(2\pi \mu t - \theta_{\nu+\mu} + \theta_{\nu})
-\Tag{(313)}
-\]
-or
-\begin{align*}
-J &= \int d\mu\, (A_{\mu} \cos 2\pi \mu t + B_{\mu} \sin 2\pi \mu t) \\
-\LeftText{where}
-A_{\mu} &= \int d\nu\, C_{\nu+\mu} C_{\nu} \cos(\theta_{\nu+\mu} - \theta_{\nu})
-\Tag{(314)}\\
-B_{\mu} &= \int d\nu\, C_{\nu+\mu} C_{\nu} \sin(\theta_{\nu+\mu} - \theta_{\nu})\Add{.}
-\end{align*}
-
-By this expression the intensity~$J$ of the exciting vibration,
-if it exists at all, is expressed by a function of the time in the form
-of a \Name{Fourier's} integral.
-
-\Section{173.} The conception of the intensity of vibration~$J$ necessarily
-contains the assumption that this quantity varies much more
-slowly with the time~$t$ than the vibration $\ssfE_{z}$ itself. The same
-follows from the calculation of~$J$ in the preceding paragraph.
-For there, according to~\Eq{(312)}, $\nu\tau$~and $\nu'\tau$ are large, but $(\nu' - \nu)\tau$
-is small for all pairs of values $C_{\nu}$~and~$C_{\nu'}$ that come into consideration;
-hence, \textit{a~fortiori},
-\[
-\frac{\nu' - \nu}{\nu} = \frac{\mu}{\nu} \quad\text{is small},
-\Tag{(315)}
-\]
-and accordingly the \Name{Fourier's} integrals~$\ssfE_{z}$ in~\Eq{(311)} and $J$ in~\Eq{(314)}
-vary with the time in entirely different ways. Hence in the
-following we shall have to distinguish, as regards dependence on
-time, two kinds of quantities, which vary in different ways:
-Rapidly varying quantities, as~$\ssfE_{z}$, and slowly varying quantities
-as $J$~and~$\ssfI$ the spectral intensity of the exciting vibration, whose
-value we shall calculate in the next paragraph. Nevertheless
-this difference in the variability with respect to time of the quantities
-%% -----File: 210.png---Folio 194-------
-named is only relative, since the absolute value of the differential
-coefficient of~$J$ with respect to time depends on the value of
-the unit of time and may, by a suitable choice of this unit, be
-made as large as we please. It is, therefore, not proper to speak
-of $J(t)$ simply as a slowly varying function of~$t$. If, in the
-following, we nevertheless employ this mode of expression for
-the sake of brevity, it will always be in the relative sense, namely,
-with respect to the different behavior of the function~$\ssfE_{z}(t)$.
-
-On the other hand, as regards the dependence of the phase
-constant~$\theta_{\nu}$ on its index~$\nu$ it necessarily possesses the property
-of rapid variability in the \emph{absolute} sense. For, although $\mu$~is
-small compared with~$\nu$, nevertheless the difference $\theta_{\nu+\mu} - \theta_{\nu}$
-is in general not small, for if it were, the quantities $A_{\mu}$~and~$B_{\mu}$
-in~\Eq{(314)} would have too special values and hence it follows that
-$\DPchg{(\dd \theta_{\nu}/\dd \nu)}{\dfrac{\dd \theta_{\nu}}{\dd \nu}} ˇ \nu$ must be large. This is not essentially modified by
-changing the unit of time or by shifting the origin of time.
-
-Hence the rapid variability of the quantities~$\theta_{\nu}$ and also $C_{\nu}$
-with~$\nu$ is, in the absolute sense, a necessary condition for the
-existence of a definite intensity of vibration~$J$, or, in other words,
-for the possibility of dividing the quantities depending on the
-time into those which vary rapidly and those which vary slowly---a
-distinction which is also made in other physical theories and
-upon which all the following investigations are based.
-
-\Section{174.} The distinction between rapidly variable and slowly
-variable quantities introduced in the preceding section has,
-at the present stage, an important physical aspect, because in
-the following we shall assume that only slow variability with
-time is capable of direct measurement. On this assumption we
-approach conditions as they actually exist in optics and heat
-radiation. Our problem will then be to establish relations between
-slowly variable quantities exclusively; for these only can
-be compared with the results of experience. Hence we shall now
-determine the most important one of the slowly variable quantities
-to be considered here, namely, the ``spectral intensity''~$\ssfI$ of
-the exciting vibration. This is effected as in~\Eq{(158)} by means of
-the equation
-\[
-J = \int_{0}^{\infty} \ssfI\, d\nu.
-\]
-%% -----File: 211.png---Folio 195-------
-By comparison with~\DPtypo{313}{\Eq{(313)}} we obtain:
-\[%[** Attn]
-\left\{
-\begin{gathered}
-\ssfI = \int d\mu\, (\ssfA_{\mu} \cos 2 \pi \mu t + \ssfB_{\mu} \sin 2 \pi \mu t) \\
-\text{where\qquad}
-\begin{aligned}[t]
-\ssfA_{\mu} &= \bar{C_{\nu + \mu} C_{\nu} \cos (\theta_{\nu + \mu} - \theta_{\nu})} \\
-\ssfB_{\mu} &= \bar{C_{\nu + \mu} C_{\nu} \sin (\theta_{\nu + \mu} - \theta_{\nu})}.
-\end{aligned}
-\end{gathered}\right.
-\Tag{(316)}
-\]
-
-By this expression the spectral intensity,~$\ssfI$, of the exciting vibration
-at a point in the spectrum is expressed as a slowly variable
-function of the time~$t$ in the form of a \Name{Fourier's} integral. The
-dashes over the expressions on the right side denote the mean
-values extended over a narrow spectral range for a given value
-of~$\mu$. If such mean values do not exist, there is no definite spectral
-intensity.
-%% -----File: 212.png---Folio 196-------
-
-
-\Chapter{II}{One Oscillator in the Field of Radiation}
-
-\Section{175.} If in any field of radiation whatever we have an ideal
-oscillator of the kind assumed above (\Sec{135}), there will take
-place between it and the radiation falling on it certain mutual
-actions, for which we shall again assume the validity of the
-elementary dynamical law introduced in the preceding section.
-The question is then, how the processes of emission and absorption
-will take place in the case now under consideration.
-
-In the first place, as regards the emission of radiant energy by
-the oscillator, this takes place, as before, according to the hypothesis
-of emission of quanta (\Sec{147}), where the probability
-quantity~$\eta$ again depends on the corresponding spectral intensity~$\ssfI$
-through the relation~\Eq{(265)}.
-
-On the other hand, the absorption is calculated, exactly as
-above, from~\Eq{(234)}, where the vibrations of the oscillator also
-take place according to the equation~\Eq{(233)}. In this way, by
-calculations analogous to those performed in the second chapter
-of the preceding part, with the difference only that instead
-of the Fourier's series~\Eq{(235)} the Fourier's integral~\Eq{(311)} is used,
-we obtain for the energy absorbed by the oscillator in the time~$\tau$
-the expression
-\[
-\frac{\tau}{4L} \int d\mu\, (\ssfA_{\mu} \cos 2\pi \mu t + \ssfB_{\mu} \sin 2\pi \mu t),
-\]
-where the constants $\ssfA_{\mu}$~and~$\ssfB_{\mu}$ denote the mean values expressed
-in~\Eq{(316)}, taken for the spectral region in the neighborhood of
-the natural frequency~$\nu_{0}$ of the oscillator. Hence the law of
-absorption will again be given by equation~\Eq{(249)}, which now
-holds also for an intensity of vibration~$\ssfI$ varying with the time.
-
-\Section{176.} There now remains the problem of deriving the expression
-for~$\ssfI$, the spectral intensity of the vibration exciting the oscillator,
-when the thermodynamic state of the field of radiation at
-%% -----File: 213.png---Folio 197-------
-the oscillator is given in accordance with the statements made in
-\Sec{17}.
-
-Let us first calculate the total intensity $J = \bar{{\ssfE_{z}^{2}}}$ of the vibration
-exciting an oscillator, from the intensities of the heat rays striking
-the oscillator from all directions.
-
-For this purpose we must also allow for the polarization of the
-monochromatic rays which strike the oscillator. Let us begin
-by considering a pencil which strikes the oscillator within a conical
-element whose vertex lies in the oscillator and whose solid
-angle,~$d\Omega$, is given by~\Eq{(5)}, where the angles $\theta$~and~$\phi$, polar coordinates,
-designate the direction of the propagation of the rays.
-The whole pencil consists of a set of monochromatic pencils,
-one of which may have the principal values of intensity $\ssfK$~and~$\ssfK'$
-(\Sec{17}). If we now denote the angle which the plane of
-vibration belonging to the principal intensity~$\ssfK$ makes with the
-plane through the direction of the ray and the $z$-axis (the axis of
-the oscillator) by~$\psi$, no matter in which quadrant it lies, then,
-according to~\Eq{(8)}, the specific intensity of the monochromatic
-pencil may be resolved into the two plane polarized components
-at right angles with each other,
-\begin{align*}
- \ssfK\cos^{2}\psi &+\ssfK'\sin^{2} \psi\\
- \ssfK\sin^{2}\psi &+\ssfK'\cos^{2} \psi,
-\end{align*}
-the first of which vibrates in a plane passing through the $z$-axis
-and the second in a plane perpendicular thereto.
-
-The latter component does not contribute anything to the
-value of~$\ssfE_{z}^{2}$, since its electric field-strength is perpendicular to
-the axis of the oscillator. Hence there remains only the first
-component whose electric field-strength makes the angle $\dfrac{\pi}{2} - \theta$
-with the $z$-axis. Now according to \Name{Poynting's} law the intensity of
-a plane polarized ray in a vacuum is equal to the product of $\dfrac{c}{4\pi}$
-and the mean square of the electric field-strength. Hence the
-mean square of the electric field-strength of the pencil here
-considered is
-\[
-\frac{4\pi}{c} (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\, d\nu\, d\Omega,
-\]
-%% -----File: 214.png---Folio 198-------
-and the mean square of its component in the direction of the
-$z$-axis is
-\[
-\frac{4\pi}{c} (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi) \sin^{2} \theta\, d\nu\, d\Omega.
