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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:12:36 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:12:36 -0700 |
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clear: right; margin-right: 0; margin-bottom: 0em; margin-top: + 1em; margin-left: 1.0em; padding: 0; text-align: center;} + + .footnote {margin-left: 10%; margin-right: 10%; font-size: 0.90em; } + .footnote .label {position: absolute; left: 14%; text-align: left; font-size: .9em; text-decoration: none;} + .fnanchor {vertical-align: super; line-height: .5em; font-size: .8em; text-decoration: none;} + + .tn {border: 1px solid black; margin-left: 25%; width: 30%; + font-size: .8em; padding: 0.5em 1em 0.5em 1em; text-align: justify;} + + .poem {margin-left: 5em; margin-right: 2em; text-align: left; font-size: 0.85em;} + .poem .stanza {margin: 1em 0em 1em 0em;} + .poem span.i0 {display: block; margin-left: 0em; padding-left: 3em; text-indent: -3em;} + + </style> +</head> +<body> + + +<pre> + +The Project Gutenberg EBook of Lord Kelvin, by Andrew Gray + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org/license + + +Title: Lord Kelvin + An account of his scientific life and work + +Author: Andrew Gray + +Release Date: April 4, 2012 [EBook #39373] + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK LORD KELVIN *** + + + + +Produced by Laura Wisewell, Turgut Dincer, Tamise Totterdell +and the Online Distributed Proofreading Team at +http://www.pgdp.net (The original copy of this book was +generously made available for scanning by the Department +of Mathematics at the University of Glasgow.) + + + + + + +</pre> + + +<div class="figcenter" style="width: 640px;"> +<div class="figleft" style="width: 101px;"> +<img src="images/spine.jpg" width="101" height="570" alt="Spine" title="Spine" /> +</div> +<div class="figright" style="width: 480px;"> +<img src="images/cover.jpg" width="480" height="572" alt="Cover" title="Cover" /> +</div> +</div> +<div style="clear: both;"></div> + + +<p> </p> + +<p style='text-align:left; margin-left:35%'>ENGLISH<br /> +MEN OF SCIENCE</p> + +<p> </p> + +<p class="center"><small>EDITED BY</small><br /> +<big>J. REYNOLDS GREEN, Sc.D.</big></p> + +<p> </p> +<p> </p> + +<p class="center"><big><b>LORD KELVIN</b></big></p> + +<p> </p> + + +<div class="tn"> +<p><big><i>ENGLISH MEN<br /> +OF SCIENCE</i></big></p> + +<p><span class="smcap">Edited by</span></p> + +<p><span class="smcap">Dr. J. REYNOLDS GREEN.</span></p> + +<p> +<i>With Photogravure Frontispiece.<br /> +Small Cr. 8vo, 2s. 6d. net per vol.</i><br /> +</p> + +<p>SPENCER. By <span class="smcap">J. Arthur +Thompson</span>.</p> + +<p>PRIESTLEY. By Dr. <span class="smcap">Thorpe</span>, +C.B., F.R.S.</p> + +<p>FLOWER. By Prof. <span class="smcap">R. Lydekker</span>, +F.R.S.</p> + +<p>HUXLEY. By Prof. <span class="smcap">Ainsworth +Davis</span>.</p> + +<p>BENTHAM. By <span class="smcap">B. Daydon +Jackson</span>, F.L.S.</p> + +<p>DALTON. By <span class="smcap">J. P. Millington</span>, +M.A.</p> + +<p><i>J. M. DENT & CO.</i></p> +</div> + +<p> </p> +<h5><i>All Rights Reserved</i></h5> + + + +<div class="figcenter" style="width: 550px; position: relative;"> +<a name="frontis" id="frontis"></a> +<img src="images/frontis.jpg" width="550" height="812" alt="Lord Kelvin" title="Lord Kelvin" /> +</div> + + + +<div class="figcenter" style="width: 550px; position: relative;"> +<img src="images/title.jpg" width="550" height="844" alt="Title" title="Title" /> +</div> + + +<hr style='width: 70%'/> + + +<h1>LORD KELVIN</h1> + +<h5><big><i>AN ACCOUNT OF HIS SCIENTIFIC<br /> +LIFE AND WORK</i></big></h5> + +<p> </p> +<h6>BY</h6> + +<h3>ANDREW GRAY</h3> +<h5>LL.D., F.R.S., V.-P.R.S.E.</h5> + +<h6>PROFESSOR OF NATURAL PHILOSOPHY IN THE<br /> +UNIVERSITY OF GLASGOW</h6> + +<div class="figcenter" style="width: 100px;"> +<a href="images/title.jpg"> + <img src="images/mark.png" width="100" height="157" alt="printer's mark" title="Click for Original Title Page" /> +</a> +</div> + +<p class="center"><small> +PUBLISHED IN LONDON BY<br /> +J. M. DENT & CO., AND IN NEW<br /> +YORK BY E. P. DUTTON & CO.<br /> +</small></p> + +<h5>1908</h5> + +<p> </p> +<hr style='width: 70%'/> +<p> </p> + +<h6> +<span class="smcap">Richard Clay & Sons</span>, <span class="smcap">Limited</span>,<br /> +BREAD STREET HILL, E.C., AND<br /> +BUNGAY, SUFFOLK.</h6> + +<p> </p> +<hr style='width: 70%'/> +<p><span class='pagenum'><a name="Page_v" id="Page_v">v</a></span></p> + +<h3>PREFACE</h3> + +<p>This book makes no claim to be a biography of Lord +Kelvin in the usual sense. It is an extension of an +article which appeared in the <i>Glasgow Herald</i> for +December 19, 1907, and has been written at the +suggestion of various friends of Lord Kelvin, in the +University of Glasgow and elsewhere, who had read +that article. The aim of the volume is to give an +account of Lord Kelvin's life of scientific activity, and +to explain to the student, and to the general reader +who takes an interest in physical science and its applications, +the nature of his discoveries. Only such a +statement of biographical facts as seems in harmony +with this purpose is attempted. But I have ventured, +as an old pupil and assistant of Lord Kelvin, to sketch +here and there the scene in his class-room and laboratory, +and to record some of the incidents of his teaching and +work.</p> + +<p>I am under obligations to the proprietors of the +<i>Glasgow Herald</i> for their freely accorded permission to +make use of their article, and to Messrs. Annan, photographers, +Glasgow, and Messrs. James MacLehose & +Sons, Glasgow, for the illustrations which are given, +and which I hope may add to the interest of the book.</p> + +<p class="quotsig"><span class="smcap">A. Gray</span>.</p> + +<p class="small"><i>The University, Glasgow,<br /> + May</i> 20, 1908.</p> + +<p><span class='pagenum'><a name="Page_vi" id="Page_vi">vi</a></span></p> + +<hr style='width: 70%'/> +<p><span class='pagenum'><a name="Page_vii" id="Page_vii">vii</a></span></p> + +<h3>CONTENTS</h3> + +<table width="70%" summary="contents" cellpadding="10" cellspacing="10"> +<tr> +<td class="right10t"><small>CHAP.</small></td> +<td class="left80"> </td> +<td class="right10"><small>PAGE</small></td> +</tr> +<tr> +<td class="right10t">I.</td> +<td class="left80">PARENTAGE AND EARLY EDUCATION</td> +<td class="right10"><a href="#Page_1">1</a></td> +</tr> +<tr> +<td class="right10t">II.</td> +<td class="left80">CLASSES AT THE UNIVERSITY OF GLASGOW. FIRST SCIENTIFIC PAPERS</td> +<td class="right10"><a href="#Page_13">13</a></td> +</tr> +<tr> +<td class="right10t">III.</td> +<td class="left80">UNIVERSITY OF CAMBRIDGE. SCIENTIFIC WORK AS UNDERGRADUATE</td> +<td class="right10"><a href="#Page_23">23</a></td> +</tr> +<tr> +<td class="right10t">IV.</td> +<td class="left80">THE MATHEMATICAL THEORY OF ELECTRICITY +IN EQUILIBRIUM. ELECTRIC IMAGES. ELECTRIC INVERSION</td> +<td class="right10"><a href="#Page_33">33</a></td> +</tr> +<tr> +<td class="right10t">V.</td> +<td class="left80">THE CHAIR OF NATURAL PHILOSOPHY AT +GLASGOW. ESTABLISHMENT OF THE +FIRST PHYSICAL LABORATORY</td> +<td class="right10"><a href="#Page_61">61</a></td> +</tr> +<tr> +<td class="right10t">VI.</td> +<td class="left80">FRIENDSHIP WITH STOKES AND JOULE. +EARLY WORK AT GLASGOW</td> +<td class="right10"><a href="#Page_79">79</a></td> +</tr> +<tr> +<td class="right10t">VII.</td> +<td class="left80">THE 'ACCOUNT OF CARNOT'S THEORY OF +THE MOTIVE POWER OF HEAT'—TRANSITION +TO THE DYNAMICAL THEORY OF HEAT</td> +<td class="right10"><a href="#Page_99">99</a></td> +</tr> +<tr> +<td class="right10t">VIII.</td> +<td class="left80">THERMODYNAMICS AND ABSOLUTE THERMOMETRY</td> +<td class="right10"><a href="#Page_114">114</a></td> +</tr> +<tr> +<td class="right10t">IX.</td> +<td class="left80">HYDRODYNAMICS—DYNAMICAL THEOREM +OF MINIMUM ENERGY—VORTEX MOTION</td> +<td class="right10"><a href="#Page_153">153</a></td> +</tr> +<tr> +<td class="right10t">X.</td> +<td class="left80">THE ENERGY THEORY OF ELECTROLYSIS—ELECTRICAL +UNITS—ELECTRICAL OSCILLATIONS</td> +<td class="right10"><a href="#Page_176">176</a></td> +</tr> +<tr> +<td class="right10t">XI.</td> +<td class="left80">THOMSON AND TAIT'S 'NATURAL PHILOSOPHY'—GYROSTATIC +ACTION—'ELECTROSTATICS +AND MAGNETISM'<span class='pagenum'><a name="Page_viii" id="Page_viii">viii</a></span></td> +<td class="right10"><a href="#Page_194">194</a></td> +</tr> +<tr> +<td class="right10t">XII.</td> +<td class="left80">THE AGE OF THE EARTH</td> +<td class="right10"><a href="#Page_229">229</a></td> +</tr> +<tr> +<td class="right10t">XIII.</td> +<td class="left80">BRITISH ASSOCIATION COMMITTEE ON +ELECTRICAL STANDARDS</td> +<td class="right10"><a href="#Page_244">244</a></td> +</tr> +<tr> +<td class="right10t">XIV.</td> +<td class="left80">THE BALTIMORE LECTURES</td> +<td class="right10"><a href="#Page_254">254</a></td> +</tr> +<tr> +<td class="right10t">XV.</td> +<td class="left80">SPEED OF TELEGRAPH SIGNALLING—LAYING +OF SUBMARINE CABLES—TELEGRAPH +INSTRUMENTS—NAVIGATIONAL INSTRUMENTS, +COMPASS AND SOUNDING MACHINE</td> +<td class="right10"><a href="#Page_264">264</a></td> +</tr> +<tr> +<td class="right10t">XVI.</td> +<td class="left80">LORD KELVIN IN HIS CLASS-ROOM AND +LABORATORY</td> +<td class="right10"><a href="#Page_279">279</a></td> +</tr> +<tr> +<td class="right10t">XVII.