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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: Spinning Tops + +Author: John Perry + +Release Date: November 9, 2010 [EBook #34268] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK SPINNING TOPS *** + + + + +Produced by Chris Curnow, Keith Edkins and the Online +Distributed Proofreading Team at https://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive) + + + + + + +</pre> + + +<table border="0" cellpadding="10" style="background-color: #ccccff;"> +<tr> +<td style="width:25%; vertical-align:top"> +Transcriber's note: +</td> +<td> +A few typographical errors have been corrected. They +appear in the text <span class="correction" title="explanation will pop up">like this</span>, and the +explanation will appear when the mouse pointer is moved over the marked +passage.<br /><br /> + +</td> +</tr> +</table> + +<p class="cenhead"><span class="sc">The Earl of Pembroke to the Abbess of Wilton.</span></p> + +<p class="cenhead">"Go spin, you jade! go spin!"</p> + + <div class="figcenter" style="width:45%;"> + <a href="images/front.jpg"><img style="width:100%" src="images/front.jpg" + alt="Frontispiece" title="Frontispiece" /></a> + MAGNETISM, LIGHT, AND MOLECULAR SPINNING TOPS. + + <p class="author"><i>Page 122.</i></p> + + <p class="poem"></p> + </div> + +<h3><i>THE ROMANCE OF SCIENCE.</i></h3> + +<h1>SPINNING TOPS.</h1> + +<h3><i>THE "OPERATIVES' LECTURE"</i><br /> +OF THE BRITISH ASSOCIATION MEETING AT LEEDS,<br /> +6th SEPTEMBER, 1890.</h3> + +<p class="cenhead">BY</p> + +<h3>PROFESSOR JOHN PERRY,<br /> +M.E., D.Sc, LL.D., F.R.S.</h3> + +<h4>With Numerous Illustrations.</h4> + +<h3><i>REPRINT OF NEW AND REVISED EDITION,</i></h3> + +<p class="cenhead"><i>With an Illustrated Appendix on the Use of Gyrostats.</i></p> + +<p class="cenhead">LONDON<br /> +SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE,<br /> +Northumberland Avenue, W.C.; 43, Queen Victoria Street, E.C.<br /> +<span class="sc">Brighton</span>: 129, North Street.<br /> +<span class="sc">N<span class="gsp"> </span>e<span class="gsp"> </span>w<span class="gsp"> </span> <span class="gsp"> </span>Y<span class="gsp"> </span>o<span class="gsp"> </span>r<span class="gsp"> </span>k</span>: E.<span class="gsp"> </span> <span class="gsp"> </span>S.<span class="gsp"> </span> <span class="gsp"> </span>G<span class="gsp"> </span>O<span class="gsp"> </span>R<span class="gsp"> </span>H<span class="gsp"> </span>A<span class="gsp"> </span>M.</p> + +<p class="cenhead">1910</p> + +<p class="cenhead">PUBLISHED UNDER THE DIRECTION OF THE GENERAL +LITERATURE COMMITTEE</p> + +<p class="cenhead">[<i>Date of last impression, April 1908</i>]</p> + +<h3>This Report of an Experimental Lecture<br /> +WAS INSCRIBED TO<br /> +THE LATE<br /> +LORD KELVIN,<br /> +BY HIS AFFECTIONATE PUPIL, THE LECTURER, WHO<br /> +HEREBY TOOK A CONVENIENT METHOD OF<br /> +ACKNOWLEDGING THE REAL AUTHOR OF<br /> +WHATEVER IS WORTH PUBLICATION<br /> +IN THE FOLLOWING<br /> +PAGES.</h3> + + <p><br style="clear:both" /></p> +<hr class="full" /> + +<h3>PREFACE.</h3> + + <p>This is not the lecture as it was delivered. Instead of two pages of + letterpress and a woodcut, the reader may imagine that for half a minute + the lecturer played with a spinning top or gyrostat, and occasionally + ejaculated words of warning, admonition, and explanation towards his + audience. A verbatim report would make rather uninteresting reading, and + I have taken the liberty of trying, by greater fullness of explanation, + to make up to the reader for his not having seen the moving apparatus. It + has also been necessary in a treatise intended for general readers to + simplify the reasoning, the lecture having been delivered to persons + whose life experiences peculiarly fitted them for understanding + scientific things. An "argument" has been added at the end to make the + steps of the reasoning clearer.</p> + + <div class="poem"> + <div class="stanza"> + <p>JOHN PERRY.</p> + </div> + </div> + + <p><br style="clear:both" /></p> +<hr class="full" /> + +<p><!-- Page 9 --><span class="pagenum"><a name="page9"></a>{9}</span></p> + +<h2>SPINNING TOPS.</h2> + +<h2>————</h2> + + <p>At a Leeds Board School last week, the master said to his class, + "There is to be a meeting of the British Association in Leeds. What is it + all about? Who are the members of the British Association? What do they + do?" There was a long pause. At length it was broken by an intelligent + shy boy: "Please, sir, I know—they spin tops!"<a name="NtA1" + href="#Nt1"><sup>[1]</sup></a></p> + + <p>Now I am sorry to say that this answer was wrong. The members of the + British Association and the Operatives of Leeds have neglected + top-spinning since they were ten years of age. If more attention were + paid to the intelligent examination of the behaviour of tops, there would + be greater advances in mechanical engineering and a great many + industries. There would be a better general knowledge of astronomy. + Geologists would not make mistakes by millions of years, and our + knowledge of Light, and Radiant Heat, and other <!-- Page 10 --><span + class="pagenum"><a name="page10"></a>{10}</span>Electro-magnetic + Phenomena would extend much more rapidly than it does.</p> + + <p>I shall try to show you towards the end of the lecture that the fact + of our earth's being a spinning body is one which would make itself known + to us even if we lived in subterranean regions like the coming race of an + ingenious novelist.<a name="NtA2" href="#Nt2"><sup>[2]</sup></a> It is + the greatest and most persistent cause of many of the phenomena which + occur around us and beneath us, and it is probable that even Terrestrial + Magnetism is almost altogether due to it. Indeed there is only one + possible explanation of the <i>Vril-ya</i> ignorance about the earth's + rotation. Their knowledge of mechanics and dynamics was immense; no + member attending the meeting of the British Association can approach them + in their knowledge of, I will not say, <i>Vril</i>, but even of quite + vulgar electricity and magnetism; and yet this great race which expresses + so strongly its contempt for Anglo-Saxon <i>Koom-Poshery</i> was actually + ignorant of the fact that it had existed for untold generations inside an + object that spins about an axis.</p> + + <p>Can we imagine for one instant that the children of that race had + never spun a top or trundled a hoop, and so had had no chance of being + led to the greatest study of nature? No; the only possible explanation + lies in the great novelist's never <!-- Page 11 --><span + class="pagenum"><a name="page11"></a>{11}</span>having done these things + himself. He had probably as a child a contempt for the study of nature, + he was a baby Pelham, and as a man he was condemned to remain in + ignorance even of the powers of the new race that he had created.</p> + + <p>The <i>Vril-ya</i> ignorance of the behaviour of spinning bodies + existing as it does side by side with their deep knowledge of magnetism, + becomes even more remarkable when it comes home to us that the phenomena + of magnetism and of light are certainly closely connected with the + behaviour of spinning bodies, and indeed that a familiar knowledge of the + behaviour of such bodies is absolutely necessary for a proper + comprehension of most of the phenomena occurring in nature. The + instinctive craving to investigate these phenomena seems to manifest + itself soon after we are able to talk, and who knows how much of the + intellectual inferiority of woman is due to her neglect of the study of + spinning tops; but alas, even for boys in the pursuit of top-spinning, + the youthful mind and muscle are left with no other guidance than that + which is supplied by the experience of young and not very scientific + companions. I remember distinctly that there were many puzzling problems + presented to me every day. There were tops which nobody seemed able to + spin, and there were others, well <!-- Page 12 --><span + class="pagenum"><a name="page12"></a>{12}</span>prized objects, often + studied in their behaviour and coveted as supremely valuable, that + behaved well under the most unscientific treatment. And yet nobody, even + the makers, seemed to know why one behaved badly and the other well.</p> + + <p>I do not disguise from myself the fact that it is rather a difficult + task to talk of spinning tops to men who have long lost that skill which + they wonder at in their children; that knowingness of touch and handling + which gave them once so much power over what I fear to call inanimate + nature. A problem which the child gives up as hopeless of solution, is + seldom attacked again in maturer years; he drives his desire for + knowledge into the obscure lumber-closets of his mind, and there it lies, + with the accumulating dust of his life, a neglected and almost forgotten + instinct. Some of you may think that this instinct only remains with + those minds so many of which are childish even to the limit of life's + span; and probably none of you have had the opportunity of seeing how the + old dust rubs off from the life of the ordinary man, and the old desire + comes back to him to understand the mysteries that surround him.</p> + + <p>But I have not only felt this desire myself, I have seen it in the + excited eyes of the crowd of people who stand by the hour under the + dropping cherry-blossoms beside the red-pillared temple of <!-- Page 13 + --><span class="pagenum"><a name="page13"></a>{13}</span>Asakusa in the + Eastern capital of Japan, watching the <i>tedzu-mashi</i> directing the + evolutions of his heavily rimmed <i>Koma</i>. First he throws away from + him his great top obliquely into the air and catches it spinning on the + end of a stick, or the point of a sword, or any other convenient + implement; he now sends it about quite carelessly, catching it as it + comes back to him from all sorts of directions; he makes it run up the + hand-rail of a staircase into a house by the door and out again by the + window; he makes it travel up a great corkscrew. Now he seizes it in his + hands, and with a few dexterous twists gives it a new stock of spinning + energy. He makes it travel along a stretched string or the edge of a + sword; he does all sorts of other curious things with his tops, and + suddenly sinks from his masterful position to beg for a few coppers at + the end of his performance.</p> + + <p>How tame all this must seem to you who more than half forget your + childish initiation into the mysteries of nature; but trust me, if I + could only make that old top-spinner perform those magical operations of + his on this platform, the delight of the enjoyment of beautiful motion + would come back. Perhaps it is only in Japan that such an exhibition is + possible; the land where the waving bamboo, and the circling hawk, and + the undulating summer sea, and every beautiful motion of nature <!-- Page + 14 --><span class="pagenum"><a name="page14"></a>{14}</span>are looked + upon with tenderness; and perhaps it is from Japan that we shall learn + the development of our childish enthusiasm.</p> + + <p>The devotees of the new emotional art of beautiful motion and changing + colour are still in the main beggars like Homer, and they live in garrets + like Johnson and Savage; but the dawn of a new era is heralded, or rather + the dawn has already come, for Sir William Thomson's achievements in the + study of spinning tops rank already as by no means the meanest of his + great career.</p> + + <p>If you will only think of it, the behaviour of the commonest spinning + top is very wonderful. When not spinning you see that it falls down at + once, I find it impossible to balance it on its peg; but what a very + different object it is when spinning; you see that it not only does not + fall down, it offers a strange resistance when I strike it, and actually + lifts itself more and more to an upright position. Once started on + scientific observation, nature gives us facts of an analogous kind in + great plenty.</p> + + <p>Those of you who have observed a rapidly moving heavy belt or rope, + know that rapid motion gives a peculiar quasi-rigidity to flexible and + even to fluid things.</p> + + <p>Here, for example, is a disc of quite thin paper (Fig. 1), and when I + set it in rapid rotation you observe that it resists the force exerted by + my <!-- Page 15 --><span class="pagenum"><a + name="page15"></a>{15}</span>hand, the blow of my fist, as if it were a + disc of steel. Hear how it resounds when I strike it with a stick. Where + has its flexibility gone?</p> + + <div class="figcenter" style="width:45%;"> + <a href="images/fig01.jpg"><img style="width:100%" src="images/fig01.jpg" + alt="Fig. 1" title="Fig. 1" /></a> + <span class="sc">Fig. 1.</span> + </div> + + <p>Here again is a ring of chain which is quite flexible. It seems + ridiculous to imagine that this <!-- Page 16 --><span class="pagenum"><a + name="page16"></a>{16}</span>could be made to stand up like a stiff hoop, + and yet you observe that when I give it a rapid rotation on this mandril + and let it slide off upon the table, it runs over the table just as if it + were a rigid ring, and when it drops on the floor it rebounds like a + boy's hoop (Fig. 2).</p> + + <div class="figcenter" style="width:47%;"> + <a href="images/fig02.jpg"><img style="width:100%" src="images/fig02.jpg" + alt="Fig. 2" title="Fig. 2" /></a> + <span class="sc">Fig. 2.</span> + </div> + + <p>Here again is a very soft hat, specially made for this sort of + experiment. You will note that it collapses to the table in a shapeless + mass when I lay it down, and seems quite incapable of resisting forces + which tend to alter its shape. In fact, there is almost a complete + absence of rigidity; but when this is spun on the end of a stick, first + note <!-- Page 17 --><span class="pagenum"><a + name="page17"></a>{17}</span>how it has taken a very easily defined + shape; secondly, note how it runs along the table as if it were made of + steel; thirdly, note how all at once it collapses again into a shapeless + heap of soft material when its rapid motion has ceased. Even so you will + see that when a drunken man is not leaning against a wall or lamp-post, + he feels that his only chance of escape from ignominious collapse is to + get up a decent rate of speed, to obtain a quasi-sobriety of demeanour by + rapidity of motion.</p> + + <p>The water inside this glass vessel (Fig. 3) is in a state of rapid + motion, revolving with the vessel itself. Now observe the piece of + paraffin wax A immersed in the water, and you will see when I push at it + with a rod that it vibrates just as if it were surrounded with a thick + jelly. Let us now apply Prof. Fitzgerald's improvement on this experiment + of Sir William Thomson's. Here is a disc B stuck on the end of the rod; + observe that when I introduce it, although it does not touch A, A is + repelled from the disc. Now observe that when I twirl the disc it seems + to attract A.</p> + + <div class="figcenter" style="width:35%;"> + <a href="images/fig03.jpg"><img style="width:100%" src="images/fig03.jpg" + alt="Fig. 3" title="Fig. 3" /></a> + <span class="sc">Fig. 3.</span><a name="NtA3" + href="#Nt3"><sup>[3]</sup></a> + </div> + + <p>At the round hole in front of this box a rapid motion is given to a + small quantity of air which is mixed with smoke that you may see it. That + smoke-ring moves through the air almost like a solid body for a + considerable distance unchanged, and I am not sure that it may not be + possible yet <!-- Page 18 --><span class="pagenum"><a + name="page18"></a>{18}</span>to send as a projectile a huge poisoned + smoke-ring, so that it may destroy or stupefy an army miles away. + Remember that it is really the same air all the time. You will observe + that two smoke rings sent from two boxes have curious actions <!-- Page + 19 --><span class="pagenum"><a name="page19"></a>{19}</span>upon one + another, and the study of these actions has given rise to Thomson's + smoke-ring or vortex theory of the constitution of matter (Fig. 4).</p> + + <div class="figcenter" style="width:57%;"> + <a href="images/fig04.jpg"><img style="width:100%" src="images/fig04.jpg" + alt="Fig. 4" title="Fig. 4" /></a> + <span class="sc">Fig. 4.</span> + </div> + + <p>It was Rankine, the great guide of all engineers, who first suggested + the idea of molecular vortices in his explanations of heat phenomena and + the phenomena of elasticity—the idea that every particle of matter + is like a little spinning top; but I am now speaking of Thomson's theory. + To imagine that an atom of matter is merely a <!-- Page 20 --><span + class="pagenum"><a name="page20"></a>{20}</span>curiously shaped + smoke-ring formed miraculously in a perfect fluid, and which can never + undergo permanent alteration, looks to be a very curious and far-fetched + hypothesis. But in spite of certain difficulties, it is the foundation of + the theory which will best explain most of the molecular phenomena + observed by philosophers. Whatever be the value of the theory, you see + from these experiments that motion does give to small quantities of fluid + curious properties of elasticity, attraction and repulsion; that each of + these entities refuses to be cut in two; that you cannot bring a knife + even near the smoke-ring; and that what may be called a collision between + two of them is not very different in any way from the collision between + two rings of india-rubber.</p> + + <p>Another example of the rigidity given to a fluid by rapid motion, is + the feeling of utter helplessness which even the strongest swimmers + sometimes experience when they get caught in an eddy underneath the + water.</p> + + <p>I could, if I liked, multiply these instances of the quasi-rigidity + which mere motion gives to flexible or fluid bodies. In Nevada a jet of + water like the jet from a fireman's hose, except that it is much more + rapid, which is nearly as easily projected in different directions, is + used in mining, and huge masses of earth and rock are rapidly + disintegrated <!-- Page 21 --><span class="pagenum"><a + name="page21"></a>{21}</span>by the running water, which seems to be + rather like a bar of steel than a jet of water in its rigidity.</p> + + <p>It is, however, probable that you will take more interest in this box + of brass which I hold in my hands. You see nothing moving, but really, + inside this case there is a fly-wheel revolving rapidly. Observe that I + rest this case on the table on its sharp edge, a sort of skate, and it + does not tumble down as an ordinary box would do, or as this box will do + after a while, when its contents come to rest. Observe that I can strike + it violent blows, and it does not seem to budge from its vertical + position; it turns itself just a little round, but does not get tilted, + however hard I strike it. Observe that if I do get it tilted a little it + does not fall down, but slowly turns with what is called a precessional + motion (Fig. 5).</p> + + <p>You will, I hope, allow me, all through this lecture, to use the term + <i>precessional</i> for any motion of this kind. Probably you will object + more strongly to the great liberty I shall take presently, of saying that + the case <i>precesses</i> when it has this kind of motion; but I really + have almost no option in the matter, as I must use some verb, and I have + no time to invent a less barbarous one.</p> + + <div class="figcenter" style="width:37%;"> + <a href="images/fig05.jpg"><img style="width:100%" src="images/fig05.jpg" + alt="Fig. 5" title="Fig. 5" /></a> + <span class="sc">Fig. 5.</span> + </div> + + <p>When I hold this box in my hands (Fig. 6), I find that if I move it + with a motion of mere translation in any direction, it feels just as it + would do <!-- Page 22 --><span class="pagenum"><a + name="page22"></a>{22}</span>if its contents were at rest, but if I try + to turn it in my hands I find the most curious great resistance to such a + motion. The result is that when you hold this in your hands, its + readiness to move so long as it is not turned round, and its great + resistance to turning round, and its unexpected tendency to turn in a + different way from that in which you try to turn it, give one the most + uncanny sensations. It seems almost as if an invisible being had hold of + the box and exercised forces capriciously. And <!-- Page 23 --><span + class="pagenum"><a name="page23"></a>{23}</span>indeed there is a + spiritual being inside, what the algebraic people call an impossible + quantity, what other mathematicians call "an operator."</p> + + <div class="figcenter" style="width:36%;"> + <a href="images/fig06.jpg"><img style="width:100%" src="images/fig06.jpg" + alt="Fig. 6" title="Fig. 6" /></a> + <span class="sc">Fig. 6.</span> + </div> + + <p>Nearly all the experiments, even the tops and other apparatus you have + seen or will see to-night, have been arranged and made by my enthusiastic + assistant, Mr. Shepherd. The following experiment is not only his in + arrangement; even the idea of it is his. He said, you may grin and + contort your body with that large gyrostat in your hands, but many of + your audience will simply say to <!-- Page 24 --><span class="pagenum"><a + name="page24"></a>{24}</span>themselves that you only <i>pretend</i> to + find a difficulty in turning the gyrostat. So he arranged this pivoted + table for me to stand upon, and you will observe that when I now try to + turn the gyrostat, it will not turn; however I may exert myself, it keeps + pointing to that particular corner of the room, and all my efforts only + result in turning round my own body and the table, but not the + gyrostat.</p> + + <p>Now you will find that in every case this box only resists having the + axis of revolution of its hidden flywheel turned round, and if you are + interested in the matter and make a few observations, you will soon see + that every spinning body like the fly-wheel inside this case resists more + or less the change of direction of its spinning axis. When the fly-wheels + of steam-engines and dynamo machines and other quick speed machines are + rotating on board ship, you may be quite sure that they offer a greater + resistance to the pitching or rolling or turning of the ship, or any + other motion which tends to turn their axes in direction, than when they + are not rotating.</p> + + <p>Here is a top lying on a plate, and I throw it up into the air; you + will observe that its motion is very difficult to follow, and nobody + could predict, before it falls, exactly how it will alight on the plate; + it may come down peg-end foremost, or hindmost, or sideways. But when I + spin it (Fig. 7), and now throw it up into the air, there is no doubt + whatever <!-- Page 25 --><span class="pagenum"><a + name="page25"></a>{25}</span>as to how it will come down. The spinning + axis keeps parallel to itself, and I can throw the top up time after + time, without disturbing much the spinning motion.</p> + + <div class="figcenter" style="width:20%;"> + <a href="images/fig07.jpg"><img style="width:100%" src="images/fig07.jpg" + alt="Fig. 7" title="Fig. 7" /></a> + <span class="sc">Fig. 7.</span> + </div> + + <div class="figcenter" style="width:26%;"> + <a href="images/fig08.jpg"><img style="width:100%" src="images/fig08.jpg" + alt="Fig. 8" title="Fig. 8" /></a> + <span class="sc">Fig. 8.</span> + </div> + + <p>If I pitch up this biscuit, you will observe that I can have no + certainty as to how it will come down, but if I give it a spin before it + leaves my hand there is no doubt whatever (Fig. 8). Here is a hat. I + throw it up, and I cannot be sure as to how it will move, but if I give + it a spin, you see that, as <!-- Page 26 --><span class="pagenum"><a + name="page26"></a>{26}</span>with the top and the biscuit, the axis about + which the spinning takes place keeps parallel to itself, and we have + perfect certainty as to the hat's alighting on the ground brim downwards + (Fig. 9).