-\Tag{(317)}
-\]
-By integration over all frequencies and all solid angles we then
-obtain the value required
-\[
-\bar{\ssfE_{z}^{2}}
- = \frac{4\pi}{c} \int \sin^{2} \theta\, d\Omega
- \int d\nu\, (\ssfK_{\nu} \cos^{2} \psi + \ssfK_{\nu}' \sin^{2} \psi) = J.
-\Tag{(318)}
-\]
-
-The space density~$u$ of the electromagnetic energy at a point
-of the field is
-\[
-u = \frac{1}{8\pi}
- (\bar{\ssfE_{x}^{2}} + \bar{\ssfE_{y}^{2}} + \bar{\ssfE_{z}^{2}}
- + \bar{\ssfH_{x}^{2}} + \bar{\ssfH_{y}^{2}} + \bar{\ssfH_{z}^{2}}),
-\]
-where $\ssfE_{x}^{2}$,~$\ssfE_{y}^{2}$,~$\ssfE_{z}^{2}$, $\ssfH_{x}^{2}$,~$\ssfH_{y}^{2}$,~$\ssfH_{z}^{2}$ denote the squares of the
-field-strengths, regarded as ``slowly variable'' quantities, and are
-hence supplied with the dash to denote their mean value. Since
-for every separate ray the mean electric and magnetic energies
-are equal, we may always write
-\[
-u = \frac{1}{4\pi} = (\bar{\ssfE_{x}^{2}} + \bar{\ssfE_{y}^{2}} + \bar{\ssfE_{z}^{2}}).
-\Tag{(319)}
-\]
-If, in particular, all rays are unpolarized and if the intensity of
-radiation is constant in all directions, $\ssfK_{\nu} = \ssfK_{\nu}'$ and, since
-\[
-\begin{gathered}
-\int \sin^{2} \theta\, d\Omega = \iint \sin^{3} \theta\, d\theta\, d\phi = \frac{8\pi}{3} \\
-\bar{\ssfE_{z}^{2}} = \frac{32\pi^{2}}{3c} \int \ssfK_{\nu}\, d\nu = \bar{\ssfE_{x}^{2}} = \bar{\ssfE_{y}^{2}}
-\end{gathered}
-\Tag{(319a)}
-\]
-and, by substitution in~\Eq{(319)},
-\[
-u = \frac{8\pi}{c} \int \ssfK_{\nu}\, d\nu,
-\]
-which agrees with \Eq{(22)}~and~\Eq{(24)}.
-
-\Section{177.} Let us perform the spectral resolution of the intensity~$J$
-according to \Sec{174}; namely,
-\[
-J = \int \ssfI_{\nu}\, d\nu.
-\]
-%% -----File: 215.png---Folio 199-------
-Then, by comparison with~\Eq{(318)}, we find for the intensity of a
-definite frequency~$\nu$ contained in the exciting vibration the value
-\[
-\ssfI = \frac{4\pi}{c} \int \sin^{2} \theta\, d\Omega\, (\ssfK_{\nu} \cos^{2} \psi + \ssfK_{\nu}' \sin^{2} \psi).
-\Tag{(320)}
-\]
-For radiation which is unpolarized and uniform in all directions
-we obtain again, in agreement with~\Eq{(160)},
-\[
-\ssfI = \frac{32\pi^{2}}{3c} \ssfK.
-\]
-
-\Section{178.} With the value~\Eq{(320)} obtained for~$\ssfI$ the total energy
-absorbed by the oscillator in an element of time~$dt$ from the
-radiation falling on it is found from~\Eq{(249)} to be
-\[
-\frac{\pi\, dt}{cL} \int \sin^{2} \theta\, d\Omega\, (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi).
-\]
-Hence the oscillator absorbs in the time~$dt$ from the pencil striking
-it within the conical element~$d\Omega$ an amount of energy equal to
-\[
-\frac{\pi\, dt}{cL} \sin^{2} \theta (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\, d\Omega.
-\Tag{(321)}
-\]
-%% -----File: 216.png---Folio 200-------
-
-\Chapter{III}{A System of Oscillators}
-
-\Section{179.} Let us suppose that a large number~$N$ of similar oscillators
-with parallel axes, acting quite independently of one another, are
-distributed irregularly in a volume-element of the field of radiation,
-the dimensions of which are so small that within it the intensities
-of radiation~$\ssfK$ do not vary appreciably. We shall investigate
-the mutual action between the oscillators and the radiation
-which is propagated freely in space.
-
-As before, the state of the field of radiation may be given by
-the magnitude and the azimuth of vibration~$\psi$ of the principal
-intensities $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ of the pencils which strike the system of
-oscillators, where $\ssfK_{\nu}$~and~$\ssfK_{\nu}'$ depend in an arbitrary way on the
-direction angles $\theta$~and~$\phi$. On the other hand, let the state of the
-system of oscillators be given by the densities of distribution
-$w_{1}$,~$w_{2}$, $w_{3},~\dots$~\Eq{(166)}, with which the oscillators are distributed
-among the different region elements, $w_{1}$,~$w_{2}$, $w_{3},~\dots$
-being any proper fractions whose sum is~$1$. Herein, as always,
-the $n$th~region element is supposed to contain the oscillators
-with energies between $(n - 1)h\nu$ and~$nh\nu$.
-
-The energy absorbed by the system in the time~$dt$ within the
-conical element~$d\Omega$ is, according to~\Eq{(321)},
-\[
-\frac{\pi N\, dt}{cL} \sin^{2} \theta (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\, d\Omega.
-\Tag{(322)}
-\]
-Let us now calculate also the energy emitted within the same
-conical element.
-
-\Section{180.} The total energy emitted in the time element~$dt$ by all $N$
-oscillators is found from the consideration that a single oscillator,
-according to~\Eq{(249)}, takes up an energy element~$h\nu$ during the time
-\[
-\frac{4h\nu L}{\ssfI} = \tau,
-\Tag{(323)}
-\]
-%% -----File: 217.png---Folio 201-------
-and hence has a chance to emit once, the probability being~$\eta$.
-We shall assume that the intensity~$\ssfI$ of the exciting vibration
-does not change appreciably in the time~$\tau$. Of the $Nw_{n}$~oscillators
-which at the time~$t$ are in the $n$th~region element a number
-$Nw_{n}\eta$ will emit during the time~$\tau$, the energy emitted by each
-being~$nh\nu$. From~\Eq{(323)} we see that the energy emitted by all
-oscillators during the time element~$dt$ is
-\[
-\sum Nw_{n}\, \eta\, nh\nu \frac{dt}{\tau} = \frac{N\eta \ssfI\, dt}{4L} \sum nw_{n},
-\]
-or, according to~\Eq{(265)},
-\[
-\frac{N(1 - \eta)\, dt}{4pL} \sum nw_{n}.
-\Tag{(324)}
-\]
-
-From this the energy emitted within the conical element~$d\Omega$
-may be calculated by considering that, in the state of thermodynamic
-equilibrium, the energy emitted in every conical element
-is equal to the energy absorbed and that, in the general case, the
-energy emitted in a certain direction is independent of the energy
-simultaneously absorbed. For the stationary state we have
-from \Eq{(160)}~and~\Eq{(265)}
-\[
-\ssfK = \ssfK'
- = \frac{3c}{32\pi^{2}} \ssfI
- = \frac{3c}{32\pi^{2}}\, \frac{1 - \eta}{p\eta}
-\Tag{(325)}
-\]
-and further from \Eq{(271)}~and~\Eq{(265)}
-\[
-w_{n} = \frac{1}{p \ssfI} \left(\frac{p \ssfI}{1 + p \ssfI}\right)^{n}
- = \eta(1 - \eta)^{n-1},
-\Tag{(326)}
-\]
-and hence
-\[
-\sum nw_{n} = \eta \sum n(1 - \eta)^{n-1} = \frac{1}{\eta}.
-\Tag{(327)}
-\]
-Thus the energy emitted~\Eq{(324)} becomes
-\[
-\frac{N(1 - \eta)\, dt}{4Lp\eta}.
-\Tag{(328)}
-\]
-This is, in fact, equal to the total energy absorbed, as may be
-found by integrating the expression~\Eq{(322)} over all conical elements~$d\Omega$
-and taking account of~\Eq{(325)}.
-%% -----File: 218.png---Folio 202-------
-
-Within the conical element~$d\Omega$ the energy emitted or absorbed
-will then be
-\[
-\frac{\pi N\, dt}{c} \sin^{2} \theta\, \ssfK\, d\Omega,
-\]
-or, from \Eq{(325)},~\Eq{(327)} and~\Eq{(268)},
-\[
-\frac{\pi h\nu^{3} (1 - \eta)N}{c^{3}L} \sum nw_{n} \sin^{2} \theta\, d\Omega\, dt,
-\Tag{(329)}
-\]
-and this is the general expression for the energy emitted by the
-system of oscillators in the time element~$dt$ within the conical
-element~$d\Omega$, as is seen by comparison with~\Eq{(324)}.
-
-\Section{181.} Let us now, as a preparation for the following deductions,
-consider more closely the properties of the different pencils
-passing the system of oscillators. From all directions rays
-strike the volume-element that contains the oscillators; if we
-again consider those which come toward it in the direction
-$(\theta, \phi)$ within the conical element~$d\Omega$, the vertex of which lies in
-the volume-element, we may in the first place think of them as
-being resolved into their monochromatic constituents, and then
-we need consider further only that one of these constituents which
-corresponds to the frequency~$\nu$ of the oscillators; for all other rays
-simply pass the oscillators without influencing them or being
-influenced by them. The specific intensity of a monochromatic
-ray of frequency~$\nu$ is
-\[
-\ssfK + \ssfK'
-\]
-where $\ssfK$~and~$\ssfK'$ represent the principal intensities which we
-assume as non-coherent. This ray is now resolved into two components
-according to the directions of its principal planes of
-vibration (\Sec{176}).