</td> +<td class="left80">PRACTICAL ACTIVITIES—HONOURS AND +DISTINCTIONS—LAST ILLNESS AND DEATH</td> +<td class="right10"><a href="#Page_299">299</a></td> +</tr> +<tr> +<td class="right10t"> </td> +<td class="left80">CONCLUSION</td> +<td class="right10"><a href="#Page_305">305</a></td> +</tr> +<tr> +<td class="right10t"> </td> +<td class="left80">INDEX</td> +<td class="right10"><a href="#Page_317">317</a></td> +</tr> +</table> + +<hr style='width: 10%'/> + +<h4>CORRIGENDUM</h4> + +<p class="center"><a href="#Page_105">Page 105</a>, line 9 from foot, for +<i>∂</i><i>e</i> + <i>O</i> read <i>∂</i><i>e</i> + o</p> +<p><span class='pagenum'><a name="Page_ix" id="Page_ix">ix</a></span></p> + +<hr style='width: 70%'/> +<p><span class='pagenum'><a name="Page_x" id="Page_x">x</a></span></p> + +<h3>LIST OF ILLUSTRATIONS</h3> + +<table width="70%" summary="illustrations" cellpadding="10" cellspacing="10"> +<tr> +<td class="right10t"> </td> +<td class="right10" colspan="2"><small><i>To face page</i></small></td> +</tr> +<tr> +<td class="left80" colspan="2"><span class="smcap">Lord Kelvin</span> (<i>photogravure</i>)</td> +<td class="right10"><a href="#frontis"><i>Frontispiece</i></a></td> +</tr> +<tr> +<td class="left80" colspan="2"><span class="smcap">Lord Kelvin in 1846</span></td> +<td class="right10"><a href="#thomson">64</a></td> +</tr> +<tr> +<td class="left80" colspan="2"><span class="smcap">View of Old College</span></td> +<td class="right10"><a href="#college">70</a></td> +</tr> +</table> + + +<hr style='width: 70%'/> +<p><span class='pagenum'><a name="Page_1" id="Page_1">1</a></span></p> + +<h1>LORD KELVIN</h1> + +<h3>CHAPTER I</h3> + +<h4>PARENTAGE AND EARLY EDUCATION</h4> + +<p>Lord Kelvin came of a stock which has helped +to give to the north of Ireland its commercial and +industrial supremacy over the rest of that distressful +country. His ancestors were county Down agriculturists +of Scottish extraction. His father was James +Thomson, the well-known Glasgow Professor of +Mathematics, and author of mathematical text-books +which at one time were much valued, and are even +now worth consulting. James Thomson was born on +November 13, 1786, near Ballynahinch, county Down. +Being the son of a small farmer he was probably +unable to enter on university studies at the usual age, +for he did not matriculate in Scotland until 1810. +The class-lists of the time show that he distinguished +himself highly in mathematics, natural philosophy, and +classics.</p> + +<p>An interesting incident of these student days of +his father was related by Lord Kelvin in his installation +address as Chancellor of the University in +1904, and is noteworthy as indicating how comparatively +recent are many of the characteristics of our +<span class='pagenum'><a name="Page_2" id="Page_2">2</a></span> +present-day life and commerce. James Thomson and +some companions, walking from Greenock to Glasgow, +on their way to join the college classes at the commencement +of the session, "saw a prodigy—a black chimney +moving rapidly beyond a field on the left-hand side of +their road. They jumped the fence, ran across the +field, and saw, to their astonishment, Henry Bell's +'Comet' (then not a year old) travelling on the Clyde +between Glasgow and Greenock."<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">1</a> Sometimes then +the passage from Belfast to Greenock took a long time. +Once James Thomson, crossing in an old lime-carrying +smack, was three or four days on the way, in the course +of which the vessel, becalmed, was carried three times +by the tide round Ailsa Craig.</p> + +<p>Mr. Thomson was elected in 1815 to the Professorship +of Mathematics in the Royal Academical +Institution of Belfast, and held the post for seventeen +years, building up for himself an excellent reputation +as a teacher, and as a clear and accurate writer. Just +then analytical methods were beginning to supersede +the processes of geometrical demonstration which the +form adopted by Newton for the <i>Principia</i> had tended +to perpetuate in this country. Laplace was at the +height of his fame in France, and was writing the great +analytical <i>Principia</i>, his <i>Mécanique Céleste</i>, applying the +whole force of his genius, and all the resources of the +differential and integral calculus invented by Newton +and improved by the mathematicians of the intervening +century, to the elucidation and extension of the +"system of the world," which had been so boldly +sketched by the founder of modern physical science.</p> +<p><span class='pagenum'><a name="Page_3" id="Page_3">3</a></span></p> +<p>In that period Fourier wrote his memoirs on the conduction +of heat, and gave to the world his immortal +book to be an inspiration to the physical philosophers +of succeeding generations. Legendre had written +memoirs which were to lead, in the hands of Jacobi +and his successors, to a new province of mathematics, +while, in Germany, Gauss had begun his stately march +of discovery.</p> + +<p>The methods and results of this period of mathematical +activity were at first hardly known in this +country: the slavish devotion of Cambridge to the +geometrical processes and the fluxional notation of +Newton, an exclusive partiality which Newton himself +would have been the first to condemn, led analytical +methods, equally Newtonian, to be stigmatised as +innovations, because clothed in the unfamiliar garb of +the continental notation. A revolt against this was +led by Sir John Herschel, Woodhouse, Peacock, and +some others at Cambridge, who wrote books which +had a great effect in bringing about a change of methods. +Sir John thus described the effect of the new movements:—"Students +at our universities, fettered by +no prejudices, entangled by no habits, and excited by +the ardour and emulation of youth, had heard of the +existence of masses of knowledge from which they +were debarred by the mere accident of position. They +required no more. The prestige which magnifies what +is unknown, and the attractions inherent in what is +forbidden, coincided in their impulse. The books +were procured and read, and produced their natural +effects. The brows of many a Cambridge examiner +were elevated, half in ire, half in admiration, at the +unusual answers which began to appear in examination<span class='pagenum'><a name="Page_4" id="Page_4">4</a></span> +papers. Even moderators are not made of impenetrable +stuff, though fenced with sevenfold Jacquier, +and tough bull-hide of Vince and Wood."</p> + +<p>The memoirs and treatises of the continental +analysts were eagerly procured and studied by James +Thomson, and as he was bound by no examination +traditions, he freely adopted their methods, so far as +these came within the scope of his teaching, and made +them known to the English reading public in his text-books. +Hence when the chair of Mathematics at +Glasgow became vacant in 1832 by the death of +Mr. James Millar, Mr. Thomson was at once chosen +by the Faculty, which at that time was the electing +body.</p> + +<p>The Faculty consisted of the Principal and the +Professors of Divinity, Church History, Oriental +Languages, Natural Philosophy, Moral Philosophy, +Mathematics, Logic, Greek, Humanity, Civil Law, +Practice of Medicine, Anatomy, and Practical Astronomy. +It administered the whole revenues and +property of the College, and possessed the patronage +of the above-named chairs with the exception of +Church History, Civil Law, Medicine, Anatomy, and +Astronomy, so that Mr. Thomson became not only +Professor of Mathematics, but also, in virtue of his +office, a member of what was really the supreme +governing body of the University. The members of +the Faculty, with the exception of the Professor of +Astronomy, who resided at the observatory, were +provided with official residences in the College. This +arrangement is still adhered to; though now the government +is in the hands of a University Court, with the +Senate (which formerly only met to confer degrees or<span class='pagenum'><a name="Page_5" id="Page_5">5</a></span> +to manage the library and some other matters) to +regulate and superintend teaching and discipline.</p> + +<p>Professor Thomson was by no means the first or +the only professor of the name in the University of +Glasgow, as the following passage quoted from a letter +of John Nichol, son of Dr. J. P. Nichol, and first +Professor of English at Glasgow, amusingly testifies:—</p> + +<p>"Niebuhr, after examining a portion of the <i>Fasti +Consulares</i>, arrived at the conclusion that the <i>senatus +populusque Romanus</i> had made a compact to elect every +year a member of the Fabian house to one of the +highest offices of state, so thickly are the records studded +with the name of the Fabii. Some future Niebuhr +of the New Zealand Macaulay imagines, turning his +attention to the annals of Glasgow College, will undoubtedly +arrive at the conclusion that the leaders of +that illustrious corporation had, during the period of +which I am writing, become bound in a similar +manner to the name of Thomson. Members of that +great <i>gens</i> filled one-half of the chairs in the University. +I will not venture to say how many I have known. +There was Tommy Thomson the chemist; William +Thomson of Materia Medica; Allen Thomson of +Anatomy, brother of the last; Dr. James Thomson +of Mathematics; William, his son, etc., etc. Old +Dr. James was one of the best of Irishmen, a good +mathematician, an enthusiastic and successful teacher, +the author of several valuable school-books, a friend of +my father's, and himself the father of a large family, +the members of which have been prosperous in the +world. They lived near us in the court, and we made +a pretty close acquaintanceship with them all."</p> + +<p>A former Professor of Natural Philosophy, Dr.<span class='pagenum'><a name="Page_6" id="Page_6">6</a></span> +Anderson,<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">2</a> who appears to have lived the closing years +of his life in almost constant warfare with his colleagues +of the Faculty, and who established science classes for +workmen in Glasgow, bequeathed a sum of money to +set up a college in Glasgow in which such classes might +be carried on. The result was the foundation of what +used to be called the "Andersonian University" in +George Street, the precursor of the magnificent Technical +College of the present day. This name, and the +large number of Thomsons who had been and were still +connected with the University of Glasgow, caused the +more ancient institution to be not infrequently referred +to as the "Thomsonian University"!