</p> + + <div class="figcenter" style="width:27%;"> + <a href="images/fig09.jpg"><img style="width:100%" src="images/fig09.jpg" + alt="Fig. 9" title="Fig. 9" /></a> + <span class="sc">Fig. 9.</span> + </div> + + <p>I need not again bring before you the very soft hat to which we gave a + quasi-rigidity a few minutes ago; but you will remember that my assistant + sent that off like a projectile through the air when it was spinning, and + that it kept its spinning axis parallel to itself just like this more + rigid hat and the biscuit.</p> + + <div class="figcenter" style="width:70%;"> + <a href="images/fig10.jpg"><img style="width:100%" src="images/fig10.jpg" + alt="Fig. 10" title="Fig. 10" /></a> + <span class="sc">Fig. 10.</span> + </div> + + <div class="figright" style="width:36%;"> + <a href="images/fig11.jpg"><img style="width:100%" src="images/fig11.jpg" + alt="Fig. 11" title="Fig. 11" /></a> + <span class="sc">Fig. 11.</span> + </div> + + <p>I once showed some experiments on spinning tops to a coffee-drinking, + tobacco-smoking audience in that most excellent institution, the Victoria + Music Hall in London. In that music hall, things are not very different + from what they are at any other <!-- Page 27 --><span class="pagenum"><a + name="page27"></a>{27}</span>music hall except in beer, wine, and spirits + being unobtainable, and in short scientific addresses being occasionally + given. Now, I impressed my audience as strongly as I could with the above + fact, that if one wants to throw a quoit with certainty as to how it will + alight, one gives it a spin; if one wants to throw a hoop or a hat to + somebody to catch upon a stick, one gives the hoop or hat a spin; the + disinclination of a spinning body to let its axis get altered in + direction can always be depended upon. I told them that this was why + smooth-bore guns cannot be depended upon for accuracy;<a name="NtA4" + href="#Nt4"><sup>[4]</sup></a> that the spin which an ordinary bullet + took depended greatly on how it chanced to touch the muzzle as it just + left the gun, whereas barrels are now rifled, that is, spiral grooves are + now cut inside the barrel of a gun, and excrescences from the bullet or + projectile fit into these grooves, so that as it is forced along the + barrel of the gun by the explosive force of the powder, it must also spin + about its axis. Hence it leaves the gun with a perfectly well-known + spinning motion about which there can be no doubt, and we know too that + Fig. 10 shows the <!-- Page 28 --><span class="pagenum"><a + name="page28"></a>{28}</span>kind of motion which it has afterwards, for, + just like the hat or the biscuit, its spinning axis keeps nearly parallel + to itself. Well, this was all I could do, for I am not skilful in + throwing hats or quoits. But after my address was finished, and after a + young lady in a spangled dress had sung a comic song, two jugglers came + upon the stage, and I could not have had better illustrations of the + above principle than were given in almost every trick performed by this + lady and gentleman. They sent hats, and hoops, and plates, and umbrellas + spinning from one to the other. One of them threw a stream of knives into + the air, catching them and throwing them up again with perfect precision + and my now educated audience shouted with delight, and showed in other + unmistakable <!-- Page 29 --><span class="pagenum"><a + name="page29"></a>{29}</span>ways that they observed the spin which that + juggler gave to every knife as it left his hand, so that he might have a + perfect knowledge as to how it would come back to him again (Fig. 11). + <!-- Page 30 --><span class="pagenum"><a name="page30"></a>{30}</span>It + struck me with astonishment at the time that, almost without exception, + every juggling trick performed that evening was an illustration of the + above principle. And now, if you doubt my statement, just ask a child + whether its hoop is more likely to tumble down when it is rapidly rolling + along, or when it is going very slowly; ask a man on a bicycle to go more + and more slowly to see if he keeps his balance better; ask a + ballet-dancer how long she could stand on one toe without balancing + herself with her arms or a pole, if she were not spinning; ask + astronomers how many months would elapse before the earth would point + ever so far away from the pole star if it were not spinning; and above + all, ask a boy whether his top is as likely to stand upright upon its peg + when it is not spinning as when it is spinning.</p> + + <div class="figcenter" style="width:28%;"> + <a href="images/fig12.jpg"><img style="width:100%" src="images/fig12.jpg" + alt="Fig. 12" title="Fig. 12" /></a> + <span class="sc">Fig. 12.</span> + </div> + + <p>We will now examine more carefully the behaviour of this common top + (Fig. 12). It is not <!-- Page 31 --><span class="pagenum"><a + name="page31"></a>{31}</span>spinning, and you observe that it tumbles + down at once; it is quite unstable if I leave it resting upright on its + peg. But now note that when it is spinning, it not only will remain + upright resting on its peg, but if I give it a blow and so disturb its + state, it goes circling round with a precessional motion which grows + gradually less and less as time goes on, and the top lifts itself to the + upright position again. I hope you do not think that time spent in + careful observation of a phenomenon of this kind is wasted. Educated + observation of the commonest phenomena occurring in our everyday life is + never wasted, and I often feel that if workmen, who are the persons most + familiar with inorganic nature, could only observe and apply simple + scientific laws to their observations, instead of a great discovery every + century we should have a great discovery every year. Well, to return to + our top; there are two very curious observations to make. Please neglect + for a short time the slight wobbling motions that occur. One observation + we make is, that the top does not at first bow down in the direction of + the blow. If I strike towards the south, the top bows towards the west; + if I strike towards the west, the top bows down towards the north. Now + the reason of this is known to all scientific men, and the principle + underlying the top's behaviour is of very great <!-- Page 32 --><span + class="pagenum"><a name="page32"></a>{32}</span>importance in many ways, + and I hope to make it clear to you. The second fact, that the top + gradually reaches its upright position again, is one known to everybody, + but the reason for it is not by any means well known, although I think + that you will have no great difficulty in understanding it.</p> + + <p>The first phenomenon will be observed in this case which I have + already shown you. This case (Fig. 5), <!-- Page 33 --><span + class="pagenum"><a name="page33"></a>{33}</span>with the fly-wheel inside + it, is called a <i>gyrostat</i>. When I push the case it does not bow + down, but slowly turns round. This gyrostat will not exhibit the second + phenomenon; it will not rise up again if I manage to get it out of its + upright position, but, on the contrary, will go precessing in wider and + wider circles, getting further and further away from its upright + position.</p> + + <div class="figleft" style="width:45%;"> + <a href="images/fig13.jpg"><img style="width:100%" src="images/fig13.jpg" + alt="Fig. 13" title="Fig. 13" /></a> + <span class="sc">Fig. 13.</span> + </div> + + <div class="figleft" style="width:35%;"> + <a href="images/fig14.jpg"><img style="width:100%" src="images/fig14.jpg" + alt="Fig. 14" title="Fig. 14" /></a> + <span class="sc">Fig. 14.</span> + </div> + +<div style="clear: both"></div> + <p>The first phenomenon is most easily studied in this balanced gyrostat + (Fig. 13). You here see the fly-wheel G in a strong brass frame F, which + is supported so that it is free to move about the vertical axis A B, or + about the horizontal axis C D. The gyrostat is balanced by a weight W. + Observe that I can increase the leverage of W or diminish it by shifting + the position of the sleeve at A so that it will tend to either lift or + lower the gyrostat, or exactly balance it as it does now. You must + observe exactly what it is that we wish to study. If I endeavour to push + F downwards, with the end of this stick (Fig. 14), it really moves + horizontally to the right; now I push it to the right (Fig. 15), and it + only rises; now push it up, and you see that it goes to the left; push it + to the left, and it only goes downwards. You will notice that if I clamp + the instrument so that it cannot move vertically, it moves at once + horizontally; if I prevent mere horizontal motion it readily moves + vertically when I push it. Leaving it free as <!-- Page 34 --><span + class="pagenum"><a name="page34"></a>{34}</span>before, I will now shift + the position of the weight W, so that it tends continually to lift the + gyrostat, and of course the instrument does not lift, it moves + horizontally with a slow precessional motion. I now again shift the + weight W, so that the gyrostat would fall if it were not spinning (Fig. + 16), and it now moves horizontally with a slow precessional motion which + is in a direction opposed to the last. These phenomena are easily + explained, but, <!-- Page 35 --><span class="pagenum"><a + name="page35"></a>{35}</span>as I said before, it is necessary first to + observe them carefully. You all know now, vaguely, the fundamental fact. + It is that if I try to make a very quickly spinning body change the + direction of its axis, the direction of the axis will change, but not in + the way I intended. It is even more curious than my countryman's pig, for + when he wanted the pig to go to Cork, he had to pretend that he was + driving the pig home. His rule was a very <!-- Page 36 --><span + class="pagenum"><a name="page36"></a>{36}</span>simple one, and we must + find a rule for our spinning body, which is rather like a crab, that will + only go along the road when you push it sidewise.</p> + + <div class="figleft" style="width:37%;"> + <a href="images/fig15.jpg"><img style="width:100%" src="images/fig15.jpg" + alt="Fig. 15" title="Fig. 15" /></a> + <span class="sc">Fig. 15.</span> + </div> + + <div class="figleft" style="width:36%;"> + <a href="images/fig16.jpg"><img style="width:100%" src="images/fig16.jpg" + alt="Fig. 16" title="Fig. 16" /></a> + <span class="sc">Fig. 16.</span><a name="NtA5" + href="#Nt5"><sup>[5]</sup></a> + </div> + +<div style="clear: both"></div> + <div class="figcenter" style="width:70%;"> + <a href="images/fig10.jpg"><img style="width:100%" src="images/fig10.jpg" + alt="Fig. 10" title="Fig. 10" /></a> + <span class="sc">Fig. 10.</span> + </div> + + <p>As an illustration of this, consider the spinning projectile of Fig. + 10. The spin tends to keep its axis always in the same direction. But + there is a defect in the arrangement, which you are now in a <!-- Page 37 + --><span class="pagenum"><a name="page37"></a>{37}</span>position to + understand. You see that at A the air must be pressing upon the + undersurface A A, and I have to explain that this pressure tends to make + the projectile turn itself broadside on to the air. A boat in a current + not allowed to move as a whole, but tied at its middle, sets itself + broadside on to the current. Observe this disc of cardboard which I drop + through the air edgewise, and note how quickly it sets itself broadside + on and falls more slowly; and some of you may have thrown over into the + water at Aden small pieces of silver for the diving boys, and you are + aware that if it were not for this slow falling of the coins with a + wobbling motion broadside on, it would be nearly impossible for any + diving boy to get possession of them. Now all this is a parenthesis. The + <!-- Page 38 --><span class="pagenum"><a + name="page38"></a>{38}</span>pressure of the air tends to make the + projectile turn broadside on, but as the projectile is spinning it does + not tilt up, no more than this gyrostat does when I try to tilt it up, it + really tilts out of the plane of the diagram, out of the plane of its + flight; and only that artillerymen know exactly what it will do, this + kind of <i>windage</i> of the projectile would give them great + trouble.</p> + + <p>You will notice that an experienced child when it wants to change the + direction of a hoop, just exerts a tilting pressure with its hoop-stick. + A man on a bicycle changes his direction by leaning over so as to be out + of balance. It is well to remind you, however, that the motion of a + bicycle and its rider is not all rotational, so that it is not altogether + the analogue of a top or gyrostat. The explanation of the swerving from a + straight path when the rider tilts his body, ultimately comes to the same + simple principle, Newton's second law of motion, but it is arrived at + more readily. It is for the same reason—put briefly, the exercise + of a centripetal force—that when one is riding he can materially + assist his horse to turn a corner quickly, if he does not mind + appearances, by inclining his body towards the side to which he wants to + turn; and the more slowly the horse is going the greater is the tendency + to turn for a given amount of tilting of one's body. Circus-riders, when + galloping in a circle, assist their horses greatly by the position of + their bodies; it is <!-- Page 39 --><span class="pagenum"><a + name="page39"></a>{39}</span>not to save themselves from falling by + centrifugal force that they take a position on a horse's back which no + riding-master would allow his pupil to imitate; and the respectable + riders of this country would not scorn to help their horses in this way + to quick turning movements, if they had to chase and collect cattle like + American cowboys.</p> + + <p>Very good illustrations of change of direction are obtained in playing + <i>bowls</i>. You know that a bowl, if it had no <i>bias</i>, that is, if + it had no little weight inside it tending to tilt it, would roll along + the level bowling-green in a straight path, its speed getting less and + less till it stopped. As a matter of fact, however, you know that at the + beginning, when it is moving fast, its path is pretty straight, but + because it always has bias the path is never quite straight, and it bends + more and more rapidly as the speed diminishes. In all our examples the + slower the spin the quicker is the precession produced by given tilting + forces.</p> + + <p>Now close observation will give you a simple rule about the behaviour + of a gyrostat. As a matter of fact, all that has been incomprehensible or + curious disappears at once, if instead of speaking of this gyrostat as + moving up or down, or to the right or left, I speak of its motions about + its various axes. It offers no resistance to mere motion of translation. + But when I spoke of its moving <!-- Page 40 --><span class="pagenum"><a + name="page40"></a>{40}</span>horizontally, I ought to have said that it + moved about the vertical axis A B (Fig. 13). Again, what I referred to as + up and down motion of F is really motion in a vertical plane about the + horizontal axis C D. In future, when I speak of trying to give motion to + F, think only of the axis about which I try to turn it, and then a little + observation will clear the ground.</p> + + <div class="figright" style="width:28%;"> + <a href="images/fig18.jpg"><img style="width:100%" src="images/fig18.jpg" + alt="Fig. 18" title="Fig. 18" /></a> + <span class="sc">Fig. 18.</span> + </div> + + <div class="figright" style="width:29%;"> + <a href="images/fig17.jpg"><img style="width:100%" src="images/fig17.jpg" + alt="Fig. 17" title="Fig. 17" /></a> + <span class="sc">Fig. 17.</span> + </div> + + <p>Here is a gyrostat (Fig. 17), suspended in gymbals so carefully that + neither gravity nor any frictional forces at the pivots constrain it; + nothing that I can do to this frame which I hold in my hand will affect + the direction of the axis E F of the gyrostat. Observe that I whirl round + on my toes like a ballet-dancer while this is in my hand. I move it about + in all sorts of ways, but if it was pointing to the pole star at the + beginning it remains pointing to the pole star; if it pointed towards the + moon at the beginning it still points <!-- Page 41 --><span + class="pagenum"><a name="page41"></a>{41}</span>towards the moon. The + fact is, that as there is almost no frictional constraint at the pivots + there are almost no forces tending to turn the axis of rotation of the + gyrostat, and I can only give it motions of translation. But now I will + clamp this vertical spindle by means of a screw and repeat my + ballet-dance whirl; you will note that I need not whirl round, a very + small portion of a whirl is enough to cause this gyrostat (Fig. 18) to + set its spinning axis vertical, to set its axis parallel to the vertical + axis of rotation which I give it. Now I whirl in the opposite direction, + the gyrostat at once turns a somersault, turns completely round and + remains again with its axis vertical, and if you were to carefully note + the direction of the spinning of the <!-- Page 42 --><span + class="pagenum"><a name="page42"></a>{42}</span>gyrostat, you would find + the following rule to be generally true:—Pay no attention to mere + translational motion, think only of rotation about axes, and just + remember that when you constrain the axis of a spinning body to rotate, + it will endeavour to set its own axis parallel to the new axis about + which you rotate it; and not only is this the case, but it will endeavour + to have the direction of its own spin the same as the direction of the + new rotation. I again twirl on my toes, holding this frame, and now I + know that to a person looking down upon the gyrostat and me from the + ceiling, as I revolved in the direction of the hands of a clock, the + gyrostat is spinning in the direction of the hands of a clock; but if I + revolve against the clock direction (Fig. 19) the gyrostat tumbles over + so as again to be revolving in the same direction as that in which I + revolve.</p> + + <div class="figcenter" style="width:20%;"> + <a href="images/fig19.jpg"><img style="width:100%" src="images/fig19.jpg" + alt="Fig. 19" title="Fig. 19" /></a> + <span class="sc">Fig. 19.</span> + </div> + + <p>This then is the simple rule which will enable you to tell beforehand + how a gyrostat will move <!-- Page 43 --><span class="pagenum"><a + name="page43"></a>{43}</span>when you try to turn it in any particular + direction. You have only to remember that if you continued your effort + long enough, the spinning axis would become parallel to your new axis of + motion, and the direction of spinning would be the same as the direction + of your new turning motion.</p> + + <p>Now let me apply my rule to this balanced gyrostat. I shove it, or + give it an impulse downwards, but observe that this really means a + rotation about the horizontal axis C D (Fig. 13), and hence the gyrostat + turns its axis as if it wanted to become parallel to C D. Thus, looking + down from above (as shown by Fig. 20), O E was the direction of the + spinning axis, O D was the axis about which I endeavoured to move it, and + the instantaneous effect was that O E altered to the position O G. A + greater impulse of the same kind would have caused the spinning axis + instantly to go to O H or O J, whereas an upward opposite impulse would + have instantly made the spinning axis point in the direction O K, O L or + O M, depending on how great the impulse was and the rate of spinning. + When one observes these phenomena for the first time, one says, "I shoved + it down, and it moved to the right; I shoved it up, and it moved to the + left;" but if the direction of the spin were opposite to what it is, one + would say, "I shoved it down, and it moved to the left; I shoved it up, + and it moved to the right." The simple <!-- Page 44 --><span + class="pagenum"><a name="page44"></a>{44}</span>statement in all cases + ought to be, "I wanted to rotate it about a new axis, and the effect was + to send its spinning axis towards the direction of the new axis." And now + if you play with this balanced gyrostat as I am doing, shoving it about + in all sorts of ways, you will find the rule to be a correct one, and + there is no difficulty in predicting what will happen.</p> + + <div class="figcenter" style="width:34%;"> + <a href="images/fig20.jpg"><img style="width:100%" src="images/fig20.jpg" + alt="Fig. 20" title="Fig. 20" /></a> + <span class="sc">Fig. 20.</span> + </div> + +<p><!-- Page 45 --><span class="pagenum"><a name="page45"></a>{45}</span></p> + + <p>If this rule is right, we see at once why precession takes place. I + put this gyrostat (Fig. 13) out of balance, and if it were not rotating + it would fall downwards; but a force acting downwards really causes the + gyrostat to move to the right, and so you see that it is continually + moving in this way, for the force is always acting downwards, and the + spinning axis is continually chasing the new axes about which gravity + tends continually to make it revolve. We see also why it is that if the + want of balance is the other way, if gravity tends to lift the gyrostat, + the precession is in the opposite direction. And in playing with this + gyrostat as I do now, giving it all sorts of pushes, one makes other + observations and sees that the above rule simplifies them all; that is, + it enables us to remember them. For example, if I use this stick to hurry + on the precession, the gyrostat moves in opposition to the force which + causes the precession. I am particularly anxious that you should remember + this. At present the balance-weight is so placed that the gyrostat would + fall if it were not spinning. But it is spinning, and so it precesses. If + gravity were greater it would precess faster, and it comes home to us + that it is this precession which enables the force of gravity to be + inoperative in mere downward motion. You see that if the precession is + hurried, it is more than sufficient to balance gravity, <!-- Page 46 + --><span class="pagenum"><a name="page46"></a>{46}</span>and the gyrostat + rises. If I retard the precession, it is unable to balance gravity, and + the gyrostat falls. If I clamp this vertical axis so that precession is + impossible, you will notice that the gyrostat falls just as if it were + not spinning. If I clamp the instrument so that it cannot move + vertically, you notice how readily I can make it move horizontally; I can + set it rotating horizontally like any ordinary body.</p> + + <p>In applying our rule to this top, observe that the axis of spinning is + the axis E F of the top (Fig. 12). As seen in the figure, gravity is + tending to make the top rotate about the axis F D, and the spinning axis + in its chase of the axis F D describes a cone in space as it precesses. + This gyrostat, which is top-heavy, rotates and precesses in much the same + way as the top; that is, if you apply our rule, or use your observation, + you will find that to an observer above the table the spinning and + precession occur in the same direction, that is, either both with the + hands of a watch, or both against the hands of a watch. Whereas, a top + like this before you (Fig. 21), supported above its centre of gravity, or + the gyrostat here (Fig. 22), which is also supported above its centre of + gravity, or the gyrostat shown in Fig. 56, or any other gyrostat + supported in such a way that it would be in stable equilibrium if it were + not spinning; in all these <!-- Page 47 --><span class="pagenum"><a + name="page47"></a>{47}</span>cases, to an observer placed above the + table, the precession is in a direction opposite to that of the + spinning.</p> + + <div class="figleft" style="width:40%;"> + <a href="images/fig21.jpg"><img style="width:100%" src="images/fig21.jpg" + alt="Fig. 21" title="Fig. 21" /></a> + <span class="sc">Fig. 21.