-
-The first component,
-\[
-\ssfK \sin^{2} \psi + \ssfK' \cos^{2} \psi,
-\]
-passes by the oscillators and emerges on the other side with no
-change whatever. Hence it gives a plane polarized ray, which
-starts from the system of oscillators in the direction $(\theta, \phi)$ within
-the solid angle~$d\Omega$ and whose vibrations are perpendicular to the
-axis of the oscillators and whose intensity is
-\[
-\ssfK \sin^{2} \psi + \ssfK' \cos^{2} \psi = K''.
-\Tag{(330)}
-\]
-%% -----File: 219.png---Folio 203-------
-The second component,
-\[
-\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi,
-\]
-polarized at right angles to the first consists again, according to
-\Sec{176}, of two parts
-\begin{alignat*}{2}
-&(\ssfK \cos^{2} \psi &&+ \ssfK' \sin^{2} \psi) \cos^{2} \theta
-\Tag{(331)} \\
-\LeftText{and}
-&(\ssfK \cos^{2} \psi &&+ \ssfK' \sin^{2} \psi) \sin^{2} \theta,
-\Tag{(332)}
-\end{alignat*}
-of which the first passes by the system without any change, since
-its direction of vibration is at right angles to the axes of the oscillators,
-while the second is weakened by absorption, say by the
-small fraction~$\beta$. Hence on emergence this component has only
-the intensity
-\[
-(1 - \beta) (\ssfK \cos^{2} \psi + K' \sin^{2} \psi) \sin^{2} \theta.
-\Tag{(333)}
-\]
-It is, however, strengthened by the radiation emitted by the system
-of oscillators~\Eq{(329)}, which has the value
-\[
-\beta' (1 - \eta) \sum nw_{n} \sin^{2} \theta,
-\Tag{(334)}
-\]
-where $\beta'$~denotes a certain other constant, which depends only
-on the nature of the system and whose value is obtained at once
-from the condition that, in the state of thermodynamic equilibrium,
-the loss is just compensated by the gain.
-
-For this purpose we make use of the relations \Eq{(325)}~and~\Eq{(327)}
-corresponding to the stationary state, and thus find that the sum
-of the expressions \Eq{(333)}~and~\Eq{(334)} becomes just equal to~\Eq{(332)};
-and thus for the constant~$\beta'$ the following value is found:
-\[
-\beta'
- = \beta \frac{3c}{32\pi^{2} p}
- = \beta \frac{h\nu^{3}}{c^{2}}.
-\]
-Then by addition of \Eq{(331)},~\Eq{(333)} and~\Eq{(334)} the total specific
-intensity of the radiation which emanates from the system of
-oscillators within the conical element~$d\Omega$, and whose plane of
-vibration is parallel to the axes of the oscillators, is found to be
-\[
-%[** TN: Set on two lines in the original]
-\ssfK''' = \ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi +
-\beta \sin^{2} \theta \bigl(\ssfK_{e} - (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\bigr)
-\Tag{(335)}
-\]
-where for the sake of brevity the term referring to the emission
-is written
-\[
-\frac{h\nu^{3}}{c^{2}}(1 - \eta) \sum nw_{n} = \ssfK_{e}.
-\Tag{(336)}
-\]
-%% -----File: 220.png---Folio 204-------
-
-Thus we finally have a ray starting from the system of oscillators
-in the direction $(\theta, \phi)$ within the conical element~$d\Omega$ and
-consisting of two components $\ssfK''$~and~$\ssfK'''$ polarized perpendicularly
-to each other, the first component vibrating at right angles
-to the axes of the oscillators.
-
-In the state of thermodynamic equilibrium
-\[
-\ssfK = \ssfK' = \ssfK'' = \ssfK''' = \ssfK_{e},
-\]
-a result which follows in several ways from the last equations.
-
-\Section{182.} The constant~$\beta$ introduced above, a small positive number,
-is determined by the spacial and spectral limits of the radiation
-influenced by the system of oscillators. If $q$~denotes the
-cross-section at right angles to the direction of the ray, $\Delta\nu$~the
-spectral width of the pencil cut out of the total incident radiation
-by the system, the energy which is capable of absorption and
-which is brought to the system of oscillators within the conical
-element~$d\Omega$ in the time~$dt$ is, according to \Eq{(332)}~and~\Eq{(11)},
-\[
-q\, \Delta\nu\, (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi) \sin^{2} \theta\, d\Omega\, dt.
-\Tag{(337)}
-\]
-Hence the energy actually absorbed is the fraction~$\beta$ of this value.
-Comparing this with~\Eq{(322)} we get
-\[
-\beta = \frac{\pi N}{q ˇ \Delta\nu ˇ cL}.
-\Tag{(338)}
-\]
-%% -----File: 221.png---Folio 205-------
-
-\Chapter[Conservation of Energy and Increase Of Entropy]
-{IV}{Conservation of Energy and Increase Of
-Entropy. Conclusion}
-
-\Section{183.} It is now easy to state the relation of the two principles of
-thermodynamics to the irreversible processes here considered.
-Let us consider first the \emph{conservation of energy}. If there is no
-oscillator in the field, every one of the elementary pencils, infinite
-in number, retains, during its rectilinear propagation, both its
-specific intensity~$\ssfK$ and its energy without change, even though it
-be reflected at the surface, assumed as plane and reflecting, which
-bounds the field (\Sec{166}). The system of oscillators, on the
-other hand, produces a change in the incident pencils and hence
-also a change in the energy of the radiation propagated in the
-field. To calculate this we need consider only those monochromatic
-rays which lie close to the natural frequency~$\nu$ of the
-oscillators, since the rest are not altered at all by the system.
-
-The system is struck in the direction $(\theta, \phi)$ within the conical
-element~$d\Omega$ which converges toward the system of oscillators by
-a pencil polarized in some arbitrary way, the intensity of which
-is given by the sum of the two principal intensities $\ssfK$~and~$\ssfK$'.
-This pencil, according to \Sec{182}, conveys the energy
-\[
-q\, \Delta\nu\, (\ssfK + \ssfK')\, d\Omega\, dt
-\]
-to the system in the time~$dt$; hence this energy is taken from the
-field of radiation on the side of the rays arriving within~$d\Omega$. As
-a compensation there emerges from the system on the other side
-in the same direction $(\theta, \phi)$ a pencil polarized in some definite
-way, the intensity of which is given by the sum of the two components
-$\ssfK''$~and~$\ssfK'''$. By it an amount of energy
-\[
-q\, \Delta\nu\, (\ssfK'' + \ssfK''')\, d\Omega\, dt,
-\]
-is added to the field of radiation. Hence, all told, the change in
-energy of the field of radiation in the time~$dt$ is obtained by subtracting
-%% -----File: 222.png---Folio 206-------
-the first expression from the second and by integrating
-with respect to~$d\Omega$. Thus we get
-\[
-dt\, \Delta\nu \int (\ssfK'' + \ssfK''' - \ssfK - \ssfK')q\, d\Omega,
-\]
-or by taking account of \Eq{(330)},~\Eq{(335)}, and~\Eq{(338)}
-\[
-\frac{\pi N\, dt}{cL} \int d\Omega \sin^{2} \theta \bigl(\ssfK_{e} - (\ssfK \cos^{2} \psi + \ssfK' \sin^{2} \psi)\bigr).
-\Tag{(339)}
-\]
-
-\Section{184.} Let us now calculate the change in energy of the system
-of oscillators which has taken place in the same time~$dt$. According
-to~\Eq{(219)}, this energy at the time~$t$ is
-\[
-E = Nh\nu \sum_{1}^{\infty} (n - \DPchg{\frac{1}{2}}{\tfrac{1}{2}})w_{n},
-\]
-where the quantities~$w_{n}$ whose total sum is equal to~$1$ represent
-the densities of distribution characteristic of the state. Hence
-the energy change in the time~$dt$ is
-\[
-dE = Nh\nu \sum_{1}^{\infty} (n - \DPchg{\frac{1}{2}}{\tfrac{1}{2}})\, dw_{n}
- = Nh\nu \sum_{1}^{\infty} n\, dw_{n}.
-\Tag{(340)}
-\]
-To calculate~$dw_{n}$ we consider the $n$th~region element. All of
-the oscillators which lie in this region at the time~$t$ have, after
-the lapse of time~$\tau$, given by~\Eq{(323)}, left this region; they have
-either passed into the $(n + 1)$st~region, or they have performed
-an emission at the boundary of the two regions. In compensation
-there have entered $(1 - \eta)Nw_{n-1}$ oscillators during the
-time~$\tau$, that is, all oscillators which, at the time~$t$, were in the
-$(n - 1)$st~region element, excepting such as have lost their energy
-by emission. Thus we obtain for the required change in the
-time~$dt$
-\[
-N\, dw_{n} = \frac{dt}{\tau} N \bigl((1 - \eta)w_{n-1} - w_{n}\bigr).
-\Tag{(341)}
-\]
-A separate discussion is required for the first region element $n = 1$.
-For into this region there enter in the time~$\tau$ all those oscillators
-which have performed an emission in this time. Their number
-is
-\[
-\eta (w_{1} + w_{2} + w_{3} + \dots)N = \eta N.
-\]
-%% -----File: 223.png---Folio 207-------
-Hence we have
-\[
-N\, dw_{1} = \frac{dt}{\tau} N(\eta - w_{1}).
-\]
-We may include this equation in the general one~\Eq{(341)} if we
-introduce as a new expression
-\[
-w_{0} = \frac{\eta}{1 - \eta}.