</p> + +<p>The Thomas Thomson (no relative of the Belfast +Thomsons) affectionately, if a little irreverently, mentioned +in the above quotation, was then the Professor +of Chemistry. He was the first to establish a chemical +laboratory for students in this country; indeed, his +laboratory preceded that of Liebig at Giessen by some +years, and it is probable that as regards experimental +chemistry Glasgow was then in advance of the rest +of the world. His pupil and life-long admirer was +destined to establish the first physical laboratory for +such students as were willing to spend some time in +the experimental investigation and verification of +physical principles, or to help the professor in his +researches. The systematic instruction of students in +methods of experimenting by practical exercises with +apparatus was a much later idea, and this fact must be +<span class='pagenum'><a name="Page_7" id="Page_7">7</a></span>taken account of when the laboratories of the present +time are contrasted with the much more meagre +provision of those early days. The laboratory is now, +as much as the lecture-room, the place where classes +are held and instruction given in experimental science +to crowds of students, and it is a change for the better.</p> + +<p>The arrival of James Thomson and his family at +Glasgow College, in 1832, was remarked at the time +as an event which brought a large reinforcement to +the <i>gens</i> already inseparably associated with the place: +how great were to be its consequences not merely to +the University but to the world at large nobody can +then have imagined. His family consisted of four sons +and two daughters: his wife, Margaret Gardner, +daughter of William Gardner, a merchant in Glasgow, +had died shortly before, and the care of the family was +undertaken by her sister, Mrs. Gall. The eldest son, +James Thomson, long after to be Rankine's successor +in the Chair of Engineering, was ten years of age and +even then an inveterate inventor; William, the future +Lord Kelvin (born June 26, 1824), was a child of eight. +Two younger sons were John (born in 1826)—who +achieved distinction in Medicine, became Resident +Assistant in the Glasgow Royal Infirmary, and died +there of a fever caught in the discharge of his duty—and +Robert, who was born in 1829, and died in +Australia in 1905. Besides these four sons there were +in all three daughters:—Elizabeth, afterwards wife of +the Rev. David King, D.D.; Anna, who was married +to Mr. William Bottomley of Belfast (these two were +the eldest of the family), and Margaret, the youngest, +who died in childhood. Thus began William Thomson's +residence in and connection with the University of<span class='pagenum'><a name="Page_8" id="Page_8">8</a></span> +Glasgow, a connection only terminated by the funeral +ceremony in Westminster Abbey on December 23, +1907.</p> + +<p>Professor Thomson himself carefully superintended +the education of his sons, which was carried out at +home. They were well grounded in the old classical +languages, and moreover received sound instruction in +what even now are called, but in a somewhat disparaging +sense, modern subjects. As John Nichol has +said in his letters, "He was a stern disciplinarian, and +did not relax his discipline when he applied it to his +children, and yet the aim of his life was their +advancement."</p> + +<p>It would appear from John Nichol's recollections +that even in childhood and youth, young James Thomson +was an enthusiastic experimentalist and inventor, +eager to describe his ideas and show his models to a +sympathetic listener.<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">3</a> And both then and in later +years his charming simplicity, his devouring passion +for accuracy of verbal expression in all his scientific +writing and teaching, and his unaffected and unconscious +genius for the invention of mechanical appliances, +all based on true and intuitively perceived physical +principles, showed that if he had had the unrelenting +power of ignoring accessories and unimportant details +which was possessed by his younger brother, he might +have accomplished far more than he did, considerable as +that was. But William had more rapid decision, and +though careful and exact in expressing his meaning, +<span class='pagenum'><a name="Page_9" id="Page_9">9</a></span>was less influenced by considerations of the errors that +might arise from the various connotations of such +scientific terms as are also words in common use; and +he quickly completed work which his brother would +have pondered over for a long time, and perhaps never +finished.</p> + +<p>It is difficult for a stranger to Glasgow, or even for +a resident in Glasgow in these days of quick and frequent +communication with England, and for that +matter with all parts of the world, to form a true idea +of life and work at the University of Glasgow seventy +years ago. The University had then its home in the +old "tounis colledge" in the High Street, where +many could have wished it to remain, and, extending +its buildings on College Green, retain the old and +include the new. Its fine old gateway, and part of one +of the courts, were still a quaint adornment of the +somewhat squalid street in 1871, after the University +had moved to its present situation on the windy top of +Gilmorehill. Deserted as it was, its old walls told +something of the history of the past, and reminded the +passer-by that learning had flourished amid the shops +and booths of the townspeople, and that students and +professors had there lived and worked within sound of +the shuttle and the forge. The old associations of a +town or a street or a building, linked as they often are +with the history of a nation, are a valuable possession, +not always placed in the account when the advantages +or disadvantages of proposed changes are discussed; +but a University which for four hundred years +has seen the tide of human life flow round it in a great +city, is instinct with memories which even the demolition +of its walls can only partially destroy. Poets and<span class='pagenum'><a name="Page_10" id="Page_10">10</a></span> +statesmen, men of thought and men of action, lords +and commoners, rich men's sons and the children of +farmers, craftsmen and labourers, had mingled in its +classes and sat together on its benches; and so had +been brought about a community of thought and feeling +which the practice of our modern and wealthy +cosmopolites, who affect to despise nationality, certainly +does nothing to encourage. In the eighteenth +century the Provosts and the Bailies of the time still +dwelt among men and women in the High Street, +and its continuation the Saltmarket, or not far off +in Virginia Street, the home of the tobacco lords +and the West India merchants. Their homely +hospitality, their cautious and at the same time splendid +generosity, their prudent courage, and their faithful +and candid friendships are depicted in the pages of +Scott; and though a change in men and manners, not +altogether for the better, has been gradually brought +about by sport and fashion, those peculiarly Scottish +virtues are still to be found in the civic statesmen and +merchant princes of the Glasgow of to-day. Seventy +years ago the great migration of the well-to-do towards +the west had commenced, but it had but little interfered +with the life of the High Street or of the College. +Now many old slums besides the Vennel and the +Havannah have disappeared, much to the credit of +the Corporation of Glasgow; and, alas, so has every +vestige of the Old College, much to the regret of +all who remember its quaint old courts. A railway +company, it is to be supposed, dare not possess an +artistic soul to be saved; and therefore, perhaps, it +is that it builds huge and ugly caravanserais of +which no one, except perhaps the shareholders, would<span class='pagenum'><a name="Page_11" id="Page_11">11</a></span> +keenly regret the disappearance. But both artists +and antiquaries would have blessed the directors—and +such a blessing would have done them no harm—if +they had been ingenious and pious enough to leave +some relic of the old buildings as a memorial of the +old days and the old life of the High Street.</p> + +<p>A picture of the College in the High Street has +recently been drawn by one who lived and worked in +it, though some thirty years after James Thomson +brought his family to live in its courts. Professor G. +G. Ramsay has thus portrayed some features of the +place, which may interest those who would like to +imagine the environment in which Lord Kelvin grew +up from childhood, until, a youth of seventeen, he left +Glasgow for Cambridge.