</span> + </div> + + <div class="figleft" style="width:38%;"> + <a href="images/fig22.jpg"><img style="width:100%" src="images/fig22.jpg" + alt="Fig. 22" title="Fig. 22" /></a> + <span class="sc">Fig. 22.</span> + </div> + +<div style="clear: both"></div> +<p><!-- Page 48 --><span class="pagenum"><a name="page48"></a>{48}</span></p> + + <p>If an impulse be given to a top or gyrostat in the direction of the + precession, it will rise in opposition to the force of gravity, and + should at any instant the precessional velocity be greater than what it + ought to be for the balance of the force of gravity, the top or gyrostat + will rise, its precessional velocity diminishing. If the precessional + velocity is too small, the top will fall, and as it falls the + precessional velocity increases.</p> + + <p>Now I say that all these facts, which are mere facts of observation, + agree with our rule. I wish I dare ask you to remember them all. You will + observe that in this wall sheet I have made a list of them. I speak of + gravity as causing the precession, but the forces may be any others than + such as are due to gravity.</p> + +<h3><span class="sc">Wall Sheet.</span></h3> + + <p>I. <span class="sc">Rule.</span> When forces act upon a spinning body, + tending to cause rotation about any other axis than the spinning axis, + the spinning axis sets itself in better agreement with the new axis of + rotation. Perfect agreement would mean perfect parallelism, the + directions of rotation being the same.</p> + + <p>II. Hurry on the precession, and the body rises in opposition to + gravity. <!-- Page 49 --><span class="pagenum"><a + name="page49"></a>{49}</span></p> + + <p>III. Delay the precession and the body falls, as gravity would make it + do if it were not spinning.</p> + + <p>IV. A common top precesses in the same direction as that in which it + spins.</p> + + <p>V. A top supported above its centre of gravity, or a body which would + be in stable equilibrium if not spinning, precesses in the opposite + direction to that of its spinning.</p> + + <p>VI. The last two statements come to this:—When the forces acting + on a spinning body tend to make the <i>angle</i> of precession greater, + the precession is in the same direction as the spinning, and <i>vice + versâ</i>.</p> + + <p>Having by observation obtained a rule, every natural philosopher tries + to make his rule a rational one; tries to explain it. I hope you know + what we mean when we say that we explain a phenomenon; we really mean + that we show the phenomenon to be consistent with other better known + phenomena. Thus when you unmask a spiritualist and show that the + phenomena exhibited by him are due to mere sleight-of-hand and trickery, + you explain the phenomena. When you show that they are all consistent + with well-observed and established mesmeric influences, you are also said + to explain the phenomena. When you show that they can be effected by + means of telegraphic messages, or by reflection of light from mirrors, + you explain the <!-- Page 50 --><span class="pagenum"><a + name="page50"></a>{50}</span>phenomena, although in all these cases you + do not really know the nature of mesmerism, electricity, light, or moral + obliquity.</p> + + <p>The meanest kind of criticism is that of the man who cheapens a + scientific explanation by saying that the very simplest facts of nature + are unexplainable. Such a man prefers the chaotic and indiscriminate + wonder of the savage to the reverence of a Sir Isaac Newton.</p> + + <div class="figcenter" style="width:21%;"> + <a href="images/fig23.jpg"><img style="width:100%" src="images/fig23.jpg" + alt="Fig. 23" title="Fig. 23" /></a> + <span class="sc">Fig. 23.</span> + </div> + + <p>The explanation of our rule is easy. Here is a gyrostat (Fig. 23) + something like the earth in shape, and it is at rest. I am sorry to say + that I am compelled to support this globe in a very visible manner by + gymbal rings. If this globe were just floating in the air, if it had no + tendency to fall, my explanation would be easier to understand, and I + could illustrate it better experimentally. Observe the point P. If I move + the globe slightly about the axis A, the point P moves to Q. But suppose + instead of this that the globe and inner gymbal <!-- Page 51 --><span + class="pagenum"><a name="page51"></a>{51}</span>ring had been moved about + the axis B; the point P would have moved to R. Well, suppose both those + rotations took place simultaneously. You all know that the point P would + move neither to Q nor to R, but it would move to S; P S being the + diagonal of the little parallelogram. The resultant motion then is + neither about the axis O A in space, nor about the axis O B, but it is + about some such axis as O C.</p> + + <p>To this globe I have given two rotations simultaneously. Suppose a + little being to exist on this globe which could not see the gymbals, but + was able to observe other objects in the room. It would say that the + direction of rotation is neither about O A nor about O B, but that the + real axis of its earth is some line intermediate, O C in fact.</p> + + <p>If then a ball is suddenly struck in two different directions at the + same instant, to understand how it will spin we must first find how much + spin each blow would produce if it acted alone, and about what axis. A + spin of three turns per second about the axis O A (Fig. 24), and a spin + of two turns per second about the axis O B, really mean that the ball + will spin about the axis O C with a spin of three and a half turns per + second. To arrive at this result, I made O A, 3 feet long (any other + scale of representation would have been right) <!-- Page 52 --><span + class="pagenum"><a name="page52"></a>{52}</span>and O B, 2 feet long, and + I found the diagonal O C of the parallelogram shown on the figure to be + 3½ feet long.</p> + + <p>Observe that if the rotation about the axis O A is <i>with</i> the + hands of a watch looking from O to A, the rotation about the axis O B + looking from O to B, must also be with the hands of a watch, and the + resultant rotation about the axis O C is also in a direction with the + hands of a watch looking from O to C. Fig. 25 shows in two diagrams how + necessary it is that on looking from O along either O A or O B, the + rotation should be in the same direction as regards the hands of a watch. + These constructions are well known to all who have studied elementary + mechanical principles. Obviously if the rotation about O A is very much + greater than the rotation about O B, then the position of the new axis O + C must be much nearer O A than O B.</p> + + <div class="figleft" style="width:21%;"> + <a href="images/fig24.jpg"><img style="width:100%" src="images/fig24.jpg" + alt="Fig. 24" title="Fig. 24" /></a> + <span class="sc">Fig. 24.</span> + </div> + + <div class="figcenter" style="width:50%;"> + <a href="images/fig25.jpg"><img style="width:100%" src="images/fig25.jpg" + alt="Fig. 25" title="Fig. 25" /></a> + <span class="sc">Fig. 25.</span> + </div> + +<div style="clear: both"></div> + <p>We see then that if a body is spinning about an axis O A, and we apply + forces to it which <!-- Page 53 --><span class="pagenum"><a + name="page53"></a>{53}</span>would, if it were at rest, turn it about the + axis O B; the effect is to cause the spinning axis to be altered to O C; + that is, the spinning axis sets itself in better agreement with the new + axis of rotation. This is the first statement on our wall sheet, the rule + from which all our other statements are derived, assuming that they were + not really derived from observation. Now I do not say that I have here + given a complete proof for all cases, for the fly-wheels in these + gyrostats are running in bearings, and the bearings constrain the axes to + take the new positions, whereas there is no such <!-- Page 54 --><span + class="pagenum"><a name="page54"></a>{54}</span>constraint in this top; + but in the limited time of a popular lecture like this it is not + possible, even if it were desirable, to give an exhaustive proof of such + a universal rule as ours is. That I have not exhausted all that might be + said on this subject will be evident from what follows.</p> + + <p>If we have a spinning ball and we give to it a new kind of rotation, + what will happen? Suppose, for example, that the earth were a homogeneous + sphere, and that there were suddenly impressed upon it a new rotatory + motion tending to send Africa southwards; the axis of this new spin would + have its pole at Java, and this spin combined with the old one would + cause the earth to have its true pole somewhere between the present pole + and Java. It would no longer rotate about its present axis. In fact the + axis of rotation would be altered, and there would be no tendency for + anything further to occur, because a homogeneous sphere will as readily + rotate about one axis as another. But if such a thing were to happen to + this earth of ours, which is not a sphere but a flattened spheroid like + an orange, its polar diameter being the one-third of one per cent. + shorter than the equatorial diameter; then as soon as the new axis was + established, the axis of symmetry would resent the change and would try + to become again the axis of rotation, and a great wobbling motion would + ensue. <!-- Page 55 --><span class="pagenum"><a + name="page55"></a>{55}</span>I put the matter in popular language when I + speak of the resentment of an axis; perhaps it is better to explain more + exactly what I mean. I am going to use the expression Centrifugal Force. + Now there are captious critics who object to this term, but all engineers + use it, and I like to use it, and our captious critics submit to all + sorts of ignominious involution of language in evading the use of it. It + means the force with which any body acts upon its constraints when it is + constrained to move in a curved path. The force is always directed away + from the centre of the curve. When a ball is whirled round in a curve at + the end of a string its centrifugal force tends to break the string. When + any body keyed to a shaft is revolving with the shaft, it may be that the + centrifugal forces of all the parts just balance one another; but + sometimes they do not, and then we say that the shaft is out of balance. + Here, for example, is a disc of wood rotating. It is in balance. But I + stop its motion and fix this piece of lead, A, to it, and you observe + when it rotates that it is so much out of balance that the bearings of + the shaft and the frame that holds them, and even the lecture-table, are + shaking. Now I will put things in balance again by placing another piece + of lead, B, on the side of the spindle remote from A, and when I again + rotate the disc (Fig. 26) there <!-- Page 56 --><span class="pagenum"><a + name="page56"></a>{56}</span>is no longer any shaking of the framework. + When the crank-shaft of a locomotive has not been put in balance by means + of weights suitably placed on the driving-wheels, there is nobody in the + train who does not feel the effects. Yes, and the coal-bill shows the + effects, for an unbalanced engine tugs the train spasmodically instead of + exerting an efficient steady pull. My friend Professor Milne, of Japan, + places earthquake measuring instruments on engines and in trains for + measuring this and other wants of balance, and he has shown unmistakably + that two engines of nearly the same general design, one balanced properly + and the other not, consume very different amounts of coal in making the + same journey at the same speed.</p> + + <div class="figcenter" style="width:27%;"> + <a href="images/fig26.jpg"><img style="width:100%" src="images/fig26.jpg" + alt="Fig. 26" title="Fig. 26" /></a> + <span class="sc">Fig. 26.</span> + </div> + + <p>If a rotating body is in balance, not only does the axis of rotation + pass through the centre of gravity (or rather centre of mass) of the + body, but <!-- Page 57 --><span class="pagenum"><a + name="page57"></a>{57}</span>the axis of rotation must be one of the + three principal axes through the centre of mass of the body. Here, for + example, is an ellipsoid of wood; A A, B B, and C C (Fig. 27) are its + three principal axes, and it would be in balance if it rotated about any + one of these three axes, and it would not be in balance if it rotated + about any other axis, unless, indeed, it were like a homogeneous sphere, + every diameter of which is a principal axis.</p> + + <div class="figcenter" style="width:15%;"> + <a href="images/fig27.jpg"><img style="width:100%" src="images/fig27.jpg" + alt="Fig. 27" title="Fig. 27" /></a> + <span class="sc">Fig. 27.</span> + </div> + + <p>Every body has three such principal axes through its centre of mass, + and this body (Fig. 27) has them; but I have here constrained it to + rotate about the axis D D, and you all observe the effect of the + unbalanced centrifugal forces, which is nearly great enough to tear the + framework in pieces. The higher the speed the more important this want of + balance is. If the speed is doubled, the centrifugal forces become four + times as great; and modern mechanical engineers with their quick speed + engines, some of which revolve, like the fan-engines of torpedo-boats, at + 1700 revolutions per minute, require to pay great attention to this + subject, which the older engineers never troubled their <!-- Page 58 + --><span class="pagenum"><a name="page58"></a>{58}</span>heads about. You + must remember that even when want of balance does not actually fracture + the framework of an engine, it will shake everything, so that nuts and + keys and other fastenings are pretty sure to get loose.</p> + + <p>I have seen, on a badly-balanced machine, a securely-fastened pair of + nuts, one supposed to be locking the other, quietly revolving on their + bolt at the same time, and gently lifting themselves at a regular but + fairly rapid rate, until they both tumbled from the end of the bolt into + my hand. If my hand had not been there, the bolts would have tumbled into + a receptacle in which they would have produced interesting but most + destructive phenomena. You would have somebody else lecturing to you + to-night if that event had come off.</p> + + <p>Suppose, then, that our earth were spinning about any other axis than + its present axis, the axis of figure. If spun about any diameter of the + equator for example, centrifugal forces would just keep things in a state + of unstable equilibrium, and no great change might be produced until some + accidental cause effected a slight alteration in the spinning axis, and + after that the earth would wobble very greatly. How long and how + violently it would wobble, would depend on a number of circumstances + about which I will not now venture to guess. If you <!-- Page 59 --><span + class="pagenum"><a name="page59"></a>{59}</span>tell me that on the + whole, in spite of the violence of the wobbling, it would not get shaken + into a new form altogether, then I know that in consequence of tidal and + other friction it would eventually come to a quiet state of spinning + about its present axis.</p> + + <p>You see, then, that although every body has three axes about which it + will rotate in a balanced fashion without any tendency to wobble, this + balance of the centrifugal forces is really an unstable balance in two + out of the three cases, and there is only one axis about which a + perfectly stable balanced kind of rotation will take place, and a + spinning body generally comes to rotate about this axis in the long run + if left to itself, and if there is friction to still the wobbling.</p> + + <p>To illustrate this, I have here a method of spinning bodies which + enables them to choose as their spinning axis that one principal axis + about which their rotation is most stable. The various bodies can be hung + at the end of this string, and I cause the pulley from which the string + hangs to rotate. Observe that at first the disc (Fig. 28 <i>a</i>) + rotates soberly about the axis A A, but you note the small beginning of + the wobble; now it gets quite violent, and now the disc is stably and + smoothly rotating about the axis B B, which is the most important of its + principal axes. <!-- Page 60 --><span class="pagenum"><a + name="page60"></a>{60}</span></p> + + <div class="figcenter" style="width:69%;"> + <a href="images/fig28.jpg"><img style="width:100%" src="images/fig28.jpg" + alt="Fig. 28" title="Fig. 28" /></a> + <span class="sc">Fig. 28.</span> + </div> + + <p>Again, this cone (Fig. 28 <i>b</i>) rotates smoothly at first about + the axis A A, but the wobble begins and gets very great, and eventually + the cone rotates smoothly about the axis B B, which is the most important + of its principal axes. Here again is a rod hung from one end (Fig. 28 + <i>d</i>).</p> + + <p>See also this anchor ring. But you may be more interested in this limp + ring of chain (Fig. 28 <i>c</i>). See how at first it hangs from the cord + vertically, and how the wobbles and vibrations end in its becoming a + perfectly circular ring lying all in a horizontal plane. This experiment + illustrates also the quasi-rigidity given to a flexible body by rapid + motion.</p> + + <p>To return to this balanced gyrostat of ours (Fig. 13). It is not + precessing, so you know that the weight W just balances the gyrostat F. + Now if I leave the instrument to itself after I give a downward impulse + to F, not exerting merely a steady pressure, you will notice that F + swings to the right for the reason already given; but it swings too fast + and too far, just like any other swinging body, and it is easy from what + I have already said, to see that this wobbling motion (Fig. 29) should be + the result, and that it should continue until friction stills it, and F + takes its permanent new position only after some time elapses.</p> + + <p>You see that I can impose this wobble or nodding <!-- Page 62 --><span + class="pagenum"><a name="page62"></a>{62}</span>motion upon the gyrostat + whether it has a motion of precession or not. It is now nodding as it + processes round and round—that is, it is rising and falling as it + precesses.</p> + + <div class="figcenter" style="width:36%;"> + <a href="images/fig29.jpg"><img style="width:100%" src="images/fig29.jpg" + alt="Fig. 29" title="Fig. 29" /></a> + <span class="sc">Fig. 29.</span> + </div> + + <p>Perhaps I had better put the matter a little more clearly. You see the + same phenomenon in this top. If the top is precessing too fast for the + force of gravity the top rises, and the precession diminishes in + consequence; the precession being now too slow to balance gravity, the + top falls a little and the <!-- Page 63 --><span class="pagenum"><a + name="page63"></a>{63}</span>precession increases again, and this sort of + vibration about a mean position goes on just as the vibration of a + pendulum goes on till friction destroys it, and the top precesses more + regularly in the mean position. This nodding is more evident in the + nearly horizontal balanced gyrostat than in a top, because in a top the + turning effect of gravity is less in the higher positions.</p> + + <p>When scientific men try to popularize their discoveries, for the sake + of making some fact very plain they will often tell slight untruths, + making statements which become rather misleading when their students + reach the higher levels. Thus astronomers tell the public that the earth + goes round the sun in an elliptic path, whereas the attractions of the + planets cause the path to be only approximately elliptic; and + electricians tell the public that electric energy is conveyed through + wires, whereas it is really conveyed by all other space than that + occupied by the wires. In this lecture I have to some small extent taken + advantage of you in this way; for example, at first you will remember, I + neglected the nodding or wobbling produced when an impulse is given to a + top or gyrostat, and, all through, I neglect the fact that the + instantaneous axis of rotation is only nearly coincident with the axis of + figure of a precessing gyrostat or top. And indeed you may generally <!-- + Page 64 --><span class="pagenum"><a name="page64"></a>{64}</span>take it + that if all one's statements were absolutely accurate, it would be + necessary to use hundreds of technical terms and involved sentences with + explanatory, police-like parentheses; and to listen to many such + statements would be absolutely impossible, even for a scientific man. You + would hardly expect, however, that so great a scientific man as the late + Professor Rankine, when he was seized with the poetic fervour, would err + even more than the popular lecturer in making his accuracy of statement + subservient to the exigencies of the rhyme as well as to the necessity + for simplicity of statement. He in his poem, <i>The Mathematician in + Love</i>, has the following lines—</p> + + <div class="poem"> + <div class="stanza"> + <p class="hg3">"The lady loved dancing;—he therefore applied</p> + <p class="i2">To the polka and waltz, an equation;</p> + <p>But when to rotate on his axis he tried,</p> + <p>His centre of gravity swayed to one side,</p> + <p class="i2">And he fell by the earth's gravitation."</p> + </div> + </div> + + <p>Now I have no doubt that this is as good "dropping into poetry" as can + be expected in a scientific man, and ——'s science is as good + as can be expected in a man who calls himself a poet; but in both cases + we have illustrations of the incompatibility of science and rhyming.</p> + + <div class="figcenter" style="width:29%;"> + <a href="images/fig17.jpg"><img style="width:100%" src="images/fig17.jpg" + alt="Fig. 17" title="Fig. 17" /></a> + <span class="sc">Fig. 17.</span> + </div> + + <p>The motion of this gyrostat can be made even more complicated than it + was when we had <!-- Page 65 --><span class="pagenum"><a + name="page65"></a>{65}</span>nutation and precession, but there is really + nothing in it which is not readily explainable by the simple principles I + have put before you. Look, for example, at this well-balanced gyrostat + (Fig. 17). When I strike this inner gymbal ring in any way you see that + it wriggles quickly just as if it were a lump of jelly, its rapid + vibrations dying away just like the rapid vibrations of any yielding + elastic body. This strange elasticity is of very great interest when we + consider it in relation to the molecular properties of matter. Here again + (Fig. 30) we have an example which is even more interesting. I have + supported the cased <!-- Page 66 --><span class="pagenum"><a + name="page66"></a>{66}</span>gyrostat of Figs. 5 and 6 upon a pair of + stilts, and you will observe that it is moving about a perfectly stable + position with a very curious staggering kind of vibratory motion; but + there is nothing in these motions, however curious, that you cannot + easily explain if you have followed me so far.</p> + + <div class="figcenter" style="width:30%;"> + <a href="images/fig30.jpg"><img style="width:100%" src="images/fig30.jpg" + alt="Fig. 30" title="Fig. 30" /></a> + <span class="sc">Fig. 30.</span> + </div> + + <p>Some of you who are more observant than the others, will have remarked + that all these precessing gyrostats gradually fall lower and lower, just + as they would do, only more quickly, if they were not spinning. And if + you cast your eye upon the third statement of our wall sheet (p. <a + href="#page49">49</a>) you will readily understand why it is so.</p> + + <p>"Delay the precession and the body falls, as gravity would make it do + if it were not spinning." <!-- Page 67 --><span class="pagenum"><a + name="page67"></a>{67}</span>Well, the precession of every one of these + is resisted by friction, and so they fall lower and lower.</p> + + <p>I wonder if any of you have followed me so well as to know already why + a spinning top rises. Perhaps you have not yet had time to think it out, + but I have accentuated several times the particular fact which explains + this phenomenon. Friction makes the gyrostats fall, what is it that + causes a top to rise? Rapid rising to the upright position is the + invariable sign of rapid rotation in a top, and I recollect that when + quite vertical we used to say, "She sleeps!" Such was the endearing way + in which the youthful experimenter thought of the beautiful object of his + tender regard.