-\Tag{(342)}
-\]
-Then~\Eq{(341)} gives, substituting $\tau$ from~\Eq{(323)},
-\[
-dw_{n} = \frac{\ssfI\, dt}{4h\nu L} \bigl((1 - \eta) w_{n-1} - w_{n}\bigr),
-\Tag{(343)}
-\]
-and the energy change~\Eq{(340)} of the system of oscillators becomes
-\[
-dE = \frac{N\ssfI\, dt}{4L} \sum_{1}^{\infty} n \bigl((1 - \eta)w_{n-1} - w_{n}\bigr).
-\]
-The sum $\sum$ may be simplified by recalling that
-\begin{align*}
-\sum_{1}^{\infty} nw_{n-1}
- &= \sum_{1}^{\infty} (n - 1)w_{n-1} + \sum_{1}^{\infty} w_{n-1}\\
- &= \sum_{1}^{\infty} nw_{n} + w_{0} + 1
- = \sum_{1}^{\infty} nw_{n} + \frac{1}{1 - \eta}.
-\end{align*}
-Then we have
-\[
-dE = \frac{N\ssfI\, dt}{4L} (1 - \eta \sum_{1}^{\infty} nw_{n}).
-\Tag{(344)}
-\]
-This expression may be obtained more readily by considering that
-$dE$~is the difference of the total energy absorbed and the total
-energy emitted. The former is found from~\Eq{(250)}, the latter from~\Eq{(324)},
-by taking account of~\Eq{(265)}.
-
-The principle of the conservation of energy demands that the
-sum of the energy change~\Eq{(339)} of the field of radiation and the
-energy change~\Eq{(344)} of the system of oscillators shall be zero,
-which, in fact, is quite generally the case, as is seen from the relations
-\Eq{(320)}~and~\Eq{(336)}.
-
-\Section{185.} We now turn to the discussion of the second principle, the
-principle of the \emph{increase of entropy}, and follow closely the above
-discussion regarding the energy. When there is no oscillator in
-the field, every one of the elementary pencils, infinite in number,
-%% -----File: 224.png---Folio 208-------
-retains during rectilinear propagation both its specific intensity
-and its entropy without change, even when reflected at the surface,
-assumed as plane and reflecting, which bounds the field.
-The system of oscillators, however, produces a change in the
-incident pencils and hence also a change in the entropy of the
-radiation propagated in the field. For the calculation of this
-change we need to investigate only those monochromatic rays
-which lie close to the natural frequency~$\nu$ of the oscillators, since
-the rest are not altered at all by the system.
-
-The system of oscillators is struck in the direction $(\theta, \phi)$ within
-the conical element~$d\Omega$ converging toward the system by a pencil
-polarized in some arbitrary way, the spectral intensity of which
-is given by the sum of the two principal intensities $\ssfK$~and~$\ssfK'$ with
-the azimuth of vibration~$\psi$ and $\dfrac{\pi}{2} + \psi$ respectively, which are
-assumed to be non-coherent. According to~\Eq{(141)} and \Sec{182}
-this pencil conveys the entropy
-\[
-q\, \Delta\nu\, [\ssfL(\ssfK) + \ssfL(\ssfK')]\, d\Omega\, dt
-\Tag{(345)}
-\]
-to the system of oscillators in the time~$dt$, where the function~$\ssfL(\ssfK)$
-is given by~\Eq{(278)}. Hence this amount of entropy is taken
-from the field of radiation on the side of the rays arriving within~$d\Omega$.
-In compensation a pencil starts from the system on the
-other side in the same direction $(\theta, \phi)$ within~$d\Omega$ having the
-components $\ssfK''$~and~$\ssfK'''$ with the azimuth of vibration $\dfrac{\pi}{2}$ and $0$
-respectively, but its entropy radiation is not represented by
-$\ssfL(\ssfK'') + \ssfL(\ssfK''')$, since $\ssfK''$~and~$\ssfK'''$ are not non-coherent, but by
-\[
-\ssfL(\ssfK_{0}) + \ssfL(\ssfK_{0}')
-\Tag{(346)}
-\]
-where $\ssfK_{0}$~and~$\ssfK_{0}'$ represent the principal intensities of the pencil.
-
-For the calculation of $\ssfK_{0}$~and~$\ssfK_{0}'$ we make use of the fact that,
-according to \Eq{(330)}~and~\Eq{(335)}, the radiation $\ssfK''$~and~$\ssfK'''$, of which
-the component~$\ssfK'''$ vibrates in the azimuth~$0$, consists of the
-following three components, non-coherent with one another:
-% [** TN: Next two equations reformatted slightly from the original]
-\[
-\ssfK_{1}
- = \ssfK \sin^{2} \psi + \ssfK \cos^{2} \psi(1 - \beta \sin^{2} \theta)
- = \ssfK(1 - \beta \sin^{2} \theta \cos^{2} \psi)
-\]
-with the azimuth of vibration $\DPchg{tg}{\tg}^{2}\psi_{1} = \dfrac{\DPchg{tg}{\tg}^{2}\psi}{1 - \beta \sin^{2} \theta}$,
-%% -----File: 225.png---Folio 209-------
-\[
-\ssfK_{2}
- = \ssfK' \cos^{2} \psi + \ssfK' \sin^{2} \psi (1 - \beta \sin^{2} \theta)
- = \ssfK'(1 - \beta \sin^{2} \theta \sin^{2} \psi)
-\]
-with the azimuth of vibration $\tg^{2} \psi_{2} = \dfrac{\cot^{2} \psi}{1 - \beta \sin^{2} \theta}$,
-and,
-\[
-\ssfK_{3} = \beta \sin^{2} \theta\, \ssfK_{e}
-\]
-with the azimuth of vibration $\tg\psi_{3} = 0$.
-
-According to~\Eq{(147)} these values give the principal intensities
-$\ssfK_{0}$~and~$\ssfK_{0}'$ required and hence the entropy radiation~\Eq{(346)}.
-Thereby the amount of entropy
-\[
-q\, \Delta\nu\, [\ssfL(\ssfK_{0}) + \ssfL(\ssfK_{0}')]\, d\Omega\, dt
-\Tag{(347)}
-\]
-is added to the field of radiation in the time~$dt$. All told, the entropy
-change of the field of radiation in the time~$dt$, as given by
-subtraction of the expression~\Eq{(345)} from~\Eq{(347)} and integration
-with respect to~$d\Omega$, is
-\[
-dt\, \Delta\nu \int q\, d\Omega\, [\ssfL(\ssfK_{0}) + \ssfL(\ssfK_{0}') - \ssfL(\ssfK) - \ssfL(\ssfK')].
-\Tag{(348)}
-\]
-
-Let us now calculate the entropy change of the system of
-oscillators which has taken place in the same time~$dt$. According
-to~\Eq{(173)} the entropy at the time~$t$ is
-\[
-S = -kN \sum_{1}^{\infty} w_{n} \log w_{n}.
-\]
-Hence the entropy change in the time~$dt$ is
-\[
-dS = -kN \sum_{1}^{\infty} \log w_{n}\, dw_{n},
-\]
-and, by taking account of~\Eq{(343)}, we have:
-\[
-dS = \frac{Nk \ssfI\, dt}{4h\nu L} \sum_{1}^{\infty}
- \bigl(w_{n} - (1 - \eta) w_{n-1}\bigr) \log w_{n}.
-\Tag{(349)}
-\]
-
-\Section{186.} The principle of increase of entropy requires that the sum
-of the entropy change~\Eq{(348)} of the field of radiation and the
-entropy change~\Eq{(349)} of the system of oscillators be always
-positive, or zero in the limiting case. That this condition is in
-fact satisfied we shall prove only for the special case when all rays
-falling on the oscillators are unpolarized, \ie, when $\ssfK' = \ssfK$.
-%% -----File: 226.png---Folio 210-------
-
-In this case we have from~\Eq{(147)} and \Sec{185}.
-\[
-\left.
-\begin{aligned}
-&\ssfK_{0} \\
-&\ssfK_{0}'
-\end{aligned}
-\right\} = \tfrac{1}{2} \{2\ssfK + \beta \sin^{2} \theta(\ssfK_{e} - \ssfK) ą \beta \sin^{2} \theta(\ssfK_{e} - \ssfK)\},
-\]
-and hence
-\[
-\ssfK_{0} = \ssfK + \beta \sin^{2} \theta (\ssfK_{e} - \ssfK),\quad
-\ssfK_{0}' = \ssfK.
-\]
-The entropy change~\Eq{(348)} of the field of radiation becomes
-\[
-dt\, \Delta\nu \int q\, d\Omega\, \{\ssfL(\ssfK_{o}) - \ssfL(\ssfK)\}
- = dt\, \Delta\nu \int q\, d\Omega\, \beta \sin^{2} \theta(\ssfK_{e} - \ssfK)\, \frac{d\ssfL(\ssfK)}{d\ssfK}
-\]
-or, by \Eq{(338)}~and~\Eq{(278)},
-\[
-= \frac{\pi kN\, dt}{hc\nu L} \int d\Omega \sin^{2} \theta(\ssfK_{e} - \ssfK) \log\left(1 + \frac{h\nu^{3}}{c^{2} \ssfK}\right).
-\]
-
-On adding to this the entropy change~\Eq{(349)} of the system of
-oscillators and taking account of~\Eq{(320)}, the total increase in entropy
-in the time~$dt$ is found to be equal to the expression
-\[
-\Squeeze[0.95]{$\displaystyle\frac{\pi kN\, dt}{ch\nu L} \int d\Omega \sin^{2} \theta
- \biggl\{
- \ssfK \sum_{1}^{\infty} (w_{n} - \zeta w_{n-1}) \log w_{n}
- + (\ssfK_{e} - \ssfK) \log\biggl(1 + \frac{h\nu^{3}}{c^{2}\ssfK}\biggr)
-\biggr\}$}
-\]
-where
-\[
-\zeta = 1 - \eta.