<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">4</a> "There was something in +the very disamenities of the old place that created a +bond of fellowship among those who lived and worked +there, and that makes all old students, to this day, look +back to it with a sort of family pride and reverence. +The grimy, dingy, low-roofed rooms; the narrow, +picturesque courts, buzzing with student-life; the +dismal, foggy mornings and the perpetual gas; the +sudden passage from the brawling, huckstering High +Street into the academic quietude, or the still more +academic hubbub, of those quaint cloisters, into which +the policeman, so busy outside, was never permitted +to penetrate; the tinkling of the 'angry bell' that +made the students hurry along to the door which +was closed the moment that it stopped; the roar +and the flare of the Saturday nights, with the cries +<span class='pagenum'><a name="Page_12" id="Page_12">12</a></span>of carouse or incipient murder which would rise +into our quiet rooms from the Vennel or the Havannah; +the exhausted lassitude of Sunday mornings, +when poor slipshod creatures might be seen, as soon as +the street was clear of churchgoers, sneaking over to the +chemist's for a dose of laudanum to ease off the debauch +of yesterday; the conversations one would have +after breakfast with the old ladies on the other side of +the Vennel, not twenty feet from one's breakfast-table, +who divided the day between smoking short cutty +pipes and drinking poisonous black tea—these sharp +contrasts bound together the College folk and the +College students, making them feel at once part of the +veritable populace of the city, and also hedged off from +it by separate pursuits and interests."</p> + +<p>The university removed in 1871 to larger and more +airily situated buildings in the western part of the city. +Round these have grown up, in the intervening thirty-eight +years, new buildings for most of the great departments +of science, including a separate Institute of +Natural Philosophy, which was opened in April 1907, +by the Prince and Princess of Wales.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_13" id="Page_13">13</a></span></p> + +<h3>CHAPTER II</h3> + +<h4>CLASSES AT THE UNIVERSITY OF GLASGOW. FIRST +SCIENTIFIC PAPERS</h4> + +<p><span class="smcap">In</span> 1834, that is at the age of ten, William Thomson +entered the University classes. Though small in +stature, and youthful even for a time when mere boys +were University students, he soon made himself conspicuous +by his readiness in answering questions, and +by his general proficiency, especially in mathematical +and physical studies. The classes met at that time twice +a day—in mathematics once for lecture and once for +oral examination and the working of unseen examples +by students of the class. It is still matter of tradition +how, in his father's class, William was conspicuous for +the brilliancy of the work he did in this second hour. +His elder brother James and he seem to have gone +through their University course together. In 1834-5 +they were bracketed third in Latin Prose Composition. +In 1835-6 William received a prize for a vacation +exercise—a translation of Lucian's <i>Dialogues of the Gods</i> +"with full parsing of the first three Dialogues." In +1836-7 and 1837-8 the brothers were in the Junior +and Senior Mathematical Classes, and in each year the +first and the second place in the prize-list fell to William +and James respectively. In the second of these years, +William appears as second prizeman in the Logic Class, +while James was third, and John Caird (afterwards<span class='pagenum'><a name="Page_14" id="Page_14">14</a></span> +Principal of the University) was fifth. William and +James Thomson took the first and second prizes in +the Natural Philosophy Class at the close of session +1838-9; and in that year William gained the Class +Prize in Astronomy, and a University Medal for an +Essay on the Figure of the Earth. In 1840-1 he +appears once more, this time as fifth prizeman in the +Senior Humanity Class.</p> + +<p>In his inaugural address as Chancellor of the +University, already quoted above, Lord Kelvin refers +to his teachers in Glasgow College in the following +words:</p> + +<p>"To this day I look back to William Ramsay's +lectures on Roman Antiquities, and readings of Juvenal +and Plautus, as more interesting than many a good +stage play that I have seen in the theatre....</p> + +<p>"Greek under Sir Daniel Sandford and Lushington, +Logic under Robert Buchanan, Moral Philosophy +under William Fleming, Natural Philosophy and +Astronomy under John Pringle Nichol, Chemistry +under Thomas Thomson, a very advanced teacher +and investigator, Natural History under William +Cowper, were, as I can testify by my experience, all +made interesting and valuable to the students of Glasgow +University in the thirties and forties of the nineteenth +century....</p> + +<p>"My predecessor in the Natural Philosophy chair, +Dr. Meikleham, taught his students reverence for the +great French mathematicians Legendre, Lagrange, and +Laplace. His immediate successor in the teaching of +the Natural Philosophy Class,<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">5</a> Dr. Nichol, added +<span class='pagenum'><a name="Page_15" id="Page_15">15</a></span>Fresnel and Fourier to this list of scientific nobles: +and by his own inspiring enthusiasm for the great +French school of mathematical physics, continually +manifested in his experimental and theoretical teaching +of the wave theory of light and of practical +astronomy, he largely promoted scientific study and +thorough appreciation of science in the University of +Glasgow....</p> + +<p>"As far back as 1818 to 1830 Thomas Thomson, +the first Professor of Chemistry in the University of +Glasgow, began the systematic teaching of practical +chemistry to students, and, aided by the Faculty of +Glasgow College, which gave the site and the money +for the building, realised a well-equipped laboratory, +which preceded, I believe, by some years Liebig's +famous laboratory of Giessen, and was, I believe, the +first established of all the laboratories in the world +for chemical research and the practical instruction of +University students in chemistry. <i>That</i> was at a time +when an imperfectly informed public used to regard +the University of Glasgow as a stagnant survival of +mediævalism, and used to call its professors the 'Monks +of the Molendinar'!</p> + +<p>"The University of Adam Smith, James Watt, and +Thomas Reid was never stagnant. For two centuries +and a half it has been very progressive. Nearly two +centuries ago it had a laboratory of human anatomy. +Seventy-five years ago it had the first chemical students' +laboratory. Sixty-five years ago it had the first Professorship +of Engineering of the British Empire. Fifty +years ago it had the first physical students' laboratory—a +deserted wine-cellar of an old professorial house, +enlarged a few years later by the annexation of a<span class='pagenum'><a name="Page_16" id="Page_16">16</a></span> +deserted examination-room. Thirty-four years ago, +when it migrated from its four-hundred-years-old site +off the High Street of Glasgow to this brighter and +airier hill-top, it acquired laboratories of physiology and +zoology; but too small and too meagrely equipped."</p> + +<p>In the summer of 1840 Professor James Thomson +and his two sons went for a tour in Germany. It was +stipulated that German should be the chief, if not the +only, subject of study during the holidays. But William +had just begun to study Fourier's famous book, <i>La +Théorie Analytique de la Chaleur</i>, and took it with him. +He read that great work, full as it was of new theorems +and processes of mathematics, with the greatest delight, +and finished it in a fortnight. The result was his first +original paper "On Fourier's Expansions of Functions in +Trigonometrical Series," which is dated "Frankfort, +July 1840, and Glasgow, April 1841," and was published +in the <i>Cambridge Mathematical Journal</i> (vol. ii, May +1841). The object of the paper is to show in what +cases a function <i>f</i>(<i>x</i>), which is to have certain arbitrary +values between certain values of <i>x</i>, can be expanded in +a series of sines and when in a series of cosines. The +conclusion come to is that, for assigned limits of <i>x</i>, +between 0 and <i>a</i>, say, and for the assigned values of +the function, <i>f</i>(<i>x</i>) can be expressed either as a series +of sines or as a series of cosines. If, however, the +function is to be calculated for any value of <i>x</i>, which +lies outside the limits of that variable between which +the values of the function are assigned, the values of +<i>f</i>(<i>x</i>) there are to be found from the expansion adopted, +by rules which are laid down in the paper.</p> + +<p>Fourier used sine-expansions or cosine-expansions +as it suited him for the function between the limits, +<span class='pagenum'><a name="Page_17" id="Page_17">17</a></span> +and his results had been pronounced to be "nearly all +erroneous." From this charge of error, which was +brought by a distinguished and experienced mathematician, +the young analyst of sixteen successfully vindicated +Fourier's work. Fourier was incontestably +right in holding, though he nowhere directly proved, +that a function given for any value of <i>x</i> between +certain limits, could be expressed either by a sine-series +or by a cosine-series. The divergence of the +values of the two expressions takes place outside these +limits, as has been stated above.</p> + +<p>The next paper is of the same final date, but +appeared in the <i>Cambridge Mathematical Journal</i> of the +following November. In his treatment of the problem +of the cooling of a sphere, given with an arbitrary +initial distribution of temperature symmetrical about +the centre, Fourier assumes that the arbitrary function +<i>F</i>(<i>x</i>), which expresses the temperature at distance <i>x</i> +from the centre, can be expanded in an infinite series +of the form</p> + +<p class="center"> +<i>a</i><sub>1</sub> sin <i>n</i><sub>1</sub><i>x</i> + <i>a</i><sub>2</sub> sin <i>n</i><sub>2</sub><i>x</i> + ... +</p> + +<p>where <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ... are multipliers to be determined +and <i>n</i><sub>1</sub>, <i>n</i><sub>2</sub>, ... are the roots, infinite in number, of +the transcendental equation (<i>tan nX</i>) ⁄ <i>nX</i> = 1 − <i>hX</i>.