</p> + + <p>All so well known as this rising tendency of a top has been ever since + tops were first spun, I question if any person in this hall knows the + explanation, and I question its being known to more than a few persons + anywhere. Any great mathematician will tell you that the explanation is + surely to be found published in <i>Routh</i>, or that at all events he + knows men at Cambridge who surely know it, and he thinks that he himself + must have known it, although he has now forgotten those elaborate + mathematical demonstrations which he once exercised his mind upon. I + believe that all such statements are made in error, but I cannot <!-- + Page 68 --><span class="pagenum"><a name="page68"></a>{68}</span>be + sure.<a name="NtA6" href="#Nt6"><sup>[6]</sup></a> A partial theory of + the phenomenon was given by Mr. Archibald Smith in the <i>Cambridge + Mathematical Journal</i> many years ago, but the problem was solved by + Sir William Thomson and Professor Blackburn when they stayed together one + year at the seaside, reading for the great Cambridge mathematical + examination. It must have alarmed a person interested in Thomson's + success to notice that the seaside holiday was really spent by him and + his friend in spinning all sorts of rounded stones which they picked up + on the beach.</p> + + <p>And I will now show you the curious phenomenon that puzzled him that + year. This ellipsoid (Fig. 31) will represent a waterworn stone. It is + lying in its most stable state on the table, and I give it a spin. You + see that for a second or two it was inclined to go on spinning about the + axis A A, but it began to wobble violently, and after a while, when these + wobbles stilled, you saw that it was spinning nicely with its axis B B + vertical; but then a new series of wobblings began and became more + violent, and when they ceased you saw that the object had at length + reached a settled state of <!-- Page 69 --><span class="pagenum"><a + name="page69"></a>{69}</span>spinning, standing upright upon its longest + axis. This is an extraordinary phenomenon to any person who knows about + the great inclination of this body to spin in the very way in which I + first started it spinning. You will find that nearly any rounded stone + when spun will get up in this way upon its longest axis, if the spin is + only vigorous enough, and in the very same way this spinning top tends to + get more and more upright.</p> + + <div class="figcenter" style="width:45%;"> + <a href="images/fig31.jpg"><img style="width:100%" src="images/fig31.jpg" + alt="Fig. 31" title="Fig. 31" /></a> + <span class="sc">Fig. 31.</span> + </div> + + <p>I believe that there are very few mathematical explanations of + phenomena which may not be given in quite ordinary language to people who + have an ordinary amount of experience. In most cases the symbolical + algebraic explanation must be given first by somebody, and then comes the + time for its translation into ordinary language. This is the foundation + of the new thing called Technical Education, which assumes that a <!-- + Page 70 --><span class="pagenum"><a name="page70"></a>{70}</span>workman + may be taught the principles underlying the operations which go on in his + trade, if we base our explanations on the experience which the man has + acquired already, without tiring him with a four years' course of study + in elementary things such as is most suitable for inexperienced children + and youths at public schools and the universities.</p> + + <div class="figcenter" style="width:35%;"> + <a href="images/fig32.jpg"><img style="width:100%" src="images/fig32.jpg" + alt="Fig. 32" title="Fig. 32" /></a> + <span class="sc">Fig. 32.</span> + </div> + + <div class="figright" style="width:33%;"> + <a href="images/fig33.jpg"><img style="width:100%" src="images/fig33.jpg" + alt="Fig. 33" title="Fig. 33" /></a> + <span class="sc">Fig. 33.</span> + </div> + + <p>With your present experience the explanation of the rising of the top + becomes ridiculously simple. If you look at statement <i>two</i> on this + wall sheet (p. 48) and reflect a little, some of you will be able, + without any elaborate mathematics, to give the simple reason for this + that Thomson gave me sixteen years ago. "Hurry on the precession, and the + body rises in opposition to gravity." Well, as I am not touching the top, + and as the body does rise, we look at once for something that is hurrying + on the precession, and we naturally look to the way in which its peg is + rubbing on the table, for, with the exception of the atmosphere this top + is touching nothing else than the table. Observe carefully how any of + these objects precesses. Fig. 32 shows the way in which a top spins. + Looked at from above, if the top is spinning in the direction of the + hands of a watch, we know from the fourth statement of our wall sheet, or + by mere observation, that it also precesses in the direction of the hands + <!-- Page 71 --><span class="pagenum"><a name="page71"></a>{71}</span>of + a watch; that is, its precession is such as to make the peg roll at B + into the paper. For you will observe that the peg is rolling round a + circular path on the table, G being nearly motionless, and the axis A G A + describing nearly a cone in space whose vertex is G, above the table. + Fig. 33 <!-- Page 72 --><span class="pagenum"><a + name="page72"></a>{72}</span>shows the peg enlarged, and it is evident + that the point B touching the table is really like the bottom of a wheel + B B', and as this wheel is rotating, the rotation causes it to roll + <i>into</i> the paper, away from us. But observe that its mere precession + is making it roll <i>into</i> the paper, and that the spin if great + enough wants to roll the top faster than the precession lets it roll, so + that it hurries on the precession, and therefore the top rises. That is + the simple explanation; the spin, so long as it is <!-- Page 73 --><span + class="pagenum"><a name="page73"></a>{73}</span>great enough, is always + hurrying on the precession, and if you will cast your recollection back + to the days of your youth, when a top was supported on your hand as this + is now on mine (Fig. 34), and the spin had grown to be quite small, and + was unable to keep the top upright, you will remember that you + dexterously helped the precession by giving your hand a circling motion + so as to get from your top the advantages as to uprightness of a slightly + longer spin.</p> + + <div class="figcenter" style="width:20%;"> + <a href="images/fig34.jpg"><img style="width:100%" src="images/fig34.jpg" + alt="Fig. 34" title="Fig. 34" /></a> + <span class="sc">Fig. 34.</span> + </div> + + <p>I must ask you now by observation, and the application of exactly the + same argument, to explain the struggle for uprightness on its longer axis + of any rounded stone when it spins on a table. I may tell you that some + of these large rounded-looking objects which I now spin before you in + illustration, are made hollow, and they are either of wood or zinc, + because I have not the skill necessary to spin large solid objects, and + yet I wanted to have objects which you would be able to see. This small + one (Fig. 31) is the largest solid one to which my fingers are able to + give sufficient spin. Here is a very interesting object (Fig. 35), + spherical <!-- Page 74 --><span class="pagenum"><a + name="page74"></a>{74}</span>in shape, but its centre of gravity is not + exactly at its centre of figure, so when I lay it on the table it always + gets to its position of stable equilibrium, the white spot touching the + table as at A. Some of you know that if this sphere is thrown into the + air it seems to have very curious motions, because one is so apt to + forget that it is the motion of its centre of gravity which follows a + simple path, and the boundary is eccentric to the centre of gravity. Its + motions when set to roll upon a carpet are also extremely curious.</p> + + <div class="figcenter" style="width:32%;"> + <a href="images/fig35.jpg"><img style="width:100%" src="images/fig35.jpg" + alt="Fig. 35" title="Fig. 35" /></a> + <span class="sc">Fig. 35.</span> + </div> + + <p>Now for the very reasons that I have already given, when this sphere + is made to spin on the table, it always endeavours to get its white spot + uppermost, as in C, Fig. 35; to get into the position in which when not + spinning it would be unstable.</p> + + <div class="figcenter" style="width:69%;"> + <a href="images/fig36.jpg"><img style="width:100%" src="images/fig36.jpg" + alt="Fig. 36" title="Fig. 36" /></a> + <span class="sc">Fig. 36.</span> + </div> + + <p>The precession of a top or gyrostat leads us at once to think of the + precession of the great spinning body on which we live. You know that the + earth <!-- Page 75 --><span class="pagenum"><a + name="page75"></a>{75}</span>spins on its axis a little more than once + every twenty-four hours, as this orange is revolving, and that it goes + round the sun once in a year, as this orange is now going round a model + sun, or as is shown in the diagram (Fig. 36). Its spinning axis points in + the direction shown, very nearly to the star which is called the pole + star, almost infinitely far away. In the figure and model I have greatly + exaggerated the elliptic nature of the earth's path, as is quite usual, + although it may be a little misleading, because the earth's path is much + more nearly circular than many people imagine. As a matter of fact the + earth is about three million miles nearer the sun in winter than it is in + summer. This seems at first paradoxical, but we get to understand it when + we reflect that, because of the slope of the earth's axis to the + ecliptic, we people who live in the northern hemisphere have the sun less + vertically above us, and have a shorter day in the winter, and hence each + square foot of our part of the earth's surface receives much less heat + every day, and so we feel colder. Now in about 13,000 years the earth + will have precessed just half a revolution (<i>see</i> Fig. 38); the axis + will then be sloped towards the sun when it is nearest, instead of away + from it as it is now; consequently we shall be much warmer in summer and + colder in winter than we are now. Indeed we shall then be much worse off + than the southern <!-- Page 77 --><span class="pagenum"><a + name="page77"></a>{77}</span>hemisphere people are now, for they have + plenty of oceanic water to temper their climate. It is easy to see the + nature of the change from figures 36, 37, and 38, or from the model as I + carry the orange and its symbolic knitting-needle round the model sun. + Let us imagine an observer placed above this model, far above the north + pole of the earth. He sees the earth rotating against the direction of + the hands of a watch, and he finds that it precesses with the hands of a + watch, so that spin and precession are in opposite directions. Indeed it + is because of this that we have the word "precession," which we now apply + to the motion of a top, although the precession of a top is in the same + direction as that of the spin.</p> + + <div class="figcenter" style="width:68%;"> + <a href="images/fig37.jpg"><img style="width:100%" src="images/fig37.jpg" + alt="Fig. 37" title="Fig. 37" /></a> + <span class="sc">Fig. 37.</span> + </div> + + <div class="figcenter" style="width:70%;"> + <a href="images/fig38.jpg"><img style="width:100%" src="images/fig38.jpg" + alt="Fig. 38" title="Fig. 38" /></a> + <span class="sc">Fig. 38.</span> + </div> + + <p>The practical astronomer, in explaining the <i>luni-solar precession + of the equinoxes</i> to you, will not probably refer to tops or + gyrostats. He will tell you that the <i>longitude</i> and <i>right + ascension</i> of a star seem to alter; in fact that the point on the + ecliptic from which he makes his measurements, namely, the spring + equinox, is slowly travelling round the ecliptic in a direction opposite + to that of the earth in its orbit, or to the apparent path of the sun. + The spring equinox is to him for heavenly measurements what the longitude + of Greenwich is to the navigator. He will tell you that aberration of + light, and parallax of the stars, <!-- Page 80 --><span + class="pagenum"><a name="page80"></a>{80}</span>but more than both, this + precession of the equinoxes, are the three most important things which + prevent us from seeing in an observatory by transit observations of the + stars, that the earth is revolving with perfect uniformity. But his way + of describing the precession must not disguise for you the physical fact + that his phenomenon and ours are identical, and that to us who are + acquainted with spinning tops, the slow conical motion of a spinning axis + is more readily understood than those details of his measurements in + which an astronomer's mind is bound up, and which so often condemn a man + of great intellectual power to the life of drudgery which we generally + associate with the idea of the pound-a-week cheap clerk.</p> + + <div class="figcenter" style="width:38%;"> + <a href="images/fig22.jpg"><img style="width:100%" src="images/fig22.jpg" + alt="Fig. 22" title="Fig. 22" /></a> + <span class="sc">Fig. 22.</span> + </div> + + <p>The precession of the earth is then of the same nature as that of a + gyrostat suspended above its centre of gravity, of a body which would be + stable and not top-heavy if it were not spinning. In fact the precession + of the earth is of the same nature as that of this large gyrostat (Fig. + 22), which is suspended in gymbals, so that it has a vibration like a + pendulum when not spinning. I will now spin it, so that looked at from + above it goes against the hands of a watch, and you observe that it + precesses with the hands of a watch. Here again is a hemispherical wooden + ship, in which there is a gyrostat with its axis vertical. It is in + stable <!-- Page 81 --><span class="pagenum"><a + name="page81"></a>{81}</span>equilibrium. When the gyrostat is not + spinning, the ship vibrates slowly when put out of equilibrium; when the + gyrostat is spinning the ship gets a motion of precession which is + opposite in direction to that of the spinning. Astronomers, beginning + with Hipparchus, have made observations of the earth's motion for us, and + we have observed the motions of gyrostats, and we naturally seek for an + explanation of the precessional motion of the earth. The equator of the + earth makes an angle of 23½° with the ecliptic, which is the plane of the + earth's orbit. Or the spinning axis of the earth is always at angle of + 23½° with a perpendicular to the ecliptic, and makes a complete + revolution in 26,000 years. The surface of the water on which this wooden + ship is floating represents the ecliptic. The axis <!-- Page 82 --><span + class="pagenum"><a name="page82"></a>{82}</span>of spinning of the + gyrostat is about 23½° to the vertical; the precession is in two minutes + instead of 26,000 years; and only that this ship does not revolve in a + great circular path, we should have in its precession a pretty exact + illustration of the earth's precession.</p> + + <p>The precessional motion of the ship, or of the gyrostat (Fig. 22), is + explainable, and in the same way the earth's precession is at once + explained if we find that there are forces from external bodies tending + to put its spinning axis at right angles to the ecliptic. The earth is a + nearly spherical body. If it were exactly spherical and homogeneous, the + resultant force of attraction upon it, of a distant body, would be in a + line through its centre. And again, if it were spherical and + non-homogeneous, but if its mass were arranged in uniformly dense, + spherical layers, like the coats of an onion. But the earth is not + spherical, and to find what is the nature of the attraction of a distant + body, it has been necessary to make pendulum observations all over the + earth. You know that if a pendulum does not alter in length as we take it + about to various places, its time of vibration at each place enables the + force of gravity at each place to be determined; and Mr. Green proved + that if we know the force of gravity at all places on the surface of the + earth, although we may know nothing about the <!-- Page 83 --><span + class="pagenum"><a name="page83"></a>{83}</span>state of the inside of + the earth, we can calculate with absolute accuracy the force exerted by + the earth on matter placed anywhere outside the earth; for instance, at + any part of the moon's orbit, or at the sun. And hence we know the equal + and opposite force with which such matter will act on the earth. Now + pendulum observations have been made at a great many places on the earth, + and we know, although of course not with absolute accuracy, the + attraction on the earth, of matter outside the earth. For instance, we + know that the resultant attraction of the sun on the earth is a force + which does not pass through the centre of the earth's mass. You may + comprehend the result better if I refer to this diagram of the earth at + midwinter (Fig. 39), and use a popular method of description. A and B may + roughly be called the protuberant parts of the earth—that + protuberant belt of matter which makes the <!-- Page 84 --><span + class="pagenum"><a name="page84"></a>{84}</span>earth orange-shaped + instead of spherical. On the spherical portion inside, assumed roughly to + be homogeneous, the resultant attraction is a force through the + centre.</p> + + <div class="figcenter" style="width:67%;"> + <a href="images/fig39.jpg"><img style="width:100%" src="images/fig39.jpg" + alt="Fig. 39" title="Fig. 39" /></a> + <span class="sc">Fig. 39.</span> + </div> + + <p>I will now consider the attraction on the protuberant equatorial belt + indicated by A and B. The sun attracts a pound of matter at B more than + it attracts a pound of matter at A, because B is nearer than A, and hence + the total resultant force is in the direction M N rather than O O, + through the centre of the earth's mass. But we know that a force in the + direction M N is equivalent to a force O O parallel to M N, together with + a tilting couple of forces tending to turn the equator edge on to the + sun. You will get the true result as to the tilting tendency by imagining + the earth to be motionless, and the sun's mass to be distributed as a + circular ring of matter 184 millions of miles in diameter, inclined to + the equator at 23½°. Under the influence of the attraction of this ring + the earth would heave like a great ship on a calm sea, rolling very + slowly; in fact, making one complete swing in about three years. But the + earth is spinning, and the tilting couple or torque acts upon it just + like the forces which are always tending to cause this ship-model to + stand upright, and hence it has a precessional motion whose complete + period is 26,000 years. When there is no spin in the ship, its complete + oscillation takes place in three seconds, and <!-- Page 85 --><span + class="pagenum"><a name="page85"></a>{85}</span>when I spin the gyrostat + on board the ship, the complete period of its precession is two minutes. + In both cases the effect of the spin is to convert what would be an + oscillation into a very much slower precession.</p> + + <p>There is, however, a great difference between the earth and the + gyrostat. The forces acting on the top are always the same, but the + forces acting on the earth are continually altering. At midwinter and + midsummer the tilting forces are greatest, and at the equinoxes in spring + and autumn there are no such forces. So that the precessional motion + changes its rate every quarter year from a maximum to nothing, or from + nothing to a maximum. It is, however, always in the same + direction—the direction opposed to the earth's spin. When we speak + then of the precessional motion of the earth, we usually think of the + mean or average motion, since the motion gets quicker and slower every + quarter year.</p> + + <p>Further, the moon is like the sun in its action. It tries to tilt the + equatorial part of the earth into the plane of the moon's orbit. The + plane of the moon's orbit is nearly the same as that of the ecliptic, and + hence the average precession of the earth is of much the same kind as if + only one of the two, the moon or the sun, alone acted. That is, the + general phenomenon of precession of the <!-- Page 86 --><span + class="pagenum"><a name="page86"></a>{86}</span>earth's axis in a conical + path in 26,000 years is the effect of the combined tilting actions of the + sun and moon.</p> + + <p>You will observe here an instance of the sort of untruth which it is + almost imperative to tell in explaining natural phenomena. Hitherto I had + spoken only of the sun as producing precession of the earth. This was + convenient, because the plane of the ecliptic makes always almost exactly + 23½° with the earth's equator, and although on the whole the moon's + action is nearly identical with that of the sun, and about twice as + great, yet it varies considerably. The superior tilting action of the + moon, just like its tide-producing action, is due to its being so much + nearer us than the sun, and exists in spite of the very small mass of the + moon as compared with that of the sun.</p> + + <p>As the ecliptic makes an angle of 23½° with the earth's equator, and + the moon's orbit makes an angle 5½° with the ecliptic, we see that the + moon's orbit sometimes makes an angle of 29° with the earth's equator, + and sometimes only 18°, changing from 29° to 18°, and back to 29° again + in about nineteen years. This causes what is called "Nutation," or the + nodding of the earth, for the tilting action due to the sun is greatly + helped and greatly modified by it. The result of the variable nature of + the moon's action is then that the earth's axis <!-- Page 87 --><span + class="pagenum"><a name="page87"></a>{87}</span>rotates in an elliptic + conical path round what might be called its mean position. We have also + to remember that twice in every lunar month the moon's tilting action on + the earth is greater, and twice it is zero, and that it continually + varies in value.</p> + + <p>On the whole, then, the moon and sun, and to a small extent the + planets, produce the general effect of a precession, which repeats itself + in a period of about 25,695 years. It is not perfectly uniform, being + performed at a speed which is a maximum in summer and winter; that is, + there is a change of speed whose period is half a year; and there is a + change of speed whose period is half a lunar month, the precession being + quicker to-night than it will be next Saturday, when it will increase for + about another week, and diminish the next. Besides this, because of 5½° + of angularity of the orbits, we have something like the nodding of our + precessing gyrostat, and the inclination of the earth's axis to the + ecliptic is not constant at 23½°, but is changing, its periodic time + being nineteen years. Regarding the earth's centre as fixed at O we see + then, as illustrated in this model and in Fig. 40, the axis of the earth + describes almost a perfect circle on the celestial sphere once in 25,866 + years, its speed fluctuating every half year and every half month. But it + is not a perfect circle, it is really a wavy <!-- Page 88 --><span + class="pagenum"><a name="page88"></a>{88}</span>line, there being a + complete wave every nineteen years, and there are smaller ripples in it, + corresponding to the half-yearly and fortnightly periods. But the very + cause of the nutation, the nineteen-yearly period of retrogression of the + moon's nodes, as it is called, is itself really produced as the + precession of a gyrostat is produced, that is, by tilting forces acting + on a spinning body.