-\Tag{(350)}
-\]
-
-We now must prove that the expression
-\begin{multline*}
-F = \int d\Omega \sin^{2} \theta \biggl\{
- \ssfK \sum_{1}^{\infty} (w_{n} - \zeta w_{n-1}) \log w_{n} \\
- + (\ssfK_{e} - \ssfK) \log\left(1 + \frac{h\nu^{3}}{c^{2}\ssfK}\right)
-\biggr\}
-\Tag{(351)}
-\end{multline*}
-is always positive and for that purpose we set down once more the
-meaning of the quantities involved. $\ssfK$~is an arbitrary positive
-function of the polar angles $\theta$~and~$\phi$. The positive proper fraction~$\zeta$
-is according to \Eq{(350)},~\Eq{(265)}, and~\Eq{(320)} given by
-\[
-\frac{\zeta}{1 - \zeta} = \frac{3c^{2}}{8\pi h\nu^{3}} \int \ssfK \sin^{2} \theta\, d\Omega.
-\Tag{(352)}
-\]
-The quantities $w_{1}$,~$w_{2}$, $w_{3},~\dots$ are any positive proper
-%% -----File: 227.png---Folio 211-------
-fractions whatever which, according to~\Eq{(167)}, satisfy the condition
-\[
-\sum_{\DPtypo{I}{1}}^{\infty} w_{n} = 1
-\Tag{(353)}
-\]
-while, according to~\Eq{(342)},
-\[
-w_{0} = \frac{1 - \zeta}{\zeta}.
-\Tag{(354)}
-\]
-Finally we have from~\Eq{(336)}
-\[
-\ssfK_{e} = \frac{h\nu^{3} \zeta}{c^{2}} \sum_{1}^{\infty} nw_{n}.
-\Tag{(355)}
-\]
-
-\Section{187.} To give the proof required we shall show that the least
-value which the function~$F$ can assume is positive or zero. For
-this purpose we consider first that positive function,~$\ssfK$, of $\theta$~and~$\phi$,
-which, with fixed values of $\zeta$, $w_{1}$,~$w_{2}$, $w_{3},~\dots$ and~$\ssfK_{e}$, will
-make $F$ a minimum. The necessary condition for this is $\delta F = 0$,
-where according to~\Eq{(352)}
-\[
-\int \delta\ssfK \sin^{2} \theta\, d\Omega = 0.
-\]
-This gives, by considering that the quantities $w$~and~$\zeta$ do not
-depend on $\theta$~and~$\phi$, as a necessary condition for the minimum,
-\[
-\delta F = 0 = \int d\Omega \sin^{2} \theta\, \delta\ssfK \left\{
- -\log\left(1 + \frac{h\nu^{3}}{c^{2}\ssfK}\right)
- - \frac{\ssfK_{e} - \ssfK}{\dfrac{c^{2}\ssfK}{h\nu^{3}} + 1} ˇ \frac{1}{\ssfK}
-\right\}
-\]
-and it follows, therefore, that the quantity in brackets, and hence
-also $\ssfK$~itself is independent of $\theta$~and~$\phi$. That in this case $F$~really
-has a minimum value is readily seen by forming the second variation
-\[
-\delta^{2} F = \int d\Omega \sin^{2} \theta\, \delta\ssfK\,
- \delta \left\{-\log\left(1 + \frac{h\nu^{3}}{c^{2}\ssfK}\right)
- - \frac{\ssfK_{e} - \ssfK}{\dfrac{c^{2}\ssfK}{h\nu^{3}} + 1} ˇ \frac{1}{\ssfK}
-\right\}
-\]
-which may by direct computation be seen to be positive under all
-circumstances.
-
-In order to form the minimum value of~$F$ we calculate the value
-of~$\ssfK$, which, from~\Eq{(352)}, is independent of $\theta$~and~$\phi$. Then it
-follows, by taking account of~\Eq{(319a)}, that
-\[
-\ssfK = \frac{h\nu^{3}}{c^{2}}\, \frac{\zeta}{1 - \zeta}
-\]
-%% -----File: 228.png---Folio 212-------
-and, by also substituting~$\ssfK_{e}$ from~\Eq{(355)},
-\[
-F = \frac{8\pi h\nu^{3}}{3c^{2}}\, \frac{\zeta}{1 - \zeta}
- \sum_{1}^{\infty} (w_{n} - \zeta w_{n-1}) \log w_{n}
- - [(1 - \zeta) n - 1] w_{n} \log \zeta.
-\]
-
-\Section{188.} It now remains to prove that the sum
-\[
-\Phi = \sum_{1}^{\infty} (w_{n} - \zeta w_{n-1}) \log w_{n} - [(1 - \zeta) n - 1] w_{n} \log\zeta,
-\Tag{(356)}
-\]
-where the quantities~$w_{n}$ are subject only to the restrictions that
-\Eq{(353)}~and~\Eq{(354)} can never become negative. For this purpose
-we determine that system of values of the~$w$'s which, with a fixed
-value of~$\zeta$, makes the sum~$\Phi$ a minimum. In this case $\delta\Phi = 0$, or
-\begin{multline*}
-\sum_{1}^{\infty} (\delta w_{n} - \zeta\, \delta w_{n-1}) \log w_{n}
- + (w_{n} - \zeta w_{n-1})\, \frac{\delta w_{n}}{w_{n}}
-\Tag{(357)} \\ %[** Attn tag on first line of broken display]
- - [(1 - \zeta) n - 1]\, \delta w_{n} \log\zeta = 0,
-\end{multline*}
-where, according to \Eq{(353)}~and~\Eq{(354)},
-\[
-\sum_{1}^{\infty} \delta w_{n} = 0 \quad\text{and}\quad \delta w_{0} = 0.
-\Tag{(358)}
-\]
-If we suppose all the separate terms of the sum to be written out,
-the equation may be put into the following form:
-\[
-\sum_{1}^{\infty} \delta w_{n} \{\log w_{n} - \zeta \log w_{n+1}
- + \frac{w_{n} - \zeta w_{n-1}}{w_{n}} - [(1 - \zeta) n - 1] \log\zeta\} = 0.
-\Tag{(359)}
-\]
-From this, by taking account of~\Eq{(358)}, we get as the condition
-for a minimum, that
-\[
-\log w_{n} - \zeta \log w_{n+1} + \frac{w_{n} - \zeta w_{n-1}}{w_{n}} - [(1 - \zeta) n - 1] \log\zeta
-\Tag{(360)}
-\]
-must be independent of~$n$.
-
-The solution of this functional equation is
-\[
-w_{n} = (1 - \zeta) \zeta^{n-1}
-\Tag{(361)}
-\]
-for it satisfies~\Eq{(360)} as well as \Eq{(353)}~and~\Eq{(354)}. With this value
-\Eq{(356)}~becomes
-\[
-\Phi = 0.
-\Tag{(362)}
-\]
-%% -----File: 229.png---Folio 213-------
-
-\Section{189.} In order to show finally that the value~\Eq{(362)} of~$\Phi$ is really
-the minimum value, we form from~\Eq{(357)} the second variation
-\[
-\delta^{2} \Phi = \sum_{1}^{\infty} (\delta w_{n} - \zeta\, \delta w_{n-1}) \frac{\delta w_{n}}{w_{n}}
- - \frac{\zeta\, \delta w_{n-1}}{w_{n}}\, \delta w_{n}
- + \frac{\zeta w_{n-1}}{w_{n}^{2}}\, \delta w_{n}^{2},
-\]
-where all terms containing the second variation~$\delta^{2}w_{n}$ have been
-omitted since their coefficients are, by~\Eq{(360)}, independent of~$n$
-and since
-\[
-\sum_{1}^{\infty} \delta^{2} w_{n} = 0.
-\]
-
-This gives, taking account of~\Eq{(361)},
-\[
-\delta^{2} \Phi
- = \sum_{1}^{\infty} \frac{2\delta w_{n}^{2}}{(1 - \zeta) \zeta^{n-1}}
- - \frac{2\zeta\, \delta w_{n-1}\, \delta w_{n}}{(1 - \zeta) \zeta^{n-1}}
-\]
-or
-\[
-\delta^{2} \Phi = \frac{2\zeta}{1 - \zeta}
- \sum_{1}^{\infty} \frac{\delta w_{n}^{2}}{\zeta^{n}}
- - \frac{\delta w_{n-1}\, \delta w_{n}}{\zeta^{n-1}}.
-\]
-That the sum which occurs here, namely,
-\[
-\frac{\delta w_{1}^{2}}{\zeta} - \frac{\delta w_{1}\, \delta w_{2}}{\zeta}
- + \frac{\delta w_{2}^{2}}{\zeta^{2}} - \frac{\delta w_{2}\, \delta w_{3}}{\zeta^{2}}
- + \frac{\delta w_{3}^{2}}{\zeta^{3}} - \frac{\delta w_{3}\, \delta w_{4}}{\zeta^{3}} + \dots
-\Tag{(363)}
-\]
-is essentially positive may be seen by resolving it into a sum of
-squares. For this purpose we write it in the form
-\[
-\sum_{1}^{\infty} \frac{1 - \alpha_{n}}{\zeta^{n}}\, \delta w_{n}^{2}
- - \frac{\delta w_{n}\, \delta w_{n+1}}{\zeta^{n}}
- + \frac{\alpha_{n+1}}{\zeta^{n+1}}\, \delta w_{n+1}^{2},
-\]
-which is identical with~\Eq{(363)} provided $\alpha_{1} = 0$. Now the $\alpha$'s
-may be so determined that every term of the last sum is a perfect
-square, \ie, that
-\[
-4 ˇ \frac{1 - \alpha_{n}}{\zeta^{n}} ˇ \frac{\alpha_{n+1}}{\zeta^{n+1}} = \left(\frac{1}{\zeta^{n}}\right)^{2}
-\]
-or
-\[
-\alpha_{n+1} = \frac{\zeta}{4(1 - \alpha_{n})}.