</p> + +<p>This equation expresses, according to a particular +solution of the differential equation of the flow of heat +in the sphere, the condition fulfilled at the surface, that +the heat reaching the surface by conduction from the +interior in any time is radiated in that time to the +surroundings. Thomson dealt in this second paper +with the possibility of the expansion. He showed that, +inasmuch as the first of the roots of the transcendental<span class='pagenum'><a name="Page_18" id="Page_18">18</a></span> +equation lies between 0 and <sup>1</sup>⁄<sub>2</sub>, the second between +1 and <sup>3</sup>⁄<sub>2</sub>, the third between 2 and <sup>5</sup>⁄<sub>2</sub>, and so on, +with very close approach to the upper limit as the +roots become of high order, the series assumed as +possible has between the given limits of <i>x</i> the same +value as the series</p> + +<p class="center"> +<i>A</i><sub>1</sub> sin <small><sup>1</sup>⁄<sub>2</sub></small> <i>x</i> + <i>A</i><sub>2</sub> sin <small><sup>3</sup>⁄<sub>2</sub></small> <i>x</i> + ... +</p> + +<p>where <i>A</i><sub>1</sub>, <i>A</i><sub>2</sub>, ... are known in terms of <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, ... +Conversely, any series of this form is capable of being +replaced by a series of the form assumed. Further, +a series of the form just written can be made to +represent any arbitrary system of values between the +given limits, and so the possibility of the expansion is +demonstrated.</p> + +<p>The next ten papers, with two exceptions, are all +on the motion of heat, and appeared in the <i>Cambridge +Mathematical Journal</i> between 1841 and 1843, and +deal with important topics suggested by Fourier's +treatise. Of the ideas contained in one or two of +them some account will be given presently.</p> + +<p>Fourier's book was called by Clerk Maxwell, himself +a man of much spirituality of feeling, and no mean +poet, a great mathematical poem. Thomson often +referred to it in similar terms. The idea of the +mathematician as poet may seem strange to some; +but the genius of the greatest mathematicians is akin +to that of the true creative artist, who is veritably +inspired. For such a book was a work of the imagination +as well as of the reason. It contained a new +method of analysis applied with sublime success to +the solution of the equations of heat conduction, an +analysis which has since been transferred to other<span class='pagenum'><a name="Page_19" id="Page_19">19</a></span> +branches of physical mathematics, and has illuminated +them with just those rays which could reveal the +texture and structure of the physical phenomena. +That method and its applications came from Fourier's +mind in full development; he trod unerringly in its +use along an almost unknown path, with pitfalls on +every side; and he reached results which have since +been verified by a criticism searching and keen, and +lasting from Fourier's day to ours. The criticism has +been minute and logical: it has not, it is needless to +say, been poetical.</p> + +<p>Two other great works of his father's collection of +mathematical books, Laplace's <i>Mécanique Céleste</i> and +Lagrange's <i>Mécanique Analytique</i>, seem also to have +been read about this time, and to have made a deep +impression on the mind of the youthful philosopher. +The effect of these books can be easily traced in +Thomson and Tail's <i>Natural Philosophy</i>.</p> + +<p>The study of Fourier had a profound influence on +Thomson's future work, an influence which has +extended to his latest writings on the theory of certain +kinds of waves. His treatment is founded on a strikingly +original use of a peculiar form of solution (given by +Fourier) of a certain fundamental differential equation +in the theory of the flow of heat. It is probable that +William Thomson's earliest predilections as regards +study were in the direction of mathematics rather than +of physics. But the studies of the young mathematician, +for such in a very real and high sense he had +become, were widened and deepened by the interest in +physical things and their explanation aroused by the +lectures of Meikleham, then Professor of Natural +Philosophy, and especially (as Lord Kelvin testified<span class='pagenum'><a name="Page_20" id="Page_20">20</a></span> +in his inaugural address as Chancellor) by the teaching +of J. P. Nichol, the Professor of Astronomy, a man +of poetical imagination and of great gifts of vivid and +clear exposition.</p> + +<p>The <i>Cyclopædia of Physical Science</i> which Dr. Nichol +published is little known now; but the first edition, +published in 1857, to which Thomson contributed +several articles, including a sketch of thermodynamics, +contained much that was new and stimulating to the +student of natural philosophy, and some idea of the +accomplishments of its compiler and author can be +gathered from its perusal. De Morgan's <i>Differential +and Integral Calculus</i> was a favourite book in Thomson's +student days, and later when he was at Cambridge, and +he delighted to pore over its pages before the fire +when the work of the day was over. Long after, +he paid a grateful tribute to De Morgan and his +great work, in the Presidential Address to the British +Association at its Edinburgh Meeting in 1870.</p> + +<p>The next paper which Thomson published, after +the two of which a sketch has been given above, was +entitled "The Uniform Motion of Heat in Homogeneous +Solid Bodies, and its Connection with the +Mathematical Theory of Electricity." It is dated +"Lamlash, August 1841," so that it followed the first +two at an interval of only four months. It appeared +in the <i>Cambridge Mathematical Journal</i> in February +1842, and is republished in the "Reprint of Papers +on Electrostatics and Magnetism." It will always +be a noteworthy paper in the history of physical +mathematics. For although, for the most part, only +known theorems regarding the conduction of heat +were discussed, an analogy was pointed out between<span class='pagenum'><a name="Page_21" id="Page_21">21</a></span> +the distribution of lines of flow and surfaces of equal +temperature in a solid and unequally heated body, with +sources of heat in its interior, and the arrangement of +lines of forces and equipotential surfaces in an insulating +medium surrounding electrified bodies, which +correspond to the sources of heat in the thermal +case. The distribution of lines of force in a space +filled with insulating media of different inductive +qualities was shown to be precisely analogous to that +of lines of flow of heat in a corresponding arrangement +of media of different heat-conducting powers. +So the whole analysis and system of solutions in the +thermal case could be at once transferred to the electrical +one. The idea of the "conduction of lines of +force," as Faraday first and Thomson afterwards called +it, was further developed in subsequent papers, and +threw light on the whole subject of electrostatic force +in the "field" surrounding an electric distribution. +Moreover, it made the subject definite and quantitative, +and not only gave a guide to the interpretation of +unexplained facts, but opened a way to new theorems +and to further investigation.</p> + +<p>This paper contains the extremely important theorem +of the equivalence, so far as external field is concerned, +of any distribution of electricity and a certain +definite distribution, over any equipotential surface, of +a quantity equal to that contained within the surface. +But this general theorem and others contained in the +paper had been anticipated in Green's "Essay on the +Application of Mathematical Analysis to the Theories +of Electricity and Magnetism," in memoirs by Chasles +in Liouville's <i>Journal</i> (vols. iii and v), and in the celebrated +memoir by Gauss "On General Theorems<span class='pagenum'><a name="Page_22" id="Page_22">22</a></span> +relating to Attractive and Repulsive Forces varying +inversely as the Square of the Distance," published in +German in Leipzig in 1840, and in English in Taylor's +<i>Scientific Memoirs</i> in 1842. These anticipations are +again referred to below.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_23" id="Page_23">23</a></span></p> + +<h3>CHAPTER III</h3> + +<h4>UNIVERSITY OF CAMBRIDGE. SCIENTIFIC WORK AS +UNDERGRADUATE</h4> + +<p><span class="smcap">Thomson</span> entered at St. Peter's College, Cambridge, +in October 1841, and began the course of study then +in vogue for mathematical honours. At that time, as +always down almost to the present day, everything +depended on the choice of a private tutor or "coach," +and the devotion of the pupil to his directions, and +on adherence to the subjects of the programme. His +private tutor was William Hopkins, "best of all private +tutors," one of the most eminent of his pupils called +him, a man of great attainment and of distinction as +an original investigator in a subject which had always +deeply interested Thomson—the internal rigidity of +the earth. But the curriculum for the tripos did not +exhaust Thomson's energy, nor was it possible to keep +him entirely to the groove of mastering and writing +out book-work, and to the solution of problems of the +kind dear to the heart of the mathematical examiner. +He wrote original articles for the <i>Cambridge Mathematical +Journal</i>, on points in pure and in applied +mathematics, and read mathematical books altogether +outside the scope of the tripos. Nor did he neglect +athletic exercises and amusements; he won the Colquhoun +Sculls as an oarsman, and was an active member, +and later, during his residence at Cambridge, president<span class='pagenum'><a name="Page_24" id="Page_24">24</a></span> +of the C.U.M.S., the Cambridge University Musical +Society.