</p> + + <div class="figcenter" style="width:34%;"> + <a href="images/fig40.jpg"><img style="width:100%" src="images/fig40.jpg" + alt="Fig. 40" title="Fig. 40" /></a> + <span class="sc">Fig. 40.</span> + </div> + + <p>Imagine the earth to be stationary, and the sun and moon revolving + round it. It was Gauss who found that the present action is the same as + if the masses of the moon and sun were distributed all <!-- Page 89 + --><span class="pagenum"><a name="page89"></a>{89}</span>round their + orbits. For instance, imagine the moon's mass distributed over her orbit + in the form of a rigid ring of 480,000 miles diameter, and imagine less + of it to exist where the present speed is greater, so that the ring would + be thicker at the moon's apogee, and thinner at the perigee. Such a ring + round the earth would be similar to Saturn's rings, which have also a + precession of nodes, only Saturn's rings are not rigid, else there would + be no equilibrium. Now if we leave out of account the earth and imagine + this ring to exist by itself, and that its centre simply had a motion + round the sun in a year, since it makes an angle of 5½° with the ecliptic + it would vibrate into the ecliptic till it made the same angle on the + other side and back again. But it revolves once about its centre in + twenty-seven solar days, eight hours, and it will no longer swing like a + ship in a ground-swell, but will get a motion of precession opposed in + direction to its own revolution. As the ring's motion is against the + hands of a watch, looking from the north down on the ecliptic, this + retrogression of the moon's nodes is in the direction of the hands of a + watch. It is exactly the same sort of phenomenon as the precession of the + equinoxes, only with a much shorter period of 6798 days instead of 25,866 + years.</p> + + <p>I told you how, if we knew the moon's mass or the sun's, we could tell + the amount of the forces, or <!-- Page 90 --><span class="pagenum"><a + name="page90"></a>{90}</span>the torque as it is more properly called, + with which it tries to tilt the earth. We know the rate at which the + earth is spinning, and we have observed the precessional motion. Now when + we follow up the method which I have sketched already, we find that the + precessional velocity of a spinning body ought to be equal to the torque + divided by the spinning velocity and by the moment of inertia<a + name="NtA7" href="#Nt7"><sup>[7]</sup></a> of the body about the polar + axis. Hence the greater the tilting forces, and the less the spin and the + less the moment of inertia, the greater is the precessional speed. Given + all of these elements except one, it is easy to calculate that unknown + element. Usually what we aim at in such a calculation is the + determination of the moon's mass, as this phenomenon of precession and + the action of the tides are the only two natural phenomena which have as + yet enabled the moon's mass to be calculated.</p> + + <p>I do not mean to apologize to you for the introduction of such terms + as <i>Moment of Inertia</i>, nor do I mean to explain them. In this + lecture I have avoided, as much as I could, the introduction of + mathematical expressions and the use of technical terms. But I want you + to <!-- Page 91 --><span class="pagenum"><a + name="page91"></a>{91}</span>understand that I am not afraid to introduce + technical terms when giving a popular lecture. If there is any offence in + such a practice, it must, in my opinion, be greatly aggravated by the + addition of explanations of the precise meanings of such terms. The use + of a correct technical term serves several useful purposes. First, it + gives some satisfaction to the lecturer, as it enables him to state, very + concisely, something which satisfies his own weak inclination to have his + reasoning complete, but which he luckily has not time to trouble his + audience with. Second, it corrects the universal belief of all popular + audiences that they know everything now that can be said on the subject. + Third, it teaches everybody, including the lecturer, that there is + nothing lost and often a great deal gained by the adoption of a casual + method of skipping when one is working up a new subject.</p> + + <p>Some years ago it was argued that if the earth were a shell filled + with liquid, if this liquid were quite frictionless, then the moment of + inertia of the shell is all <span class="correction" title="Original reads 'that that'." + >that</span> we should have to take into account in considering + precession, and that if it were viscous the precession would very soon + disappear altogether. To illustrate the effect of the moment of inertia, + I have hung up here a number of glasses—one <i>a</i> filled with + sand, another <i>b</i> with treacle, a third <i>c</i> with oil, the + fourth <i>d</i> with water, <!-- Page 92 --><span class="pagenum"><a + name="page92"></a>{92}</span></p> + + <div class="figcenter" style="width:69%;"> + <a href="images/fig41.jpg"><img style="width:100%" src="images/fig41.jpg" + alt="Fig. 41" title="Fig. 41" /></a> + <span class="sc">Fig. 41.</span> + </div> + +<p><!-- Page 93 --><span class="pagenum"><a name="page93"></a>{93}</span></p> + + <p>and the fifth <i>e</i> is empty (Fig. 41). You see that if I twist + these suspending wires and release them, a vibratory motion is set up, + just like that of the balance of a watch. Observe that the glass with + water vibrates quickly, its effective moment of inertia being merely that + of the glass itself, and you see that the time of swing is pretty much + the same as that of the empty glass; that is, the water does not seem to + move with the glass. Observe that the vibration goes on for a fairly long + time.</p> + + <p>The glass with sand vibrates slowly; here there is great moment of + inertia, as the sand and glass behave like one rigid body, and again the + vibration goes on for a long time.</p> + + <p>In the oil and treacle, however, there are longer periods of vibration + than in the case of the water or empty glass, and less than would be the + case if the vibrating bodies were all rigid, but the vibrations are + stilled more rapidly because of friction.</p> + + <p>Boiled (<i>f</i>) and unboiled (<i>g</i>) eggs suspended from wires in + the same way will exhibit the same differences in the behaviour of + bodies, one of which is rigid and the other liquid inside; you see how + much slower an oscillation the boiled has than the unboiled.</p> + + <p>Even on the table here it is easy to show the difference between + boiled and unboiled eggs. <!-- Page 94 --><span class="pagenum"><a + name="page94"></a>{94}</span>Roll them both; you see that one of them + stops much sooner than the other; it is the unboiled one that stops + sooner, because of its internal friction.</p> + + <p>I must ask you to observe carefully the following very distinctive + test of whether an egg is boiled or not. I roll the egg or spin it, and + then place my finger on it just for an instant; long enough to stop the + motion of the shell. You see that the boiled egg had quite finished its + motion, but the unboiled egg's shell alone was stopped; the liquid inside + goes on moving, and now renews the motion of the shell when I take my + finger away.</p> + + <p>It was argued that if the earth were fluid inside, the effective + moment of inertia of the shell being comparatively small, and having, as + we see in these examples, nothing whatever to do with the moment of + inertia of the liquid, the precessional motion of the earth ought to be + enormously quicker than it is. This was used as an argument against the + idea of the earth's being fluid inside.</p> + + <p>We know that the observed half-yearly and half-monthly changes of the + precession of the earth would be much greater than they are if the earth + were a rigid shell containing much liquid, and if the shell were not + nearly infinitely rigid the phenomena of the tides would not occur, but + in regard to the general precession of the earth there is now <!-- Page + 95 --><span class="pagenum"><a name="page95"></a>{95}</span>no doubt that + the old line of argument was wrong. Even if the earth were liquid inside, + it spins so rapidly that it would behave like a rigid body in regard to + such a slow phenomenon as precession of the equinoxes. In fact, in the + older line of argument the important fact was lost sight of, that rapid + rotation can give to even liquids a quasi-rigidity. Now here (Fig. 42 + <i>a</i>) is a hollow brass top filled with water. The frame is light, + and the water inside has much more mass than the outside frame, and if + you test this carefully you will find that the top spins in almost + exactly the same way as if the water were quite rigid; in fact, as if the + whole top were rigid. Here you see it spinning and precessing just like + any rigid top. This top, I know, is not filled with water, it is only + partially filled; but whether partially or wholly filled it spins very + much like a rigid top.</p> + + <div class="figcenter" style="width:38%;"> + <a href="images/fig42.jpg"><img style="width:100%" src="images/fig42.jpg" + alt="Fig. 42" title="Fig. 42" /></a> + <span class="sc">Fig. 42.</span> + </div> + +<p><!-- Page 96 --><span class="pagenum"><a name="page96"></a>{96}</span></p> + + <p>This is not the case with a long hollow brass top with water inside. I + told you that all bodies have one axis about which they prefer to rotate. + The outside metal part of a top behaves in a way that is now well known + to you; the friction of its peg on the table compels it to get up on its + longer axis. But the fluid inside a top is not constrained to spin on its + longer axis of figure, and as it prefers its shorter axis like all these + bodies I showed you, it spins in its own way, and by friction and + pressure against the case constrains the case to spin about the shorter + axis, annulling completely the tendency of the outside part to rise or + keep up on its long axis. Hence it is found to be simply impossible to + spin a long hollow top when filled with water.</p> + + <div class="figright" style="width:36%;"> + <a href="images/fig44.jpg"><img style="width:100%" src="images/fig44.jpg" + alt="Fig. 44" title="Fig. 44" /></a> + <span class="sc">Fig. 44.</span> + </div> + + <div class="figright" style="width:38%;"> + <a href="images/fig43.jpg"><img style="width:100%" src="images/fig43.jpg" + alt="Fig. 43" title="Fig. 43" /></a> + <span class="sc">Fig. 43.</span> + </div> + + <p>Here, for example, is one (Fig. 42 <i>b</i>) that only differs from + the last in being longer. It is filled, or partially filled, with water, + and you observe that if <!-- Page 97 --><span class="pagenum"><a + name="page97"></a>{97}</span>I slowly get up a great spin when it is + mounted in this frame, and I let it out on the table as I did the other + one, this one lies down at once and refuses to spin on its peg. This + difference of behaviour is most remarkable in the two hollow tops you see + before you (Fig. 43). They are both nearly spherical, both filled with + water. They look so nearly alike that few persons among the audience are + able to detect any difference in their shape. But one of them (<i>a</i>) + is really slightly oblate like an orange, and the other (<i>b</i>) is + slightly prolate like a lemon. I will give them both a gradually + increasing rotation in this frame <!-- Page 98 --><span + class="pagenum"><a name="page98"></a>{98}</span>(Fig. 44) for a time + sufficient to insure the rotation of the water inside. When just about to + be set free to move like ordinary tops on the table, water and brass are + moving like the parts of a rigid top. You see that the orange-shaped one + continues to spin and precess, and gets itself upright when disturbed, + like an ordinary rigid top; indeed I have seldom seen a better behaved + top; whereas the lemon-shaped one lies down on its side at once, and + quickly ceases to move in any way.</p> + + <div class="figcenter" style="width:31%;"> + <a href="images/fig45.jpg"><img style="width:100%" src="images/fig45.jpg" + alt="Fig. 45" title="Fig. 45" /></a> + <span class="sc">Fig. 45.</span> + </div> + + <p>And now you will be able to appreciate a fourth test of a boiled egg, + which is much more easily seen by a large audience than the last. Here is + the unboiled one (Fig. 45 <i>b</i>). I try my best to spin it as it lies + on the table, but you see that I cannot give it much spin, and so there + is nothing of any importance to look at. But you observe that it is quite + easy to spin the boiled <!-- Page 99 --><span class="pagenum"><a + name="page99"></a>{99}</span>egg, and that for reasons now well known to + you it behaves like the stones that Thomson spun on the sea-beach; it + gets up on its longer axis, a very pretty object for our educated eyes to + look at (Fig. 45 <i>a</i>). You are all aware, from the behaviour of the + lemon-shaped top, that even if, by the use of a whirling table suddenly + stopped, or by any other contrivance, I could get up a spin in this + unboiled egg, it would never make the slightest effort to rise on its end + and spin about its longer axis.</p> + + <p>I hope you don't think that I have been speaking too long about + astronomical matters, for there is one other important thing connected + with astronomy that I must speak of. You see, I have had almost nothing + practically to do with astronomy, and hence I have a strong interest in + the subject. It is very curious, but quite true, that men practically + engaged in any pursuit are almost unable to see the romance of it. This + is what the imaginative outsider sees. But the overworked astronomer has + a different point of view. As soon as it becomes one's duty to do a + thing, and it is part of one's every-day work, the thing loses a great + deal of its interest. We have been told by a great American philosopher + that the only coachmen who ever saw the romance of coach-driving are + those titled individuals who pay nowadays so largely for the <!-- Page + 100 --><span class="pagenum"><a name="page100"></a>{100}</span>privilege. + In almost any branch of engineering you will find that if any invention + is made it is made by an outsider; by some one who comes to the study of + the subject with a fresh mind. Who ever heard of an old inhabitant of + Japan or Peru writing an interesting book about those countries? At the + end of two years' residence he sees only the most familiar things when he + takes his walks abroad, and he feels unmitigated contempt for the + ingenuous globe-trotter who writes a book about the country after a + month's travel over the most beaten tracks in it. Now the experienced + astronomer has forgotten the difficulties of his predecessors and the + doubts of outsiders. It is a long time since he felt that awe in gazing + at a starry sky that we outsiders feel when we learn of the sizes and + distances apart of the hosts of heaven. He speaks quite coolly of + millions of years, and is nearly as callous when he refers to the ancient + history of humanity on our planet as a weather-beaten geologist. The + reason is obvious. Most of you know that the <i>Nautical Almanac</i> is + as a literary production one of the most uninteresting works of reference + in existence. It is even more disconnected than a dictionary, and I + should think that preparing census-tables must be ever so much more + romantic as an occupation than preparing the tables of the <i>Nautical + Almanac</i>. And yet <!-- Page 101 --><span class="pagenum"><a + name="page101"></a>{101}</span>a particular figure, one of millions set + down by an overworked calculator, may have all the tragic importance of + life or death to the crew and passengers of a ship, when it is heading + for safety or heading for the rocks under the mandate of that single + printed character.</p> + + <p>But this may not be a fair sort of criticism. I so seldom deal with + astronomical matters, I know so little of the wear and tear and monotony + of the every-day life of the astronomer, that I do not even know that the + above facts are specially true about astronomers. I only know that they + are very likely to be true because they are true of other professional + men.</p> + + <p>I am happy to say that I come in contact with all sorts and conditions + of men, and among others, with some men who deny many of the things + taught in our earliest school-books. For example, that the earth is + round, or that the earth revolves, or that Frenchmen speak a language + different from ours. Now no man who has been to sea will deny the + roundness of the earth, however greatly he may wonder at it; and no man + who has been to France will deny that the French language is different + from ours; but many men who learnt about the rotation of the earth in + their school-days, and have had a plentiful opportunity of observing the + heavenly bodies, deny the rotation of the earth. <!-- Page 102 --><span + class="pagenum"><a name="page102"></a>{102}</span>They tell you that the + stars and moon are revolving about the earth, for they see them revolving + night after night, and the sun revolves about the earth, for they see it + do so every day. And really if you think of it, it is not so easy to + prove the revolution of the earth. By the help of good telescopes and the + electric telegraph or good chronometers, it is easy to show from the want + of parallax in stars that they must be very far away; but after all, we + only know that either the earth revolves or else the sky revolves.<a + name="NtA8" href="#Nt8"><sup>[8]</sup></a> Of course, it seems infinitely + more likely that the small earth should revolve than that the whole + heavenly host should turn about the earth as a centre, and infinite + likelihood is really absolute proof. Yet there is nobody who does not + welcome an independent kind of proof. The phenomena of the tides, and + nearly every new astronomical fact, may be said to be an addition to the + proof. Still there is the absence of perfect certainty, and when we are + told that these spinning-top phenomena give us a real proof of the + rotation of the earth without our leaving the room, we welcome <!-- Page + 103 --><span class="pagenum"><a name="page103"></a>{103}</span>it, even + although we may sneer at it as unnecessary after we have obtained it.</p> + + <div class="figcenter" style="width:29%;"> + <a href="images/fig17.jpg"><img style="width:100%" src="images/fig17.jpg" + alt="Fig. 17" title="Fig. 17" /></a> + <span class="sc">Fig. 17.</span> + </div> + + <p>You know that a gyrostat suspended with perfect freedom about axes, + which all pass through its centre of gravity, maintains a constant + direction in space however its support may be carried. Its axis is not + forced to alter its direction in any way. Now this gyrostat (Fig. 17) has + not the perfect absence of friction at its axes of which I speak, and + even the slightest friction will produce some constraint which is + injurious to the experiment I am about to describe. It must be + remembered, that if there were absolutely no constraint, then, even if + the <!-- Page 104 --><span class="pagenum"><a + name="page104"></a>{104}</span>gyrostat were <i>not</i> spinning, its + axis would keep a constant direction in space. But the spinning gyrostat + shows its superiority in this, that any constraint due to friction is + less powerful in altering the axis. The greater the spin, then, the + better able are we to disregard effects due to friction. You have seen + for yourselves the effect of carrying this gyrostat about in all sorts of + ways—first, when it is not spinning and friction causes quite a + large departure from constancy of direction of the axis; second, when it + is spinning, and you see that although there is now the same friction as + before, and I try to disturb the instrument more than before, the axis + remains sensibly parallel to itself all the time. Now when this + instrument is supported by the table it is really being carried round by + the earth in its daily rotation. If the axis kept its direction + perfectly, and it were now pointing horizontally due east, six hours + after this it will point towards the north, but inclining downwards, six + hours afterwards it will point due west horizontally, and after one + revolution of the earth it will again point as it does now. Suppose I try + the experiment, and I see that it points due east now in this room, and + after a time it points due west, and yet I know that the gyrostat is + constantly pointing in the same direction in space all the time, surely + it is obvious that the room must <!-- Page 105 --><span + class="pagenum"><a name="page105"></a>{105}</span>be turning round in + space. Suppose it points to the pole star now, in six hours, or twelve, + or eighteen, or twenty-four, it will still point to the pole star.</p> + + <p>Now it is not easy to obtain so frictionless a gyrostat that it will + maintain a good spin for such a length of time as will enable the + rotation of the room to be made visible to an audience. But I will + describe to you how forty years ago it was proved in a laboratory that + the earth turns on its axis. This experiment is usually connected with + the name of Foucault, the same philosopher who with Fizeau showed how in + a laboratory we can measure the velocity of light, and therefore measure + the distance of the sun. It was suggested by Mr. Lang of Edinburgh in + 1836, although only carried out in 1852 by Foucault. By these + experiments, if you were placed on a body from which you could see no + stars or other outside objects, say that you were living in underground + regions, you could discover—first, whether there is a motion of + rotation, and the amount of it; second, the meridian line or the + direction of the true north; third, your latitude. Obtain a gyrostat like + this (Fig. 46) but much larger, and far more frictionlessly suspended, so + that it is free to move vertically or horizontally. For the vertical + motion your gymbal pivots ought to be hard steel knife-edges. <!-- Page + 106 --><span class="pagenum"><a name="page106"></a>{106}</span></p> + + <div class="figcenter" style="width:43%;"> + <a href="images/fig46.jpg"><img style="width:100%" src="images/fig46.jpg" + alt="Fig. 46" title="Fig. 46" /></a> + <span class="sc">Fig. 46.</span> + </div> + + <p>As for the horizontal freedom, Foucault used a fine steel wire. Let + there be a fine scale engraved crosswise on the outer gymbal ring, and + try to discover if it moves horizontally by means of a microscope with + cross wires. When this is carefully done we find that there is a motion, + <!-- Page 107 --><span class="pagenum"><a + name="page107"></a>{107}</span>but this is not the motion of the + gyrostat, it is the motion of the microscope. In fact, the microscope and + all other objects in the room are going round the gyrostat frame.</p> + + <p>Now let us consider what occurs. The room is rotating about the + earth's axis, and we know the rate of rotation; but we only want to know + for our present purpose how much of the total rotation is about a + vertical line in the room. If the room were at the North Pole, the whole + rotation would be about the vertical line. If the room were at the + equator, none of its rotation would be about a vertical line. In our + latitude now, the horizontal rate of rotation about a vertical axis is + about four-fifths of the whole rate of rotation of the earth on its axis, + and this is the amount that would be measured by our microscope. This + experiment would give no result at a place on the equator, but in our + latitude you would have a laboratory proof of the rotation of the earth. + Foucault made the measurements with great accuracy.</p> + + <p>If you now clamp the frame, and allow the spinning axis to have no + motion except in a horizontal plane, the motion which the earth tends to + give it about a vertical axis cannot now affect the gyrostat, but the + earth constrains it to move about an axis due north and south, and + consequently the spinning axis tries to put itself parallel <!