-\Tag{(364)}
-\]
-By means of this formula the $\alpha$'s may be readily calculated. The
-first values are:
-\[
-\alpha_{1} = 0,\quad
-\alpha_{2} = \frac{\zeta}{4},\quad
-\alpha_{3} = \frac{\zeta}{4 - \zeta},\ \dots.
-\]
-%% -----File: 230.png---Folio 214-------
-{\Loosen Continuing the procedure $\alpha_{n}$~remains always positive and less
-than $\alpha' = \DPchg{\dfrac{1}{2}}{\frac{1}{2}} \left(1 - \sqrt{1 - \zeta}\right)$.} To prove the correctness of this statement
-we show that, if it holds for~$\alpha_{n}$, it holds also for~$\alpha_{n+1}$.
-We assume, therefore, that $\alpha_{n}$~is positive and~$< \alpha'$. Then from~\Eq{(364)}
-$\alpha_{n+1}$~is positive and $< \dfrac{\zeta}{4(1 - \alpha')}$. But $\dfrac{\zeta}{4(1 - \alpha')} = \alpha'$.
-Hence $\alpha_{n+1} < \alpha'$. Now, since the assumption made does actually
-hold for $n = 1$, it holds in general. The sum~\Eq{(363)} is thus
-essentially positive and hence the value~\Eq{(362)} of~$\Phi$ really is a
-minimum, so that the increase of entropy is proven generally.
-
-The limiting case~\Eq{(361)}, in which the increase of entropy
-vanishes, corresponds, of course, to the case of thermodynamic
-equilibrium between radiation and oscillators, as may also be
-seen directly by comparison of~\Eq{(361)} with \Eq{(271)},~\Eq{(265)}, and~\Eq{(360)}.
-
-\Section[190.]{190. Conclusion.}---The theory of irreversible radiation processes
-here developed explains how, with an arbitrarily assumed
-initial state, a stationary state is, in the course of time, established
-in a cavity through which radiation passes and which contains
-oscillators of all kinds of natural vibrations, by the intensities
-and polarizations of all rays equalizing one another as regards
-magnitude and direction. But the theory is still incomplete in
-an important respect. For it deals only with the mutual actions
-of rays and vibrations of oscillators of the same period. For a
-definite frequency the increase of entropy in every time element
-until the maximum value is attained, as demanded by the second
-principle of thermodynamics, has been proven directly. But, for
-all frequencies taken together, the maximum thus attained does
-not yet represent the absolute maximum of the entropy of the
-system and the corresponding state of radiation does not, in general,
-represent the absolutely stable equilibrium (compare \Sec{27}).
-For the theory gives no information as to the way in which
-the intensities of radiation corresponding to different frequencies
-equalize one another, that is to say, how from any arbitrary
-initial spectral distribution of energy the normal energy distribution
-corresponding to black radiation is, in the course of time,
-developed. For the oscillators on which the consideration was
-based influence only the intensities of rays which correspond
-%% -----File: 231.png---Folio 215-------
-to their natural vibration, but they are not capable of changing
-their frequencies, so long as they exert or suffer no other action
-than emitting or absorbing radiant energy.\footnote
- {Compare \Name{P.~Ehrenfest}, Wien.\ Ber.\ \textbf{114}~[2a], p.~1301, 1905. Ann.\ d.\ Phys.\ \textbf{36}, p.~91,
- 1911. \Name{H.~A. Lorentz}, Phys.\ Zeitschr.\ \textbf{11}, p.~1244, 1910. \Name{H.~Poincaré}, Journ.\ de Phys.~(5)
- \textbf{2}, p.~5, p.~347, 1912.}
-
-To get an insight into those processes by which the exchange of
-energy between rays of different frequencies takes place in nature
-would require also an investigation of the influence which the
-motion of the oscillators and of the electrons flying back and
-forth between them exerts on the radiation phenomena. For, if
-the oscillators and electrons are in motion, there will be impacts
-between them, and, at every impact, actions must come into play
-which influence the energy of vibration of the oscillators in a
-quite different and much more radical way than the simple emission
-and absorption of radiant energy. It is true that the final
-result of all such impact actions may be anticipated by the aid
-of the probability considerations discussed in the third section,
-but to show in detail how and in what time intervals this result
-is arrived at will be the problem of a future theory. It is certain
-that, from such a theory, further information may be expected
-as to the nature of the oscillators which really exist in nature,
-for the very reason that it must give a closer explanation of
-the physical significance of the universal elementary quantity of
-action, a significance which is certainly not second in importance
-to that of the elementary quantity of electricity.
-%% -----File: 232.png---Folio 216-------
-
-\BackMatter
-
-\Bibliography
-
-List of the papers published by the author on heat radiation and the hypothesis
-of quanta, with references to the sections of this book where the
-same subject is treated.
-\bigskip
-
-\selectlanguage{german}
-Absorption und Emission elektrischer Wellen durch Resonanz. Sitzungsber.\
-d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~21. März~1895, p.~289--301.
-\textsc{Wied.}\ Ann.\ \textbf{57}, p.~1--14, 1896.
-
-Ueber elektrische Schwingungen, welche durch Resonanz erregt und
-durch Strahlung gedämpft werden. Sitzungsber.\ d.~k.\ preuss.\ Akad.\ d.~Wissensch.\
-vom~20. Februar~1896, p.~151--170. \textsc{Wied.}\ Ann.\ \textbf{60}, p.~577--599,
-1897.
-
-Ueber irreversible Strahlungsvorgänge. (Erste Mitteilung.) Sitzungsber.\
-d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~4. Februar~1897, p.~57--68.
-
-Ueber irreversible Strahlungsvorgänge. (Zweite Mitteilung.) Sitzungsber.\
-d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~8. Juli~1897, p.~715--717.
-
-Ueber irreversible Strahlungsvorgänge. (Dritte Mitteilung.) Sitzungsber.\
-d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~16. Dezember~1897, p.~1122--1145.
-
-Ueber irreversible Strahlungsvorgänge. (Vierte Mitteilung.) Sitzungsber.\
-d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~7. Juli~1898, p.~449--476.
-
-Ueber irreversible Strahlungsvorgänge. (Fünfte Mitteilung.) Sitzungsber.\
-d.~k.\ preuss.\ Akad.\ d.~Wissensch.\ vom~18. Mai~1899, p.~440--480.
-(§§\;144 bis 190. §\;164.)
-
-Ueber irreversible Strahlungsvorgänge. Ann.\ d.\ Phys.\ \textbf{1}, p.~69--122, 1900.
-(§§\;144--190. §\;164.)
-
-Entropie und Temperatur strahlender Wärme. Ann.\ d.\ Phys.\ \textbf{1}, p.~719
-bis 737, 1900. (§\;101. §\;166.)
-
-Ueber eine Verbesserung der \Name{Wien}schen Spektralgleichung. Verhandlungen
-der Deutschen Physikalischen Gesellschaft \textbf{2}, p.~202--204, 1900.
-(§\;156.)
-
-Ein vermeintlicher Widerspruch des magneto-optischen \Name{Faraday}-Effektes
-mit der Thermodynamik. Verhandlungen der Deutschen Physikalischen
-Gesellschaft \textbf{2}, p.~206--210, 1900.
-
-Kritikzweier Sätze des Herrn \Name{W.~Wien}. Ann.\ d.\ Phys.\ \textbf{3}, p.~764--766, 1900.
-
-Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum.
-Verhandlungen der Deutschen Physikalischen Gesellschaft \textbf{2}, p.~237--245,
-1900. (§§\;141--143. §\;156 f. §\;163.)
-
-Ueber das Gesetz der Energieverteilung im Normalspektrum. Ann.\ d.\
-Phys.\ \textbf{4}, p.~553--563, 1901. (§§\;141--143. §§\;156--162.)
-
-Ueber die Elementarquanta der Materie und der Elektrizität. Ann.\ d.\
-Phys.\ \textbf{4}, p.~564--566, 1901. (§\;163.)
-
-Ueber irreversible Strahlungsvorgänge (Nachtrag). Sitzungsber.\ d.~k.\
-preuss.\ Akad.\ d.~Wissensch.\ vom~9. Mai~1901, p.~544--555. Ann.\ d.\
-Phys.\ \textbf{6}, p.~818--831, 1901. (§§\;185--189.)
-
-Vereinfachte Ableitung der Schwingungsgesetze eines linearen Resonators
-im stationär durchstrahlten Felde. Physikalische Zeitschrift \textbf{2}, p.~530
-bis p.~534, 1901.
-
-Ueber die Natur des weissen Lichtes.\ Ann.\ d.\ Phys.\ \textbf{7}, p.~390--400, 1902.
-(§§\;107--112. §§\;170--174.)
-
-Ueber die von einem elliptisch schwingenden Ion emittierte und absorbierte
-%% -----File: 233.png---Folio 217-------
-Energie. Archives Néerlandaises, Jubelband für \Name{H.~A. Lorentz}, 1900,
-p.~164--174. Ann.\ d.\ Phys.\ \textbf{9}, p.~619--628, 1902.
-
-Ueber die Verteilung der Energie zwischen Aether und Materie. Archives
-Néerlandaises, Jubelband für \Name{J.~Bosscha}, 1901, p.~55--66. Ann.\ d.\ Phys.\
-\textbf{9}, p.~629--641, 1902. (§§\;121--132.)
-
-Bemerkung über die Konstante des \Name{Wien}schen Verschiebungsgesetzes.
-Verhandlungen der Deutschen Physikalischen Gesellschaft \textbf{8}, p.~695--696,
-1906. (§\;161.)
-
-Zur Theorie der Wärmestrahlung. Ann.\ d.\ Phys.\ \textbf{31}, p.~758--768, 1910.