<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">6</a> The musical instruments he favoured were +the cornet and especially the French horn—he was +second horn in the original Peterhouse band—but +nothing seems to be on record as to the difficulties or +incidents of his practice! Long afterwards, in a few +extremely interesting lectures which he gave annually +on sound, he discoursed on the vibrations of columns +of air in wind instruments, and sometimes illustrated +his remarks by showing how notes were varied in pitch +on the old-fashioned French horn, played with the +hand in the bell, a performance which always intensely +delighted the Natural Philosophy Class.</p> + +<p>At the Jubilee commemoration of the society, 1893, +Lord Kelvin recalled that Mendelssohn, Weber and +Beethoven were the "gods" of the infant association. +Those of his pupils who came more intimately in +contact with him will remember his keen admiration +for these and other great composers, especially Bach, +Mozart, and Beethoven, and his delight in hearing their +works. The Waldstein sonata was a special favourite. +It has been remarked before now, and it seems to be +true, that the music of Bach and Beethoven has had +special attractions for many great mathematicians.</p> + +<p>At Cambridge Thomson made the acquaintance of +George Gabriel Stokes, who graduated as Senior +<span class='pagenum'><a name="Page_25" id="Page_25">25</a></span>Wrangler and First Smith's Prizeman in 1841, and +eight years later became Lucasian Professor of Mathematics +in the University of Cambridge. Their acquaintance +soon ripened into a close friendship, which +lasted until the death of Stokes in 1903. The Senior +Wrangler and the Peterhouse Undergraduate undertook +the composition of a series of notes and papers on +points in pure and physical mathematics which required +clearing up, or putting in a new point of view; +and so began a life-long intercourse and correspondence +which was of great value to science.</p> + +<p>Thomson's papers of this period are on a considerable +variety of subjects, including his favourite subject +of the flux of heat. There are sixteen in all that seem +to have been written and published during his undergraduate +residence at Cambridge. Most of them +appeared in the <i>Cambridge Mathematical Journal</i> between +1842 and 1845; but three appeared in 1845 in +Liouville's <i>Journal de Mathématiques</i>. Four are on +subjects of pure mathematics, such as Dupin's theorem +regarding lines of curvature of orthogonally intersecting +surfaces, the reduction of the general equation +of surfaces of the second order (now called second +degree), six are on various subjects of the theory of +heat, one is on attractions, five are on electrical theory, +and one is on the law of gravity at the surface of a +revolving homogeneous fluid. It is impossible to give +an account of all these papers here. Some of them are +new presentations or new proofs of known theorems, +one or two are fresh and clear statements of fundamental +principles to be used later as the foundation of +more complete statements of mathematical theory; but +all are marked by clearness and vigour of treatment.<span class='pagenum'><a name="Page_26" id="Page_26">26</a></span></p> + +<p>Another paper, published in the form of a letter, of +date October 8, 1845, to M. Liouville, and published +in the <i>Journal de Mathématiques</i> in the same year, +indicates that either before or shortly after taking his +degree, Thomson had invented his celebrated method +of "Electric Images" for the solution of problems of +electric distribution. Of this method, which is one +of the most elegant in the whole range of physical +mathematics, and solves at a stroke some problems, +otherwise almost intractable, we shall give some account +in the following chapter.</p> + +<p>This record of work is prodigious for a student +reading for the mathematical tripos; and it is somewhat +of an irony of fate that such scientific activity is, +on the whole, rather a hindrance than a help in the +preparation for that elaborate ordeal of examination. +Great expectations had been formed regarding Thomson's +performance; hardly ever before had a candidate +appeared who had done so much and so brilliant +original work, and there was little doubt that he would +be easily first in any contest involving real mathematical +power, that is, ability to deal with new problems +and to express new relations of facts in mathematical +language. But the tripos was not a test of +power merely; it was a test also of acquisition, and, to +candidates fairly equal in this respect, also of memory +and of quickness of reproduction on paper of acquired +knowledge.</p> + +<p>The moderators on the occasion were Robert Leslie +Ellis and Harvey Goodwin, both distinguished men. +Ellis had been Senior Wrangler and first Smith's +Prizeman a few years before, and was a mathematician +of original power and promise, who had already<span class='pagenum'><a name="Page_27" id="Page_27">27</a></span> +written memoirs of great merit. Goodwin had been +Second Wrangler when Ellis was Senior, and became +known to a later generation as Bishop of Carlisle. In +a life of Ellis prefixed to a volume of his collected +papers, Goodwin says:—"It was in this year that +Professor W. Thomson took his degree; great expectations +had been excited concerning him, and I remember +Ellis remarking to me, with a smile, 'You and I are +just about fit to mend his pens.'" Surely never was +higher tribute paid to candidate by examiner!</p> + +<p>Another story, which, however, does not seem +capable of such complete authentication, is told of the +same examination, or it may be of the Smith's Prize +Examination which followed. A certain problem was +solved, so it is said, in practically identical terms by +both the First and Second Wranglers. The examiners +remarked the coincidence, and were curious as to +its origin. On being asked regarding it, the Senior +Wrangler replied that he had seen the solution he gave +in a paper which had appeared in a recent number of +the <i>Cambridge Mathematical Journal</i>; Thomson's answer +was that he was the author of the paper in +question! Thomson was Second Wrangler, and +Parkinson, of St. John's College, afterwards. Dr. +Parkinson, tutor of St. John's and author of various +mathematical text-books, was Senior. These positions +were reversed in the examination for Smith's Prizes, +which was very generally regarded as a better test of +original ability than the tripos, so that the temporary +disappointment of Thomson's friends was quickly +forgotten in this higher success.</p> + +<p>The Tripos Examination was held in the early part +of January. On the 25th of that month Thomson<span class='pagenum'><a name="Page_28" id="Page_28">28</a></span> +met his private tutor Hopkins in the "Senior Wranglers' +Walk" at Cambridge, and in the course of conversation +referred to his desire to obtain a copy of Green's +'Essay' (supra, p. <a href="#Page_21">21</a>). Hopkins at once took him +to the rooms where he had attended almost daily for a +considerable time as a pupil, and produced no less than +three copies of the Essay, and gave him one of them. +A hasty perusal showed Thomson that all the general +theorems of attractions contained in his paper "On the +Uniform Motion of Heat," etc., as well as those of Gauss +and Chasles, had been set forth by Green and were +derivable from a general theorem of analysis whereby +a certain integral taken throughout a space bounded +by surfaces fulfilling a certain condition is expressed as +two integrals, one taken throughout the space, the other +taken over the bounding surface or surfaces.</p> + +<p>It has been stated in the last chapter that Thomson +had established, as a deduction from the flow of heat +in a uniform solid from sources distributed within it, +the remarkable theorem of the replacement, without +alteration of the external flow, of these sources by a +certain distribution over any surface of uniform temperature, +and had pointed out the analogue of this theorem +in electricity. This method of proof was perfectly +original and had not been anticipated, though the +theorem, as has been stated, had already been given by +Green and by Gauss. In the paper entitled "Propositions +in the Theory of Attraction," published in the +<i>Cambridge Mathematical Journal</i> in November 1842, +Thomson gave an analytical proof of this great theorem, +but afterwards found that this had been done almost +contemporaneously by Sturm in Liouville's <i>Journal</i>.</p> + +<p>Soon after the Tripos and Smith's Prize Examinations<span class='pagenum'><a name="Page_29" id="Page_29">29</a></span> +were over, Thomson went to London, and visited +Faraday in his laboratory in the Royal Institution. +Then he went on to Paris with his friend Hugh +Blackburn, and spent the summer working in Regnault's +famous laboratory, making the acquaintance +of Liouville, Sturm, Chasles, and other French mathematicians +of the time, and attending meetings of the +Académie des Sciences. He made known to the mathematicians +of Paris Green's 'Essay,' and the treasures +it contained, and frequently told in after years with +what astonishment its results were received. He used +to relate that one day, while he and Blackburn sat +in their rooms, they heard some one come panting +up the stair. Sturm burst in upon them in great +excitement, and exclaimed, "<i>Vous avez un Mèmoire +de Green! M. Liouville me l'a dit.</i>" He sat down +and turned over the pages of the 'Essay,' looking at +one result after another, until he came to a complete +anticipation of his proof of the replacement theorem. +He jumped up, pointed to the page, and cried out, +"<i>Voila mon affaire!