-- Page 108 + --><span class="pagenum"><a name="page108"></a>{108}</span>to the north + and south direction (Fig. 47). Hence with such an instrument it is easy + to find the true north. If there were absolutely no friction the + instrument would vibrate about the true north position like the compass + needle (Fig. 50), although with an exceedingly slow swing.</p> + + <div class="figcenter" style="width:34%;"> + <a href="images/fig47.jpg"><img style="width:100%" src="images/fig47.jpg" + alt="Fig. 47" title="Fig. 47" /></a> + <span class="sc">Fig. 47.</span> + </div> + + <p>It is with a curious mixture of feelings that one first recognizes the + fact that all rotating bodies, fly-wheels of steam-engines and the like, + are always tending to turn themselves towards the pole star; gently and + vainly tugging at their foundations <!-- Page 109 --><span + class="pagenum"><a name="page109"></a>{109}</span>to get round towards + the object of their adoration all the time they are in motion.</p> + + <div class="figcenter" style="width:30%;"> + <a href="images/fig48.jpg"><img style="width:100%" src="images/fig48.jpg" + alt="Fig. 48" title="Fig. 48" /></a> + <span class="sc">Fig. 48.</span> + </div> + + <p>Now we have found the meridian as in Fig. 47, we can begin a third + experiment. Prevent motion horizontally, that is, about a vertical axis, + but give the instrument freedom to move vertically in the meridian, like + a transit instrument in an observatory <!-- Page 110 --><span + class="pagenum"><a name="page110"></a>{110}</span>about its horizontal + axis. Its revolution with the earth will tend to make it change its + angular position, and therefore it places itself parallel to the earth's + axis; when in this position the daily rotation no longer causes any + change in its direction in space, so it continues to point to the pole + star (Fig. 48). It would be an interesting experiment to measure with a + delicate chemical balance the force with which the axis raises itself, + and in this way <i>weigh</i> the rotational motion of the earth.<a + name="NtA9" href="#Nt9"><sup>[9]</sup></a></p> + + <p>Now let us turn the frame of the instrument G B round a right angle, + so that the spinning axis can only move in a plane at right angles to the + meridian; obviously it is constrained by the vertical component of the + earth's rotation, and points vertically downwards.</p> + + <div class="figright" style="width:25%;"> + <a href="images/fig50.jpg"><img style="width:100%" src="images/fig50.jpg" + alt="Fig. 50" title="Fig. 50" /></a> + <span class="sc">Fig. 50.</span> + </div> + + <div class="figright" style="width:18%;"> + <a href="images/fig49.jpg"><img style="width:100%" src="images/fig49.jpg" + alt="Fig. 49" title="Fig. 49" /></a> + <span class="sc">Fig. 49.</span> + </div> + + <p>This last as well as the other phenomena of which I have spoken is + very suggestive. Here is a magnetic needle (Fig. 49), sometimes called a + dipping needle from the way in which it is suspended. If I turn its <!-- + Page 111 --><span class="pagenum"><a name="page111"></a>{111}</span>frame + so that it can only move at right angles to the meridian, you see that it + points vertically. You may reflect upon the analogous properties of this + magnetic needle (Fig. 50) and of the gyrostat (Fig. 47); they both, when + only capable of moving horizontally, point to the north; and you see that + a very frictionless gyrostat might be used as a compass, or at all events + as a corrector of compasses.<a name="NtA10" + href="#Nt10"><sup>[10]</sup></a> I have just put before you another + analogy, and I want you to understand that, although these are only + analogies, they are not mere chance analogies, for there is undoubtedly a + dynamical connection between the magnetic and the gyrostatic phenomena. + Magnetism depends on rotatory motion. The molecules of matter are in + actual rotation, and a certain allineation of the axes of the rotations + produces what we call magnetism. In a steel bar not magnetized the little + axes of rotation are all in different directions. The process <!-- Page + 112 --><span class="pagenum"><a name="page112"></a>{112}</span>of + magnetization is simply bringing these rotations to be more or less round + parallel axes, an allineation of the axes. A honey-combed mass with a + spinning gyrostat in every cell, with all the spinning axes parallel, and + the spins in the same direction, would—I was about to say, would be + a magnet, but it would not be a magnet in all its properties, and yet it + would resemble a magnet in many ways.<a name="NtA11" + href="#Nt11"><sup>[11]</sup></a></p> + + <div class="figleft" style="width:21%;"> + <a href="images/fig51.jpg"><img style="width:100%" src="images/fig51.jpg" + alt="Fig. 51" title="Fig. 51" /></a> + <span class="sc">Fig. 51.</span> + </div> + + <div class="figleft" style="width:49%;"> + <a href="images/fig52.jpg"><img style="width:100%" src="images/fig52.jpg" + alt="Fig. 52" title="Fig. 52" /></a> + <span class="sc">Fig. 52.</span> + </div> + +<div style="clear: both"></div> + <p>Some of you, seeing electromotors and other electric contrivances near + this table, may think that they have to do with our theories and + explanations of magnetic phenomena. But I must explain that this + electromotor which I hold in my hand (Fig. 51) is used by me merely as + the <!-- Page 113 --><span class="pagenum"><a + name="page113"></a>{113}</span>most convenient means I could find for the + spinning of my tops and gyrostats. On the spindle of the motor is + fastened a circular piece of wood; by touching this key I can supply the + motor with electric energy, and the wooden disc is now rotating very + rapidly. I have only to bring its rim in contact with any of these tops + or gyrostats to set them spinning, and you see that I can set half a + dozen gyrostats a-spinning in a few seconds; this chain of gyrostats, for + instance. Again, this larger motor (Fig. 52), too large to move about in + my hand, is fastened to the table, and I have used <!-- Page 114 --><span + class="pagenum"><a name="page114"></a>{114}</span>it to drive my larger + contrivances; but you understand that I use these just as a barber might + use them to brush your hair, or Sarah Jane to clean the knives, or just + as I would use a little steam-engine if it were more convenient for my + purpose. It was more convenient for me to bring from London this battery + of accumulators and these motors than to bring sacks of coals, and + boilers, and steam-engines. But, indeed, all this has the deeper meaning + that we can give to it if we like. Love is as old as the hills, and every + day Love's messages are carried by the latest servant of man, the + telegraph. These spinning tops were known probably to primeval man, and + yet we have not learnt from them more than the most fractional portion of + the lesson that they are always sending out to an unobservant world. Toys + like these were spun probably by the builders of the Pyramids when they + were boys, and here you see them side by side with the very latest of + man's contrivances. I feel almost as Mr. Stanley might feel if, with the + help of the electric light and a magic-lantern, he described his + experiences in that dreadful African forest to the usual company of a + London drawing-room.</p> + + <p>The phenomena I have been describing to you play such a very important + part in nature, that if time admitted I might go on expounding and <!-- + Page 115 --><span class="pagenum"><a + name="page115"></a>{115}</span>explaining without finding any great + reason to stop at one place rather than another. The time at my disposal + allows me to refer to only one other matter, namely, the connection + between light and magnetism and the behaviour of spinning tops.</p> + + <p>You are all aware that sound takes time to travel. This is a matter of + common observation, as one can see a distant woodchopper lift his axe + again before one hears the sound of his last stroke. A destructive sea + wave is produced on the coast of Japan many hours after an earthquake + occurs off the coast of America, the wave motion having taken time to + travel across the Pacific. But although light travels more quickly than + sound or wave motion in the sea, it does not travel with infinite + rapidity, and the appearance of the eclipse of one of Jupiter's + satellites is delayed by an observable number of minutes because light + takes time to travel. The velocity has been measured by means of such + observations, and we know that light travels at the rate of about 187,000 + miles per second, or thirty thousand millions of centimetres per second. + There is no doubt about this figure being nearly correct, for the + velocity of light has been measured in the laboratory by a perfectly + independent method.</p> + + <p>Now the most interesting physical work done since Newton's time is the + outcome of the experiments of Faraday and the theoretical deductions of + <!-- Page 116 --><span class="pagenum"><a + name="page116"></a>{116}</span>Thomson and Maxwell. It is the theory that + light and radiant heat are simply electro-magnetic disturbances + propagated through space. I dare not do more than just refer to this + matter, although it is of enormous importance. I can only say, that of + all the observed facts in the sciences of light, electricity, and + magnetism, we know of none that is in opposition to Maxwell's theory, and + we know of many that support it. The greatest and earliest support that + it had was this. If the theory is correct, then a certain + electro-magnetic measurement ought to result in exactly the same quantity + as the velocity of light. Now I want you to understand that the electric + measurement is one of quantities that seem to have nothing whatever to do + with light, except that one uses one's eyes in making the measurement; it + requires the use of a two-foot rule and a magnetic needle, and coils of + wire and currents of electricity. It seemed to bear a relationship to the + velocity of light, which was not very unlike the fabled connection + between Tenterden Steeple and the Goodwin Sands. It is a measurement + which it is very difficult to make accurately. A number of skilful + experimenters, working independently, and using quite different methods, + arrived at results only one of which is as much as five per cent. + different from the observed velocity of light, and some of them, <!-- + Page 117 --><span class="pagenum"><a name="page117"></a>{117}</span>on + which the best dependence may be placed, agree exactly with the average + value of the measurements of the velocity of light.</p> + + <p>There is then a wonderful agreement of the two measurements, but + without more explanation than I can give you now, you cannot perhaps + understand the importance of this agreement between two seemingly + unconnected magnitudes. At all events we now know, from the work of + Professor Hertz in the last two years, that Maxwell's theory is correct, + and that light is an electro-magnetic disturbance; and what is more, we + know that electro-magnetic disturbances, incomparably slower than + red-light or heat, are passing now through our bodies; that this now + recognized kind of radiation may be reflected and refracted, and yet will + pass through brick and stone walls and foggy atmospheres where light + cannot pass, and that possibly all military and marine and lighthouse + signalling may be conducted in the future through the agency of this new + and wonderful kind of radiation, of which what we call light is merely + one form. Why at this moment, for all I know, two citizens of Leeds may + be signalling to each other in this way through half a mile of houses, + including this hall in which we are present.<a name="NtA12" + href="#Nt12"><sup>[12]</sup></a></p> + +<p><!-- Page 118 --><span class="pagenum"><a name="page118"></a>{118}</span></p> + + <p>I mention this, the greatest modern philosophical discovery, because + the germ of it, which was published by Thomson in 1856, makes direct + reference to the analogy between the behaviour of our spinning-tops and + magnetic and electrical phenomena. It will be easier, however, for us to + consider here a mechanical illustration of the rotation of the plane of + polarized light by magnetism which Thomson elaborated in 1874. This + phenomenon may, I think, be regarded as the most important of all + Faraday's discoveries. It was of enormous scientific importance, because + it was made in a direction where a new phenomenon was not even suspected. + Of his discovery of induced currents of electricity, to which all + electric-lighting companies and transmission of power companies of the + present day owe their being, Faraday himself said that it was a natural + consequence of the discoveries of an earlier experimenter, Oersted. But + this magneto-optic discovery was quite unexpected. I will now describe + the phenomenon.</p> + + <p>Some of you are aware that when a beam of light is sent through this + implement, called a Nichol's Prism, it becomes polarized, or + one-sided—that is, all the light that comes through is known to be + propagated by vibrations which occur all in one plane. This rope (Fig. + 53) hanging from the ceiling <!-- Page 119 --><span class="pagenum"><a + name="page119"></a>{119}</span>illustrates the nature of plane polarized + light. All points in the rope are vibrating in the same plane. Well, this + prism A, Fig. 54, only lets through it light that is polarized in a + vertical plane. And here at B I have a similar implement, and I place it + so that it also will only allow light to pass through it which is + polarized in a vertical plane. Hence most of the light coming through the + polarizer, as the first prism is called, will pass readily through the + analyzer, as the second is called, and I am now letting this light enter + my eye. But when I turn the analyzer round through a right angle, I find + that I see no light; there was a gradual darkening as I rotated the + analyzer. The analyzer will now only allow light to pass through which is + polarized in a horizontal plane, and it receives no such light.</p> + + <div class="figcenter" style="width:11%;"> + <a href="images/fig53.jpg"><img style="width:100%" src="images/fig53.jpg" + alt="Fig. 53" title="Fig. 53" /></a> + <span class="sc">Fig. 53.</span> + </div> + + <div class="figcenter" style="width:46%;"> + <a href="images/fig54.jpg"><img style="width:100%" src="images/fig54.jpg" + alt="Fig. 54" title="Fig. 54" /></a> + <span class="sc">Fig. 54.</span> + </div> + + <p>You will see in this model (Fig. 55) a good illustration of polarized + light. The white, brilliantly illuminated thread M N is <!-- Page 120 + --><span class="pagenum"><a name="page120"></a>{120}</span>pulled by a + weight beyond the pulley M, and its end N is fastened to one limb of a + tuning-fork. Some ragged-looking pieces of thread round the portion N A + prevent its vibrating in any very determinate way, but from A to M the + thread is free from all encumbrance. A vertical slot at A, through which + the thread passes, determines the nature of the vibration of the part A + B; every part of the thread between A and B is vibrating in up and down + directions only. A vertical slot in B allows the vertical vibration to be + communicated through it, and so we see the part B M vibrating in the same + way as A B. I might point out quite a lot of ways in which this is not a + perfect illustration of what occurs with light in Fig. 54. But it is + quite good enough for my present purpose. A is a polarizer of vibration; + it only allows up and down motion to pass through it, and B also allows + up and down motion to pass through. But now, as B is turned round, it + lets less and less of the up and down motion pass through it, until when + it is in the second position shown in the lower part of the figure, it + allows no up and down motion to pass through, and there is no visible + motion of the thread between B and M. You will observe that if we did not + know in what plane (in the present case the plane is vertical) the + vibrations of the thread between A and B occurred, we should only have to + turn B round until we found no vibration <!-- Page 122 --><span + class="pagenum"><a name="page122"></a>{122}</span>passing through, to + obtain the information. Hence, as in the light case, we may call A a + polarizer of vibrations, and B an analyzer.</p> + + <div class="figcenter" style="width:69%;"> + <a href="images/fig55.jpg"><img style="width:100%" src="images/fig55.jpg" + alt="Fig. 55" title="Fig. 55" /></a> + <span class="sc">Fig. 55.</span> + </div> + + <p>Now if polarized light is passing from A to B (Fig. 54) through the + air, say, and we have the analyzer placed so that there is darkness, we + find that if we place in the path of the ray some solution of sugar we + shall no longer have darkness at B; we must turn B round to get things + dark again; this is evidence of the sugar solution having twisted round + the plane of polarization of the light. I will now assume that you know + something about what is meant by twisting the plane of polarization of + light. You know that sugar solution will do it, and the longer the path + of the ray through the sugar, the more twist it gets. This phenomenon is + taken advantage of in the sugar industries, to find the strengths of + sugar solutions. For the thread illustration I am indebted to Professor + Silvanus Thomson, and the next piece of apparatus which I shall show also + belongs to him.</p> + + <p>I have here (<i>see</i> Frontispiece) a powerful armour-clad coil, or + electro-magnet. There is a central hole through it, through which a beam + of light may be passed from an electric lamp, and I have a piece of + Faraday's heavy glass nearly filling this hole. I have a polarizer at one + end, and an analyzer at the other. You see now that the <!-- Page 123 + --><span class="pagenum"><a name="page123"></a>{123}</span>polarized + light passes through the heavy glass and the analyzer, and enters the eye + of an observer. I will now turn B until the light no longer passes. Until + now there has been no magnetism, but I have the means here of producing a + most intense magnetic field in the direction in which the ray passes, and + if your eye were here you would see that there is light passing through + the analyzer. The magnetism has done something to the light, it has made + it capable of passing where it could not pass before. When I turn the + analyzer a little I stop the light again, and now I know that what the + magnetism did was to convert the glass into a medium like the sugar, a + medium which rotates the plane of polarization of light.</p> + + <p>In this experiment you have had to rely upon my personal measurement + of the actual rotation produced. But if I insert between the polarizer + and analyzer this disc of Professor Silvanus Thomson's, built up of + twenty-four radial pieces of mica, I shall have a means of showing to + this audience the actual rotation of the plane of polarization of light. + You see now on the screen the light which has passed through the analyzer + in the form of a cross, and if the cross rotates it is a sign of the + rotation of the plane of polarization of the light. By means of this + electric key I can create, destroy, and reverse the magnetic <!-- Page + 124 --><span class="pagenum"><a name="page124"></a>{124}</span>field in + the glass. As I create magnetism you see the twisting of the cross; I + destroy the magnetism, and it returns to its old position; I create the + opposite kind of magnetism, and you see that the cross twists in the + opposite way. I hope it is now known to you that magnetism rotates the + plane of polarization of light as the solution of sugar did.</p> + + <div class="figleft" style="width:35%;"> + <a href="images/fig56.jpg"><img style="width:100%" src="images/fig56.jpg" + alt="Fig. 56" title="Fig. 56" /></a> + <span class="sc">Fig. 56.</span> + </div> + + <div class="figleft" style="width:24%;"> + <a href="images/fig57.jpg"><img style="width:100%" src="images/fig57.jpg" + alt="Fig. 57" title="Fig. 57" /></a> + <span class="sc">Fig. 57.</span> + </div> + + <p>As an illustration of what occurs between polarizer and analyzer, look + again at this rope (Fig. 53) fastened to the ceiling. I move the bottom + end sharply from east to west, and you see that every part of the rope + moves from east to west. Can you imagine a rope such that when the bottom + end was moved from east to west, a point some yards up moved from + east-north-east to west-sou'-west, that a higher point moved from + north-east to south-west, and so on, the direction gradually changing for + higher and higher points? Some of you, knowing what I have done, may be + able to imagine it. We should have what we want if this rope were a chain + of gyrostats such as you see figured in the diagram; gyrostats all + spinning in the same way looked at from below, with frictionless hinges + between them. Here is such a chain (Fig. 56), one of many that I have + tried to use in this way for several years. But although I have often + believed that I saw the phenomenon occur in <!-- Page 126 --><span + class="pagenum"><a name="page126"></a>{126}</span>such a chain, I must + now confess to repeated failures. The difficulties I have met with are + almost altogether mechanical ones. You see that by touching all the + gyrostats in succession with this rapidly revolving disc driven by the + little electromotor, I can get them all to spin at the same time; but you + will notice that what with bad mechanism and bad calculation on my part, + and want of skill, the phenomenon is completely masked by wild movements + of the gyrostats, the causes of which are better known than capable of + rectification. The principle of the action is very visible in this + gyrostat suspended as the bob of a pendulum (Fig. 57). You may imagine + this to represent a particle of the <!-- Page 127 --><span + class="pagenum"><a name="page127"></a>{127}</span>substance which + transmits light in the magnetic field, and you see by the trickling thin + stream of sand which falls from it on the paper that it is continually + changing the plane of polarization. But I am happy to say that I can show + you to-night a really successful illustration of Thomson's principle; it + is the very first time that this most suggestive experiment has been + shown to an audience. I have a number of double gyrostats (Fig. 58) + placed on the same line, joined end to end by short pieces of elastic. + Each instrument is supported at its centre of gravity, and it can rotate + both in horizontal and in vertical planes.</p> + + <div class="figcenter" style="width:70%;"> + <a href="images/fig58.jpg"><img style="width:100%" src="images/fig58.jpg" + alt="Fig. 58" title="Fig. 58" /></a> + <span class="sc">Fig. 58.</span> + </div> + + <p>The end of the vibrating lever A can only get a horizontal motion from + my hand, and the motion is transmitted from one gyrostat to the next, + until it has travelled to the very end one. Observe that when the + gyrostats are not spinning, the motion is <!-- Page 128 --><span + class="pagenum"><a name="page128"></a>{128}</span>everywhere horizontal. + Now it is very important not to have any illustration here of a reflected + ray of light, and so I have introduced a good deal of friction at all the + supports. I will now spin all the gyrostats, and you will observe that + when A moves nearly straight horizontally, the next gyrostat moves + straight but in a slightly different plane, the second gyrostat moves in + another plane, and so on, each gyrostat slightly twisting the plane in + which the motion occurs; and you see that the end one does not by any + means receive the horizontal motion of A, but a motion nearly vertical. + This is a mechanical illustration, the first successful one I have made + after many trials, of the effect on light of magnetism. The reason for + the action that occurs in this model must be known to everybody who has + tried to follow me from the beginning of the lecture.