-Eine neue Strahlungshypothese. Verhandlungen der Deutschen Physikalischen
-Gesellschaft \textbf{13}, p.~138--148, 1911. (§\;147.)
-
-Zur Hypothese der Quantenemission. Sitzungsber.\ d.~k.\ preuss.\ Akad.\
-d.~Wissensch.\ vom~13. Juli~1911, p.~723--731. (§§\;150--152.)
-
-{\Loosen Ueber neuere thermodynamische Theorien (\Name{Nernst}sches Wärmetheorem
-und Quantenhypothese). Ber.\ d.~Deutschen Chemischen Gesellschaft \textbf{45},
-p.~5--23, 1912. Physikalische Zeitschrift \textbf{13}, p.~165--175, 1912. Akademische
-Verlagsgesellschaft m.~b.~H., Leipzig 1912. (§§\;120--125.)}
-
-Ueber die Begründung des Gesetzes der schwarzen Strahlung. Ann.\ d.\
-Phys.\ \textbf{37}, p.~642--656, 1912. (§§\;145--156.)
-\selectlanguage{english}
-%% -----File: 234.png---Folio 218-------
-
-\Appendix{I}{On Deductions from Stirling's Formula}
-
-The formula is
-\Label{218}% [** TN: Page label]
-\[
-\Tag{(a)}
-\lim_{n=\infty} \frac{n!}{n^{n} e^{-n} \sqrt{2\pi n}} = 1,
-\]
-or, to an approximation quite sufficient for all practical purposes,
-provided that $n$~is larger than~$7$
-\[
-\Tag{(b)}
-n! = \left(\frac{n}{e}\right)^{n} \sqrt{2\pi n}.
-\]
-
-For a proof of this relation and a discussion of its limits of
-accuracy a treatise on probability must be consulted.
-
-%[** F2: Check par break]
-On substitution in~\Eq{(170)} this gives
-\[
-W = \frac{\left(\dfrac{N}{e}\right)^{N}}{\left(\dfrac{N_{1}}{e}\right)^{N_{1}} ˇ \left(\dfrac{N_{2}}{e}\right)^{N_{2}} \dots}
- ˇ \frac{\sqrt{2\pi N}}{\sqrt{2\pi N_{1}} ˇ \sqrt{2\pi N_{2}} \dots}\Add{.}
-\]
-On account of~\Eq{(165)} this reduces at once to
-\[
-\frac{N_{N}}{N_{1}^{N_{1}} N_{2}^{N_{2}} \dots}
- ˇ \frac{\sqrt{2\pi N}}{\sqrt{2\pi N_{1}} ˇ \sqrt{2\pi N_{2}} \dots}\Add{.}
-\]
-Passing now to the logarithmic expression we get
-\begin{multline*}
-S = k \log W
- = k [N \log N - N_{1} \log N_{1} - N_{2} \log N_{2} - \dots \\
- + \log\sqrt{2\pi N} - \log\sqrt{2\pi N_{1}} - \log\sqrt{2\pi N_{2}} - \dots],
-\end{multline*}
-or,
-\begin{multline*}
-\DPtypo{S = k \log W
- = k [(N \log N - \log\sqrt{2\pi N})
- + (N_{1} \log N_{1} - \log\sqrt{2\pi N_{1}}) +\\
- (N_{2} \log N_{2} - \log\sqrt{2\pi N_{2}}) + \dots].}
-{S = k \log W
-%[** TN: Re-broken to keep terms of same sign on same line]
- = k [(N \log N + \log\sqrt{2\pi N}) \\
- - (N_{1} \log N_{1} + \log\sqrt{2\pi N_{1}})
- - (N_{2} \log N_{2} + \log\sqrt{2\pi N_{2}}) - \dots].}
-\end{multline*}
-
-{\Loosen Now, for a large value of~$N_{i}$, the term $N_{i} \log N_{i}$ is very much
-larger than $\log\sqrt{2\pi N_{i}}$, as is seen by writing the latter in the form
-$\frac{1}{2} \log 2\pi + \frac{1}{2} \log N_{i}$. Hence the last expression will, with a fair
-approximation, reduce to}
-\[
-S = k \log W
- = k [N \log N - N_{1} \log N_{1} - N_{2} \log N_{2} - \dots].
-\]
-%% -----File: 235.png---Folio 219-------
-Introducing now the values of the densities of distribution~$w$
-by means of the relation
-\[
-N_{i} = w_{i}N
-\]
-we obtain
-\[
-S = k \log W = kN[\log N - w_{1} \log N_{1} - w_{2} \log N_{2} - \dots],
-\]
-or, since
-\[
-w_{1} + w_{2} + w_{3} + \dots = 1,
-\]
-and hence
-\[
-(w_{1} + w_{2} + w_{3} + \dots) \log N = \log N,
-\]
-and
-\[
-\log N - \log N_{1} = \log \frac{N}{N_{1}} = \log \frac{1}{w_{1}} = -\log w_{1},
-\]
-we obtain by substitution, after one or two simple transformations
-\[
-S = k \log W = -kN\sum w_{1} \log w_{1},
-\]
-a relation which is identical with~\Eq{(173)}.
-
-The statements of \Sec{143} may be proven in a similar manner.
-From~\Eq{(232)} we get at once
-\[
-S = k \log W_{m} = k \log\frac{(N + P - 1)!}{(N - 1)!\, P!}
-\]
-Now
-\[
-\log (N - 1)! = \log N! - \log N,
-\]
-and, for large values of~$N$, $\log N$~is negligible compared with
-$\log N!$\Add{.} Applying the same reasoning to the numerator we
-may without appreciable error write
-\[
-S = k \log W_{m} = k \log\frac{(N + P)!}{N!\, P!}\Add{.}
-\]
-Substituting now for $(N + P)!$, $N!$, and $P!$ their values from~\Eq{(b)}
-and omitting, as was previously shown to be approximately
-correct, the terms arising from the $\sqrt{2\pi (N + P)}$~etc.,\DPnote{** Italicized in orig} we get,
-since the terms containing $e$ cancel out
-\begin{align*}
-S &= k[(N + P) \log (N + P) - N \log N - P \log P]\\
- &= k[(N + P) \log \frac{N + P}{N} + P \log N - P \log P]\\
- &= kN\left[\left(\frac{P}{N} + 1\right) \log \left(\frac{P}{N} + 1\right) - \frac{P}{N} \log \frac{P}{N}\right].
-\end{align*}
-
-This is the relation of \Sec{143}.
-%% -----File: 236.png---Folio 220-------
-
-\Appendix{II}{References}
-
-Among general papers treating of the application of the theory
-of quanta to different parts of physics are:
-
-1. \Name{A.~Sommerfeld}, Das Planck'sche Wirkungsquantum und
-seine allgemeine Bedeutung für die Molekularphysik, Phys.\
-Zeitschr., \textbf{12}, p.~1057. Report to the Versammlung Deutscher
-Naturforscher und Aerzte. Deals especially with applications to
-the theory of specific heats and to the photoelectric effect.
-Numerous references are quoted.
-
-2. Meeting of the British Association, Sept., 1913. See
-Nature, \textbf{92}, p.~305, Nov.~6, 1913, and Phys.\ Zeitschr., \textbf{14}, p.~1297.
-Among the principal speakers were \Name{J.~H. Jeans} and \Name{H.~A. Lorentz}.
-
-(Also American Phys.\ Soc., Chicago Meeting, 1913.\footnote
- {Not yet published (Jan.~26, 1914. Tr.)})
-
-3. \Name{R.~A. Millikan}, Atomic Theories of Radiation, Science,
-\textbf{37}, p.~119, Jan.~24, 1913. A non-mathematical discussion.
-
-4. \Name{W.~Wien}, Neuere Probleme der Theoretischen Physik,
-1913. (\Name{Wien's} Columbia Lectures, in German.) This is perhaps
-the most complete review of the entire theory of quanta.
-
-\Name{H.~A. Lorentz}, Alte und Neue Probleme der Physik, Phys.\
-Zeitschr., \textbf{11}, p.~1234. Address to the Versammlung Deutscher
-Naturforscher und Aerzte, Königsberg, 1910, contains also some
-discussion of the theory of quanta.
-
-Among the papers on radiation are:
-
-\Name{E.~Bauer}, Sur la théorie du rayonnement, Comptes Rendus,
-\textbf{153}, p.~1466. Adheres to the quantum theory in the original
-form, namely, that emission and absorption both take place in a
-discontinuous manner.
-
-\Name{E.~Buckingham}, Calculation of~$c_{2}$ in Planck's equation, Bull.\
-Bur.\ Stand.\ \textbf{7}, p.~393.
-
-\Name{E.~Buckingham}, On \Name{Wien's} Displacement Law, Bull.\ Bur.\
-Stand.\ \textbf{8}, p.~543. Contains a very simple and clear proof of the
-displacement law.
-%% -----File: 237.png---Folio 221-------
-
-\Name{P.~Ehrenfest}, Strahlungshypothesen, Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{36}, p.~91.
-
-\Name{A.~Joffé}, Theorie der Strahlung, Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{36}, p.~534.
-
-Discussions of the method of derivation of the radiation formula
-are given in many papers on the subject. In addition to those
-quoted elsewhere may be mentioned:
-
-\Name{C.~Benedicks}, Ueber die Herleitung von \Name{Planck's} Energieverteilungsgesetz
-Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{42}, p.~133. Derives \Name{Planck's} law
-without the help of the quantum theory. The law of equipartition
-of energy is avoided by the assumption that solids are not
-always monatomic, but that, with decreasing temperature, the
-atoms form atomic complexes, thus changing the number of
-degrees of freedom. The equipartition principle applies only
-to the free atoms.
-
-\Name{P.~Debye}, \Name{Planck's} Strahlungsformel, Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{33}, p.~1427.
-This method is fully discussed by \Name{Wien} (see~4, above).