</i>"</p> + +<p>To this visit to Paris Thomson often referred in later +life with grateful recognition of Regnault's kindness, +and admiration of his wonderful experimental skill. +The great experimentalist was then engaged in his +researches on the thermal constants of bodies, with the +elaborate apparatus which he designed for himself, and +with which he was supplied by the wise liberality of +the French Government. This initiation into laboratory +work bore fruit not long after in the establishment +of the Glasgow Physical Laboratory, the first physical +laboratory for students in this country.</p> + +<p>It is a striking testimony to Thomson's genius that,<span class='pagenum'><a name="Page_30" id="Page_30">30</a></span> +at the age of only seventeen, he had arrived at such a +fundamental and general theorem of attractions, and +had pointed out its applications to electrical theory. +And it is also very remarkable that the theorem should +have been proved within an interval of two or three +years by three different authors, two of them—Sturm +and Gauss—already famous as mathematicians. +Green's treatment of the subject was, however, the +most general and far-reaching, for, as has been stated, +the theorem of Gauss, Sturm, and Thomson was merely +a particular case of a general theorem of analysis contained +in Green's 'Essay.' It has been said in jest, but +not without truth, that physical mathematics is made up +of continued applications of Green's theorem. Of +this enormously powerful relation, a more lately discovered +result, which is very fundamental in the theory +of functions of a complex variable, and which is generally +quoted as Riemann's theorem, is only a particular case.</p> + +<p>Thomson had the greatest reverence for the genius +of Green, and found in his memoirs, and in those of +Cauchy on wave propagation, the inspiration for much +of his own later work.<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">7</a> In 1850 he obtained the +<span class='pagenum'><a name="Page_31" id="Page_31">31</a></span>republication of Green's 'Essay' in Crelle's <i>Journal</i>; +in later years he frequently expressed regret that it had +not been published in England.</p> + +<p>In the commencement of 1845 Thomson told +Liouville of the method of <i>Electric Images</i> which he +had discovered for the solution of problems of electric +distribution. On October 8, 1845, after his return to +Cambridge, he wrote to Liouville a short account of +the results of the method in a number of different +cases, and in two letters written on June 26 and September +16 of the following year, he stated some further +results, including the solution of the problem of the +distribution upon a spherical bowl (a segment of a +spherical conducting shell made by a plane section) +insulated and electrified. This last very remarkable +result was given without proof, and remained unproved +until Thomson published his demonstration twenty-three +years later in the <i>Philosophical Magazine</i>.<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">8</a> This +had been preceded by a series of papers in March, +May, and November 1848, November 1849, and +February 1850, in the <i>Cambridge and Dublin Mathematical +Journal</i>, on various parts of the mathematical +theory of electricity in equilibrium,<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">9</a> in which the +theory of images is dealt with. The letters to Liouville +promptly appeared in the <i>Journal</i>, and the veteran +analyst wrote a long Note on their subject, which +concludes as follows: "Mon but sera rempli, je le +répéte, s'ils [ces développements] peuvent aider à bien +faire comprendre la haute importance du travail de ce +jeune géomètre, et si M. Thomson lui-même veut bien +y voir une preuve nouvelle de l'amitié que je lui porte +et de l'estime qui j'ai pour son talent."</p> + +<p><span class='pagenum'><a name="Page_32" id="Page_32">32</a></span></p><p>The method of images may be regarded as a development +in a particular direction of the paper "On the +Uniform Motion of Heat" already referred to, and, taken +along with this latter paper, forms the most striking +indication afforded by the whole range of Thomson's +earlier work of the strength and originality of his +mathematical genius. Accordingly a chapter is here +devoted to a more complete explanation of the first +paper and the developments which flowed from it. +The general reader may pass over the chapter, and +return to it from time to time as he finds opportunity, +until it is completely understood.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_33" id="Page_33">33</a></span></p> + +<h3>CHAPTER IV</h3> + +<h4>THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM. +ELECTRIC IMAGES. ELECTRIC INVERSION</h4> + +<p><span class="smcap">In</span> describing Thomson's early electrical researches we +shall not enter into detailed calculations, but merely +explain the methods employed. The meaning of certain +technical terms may be recalled in the first place.</p> + +<p>The whole space in which a distribution of electricity +produces any action on electrified bodies is called +the <i>electrical field</i> of the distribution. The force +exerted on a very small insulated trial conductor, on +which is an electric charge of amount equal to that +taken as the unit quantity of electricity, measures the +<i>field-intensity</i> at any point at which the conductor is +placed. The direction of the field-intensity at the +point is that in which the small conductor is there +urged. If the charge on the small conductor were a +negative unit, instead of a positive, the direction of +the force would be reversed; the magnitude of the +force would remain the same. To make the field-intensity +quite definite, a positive unit is chosen for its +specification. For a charge on the trial-conductor +consisting of any number of units, the force is that +number of times the field-intensity. The field-intensity +is often specified by its components, <i>X</i>, <i>Y</i>, <i>Z</i> +in three chosen directions at right angles to one +another.</p> + +<p><span class='pagenum'><a name="Page_34" id="Page_34">34</a></span>Now in all cases in which the action, whether +attraction or repulsion, between two unit quantities of +matter concentrated at points is inversely as the square +of the distance between the charges, the field-intensity, +or its components, can be found from a certain function +<i>V</i> of the charges forming the acting distribution [which +is always capable of being regarded for mathematical +purposes as a system of small charges existing at points +of space, <i>point-charges</i> we shall call them], their positions, +and the position of the point at which the field-intensity +is to be found. If <i>q</i><sub>1</sub>, <i>q</i><sub>2</sub>, ... be the point-charges, +and be positive when the charges are positive +and negative when the charges are negative, and +<i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, ... be their distances from the point <i>P</i>, <i>V</i> is +<i>q</i><sub>1</sub> ⁄ <i>r</i><sub>1</sub> + <i>q</i><sub>2</sub> ⁄ <i>r</i><sub>2</sub> + ... +The field-intensity is the rate of diminution of the value of <i>V</i> at <i>P</i>, taken along +the specified direction. The three gradients parallel to +the three chosen coordinate directions are <i>X</i>, <i>Y</i>, <i>Z</i>; but +for their calculation it is necessary to insert the values +of <i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>, ... in terms of the coordinates which +specify the positions of the point-charges, and the +coordinates <i>x</i>, <i>y</i>, <i>z</i> which specify the position of <i>P</i>. +Once this is done, <i>X</i>, <i>Y</i>, <i>Z</i> are obtained by a simple +systematic process of calculation, namely, differentiation +of the function <i>V</i> with respect to <i>x</i>, <i>y</i>, <i>z</i>.</p> + +<p>This function <i>V</i> seems to have been first used by +Laplace for gravitational matter in the <i>Mécanique +Céleste</i>; its importance for electricity and magnetism +was recognised by Green, who named it the <i>potential</i>. +It has an important physical signification. It represents +the work which would have to be done to bring +a unit of positive electricity, against the electrical repulsion +of the distribution, up to the point <i>P</i> from a point<span class='pagenum'><a name="Page_35" id="Page_35">35</a></span> +at an infinite distance from every part of the distribution; +or, in other words, what we now call the +<i>potential energy</i> of a charge <i>q</i> situated at <i>P</i> is <i>qV</i>. +The excess of the potential at <i>P</i>, over the potential at +any other point <i>Q</i> in the field, is the work which +must be spent in carrying a positive unit from <i>Q</i> to <i>P</i> +against electrical repulsion. Of course, if the force to +be overcome from <i>Q</i> to <i>P</i> is on the whole an attraction, +work has not been spent in effecting the transference, +but gained by allowing it to take place. The +difference of potential is then negative, that is, the +potential of <i>Q</i> is higher than that of <i>P</i>.</p> + +<p>The difference of potential depends only on the +points <i>P</i> and <i>Q</i>, and not at all on the path pursued +between them. Thus, if a unit of electricity be +carried from <i>P</i> to <i>Q</i> by any path, and back by any +other, no work is done on the whole by the agent +carrying the unit. This simple fact precludes the +possibility of obtaining a so-called perpetual motion (a +self-acting machine doing useful work) by means of +electrical action. The same thing is true <i>mutatis +mutandis</i> of gravitational action.