</p> + + <p>And you can all see that we have only to imagine that many particles + of the glass are rotating like gyrostats, and that magnetism has + partially caused an allineation of their axes, to have a dynamical theory + of Faraday's discovery. The magnet twists the plane of polarization, and + so does the solution of sugar; but it is found by experiment that the + magnet does it indifferently for coming and going, whereas the sugar does + it in a way that corresponds with a spiral structure of molecules. You + see that in this important <!-- Page 129 --><span class="pagenum"><a + name="page129"></a>{129}</span>particular the gyrostat analogue must + follow the magnetic method, and not the sugar method. We must regard this + model, then, the analogue to Faraday's experiment, as giving great + support to the idea that magnetism consists of rotation.</p> + + <p>I have already exceeded the limits of time usually allowed to a + popular lecturer, but you see that I am very far from having exhausted + our subject. I am not quite sure that I have accomplished the object with + which I set out. My object was, starting from the very different + behaviour of a top when spinning and when not spinning, to show you that + the observation of that very common phenomenon, and a determination to + understand it, might lead us to understand very much more complex-looking + things. There is no lesson which it is more important to learn than + this—That it is in the study of every-day facts that all the great + discoveries of the future lie. Three thousand years ago spinning tops + were common, but people never studied them. Three thousand years ago + people boiled water and made steam, but the steam-engine was unknown to + them. They had charcoal and saltpetre and sulphur, but they knew nothing + of gunpowder. They saw fossils in rocks, but the wonders of geology were + unstudied by them. They had bits of iron and copper, but not one of them + thought of any one of the fifty simple <!-- Page 130 --><span + class="pagenum"><a name="page130"></a>{130}</span>ways that are now known + to us of combining those known things into a telephone. Why, even the + simplest kind of signalling by flags or lanterns was unknown to them, and + yet a knowledge of this might have changed the fate of the world on one + of the great days of battle that we read about. We look on Nature now in + an utterly different way, with a great deal more knowledge, with a great + deal more reverence, and with much less unreasoning superstitious fear. + And what we are to the people of three thousand years ago, so will be the + people of one hundred years hence to us; for indeed the acceleration of + the rate of progress in science is itself accelerating. The army of + scientific workers gets larger and larger every day, and it is my belief + that every unit of the population will be a scientific worker before + long. And so we are gradually making time and space yield to us and obey + us. But just think of it! Of all the discoveries of the next hundred + years; the things that are unknown to us, but which will be so well known + to our descendants that they will sneer at us as utterly ignorant, + because these things will seem to them such self-evident facts; I say, of + all these things, if one of us to-morrow discovered one of them, he would + be regarded as a great discoverer. And yet the children of a hundred + years hence will know it: it will be brought home to <!-- Page 131 + --><span class="pagenum"><a name="page131"></a>{131}</span>them perhaps + at every footfall, at the flapping of every coat-tail.</p> + + <p>Imagine the following question set in a school examination paper of + 2090 <span class="scac">A.D.</span>—"Can you account for the crass + ignorance of our forefathers in not being able to see from England what + their friends were doing in Australia?"<a name="NtA13" + href="#Nt13"><sup>[13]</sup></a> Or this—"Messages are being + received every minute from our friends on the planet Mars, and are now + being answered: how do you account for our ancestors being utterly + ignorant that these messages were occasionally sent to them?" Or + this—"What metal is as strong compared with steel as steel is + compared with lead? and explain why the discovery of it was not made in + Sheffield."</p> + + <p>But there is one question that our descendants will never ask in + accents of jocularity, for to their bitter sorrow every man, woman, and + child of them will know the answer, and that question is this—"If + our ancestors in the matter of coal economy were not quite as ignorant as + a baby who takes a penny <!-- Page 132 --><span class="pagenum"><a + name="page132"></a>{132}</span>as equivalent for a half-crown, why did + they waste our coal? Why did they destroy what never can be + replaced?"</p> + + <p>My friends, let me conclude by impressing upon you the value of + knowledge, and the importance of using every opportunity within your + reach to increase your own store of it. Many are the glittering things + that seem to compete successfully with it, and to exercise a stronger + fascination over human hearts. Wealth and rank, fashion and luxury, power + and fame—these fire the ambitions of men, and attract myriads of + eager worshippers; but, believe it, they are but poor things in + comparison with knowledge, and have no such pure satisfactions to give as + those which it is able to bestow. There is no evil thing under the sun + which knowledge, when wielded by an earnest and rightly directed will, + may not help to purge out and destroy; and there is no man or woman born + into this world who has not been given the capacity, not merely to gather + in knowledge for his own improvement and delight, but even to add + something, however little, to that general stock of knowledge which is + the world's best wealth.</p> + + <p><br style="clear:both" /></p> +<hr class="full" /> + +<p><!-- Page 133 --><span class="pagenum"><a name="page133"></a>{133}</span></p> + +<h3>ARGUMENT.</h3> + +<blockquote class="b1n"> + + <p>1. <i>Introduction</i>, pages <a href="#page9">9</a>-<a + href="#page14">14</a>, showing the importance of the study of + spinning-top behaviour.</p> + + <p>2. <i>Quasi-rigidity induced even in flexible and fluid bodies by + rapid motion</i>, <a href="#page14">14</a>-<a href="#page21">21</a>.</p> + + <p>Illustrations: Top, <a href="#page14">14</a>; belt or rope, <a + href="#page14">14</a>; disc of thin paper, <a href="#page14">14</a>; ring + of chain, <a href="#page15">15</a>; soft hat, <a href="#page16">16</a>; + drunken man, <a href="#page16">16</a>; rotating water, <a + href="#page16">16</a>; smoke rings, <a href="#page17">17</a>; Thomson's + Molecular Theory, <a href="#page19">19</a>; swimmer caught in an eddy, <a + href="#page20">20</a>; mining water jet, <a href="#page20">20</a>; cased + gyrostat, <a href="#page21">21</a>.</p> + + <p>3. <i>The nature of this quasi-rigidity in spinning bodies is a + resistance to change of direction of the axis of spinning</i>, <a + href="#page21">21</a>-<a href="#page30">30</a>.</p> + + <p>Illustrations: Cased gyrostat, <a href="#page21">21</a>-<a + href="#page24">24</a>; tops, biscuits, hats, thrown into the air, <a + href="#page24">24</a>-<a href="#page26">26</a>; quoits, hoops, + projectiles from guns, <a href="#page27">27</a>; jugglers at the Victoria + Music Hall, <a href="#page26">26</a>-<a href="#page30">30</a>; child + trundling hoop, man on bicycle, ballet-dancer, the earth pointing to pole + star, boy's top, <a href="#page30">30</a>.</p> + + <p>4. <i>Study of the crab-like behaviour of a spinning body</i>, <a + href="#page30">30</a>-<a href="#page49">49</a>.</p> + + <p>Illustrations: Spinning top, <a href="#page31">31</a>; cased gyrostat, + <a href="#page32">32</a>; balanced gyrostat, <a href="#page33">33</a>-<a + href="#page36">36</a>; windage of projectiles from <!-- Page 134 --><span + class="pagenum"><a name="page134"></a>{134}</span>rifled guns, <a + href="#page36">36</a>-<a href="#page38">38</a>; tilting a hoop or + bicycle, turning quickly on horseback, <a href="#page38">38</a>; bowls, + <a href="#page39">39</a>; how to simplify one's observations, <a + href="#page39">39</a>, <a href="#page40">40</a>; the illustration which + gives us our simple universal rule, <a href="#page40">40</a>-<a + href="#page42">42</a>; testing the rule, <a href="#page42">42</a>-<a + href="#page44">44</a>; explanation of precession of gyrostat, <a + href="#page44">44</a>, <a href="#page45">45</a>; precession of common + top, <a href="#page46">46</a>; precession of overhung top, <a + href="#page46">46</a>; list of our results given in a wall sheet, <a + href="#page48">48</a>, <a href="#page49">49</a>.</p> + + <p>5. <i>Proof or explanation of our simple universal rule</i>, <a + href="#page50">50</a>-<a href="#page54">54</a>.</p> + + <p>Giving two independent rotations to a body, <a href="#page50">50</a>, + <a href="#page51">51</a>; composition of rotations, <a + href="#page52">52</a>, <a href="#page53">53</a>.</p> + + <p>6. <i>Warning that the rule is not, after all, so simple</i>, <a + href="#page54">54</a>-<a href="#page66">66</a>.</p> + + <p>Two independent spins given to the earth, <a href="#page54">54</a>; + centrifugal force, <a href="#page55">55</a>; balancing of quick speed + machinery, <a href="#page56">56</a>, <a href="#page57">57</a>; the + possible wobbling of the earth, <a href="#page58">58</a>; the three + principal axes of a body, <a href="#page59">59</a>; the free spinning of + discs, cones, rods, rings of chain, <a href="#page60">60</a>; nodding + motion of a gyrostat, <a href="#page62">62</a>; of a top, <a + href="#page63">63</a>; parenthesis about inaccuracy of statement and + Rankine's rhyme, <a href="#page63">63</a>, <a href="#page64">64</a>; + further complications in gyrostatic behaviour, <a href="#page64">64</a>; + strange elastic, jelly-like behaviour, <a href="#page65">65</a>; gyrostat + on stilts, <a href="#page66">66</a>.</p> + + <p>7. <i>Why a gyrostat falls</i>, <a href="#page66">66</a>, <a + href="#page67">67</a>.</p> + + <p>8. <i>Why a top rises</i>, <a href="#page67">67</a>-<a + href="#page74">74</a>.</p> + + <p>General ignorance, <a href="#page67">67</a>; Thomson preparing for the + mathematical tripos, <a href="#page68">68</a>; behaviour of a water-worn + stone when spun on a table, <a href="#page68">68</a>, <a + href="#page69">69</a>; parenthesis on technical education, <a + href="#page70">70</a>; simple explanation of why a top rises, <a + href="#page70">70</a>-<a href="#page73">73</a>; behaviour of + heterogeneous sphere when spun, <a href="#page74">74</a>.</p> + + <p>9. <i>Precessional motion of the earth</i>, <a + href="#page74">74</a>-<a href="#page91">91</a>.</p> + + <p>Its nature and effects on climate, <a href="#page75">75</a>-<a + href="#page80">80</a>; resemblance of the precessing earth to certain + models, <a href="#page80">80</a>-<a href="#page82">82</a>; tilting forces + exerted by the sun and moon on the <!-- Page 135 --><span + class="pagenum"><a name="page135"></a>{135}</span>earth, <a + href="#page82">82</a>-<a href="#page84">84</a>; how the earth's + precessional motion is always altering, <a href="#page85">85</a>-<a + href="#page88">88</a>; the retrogression of the moon's nodes is itself + another example, <a href="#page88">88</a>, <a href="#page89">89</a>; an + exact statement made and a sort of apology for making it, <a + href="#page90">90</a>, <a href="#page91">91</a>.</p> + + <p>10. <i>Influence of possible internal fluidity of the earth on its + precessional motion</i>, <a href="#page91">91</a>-<a + href="#page98">98</a>.</p> + + <p>Effect of fluids and sand in tumblers, <a href="#page91">91</a>-<a + href="#page93">93</a>; three tests of the internal rigidity of an egg, + that is, of its being a boiled egg, <a href="#page93">93</a>, <a + href="#page94">94</a>; quasi-rigidity of fluids due to rapid motion, + forgotten in original argument, <a href="#page95">95</a>; beautiful + behaviour of hollow top filled with water, <a href="#page95">95</a>; + striking contrasts in the behaviour of two tops which are very much + alike, <a href="#page97">97</a>, <a href="#page98">98</a>; fourth test of + a boiled egg, <a href="#page98">98</a>.</p> + + <p>11. Apology for dwelling further upon astronomical matters, and + impertinent remarks about astronomers, <a href="#page99">99</a>-<a + href="#page101">101</a>.</p> + + <p>12. How a gyrostat would enable a person living in subterranean + regions to know, <i>1st, that the earth rotates</i>; <i>2nd, the amount + of rotation</i>; <i>3rd, the direction of true north</i>; <i>4th, the + latitude</i>, <a href="#page101">101</a>-<a href="#page111">111</a>.</p> + + <p>Some men's want of faith, <a href="#page101">101</a>; disbelief in the + earth's rotation, <a href="#page102">102</a>; how a free gyrostat + behaves, <a href="#page103">103</a>, <a href="#page104">104</a>; + Foucault's laboratory measurement of the earth's rotation, <a + href="#page105">105</a>-<a href="#page107">107</a>; to find the true + north, <a href="#page108">108</a>; all rotating bodies vainly + endeavouring to point to the pole star, <a href="#page108">108</a>; to + find the latitude, <a href="#page110">110</a>; analogies between the + gyrostat and the mariner's compass and the dipping needle, <a + href="#page110">110</a>, <a href="#page111">111</a>; dynamical connection + between magnetism and gyrostatic phenomena, <a + href="#page111">111</a>.</p> + + <p>13. How the lecturer spun his tops, using electro-motors, <a + href="#page112">112</a>-<a href="#page114">114</a>.</p> + + <p>14. <i>Light</i>, <i>magnetism</i>, <i>and molecular spinning + tops</i>, <a href="#page115">115</a>-<a href="#page128">128</a>.</p> + + <p>Light takes time to travel, <a href="#page115">115</a>; the + electro-magnetic <!-- Page 136 --><span class="pagenum"><a + name="page136"></a>{136}</span>theory of light, <a + href="#page116">116</a>, <a href="#page117">117</a>; signalling through + fogs and buildings by means of a new kind of radiation, <a + href="#page117">117</a>; Faraday's rotation of the plane of polarization + by magnetism, with illustrations and models, <a + href="#page118">118</a>-<a href="#page124">124</a>; chain of gyrostats, + <a href="#page124">124</a>; gyrostat as a pendulum bob, <a + href="#page126">126</a>; Thomson's mechanical illustration of Faraday's + experiment, <a href="#page127">127</a>, <a href="#page128">128</a>.</p> + + <p>15. <i>Conclusion</i>, <a href="#page129">129</a>-<a + href="#page132">132</a>.</p> + + <p>The necessity for cultivating the observation, <a + href="#page129">129</a>; future discovery, <a href="#page130">130</a>; + questions to be asked one hundred years hence, <a + href="#page131">131</a>; knowledge the thing most to be wished for, <a + href="#page132">132</a>.</p> + +</blockquote> + + <p><br style="clear:both" /></p> +<hr class="full" /> + +<p><!-- Page 137 --><span class="pagenum"><a name="page137"></a>{137}</span></p> + +<h3>APPENDIX I.</h3> + +<p class="cenhead">THE USE OF GYROSTATS.</p> + + <p>In 1874 two famous men made a great mistake in endeavouring to prevent + or diminish the rolling motion of the saloon of a vessel by using a + rapidly rotating wheel. Mr. Macfarlane Gray pointed out their mistake. It + is only when the wheel is allowed to <i>precess</i> that it can exercise + a steadying effect; the moment which it then exerts is equal to the + angular speed of the precession multiplied by the moment of momentum of + the spinning wheel.</p> + + <p>It is astonishing how many engineers who know the laws of motion of + mere translation, are ignorant of angular motion, and yet the analogies + between the two sets of laws are perfectly simple. I have set out these + analogies in my book on <i>Applied Mechanics</i>. The last of them + between centripetal force on a body moving in a curved path, and torque + or moment on a rotating body is the simple key to all gyrostatic or top + calculation. When the spin of a top is greatly reduced it is necessary to + remember that the total moment of momentum is not about the spinning axis + (see my <i>Applied Mechanics</i>, page 594); correction for this is, I + suppose, what introduces the complexity which scares students from + studying the vagaries of tops; but in all cases that are likely to come + before an engineer it would be absurd to study <!-- Page 138 --><span + class="pagenum"><a name="page138"></a>{138}</span>such a small + correction, and consequently calculation is exceedingly simple.</p> + + <p>Inventors using gyrostats have succeeded in doing the following + things—</p> + + <p>(1) Keeping the platform of a gun level on board ship, however the + ship may roll or pitch. Keeping a submarine vessel or a flying machine + with any plane exactly horizontal or inclined in any specified way.<a + name="NtA14" href="#Nt14"><sup>[14]</sup></a> It is easy to effect such + objects without the use of a gyrostat, as by means of spirit levels it is + possible to command powerful electric or other motors to keep anything + always level. The actual methods employed by Mr. Beauchamp Tower (an + hydraulic method), and by myself (an electric method), depend upon the + use of a gyrostat, which is really a pendulum, the axis being + vertical.</p> + + <p>(2) Greatly reducing the rolling (or pitching) of a ship, or the + saloon of a ship. This is the problem which Mr. Schlick has solved with + great success, at any rate in the case of torpedo boats.</p> + + <p>(3) In Mr. Brennan's Mono-rail railway, keeping the resultant force + due to weight, wind pressure, centrifugal force, etc., exactly in line + with the rail, so that, however the load on a wagon may alter in + position, and although the wagon may be going round a curve, it is + quickly brought to a position such that there are no forces tending to + alter its angular position. The wagon leans over towards the wind or + towards the centre of the curve of the rail so as to be in + equilibrium.</p> + + <p>(4) I need not refer to such matters as the use of gyrostats for the + correction of compasses on board ship, referred to in page <a + href="#page111">111</a>.</p> + +<p><!-- Page 139 --><span class="pagenum"><a name="page139"></a>{139}</span></p> + + <div class="figcenter" style="width:71%;"> + <a href="images/figapp1.jpg"><img style="width:100%" src="images/figapp1.jpg" + alt="Fig. 1" title="Fig. 1" /></a> + <span class="sc">Fig. 1.</span> + </div> + +<p><!-- Page 140 --><span class="pagenum"><a name="page140"></a>{140}</span></p> + + <p>Problems (2) and (3) are those to which I wish to refer. For a ship of + 6,000 tons Mr. Schlick would use a large wheel of 10 to 20 tons, + revolving about an axis E F (fig. 1) whose mean position is vertical. Its + bearings are in a frame E C F D which can move about a thwart-ship axis + C D with a precessional motion. Its centre of gravity is below this axis, + so that like the ship itself the frame is in stable equilibrium. Let the + ship have rolled through an angle R from its upright position, and + suppose the axis E F to have precessed through the angle P from a + vertical position. Let the angular velocity of rolling be called R<span + class="x1"><span class="x7">˙</span></span>, and the angular + velocity of precession P<span class="x1"><span + class="x7">˙</span></span>; let the moment of momentum of the wheel + be <i>m</i>. For any vibrating body like a ship it is easy to write out + the equation of motion; into this equation we have merely to introduce + the moment <i>m</i> P<span class="x1"><span + class="x7">˙</span></span> diminishing R; into the equation for P + we merely introduce the moment <i>m</i> R<span class="x1"><span + class="x7">˙</span></span> increasing P. As usual we introduce + frictional terms; in the first place F R<span class="x1"><span + class="x7">˙</span></span> (F being a constant co-efficient) + stilling the roll of the ship; in the second case <i>f</i> P<span + class="x1"><span class="x7">˙</span></span> a fluid friction + introduced by a pair of dash pots applied at the pins A and B to still + the precessional vibrations of the frame. It will be found that the + angular motion P is very much greater than the roll R. Indeed, so great + is P that there are stops to prevent its exceeding a certain amount. Of + course so long as a stop acts, preventing precession, the roll of the + ship proceeds as if the gyrostat wheel were not rotating. Mr. Schlick + drives his wheels by steam; he will probably in future do as Mr. Brennan + does, drive them by electromotors, and keep them in air-tight cases in + good vacuums, because the loss of energy by friction against an + atmosphere is proportional to the density of the atmosphere. The solution + of the equations to find the nature of the R and P motions is sometimes + tedious, but requires no great amount of mathematical knowledge. In a + case considered by me of <!-- Page 141 --><span class="pagenum"><a + name="page141"></a>{141}</span>a 6,000 ton ship, the period of a roll was + increased from 14 to 20 seconds by the use of the gyrostat, and the roll + rapidly diminished in amount. There was accompanying this slow periodic + motion, one of a two seconds' period, but if it did appear it was damped + out with great rapidity. Of course it is assumed that, by the use of + bilge keels and rolling chambers, and as low a metacentre as is + allowable, we have already lengthened the time of vibration, and damped + the roll R as much as possible, before applying the gyrostat. I take it + that everybody knows the importance of lengthening the period of the + natural roll of a ship, although he may not know the reason. The reason + why modern ships of great tonnage are so steady is because their natural + periodic times of rolling vibration are so much greater than the probable + periods of any waves of the sea, for if a series of waves acts upon a + ship tending to make it roll, if the periodic time of each wave is not + very different from the natural periodic time of vibration of the ship, + the rolling motion may become dangerously great.</p> + + <p>If we try to apply Mr. Schlick's method to Mr. Brennan's car it is + easy to show that there is instability of motion, whether there is or is + not friction. If there is no friction, and we make the gyrostat frame + unstable by keeping its centre of gravity above the axis C D, there will + be vibrations, but the smallest amount of friction will cause these + vibrations to get greater and greater. Even without friction there will + be instability if <i>m</i>, the moment of momentum of the wheel, is less + than a certain amount. We see, then, that no form of the Schlick method, + or modification of it, can be applied to solve the Brennan problem.</p> + +<p><!-- Page 142 --><span class="pagenum"><a name="page142"></a>{142}</span></p> + + <div class="figcenter" style="width:68%;"> + <a href="images/figapp2.jpg"><img style="width:100%" src="images/figapp2.jpg" + alt="Fig. 2" title="Fig. 2" /></a> + <span class="sc">Fig. 2.</span> + </div> + +<p><!-- Page 143 --><span class="pagenum"><a name="page143"></a>{143}</span></p> + + <p>Mr. Brennan's method of working is quite different from that of Mr. + Schlick. Fig. 2 shows his model car (about six feet long); it is driven + by electric accumulators carried by the car. His gyrostat wheels are + driven by electromotors (not shown in fig. 3); as they are revolving in + nearly vacuous spaces they consume but little power, and even if the + current were stopped they would continue running at sufficiently high + speeds to be effective for a length of time. Still it must not be + forgotten that energy is wasted in friction, and work has to be done in + bringing the car to a new position of equilibrium, and this energy is + supplied by the electromotors. Should the gyrostats really stop, or fall + to a certain low speed, two supports are automatically dropped, one on + either side of the car; each of them drops till it reaches the ground; + one of them dropping, perhaps, much farther than the other.