-It somewhat resembles \Name{Jeans'} method (\Sec{169}) since it avoids
-all reference to resonators of any particular kind and merely
-establishes the most probable energy distribution. It differs,
-however, from \Name{Jeans'} method by the assumption of discrete
-energy quanta~$h\nu$. The physical nature of these units is not
-discussed at all and it is also left undecided whether it is a
-property of matter or of the ether or perhaps a property of the
-energy exchange between matter and the ether that causes their
-existence. (Compare also some remarks of \Name{Lorentz} in~2.)
-
-\Name{P.~Frank}, Zur Ableitung der Planckschen Strahlungsformel,
-Phys.\ Zeitschr., \textbf{13}, p.~506.
-
-\Name{L.~Natanson}, Statistische Theorie der Strahlung, Phys.\ Zeitschr.,
-\textbf{12}, p.~659.
-
-\Name{W.~Nernst}, Zur Theorie der \DPtypo{Specifischen}{specifischen} Wärme und über die
-Anwendung\DPtypo{,}{} der Lehre von den Energiequanten auf \DPtypo{Physikalisch-chemische}{physikalisch-chemische}
-Fragen überhaupt, Zeitschr.\ f.\ \DPtypo{Elektochemie}{Elektrochemie}, \textbf{17}, p.~265.
-
-The experimental facts on which the recent theories of specific
-heat (quantum theories) rely, were discovered by \Name{W.~Nernst}
-and his fellow workers. The results are published in a large
-number of papers that have appeared in different periodicals.
-See, \eg, \Name{W.~Nernst}, Der Energieinhalt fester Substanzen, Ann.\
-d.\ Phys.\DPchg{,}{}\ \textbf{36}, p.~395, where also numerous other papers are quoted.
-(See also references given in~1.) These experimental facts give
-very strong support to the heat theorem of \Name{Nernst} (\Sec{120}),
-%% -----File: 238.png---Folio 222-------
-according to which the entropy approaches a definite limit
-(perhaps the value zero, see \Name{Planck's} Thermodynamics, 3.~ed.,
-sec.~282, \textit{et~seq.})\ at the absolute zero of temperature, and which
-is consistent with the quantum theory. This work is in close
-connection with the recent attempts to develop an equation of
-state applicable to the solid state of matter. In addition to
-the papers by \Name{Nernst} and his school there may be mentioned:
-
-\Name{K.~Eisenmann}, Canonische Zustandsgleichung einatomiger
-fester Körper und die Quantentheorie, Verhandlungen der
-Deutschen Physikalischen Gesellschaft, \textbf{14}, p.~769.
-
-\Name{W.~H. Keesom}, Entropy and the Equation of State, Konink.\
-Akad.\ Wetensch.\ Amsterdam Proc., \textbf{15}, p.~240.
-
-\Name{L.~Natanson}, Energy Content of Bodies, Acad.\ Science Cracovie
-Bull.\ Ser.~A, p.~95. In \Name{Einstein's} theory of specific heats
-(\Sec{140}) the atoms of actual bodies in nature are apparently
-identified with the ideal resonators of \Name{Planck}. In this paper it
-is pointed out that this is implying too special features for the
-atoms of real bodies, and also, that such far-reaching specializations
-do not seem necessary for deriving the laws of specific heat
-from the quantum theory.
-
-\Name{L.~S. Ornstein}, Statistical Theory of the Solid State, Konink.\
-Akad.\ Wetensch.\ Amsterdam Proc., \textbf{14}, p.~983.
-
-\Name{S.~Ratnowsky}, Die Zustandsgleichung einatomiger fester
-Körper und die Quantentheorie, Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{38}, p.~637.
-
-Among papers on the law of equipartition of energy (\Sec{169})
-are:
-
-\Name{J.~H. Jeans}, Planck's Radiation Theory and \Name{Non-Newtonian}
-Mechanics, Phil.\ Mag., \textbf{20}, p.~943.
-
-\Name{S.~B. McLaren}, Partition of Energy between Matter and
-Radiation, Phil.\ Mag., \textbf{21}, p.~15.
-
-\Name{S.~B. McLaren}, Complete Radiation, Phil.\ Mag.\Add{,} \textbf{23}, p.~513.
-This paper and the one of Jeans deal with the fact that from
-%[** Unitalicized names]
-Newtonian Mechanics (Hamilton's Principle) the equipartition
-principle necessarily follows, and that hence either Planck's law
-or the fundamental principles of mechanics need a modification.
-
-For the law of equipartition compare also the discussion at the
-meeting of the British Association (see~2).
-
-In many of the papers cited so far deductions from the quantum
-%% -----File: 239.png---Folio 223-------
-theory are compared with experimental facts. This is also
-done by:
-
-\Name{F.~Haber}, Absorptionsspectra fester Körper und die Quantentheorie,
-Verhandlungen der Deutschen Physikalischen Gesellschaft,
-\textbf{13}, p.~1117.
-
-\Name{J.~Franck} \emph{und} \Name{G.~Hertz}, Quantumhypothese und Ionisation,
-Ibid., \textbf{13}, p.~967.
-
-Attempts of giving a concrete physical idea of Planck's constant~$h$
-are made by:
-
-\Name{A.~Schidlof}, Zur Aufklärung der universellen electrodynamischen
-Bedeutung der Planckschen Strahlungsconstanten~$h$,
-Ann.\ d.\ Phys.\DPchg{,}{}\ \textbf{35}, p.~96.
-
-\Name{D.~A. Goldhammer}, Ueber die Lichtquantenhypothese, Phys.\
-Zeitschr., \textbf{13}, p.~535.
-
-\Name{J.~J. Thomson}, On the Structure of the Atom, Phil.\ Mag., \textbf{26},
-p.~792.
-
-\Name{N.~Bohr}, On the Constitution of the Atom, Phil.\ Mag., \textbf{26},
-p.~1.
-
-\Name{S.~B. McLaren}, The Magneton and Planck's Universal Constant,
-Phil.\ Mag., \textbf{26}, p.~800.
-
-The line of reasoning may be briefly stated thus: Find some
-quantity of the same dimension as~$h$, and then construct a model
-of an atom where this property plays an important part and can
-be made, by a simple hypothesis, to vary by finite amounts instead
-of continuously. The simplest of these is \Name{Bohr's}, where $h$~is
-interpreted as angular momentum.
-
-The logical reason for the quantum theory is found in the
-fact that the \Name{Rayleigh-Jeans} radiation formula does not agree
-with experiment. Formerly \Name{Jeans} attempted to reconcile
-theory and experiment by the assumption that the equilibrium
-of radiation and a black body observed and agreeing with
-\Name{Planck's} law rather than his own, was only apparent, and that
-the true state of equilibrium which really corresponds to his law
-and the equipartition of energy among all variables, is so slowly
-reached that it is never actually observed. This standpoint,
-which was strongly objected to by authorities on the experimental
-side of the question (see, \eg, \Name{E.~Pringsheim} in~2), he
-has recently abandoned. \Name{H.~Poincaré}, in a profound mathematical
-investigation (\Name{H.~Poincaré}, Sur\DPtypo{.}{} la Théorie des Quanta,
-%% -----File: 240.png---Folio 224-------
-Journal de Physique~(5), \textbf{2}, p.~1, 1912) reached the conclusion
-that whatever the law of radiation may be, it must always, if
-the total radiation is assumed as finite, lead to a function presenting
-similar \DPtypo{discontinuites}{discontinuities} as the one obtained from the
-hypothesis of quanta.
-
-While most authorities have accepted the quantum theory for
-good (see \Name{J.~H. Jeans} and \Name{H.~A. Lorentz} in~2), a few still entertain
-%[** More non-italic names; maybe italics used only for authors in the appendix?]
-doubts as to the general validity of Poincaré's conclusion
-(see above \Name{C.~Benedicks} and \Name{R.~A\Add{.} Millikan}~3). Others still
-reject the quantum theory on account of the fact that the experimental
-evidence in favor of \Name{Planck's} law is not absolutely
-conclusive (see \Name{R.~A. Millikan}~3); among these is \Name{A.~E.~H. Love}~(2),
-who suggests that \Name{Korn's} (\Name{A.~Korn}, Neue Mechanische
-Vorstellungen \DPtypo{uber}{über} die \DPtypo{Schwarze}{schwarze} Strahlung und eine sich aus
-denselben ergebende Modification des Planckschen Verteilungsgesetzes,
-Phys.\ Zeitschr., \textbf{14}, p.~632) radiation formula fits the
-facts as well as that of Planck.
-
-\Name{H.~A. Callendar}, Note on Radiation and Specific Heat, Phil.\
-Mag., \textbf{26}, p.~787, has also suggested a radiation formula that fits
-the data well. Both Korn's and Callendar's formulć conform
-to \Name{Wien's} displacement law and degenerate for large values of~$\lambda T$
-into the Rayleigh-Jeans, and for small values of~$\lambda T$ into
-Wien's radiation law. Whether Planck's law or one of these
-is the correct law, and whether, if either of the others should
-prove to be right, it would eliminate the necessity of the adoption
-of the quantum theory, are questions as yet undecided.
-Both Korn and Callendar have promised in their papers to follow
-them by further ones.
-%% -----File: 241.png---Folio 225-------
-\iffalse
-%[** TN: Errata have been applied to the text body]
-ERRATA
-
-Page~77. The last sentence of \Sec{77} should be replaced by: \\
-The corresponding additional terms may, however, be omitted
-here without appreciable error, since the correction caused by
-them would consist merely of the addition to the energy change
-here calculated of a comparatively infinitesimal energy change of
-the same kind with an external work that is infinitesimal of the
-second order.
-
-Page~83. Insert at the end of \Sec{84a}: \\
-These laws hold for any original distribution of energy whatever;
-hence, \eg, an originally monochromatic radiation remains
-monochromatic during the process described, its color changing
-in the way stated.
-\fi
-
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