</p> + + <p>The real full-size car, which he is now constructing, may be pulled + with other cars by any kind of locomotive using electricity or petrol or + steam, or each of the wheels may be a driving wheel. He would prefer to + generate electropower on his train, and to drive every wheel with an + electric motor. His wheels are so independent of one another that they + can take very quick curves and vertical inequalities of the rail. The + rail is fastened to sleepers lying on ground that may have sidelong + slope. The model car is supported by a mono-rail bogie at each end; each + bogie has two wheels pivoted both vertically and horizontally; it runs on + a round iron gas pipe, and sometimes on steel wire rope; the ground is + nowhere levelled or cut, and at one place the rail is a steel wire rope + spanning a gorge, as shown in fig. 2. It is interesting to stop the car + in the middle of this rope and to swing the rope sideways to see the + automatic balancing of the car. The car may be left here or elsewhere + balancing itself with nobody in charge of it. If the load on the + car—great lead weights—be dumped about into new positions, + the car adjusts itself to the new conditions with great <!-- Page 144 + --><span class="pagenum"><a name="page144"></a>{144}</span>quickness. + When the car is stopped, if a person standing on the ground pushes the + car sidewise, the car of course pushes in opposition, like an indignant + animal, and by judicious pushing and yielding it is possible to cause a + considerable tilt. Left now to itself the car rights itself very + quickly.</p> + + <div class="figcenter" style="width:53%;"> + <a href="images/figapp3.jpg"><img style="width:100%" src="images/figapp3.jpg" + alt="Fig. 3" title="Fig. 3" /></a> + <span class="sc">Fig. 3.</span> + </div> + +<p><!-- Page 145 --><span class="pagenum"><a name="page145"></a>{145}</span></p> + + <div class="figcenter" style="width:72%;"> + <a href="images/figapp3b.jpg"><img style="width:100%" src="images/figapp3b.jpg" + alt="Fig. 3b" title="Fig. 3b" /></a> + <span class="sc">Fig.</span> <i>3<sup>b</sup></i> (showing the + ground-plan of Fig. 3). + </div> + +<p><!-- Page 146 --><span class="pagenum"><a name="page146"></a>{146}</span></p> + + <p>Fig. 3 is a diagrammatic representation of Mr. Brennan's pair of + gyrostats in sectional elevation and plan. The cases G and G', inside + which the wheels F and F' are rotating <i>in vacuo</i> at the same speed + and in opposite directions (driven by electromotors not shown in the + figure), are pivoted about vertical axes E J and E' J'. They are + connected by spur-toothed segments J J and J' J', so that their + precessional motions are equal and opposite. The whole system is pivoted + about C, a longitudinal axis. Thus when precessing so that H comes out of + the paper, so will H', and when H goes into the paper, so does H'. When + the car is in equilibrium the axes K H and K' H' are in line N O O' N' + across the car in the plane of the paper. They are also in a line which + is at right angles to the total resultant (vertical or nearly vertical) + force on the car. I will call N O O' N' the mid position. Let ½<i>m</i> + be the moment of momentum of either wheel. Let us suppose that suddenly + the car finds that it is not in equilibrium because of a gust of wind, or + centrifugal force, or an alteration of loading, so that the shelf D comes + up against H, the spinning axis (or a roller revolving with the spinning + axis) of the gyrostat. H begins to roll away from me, and if no slipping + occurred (but there always is slipping, and, indeed, slipping is a + necessary condition) it would roll, that is, the gyrostats would precess + with a constant angular velocity <span class="grk">α</span>, and + exert the moment <i>m</i><span class="grk">α</span> upon the shelf + D, and therefore on the car. It is to be observed that this is greater as + the diameter of the rolling part is greater. This precession continues + until the roller and the shelf cease to touch. At first H lifts with the + shelf, and afterwards the shelf moving downwards is followed for some + distance by the roller. If the tilt had been in the opposite direction + the shelf D' would have acted upwards upon the roller H', and caused just + the opposite kind of precession, and a moment of the opposite kind.</p> + + <p>We now have the spindles out of their mid position; how are they + brought back from O Q and O' Q' to O N and O' N', <!-- Page 147 --><span + class="pagenum"><a name="page147"></a>{147}</span>but with H permanently + lowered just the right amount? It is the essence of Mr. Brennan's + invention that after a restoring moment has been applied to the car the + spindles shall go back to the position N O O' N' (with H permanently + lowered), so as to be ready to act again. He effects this object in + various ways. Some ways described in his patents are quite different from + what is used on the model, and the method to be used on the full-size + wagon will again be quite different. I will describe one of the methods. + Mr. Brennan tells me that he considers this old method to be crude, but + he is naturally unwilling to allow me to publish his latest method.</p> + + <p>D' is a circular shelf extending from the mid position in my + direction; D is a similar shelf extending from the mid position into the + paper, or away from me. It is on these shelves that H' and H roll, + causing precession away from N O O' N', as I have just described. When H' + is inside the paper, or when H is outside the paper, they find no shelf + to roll upon. There are, however, two other shelves L and L', for two + other rollers M and M', which are attached to the frames concentric with + the spindles; they are free to rotate, but are not rotated by the + spindles. When they are pressed by their shelves L or L' this causes + negative precession, and they roll towards the N O O' N' position. There + is, of course, friction at their supports, retarding their rotation, and + therefore the precession. The important thing to remember is that H and + H', when they touch their shelves (when one is touching the other is not + touching) cause a precession away from the mid position N O O' N' at a + rate <span class="grk">α</span>, which produces a restoring moment + <i>m</i><span class="grk">α</span> of nearly constant amount + (except for slipping), whereas where M or M' touches its shelf L or L' + (when one is touching the other is not touching) the pressure on the + shelf and friction determine the rate of the precession towards the mid + position N O O' N', <!-- Page 148 --><span class="pagenum"><a + name="page148"></a>{148}</span>as well as the small vertical motion. The + friction at the supports of M and M' is necessary.</p> + + <p>Suppose that the tilt from the equilibrium position to be corrected is + R, when D presses H upward. The moment <i>m</i><span + class="grk">α</span>, and its time of action (the total momental + impulse) are too great, and R is over-corrected; this causes the roller + M' to act on L', and the spindles return to the mid position; they go + beyond the mid position, and now the roller H' acts on D', and there is a + return to the mid position, and beyond it a little, and so it goes on, + the swings of the gyrostats out of and into the mid position, and the + vibrations of the car about its position of equilibrium getting rapidly + less and less until again neither H nor H', nor M nor M' is touching a + shelf. It is indeed marvellous to see how rapidly the swings decay. + Friction accelerates the precession away from N O O' N'. Friction retards + the precession towards the middle position.</p> + + <p>It will be seen that by using the two gyrostats instead of one when + there is a curve on the line, although the plane N O O' N' rotates, and + we may say that the gyrostats precess, the tilting couples which they + might exercise are equal and opposite. I do not know if Mr. Brennan has + tried a single gyrostat, the mid position of the axis of the wheel being + vertical, but even in this case a change of slope, or inequalities in the + line, might make it necessary to have a pair.</p> + + <p>It is evident that this method of Mr. Brennan is altogether different + in character from that of Mr. Schlick. Work is here actually done which + must be supplied by the electromotors.</p> + + <p>One of the most important things to know is this: the Brennan model is + wonderfully successful; the weight of the apparatus is not a large + fraction of the weight of the wagon; will this also be the case with a + car weighing 1,000 times as <!-- Page 149 --><span class="pagenum"><a + name="page149"></a>{149}</span>much? The calculation is not difficult, + but I may not give it here. If we assume that suddenly the wagon finds + itself at the angle R from its position of equilibrium, it may be taken + that if the size of each dimension of the wagon be multiplied by + <i>n</i>, and the size of each dimension of the apparatus be multiplied + by <i>p</i>, then for a sudden gust of wind, or suddenly coming on a + curve, or a sudden shift of position of part of the cargo, R may be taken + as inversely proportional to <i>n</i>. I need not state the reasonable + assumption which underlies this calculation, but the result is that if + <i>n</i> is 10, <i>p</i> is 7.5. That is, if the weight of the wagon is + multiplied by 1,000, the weight of the apparatus is only multiplied by + 420. In fact, if, in the model, the weight of the apparatus is 10 per + cent. of that of the wagon, in the large wagon the weight of the + apparatus is only about 4 per cent. of that of the wagon. This is a very + satisfactory result.<a name="NtA15" href="#Nt15"><sup>[15]</sup></a></p> + + <p>My calculations seem to show that Mr. Schlick's apparatus will form a + larger fraction of the whole weight of a ship, as the ship is larger, but + in the present experimental stage of the subject it is unfair to say more + than that this seems probable. My own opinion is that large ships are + sufficiently steady already.</p> + + <p>In both cases it has to be remembered that if the <i>diameter</i> of + the wheel can be increased in greater proportion than the dimensions of + ship or wagon, the proportional weight of the apparatus may be + diminished. A wheel of twice the diameter, but of the same weight, may + have twice the moment of momentum, and may therefore be twice as + effective. I assume the stresses in the material to be the same.</p> + + <p><br style="clear:both" /></p> +<hr class="full" /> + +<p><!-- Page 150 --><span class="pagenum"><a name="page150"></a>{150}</span></p> + +<h3>APPENDIX II.</h3> + + <p>Page <a href="#page23">23</a>; note at line 3. Prof. Osborne Reynolds + made the interesting remark (<i>Collected Papers</i>, Vol. ii., p. 154), + "That if solid matter had certain kinds of internal motions, such as the + box has, pears differing, say, from apples, the laws of motion would not + have been discovered; if discovered for pears they would not have applied + to <span class="correction" title="Original reads 'applies'." + >apples</span>."</p> + + <p>Page <a href="#page38">38</a>; note at line 8. The motion of a rifle + bullet is therefore one of precession about the tangent to the path. The + mathematical solution is difficult, but Prof. Greenhill has satisfied + himself mathematically that air friction damps the precession, and causes + the axis of the shot to get nearer the tangential direction, so that fig. + 10 illustrates what would occur in a vacuum, but not in air. It is + probable that this statement applies only to certain proportions of + length to diameter.</p> + + <p>Page <a href="#page129">129</a>; note at line 5. Many men wonder how + the ether can have the enormous rigidity necessary for light + transmission, and yet behave like a frictionless fluid. One way of seeing + how this may occur is to imagine that when ordinary matter moves in the + ether it only tends to produce motion of translation of the ether + particles, and therefore no resistance. But anything such as light, which + must operate in turning axes of rotating parts, may encounter enormous + elastic resistance.</p> + + <p><i>Richard Clay & Sons, Limited, London and Bungay.</i></p> + + <p><br style="clear:both" /></p> +<hr class="full" /> + +<h3>PUBLICATIONS</h3> + +<p class="cenhead">OF THE</p> + +<h3><b>Society for Promoting Christian Knowledge</b>.</h3> + +<h3>THE ROMANCE OF SCIENCE.</h3> + +<p class="cenhead">Small post 8vo, cloth boards.</p> + +<blockquote class="b1n"> + + <p><b>Coal, and What We Get From It.</b> By Professor <span class="sc">R. + Meldola</span>, F.R.S., F.I.C. With several Illustrations. 2<i>s.</i> + 6<i>d.</i></p> + + <p><b>Colour Measurement and Mixture.</b> By Sir <span class="sc">W. de + W.</span> <span class="sc">Abney</span>, K.C.B., R.E., F.R.S. Numerous + Illustrations. 2<i>s.</i> 6<i>d.</i></p> + + <p><b>Diseases of Plants.</b> By Professor <span class="sc">Marshall + Ward</span>. 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Pearman</span>. + With Map. <i>4s.</i></p> + + <p><b>Salisbury.</b> By the Rev. <span class="sc">W. H. Jones</span>. + With Map. <i>2s. 6d.</i></p> + + <p><b>Sodor and Man.</b> By <span class="sc">A. W. Moore</span>, M.A. + <i>3s.</i></p> + + <p><b>St. Asaph.</b> By the Ven. Archdeacon <span + class="sc">Thomas</span>. <i>2s.</i></p> + + <p><b>St. David's.</b> By the Rev. Canon <span class="sc">Bevan</span>. + With Map. <i>2s. 6d.</i></p> + + <p><b>Winchester.</b> By the Rev. <span class="sc">W. Benham</span>, B.D. + <i>3s.</i></p> + + <p><b>Worcester.</b> By the Rev. <span class="sc">I. Gregory Smith</span> + and Rev. <span class="sc">Phipps Onslow</span>. <i>3s. 6d.</i></p> + + <p><b>York.</b> By the Rev. Canon <span class="sc">Ornsby</span>, M.A., + F.S.A. <i>3s. 6d.</i></p> + +</blockquote> + + <p><br style="clear:both" /></p> +<hr class="short" /> + +<h3>EARLY CHURCH CLASSICS.</h3> + +<p class="cenhead"><b>Small post 8vo, cloth boards.</b></p> + +<blockquote class="b1n"> + + <p><b>A Homily of Clement of Alexandria,</b> entitled, Who is the Rich + Man that is Being Saved? By Rev. <span class="sc">P. Mordaunt + Barnard</span>. <i>1s.</i></p> + + <p><b>Bishop Sarapion's Prayer-Book:</b> An Egyptian Pontifical dated + probably about 350-356 A.D. Translated from the Edition of Dr. <span + class="sc">G. Wobbermin</span>. With Introduction, Notes, and Indices, by + the Right Rev. <span class="sc">John Wordsworth</span>, D.D. <i>1s. + 6d.</i></p> + + <p><b>Origen the Teacher.</b> Being the Address of Gregory the + Wonder-Worker to Origen, together with Origen's Letter to Gregory. + Translated, with an Introduction and Notes, by the Rev. <span + class="sc">W. Metcalfe</span>, D.D. <i>1s. 6d.</i></p> + + <p><b>St. Cyprian on the Lord's Prayer.</b> An English Translation, with + Introduction by the Ven. Archdeacon <span class="sc">T. H. + Bindley</span>, D.D. <i>1s. 6d.</i></p> + + <p><b>St. Polycarp, Bishop of Smyrna</b>. By the late Rev. <span + class="sc">Blomfield Jackson</span>, M.A. <i>1s.</i></p> + + <p><b>The Doctrine of the Twelve Apostles.</b> Translated into English, + with Introduction and Notes, by the Rev. <span class="sc">Charles + Bigg</span>, D.D. <i>1s.</i></p> + + <p><b>The Epistle of St. Clement, Bishop of Rome.</b> By the Rev. <span + class="sc">John A. F. Gregg</span>, M.A. <i>1s.</i></p> + + <p><b>St. Augustine's Treatise on the City of God.</b> By Rev. <span + class="sc">F. R. M. Hitchcock</span>, M.A., B.D. <i>1s. 6d.</i></p> + + <p><b>St. Chrysostom on the Priesthood.</b> By the Rev. <span + class="sc">T. Allen Moxom</span>, M.A. <i>2s.</i></p> + + <p><b>The Apostolical Constitutions and Cognate Documents,</b> with + special reference to their Liturgical Elements. By the Rev. <span + class="sc">De Lacy O'Leary</span>, M.A. <i>1s.</i></p> + + <p><b>The Epistle of Diognetus.</b> By the Rev. <span class="sc">L. B. + Radford</span>, M.A. <i>1s. 6d.</i></p> + + <p><b>The Epistle of the Galilean Churches:</b> Lugdunum and Vienne. With + an Appendix containing Tertullian's Address to Martyrs and the Passion of + St. Perpetua. Translated, with Introduction and Notes, by Ven. Archdeacon + <span class="sc">T. H. Bindley</span>, D.D. <i>1s.</i></p> + + <p><b>The Epistles of St. Ignatius, Bishop of Antioch.</b> By Rev. <span + class="sc">J. H. Srawley</span>, M.A. In two volumes. <i>1s.</i> + each.</p> + + <p><b>The Liturgy of the Eighth Book of "the Apostolic + Constitutions,"</b> commonly called the Clementine Liturgy. Translated + into English, with Introduction and Notes, by Rev. <span class="sc">R. H. + Cresswell</span>, M.A. <i>1s. 6d.</i></p> + + <p><b>The Shepherd of Hermas.</b> By the late Rev. <span class="sc">C. + Taylor</span>, D.D. Vols. I. and II. <i>2s.</i> each.</p> + +</blockquote> + + <p><br style="clear:both" /></p> +<hr class="short" /> + +<h3>NON-CHRISTIAN RELIGIOUS SYSTEMS.</h3> + +<p class="cenhead">Fcap. 8vo, cloth boards, 2s. 6d. each.</p> + +<blockquote class="b1n"> + + <p><b>Buddhism:</b> being a sketch of the Life and Teachings of Gautama, + the Buddha. By <span class="sc">T. W. Rhys Davids</span>, M.A. With + Map.</p> + + <p><b>Buddhism in China.</b> By the Rev. <span class="sc">S. Beal</span>. + With Map.</p> + + <p><b>Confucianism and Taouism.</b> By Sir <span class="sc">Robert K. + Douglas</span>, of the British Museum. With Map.</p> + + <p><b>Hinduism.</b> By the late Sir <span class="sc">M. + Monier-Williams</span>, M.A., D.C.L. With Map.</p> + + <p><b>Islam and its Founder.</b> By <span class="sc">J. W. H. + Stobart</span>. With Map.</p> + + <p><b>Islam as a Missionary Religion.</b> By <span class="sc">Charles R. + Haines</span>. <i>2s.</i></p> + + <p><b>The Coran:</b> its Composition and Teaching, and the Testimony it + bears to the Holy Scriptures. By <span class="sc">Sir William + Muir</span>, K.C.S.I.</p> + + <p><b>The Historical Development of the Qurán.</b> By the Rev. <span + class="sc">Edward Sell</span>, D.D., M.R.A.S.</p> + + <p><b>The Religion of the Crescent, or Islam:</b> its Strength, its + Weakness, its Origin, its Influence. By the Rev. <span class="sc">W. St. + Clair Tisdall</span>, M.A. <i>4s.</i></p> + + <p><b>Studies of Non-Christian Religions.</b> By <span class="sc">Eliot + Howard</span>.</p> + +</blockquote> + + <p><br style="clear:both" /></p> +<hr class="short" /> + +<h3>COLONIAL CHURCH HISTORIES.</h3> + +<p class="cenhead">Fcap. 8vo, with Map, cloth boards.</p> + +<blockquote class="b1n"> + + <p><b>Diocese of Mackenzie River,</b> by the Right Rev. <span + class="sc">W. C. Bompas</span>, D.D., Bishop of the Diocese. + <i>2s.</i></p> + + <p><b>New Zealand,</b> by the late Very Rev. <span class="sc">Henry + Jacobs</span>, D.D., Dean of Christchurch. Containing the Dioceses of + Auckland, Christchurch, Dunedin, Nelson, Waiapu, Wellington and + Melanesia. <i>5s.</i></p> + + <p><b>History of the Church in Eastern Canada and Newfoundland,</b> by + the Rev. <span class="sc">J. Langtry</span>. <i>3s.</i></p> + + <p><b>The Church in the West Indies,</b> by the Rev. <span class="sc">A. + Caldecott</span>, B.D. <i>3s. 6d.</i></p> + + <p><b>The Story of the Australian Church,</b> by the Rev. <span + class="sc">E. Symonds</span>. <i>2s. 6d.</i></p> + +</blockquote> + + <p><br style="clear:both" /></p> +<hr class="short" /> + +<p class="cenhead">LONDON: NORTHUMBERLAND AVENUE, W.C.<br /> +43 QUEEN VICTORIA STREET, E.C.</p> + + <p><br style="clear:both" /></p> +<hr class="full" /> + +<h3>Notes</h3> + +<div class="note"> + <p><a name="Nt1" href="#NtA1">[1]</a> The <i>Operatives' Lecture</i> is + always well advertised in the streets beforehand by large posters.</p> + + <p><a name="Nt2" href="#NtA2">[2]</a> Bulwer Lytton's <i>Coming + Race</i>.</p> + + <p><a name="Nt3" href="#NtA3">[3]</a> The glass vessel ought to be + broader in comparison with its height.</p> + + <p><a name="Nt4" href="#NtA4">[4]</a> In 1746 Benjamin Robins taught the + principles of rifling as we know them now. He showed that the <i>spin</i> + of the round bullet was the most important thing to consider. He showed + that even the bent barrel of a gun did not deflect the bullet to anything + like the extent that the spin of the bullet made it deflect in the + opposite direction.</p> + + <p><a name="Nt5" href="#NtA5">[5]</a> <span + class="sc">Note.</span>—In Fig. 16 the axis is shown inclined, but, + only that it would have been more troublesome to illustrate, I should + have preferred to show the precession occurring when the axis keeps + horizontal.</p> + + <p><a name="Nt6" href="#NtA6">[6]</a> When this lecture containing the + above statement was in the hands of the printers, I was directed by Prof. + Fitzgerald to the late Prof. Jellet's <i>Treatise on the Theory of + Friction</i>, published in 1872, and there at page 18 I found the + mathematical explanation of the rising of a top.</p> + + <p><a name="Nt7" href="#NtA7">[7]</a> Roughly, the <i>Inertia</i> or + <i>Mass</i> of a body expresses its resistance to change of mere + translational velocity, whereas, the <i>Moment of Inertia</i> of a body + expresses its resistance to change of rotational velocity.</p> + + <p><a name="Nt8" href="#NtA8">[8]</a> It is a very unlikely, and + certainly absurd-looking, hypothesis, but it seems that it is not + contradicted by any fact in spectrum analysis, or even by any probable + theory of the constitution of the interstellar ether, that the stars are + merely images of our own sun formed by reflection at the boundaries of + the ether.</p> + + <p><a name="Nt9" href="#NtA9">[9]</a> Sir William Thomson has performed + this.</p> + + <p><a name="Nt10" href="#NtA10">[10]</a> It must be remembered that in + one case I speak of the true north, and in the other of the magnetic + north.</p> + + <p><a name="Nt11" href="#NtA11">[11]</a> Rotating a large mass of iron + rapidly in one direction and then in the other in the neighbourhood of a + delicately-suspended magnetic needle, well protected from air currents, + ought, I think, to give rise to magnetic phenomena of very great interest + in the theory of magnetism. I have hitherto failed to obtain any trace of + magnetic action, but I attribute my failure to the comparatively slow + speed of rotation which I have employed, and to the want of delicacy of + my magnetometer.</p> + + <p><a name="Nt12" href="#NtA12">[12]</a> I had applied for a patent for + this system of signalling some time before the above words were spoken, + but although it was valid I allowed it to lapse in pure shame that I + should have so unblushingly patented the use of the work of Fitzgerald, + Hertz, and Lodge.</p> + + <p><a name="Nt13" href="#NtA13">[13]</a> How to see by electricity is + perfectly well known, but no rich man seems willing to sacrifice the few + thousands of pounds which are necessary for making the apparatus. If I + could spare the money and time I would spend them in doing this + thing—that is, I think so—but it is just possible that if I + could afford to throw away three thousand pounds, I might feel greater + pleasure in the growth of a great fortune than in any other natural + process.</p> + + <p><a name="Nt14" href="#NtA14">[14]</a> Probably first described by Mr. + Brennan.</p> + + <p><a name="Nt15" href="#NtA15">[15]</a> The weight of Mr. Brennan's + loaded wagon is 313 lb., including gyrostats and storage cells. His two + wheels weigh 13 lb. If made of nickel steel and run at their highest safe + speed they would weigh much less.</p> + +</div> + + + + + + + +<pre> + + + + + +End of the Project Gutenberg EBook of Spinning Tops, by John Perry + +*** END OF THIS PROJECT GUTENBERG EBOOK SPINNING TOPS *** + +***** This file should be named 34268-h.htm or 34268-h.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/3/4/2/6/34268/ + +Produced by Chris Curnow, Keith Edkins and the Online +Distributed Proofreading Team at https://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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