</p> + +<p>In the thermal analogy explained by Thomson in his +first paper, the positive point-charges are point-sources +of heat, which is there poured at constant rate into the +medium (supposed of uniform quality) to be drawn off +in part from the medium at constant rate where there +are <i>sinks</i> (or negative sources),—the negative point-charges +in the electrical case,—while the remainder +is conducted away to more and more distant parts of +the conducting medium supposed infinitely extended. +Whenever a point-source, or a point-sink, exists at a +distance from other sources or sinks, the flow in the<span class='pagenum'><a name="Page_36" id="Page_36">36</a></span> +vicinity is in straight lines from or to the point, and +these straight lines would be indefinitely extended if +either source or sink existed by itself. As it is, the +direction and amount of flow everywhere depends on +the flow resulting from the whole arrangement of +sources and sinks. Lines can be drawn in the medium +which show the direction of the resultant flow from +point to point, and these lines of flow can be so spaced +as to indicate, by their closeness together or their distance +apart, where the rate of flow is greater or smaller; +and such lines start from sources, and either end in +sinks or continue their course to infinity. In the +electrical case these lines are the analogues of the lines +of electric force (or field-intensity) in the insulating +medium, which start from positive charges and end in +negative, or are prolonged to infinity.</p> + +<p>Across such lines of flow can be drawn a family of +surfaces, to each of which the lines met by the surface +are perpendicular. These surfaces are the equitemperature +surfaces, or, as they are usually called, the isothermal +surfaces. They can be drawn more closely +crowded together, or more widely separated, so as to +indicate where the rate of falling off of temperature +(the "temperature slope") is greater or less, just as the +contour lines in a map show the slopes on a hill-side.</p> + +<p>Instead of the thermal analogy might have been +used equally well that of steady flow in an indefinitely +extended mass of homogeneous frictionless and incompressible +fluid, into which fluid is being poured at a +constant rate by sources and withdrawn by sinks. +The isothermal surfaces are replaced by surfaces of +equal pressure, while lines of flow in one are also lines +of flow in the other.<span class='pagenum'><a name="Page_37" id="Page_37">37</a></span></p> + +<div class="figright" style="width: 300px; position: relative;"> +<a name="f1" id="f1"></a> +<img src="images/fig01.png" width="300" height="179" alt="Fig. 1." title="" /> +<p class="caption"><span class="smcap">Fig. 1.</span></p> +</div> + +<p>Now let heat be poured into the medium at constant +rate by a single point-source <i>P</i> (Fig. <a href="#f1">1</a>), and drawn off +at a smaller rate by a single point-sink <i>P'</i>, while the +remainder flows to more and more remote parts of the +medium, supposed infinite in extent in every direction. +After a sufficient time +from the beginning of +the flow a definite +system of lines of flow +and isothermal surfaces +can be traced for +this case in the manner +described above. +One of the isothermal +surfaces will be a sphere <i>S</i> surrounding the sink, +which, however, will not be at the centre of the +sphere, but so situated that the source, sink, and centre +are in line, and that the radius of the sphere is a +mean proportional between the distances of the source +and sink from the centre. If <i>a</i> be the radius of the +sphere and <i>f</i> the distance of the source from the centre +of the sphere, the heat carried off by the sink is the +fraction <i>a</i> ⁄ <i>f</i> of that given out by the source.</p> + +<p>In the electrical analogue, the source and sink are +respectively a point-charge and what is called the +"electric image" of that charge with respect to the +sphere, which is in this case an equipotential surface. +And just as the lines of flow of heat meet the spherical +isothermal surface at right angles, so the lines of force +in the electrical case meet the equipotential surface +also at right angles. Now obviously in the thermal +case a spherical sink could be arranged coinciding with +the spherical surface so as to receive the flow there<span class='pagenum'><a name="Page_38" id="Page_38">38</a></span> +arriving and carry off the heat from the medium, without +in the least disturbing the flow outside the sphere. +The whole amount of heat arriving would be the +same: the amount received per unit area at any point +on the sphere would evidently be proportional to the +gradient of temperature there towards the surface. Of +course the same thing could be done at any isothermal +surface, and the same proportionality would hold in +that case.</p> + +<p>Similarly the source could be replaced by a surface-distribution +of sources over any surrounding isothermal +surface; and the condition to be fulfilled in that case +would be that the amount of heat given out per unit +area anywhere should be exactly that which flows +out along the lines of flow there in the actual case. +Outside the surface the field of flow would not be +affected by this replacement. It is obvious that in +this case the outflow per unit area must be proportional +to the temperature slope outward from the +surface.</p> + +<p>The same statements hold for any complex system +of sources and sinks. There must be the same outflow +from the isothermal surface or inflow towards it, as +there is in the actual case, and the proportionality to +temperature slope must hold.</p> + +<p>This is exactly analogous to the replacement by a +distribution on an equipotential surface of the electrical +charge or charges within the surface, by a distribution +over the surface, with fulfilment of Coulomb's theorem +(p. <a href="#Page_43">43</a> below) at the surface. Thomson's paper on the +"Uniform Motion of Heat" gave an intuitive proof of +this great theorem of electrostatics, which the statements +above may help to make clear to those who have, or<span class='pagenum'><a name="Page_39" id="Page_39">39</a></span> +are willing to acquire, some elementary knowledge of +electricity.</p> + +<p>Returning to the distribution on any isothermal surface +surrounding the sink (or sinks) we see that it represents +a surface-sink in equilibrium with the flow in the field. +The distribution on a metal shell, coinciding with the +surface, which keeps the surface at a potential which is +the analogue of the temperature at the isothermal surface, +while the shell is under the influence of a point-charge +of electricity—the analogue of the thermal +source—is the distribution as affected by the induction +of the point-charge. If the shell coincide with the +spherical equipotential surface referred to above, and +the distribution given by the theorem of replacement +be made upon it, the shell will be at zero potential, and +the charge will be that which would exist if the shell +were uninsulated, that is, the "induced charge."</p> + +<p>The consideration of the following simple problem +will serve to make clear the meaning of an electric +image, and form a suitable introduction to a description +of the application of the method to the electrification +of spherical surfaces. Imagine a very large plane sheet +of tinfoil connected by a conducting wire with the +earth. If there are no electrified bodies near, the sheet +will be unelectrified. But let a very small metallic ball +with a charge of positive electricity upon it be brought +moderately close to one face of the tinfoil. The tinfoil +will be electrified negatively by induction, and the +distribution of the negative charge will depend on the +position of the ball. Now, it can be shown that the +field of electric force, on the same side of the tinfoil as +the ball, is precisely the same as would be produced if +the foil (and everything behind it) were removed, and<span class='pagenum'><a name="Page_40" id="Page_40">40</a></span> +an equal negative charge of electricity placed behind +the tinfoil on the prolonged perpendicular from the ball +to the foil, and as far from the foil behind as the ball is +from it in front. Such a negative charge behind the +tinfoil sheet is called an electric image of the positive +charge in front. It is situated, as will be seen at what +would be, if the tinfoil were a mirror, the optical +image of the ball in the mirror.</p> + +<div class="figcenter" style="width: 550px;"> +<a name="f2" id="f2"></a> +<img src="images/fig02.png" width="550" height="276" alt="Fig. 2." title="" /> +<p class="caption"><span class="smcap">Fig. 2.</span></p> +</div> + + +<div class="figright" style="width: 300px; position: relative;"> +<a name="f3" id="f3"></a> +<img src="images/fig03.png" width="300" height="224" alt="Fig. 3." title="" /> +<p class="caption"><span class="smcap">Fig. 3.</span></p> +</div> + +<p>Now, suppose a second very large sheet of tinfoil to +be placed parallel to the first sheet, so that the small +electrified sphere is between the two sheets, and that +this second sheet is also connected to the earth. The +charge on the ball induces negative electricity on both +sheets, but besides this each sheet by its charge influences +the other. The problem of distribution is much +more complicated than in the case of a single sheet, +but its solution is capable of very simple statement. +Let us call the two sheets <i>A</i> and <i>B</i> (Fig. <a href="#f2">2</a>), and +regard them for the moment as mirrors. A first image +of an object <i>P</i> between the two mirrors is produced +directly by each, but the image <i>I</i><sub>1</sub> in <i>A</i> is virtually an +object in front of <i>B</i>, and the image <i>J</i><sub>1</sub> in <i>B</i> an object<span class='pagenum'><a name="Page_41" id="Page_41">41</a></span> +in front of <i>A</i>, so that a second image more remote +from the mirror than the first is produced in each case. +These second images <i>I</i><sub>2</sub> and <i>J</i><sub>2</sub> in the same way produce +third images still more remote, and so on. The +positions are determined just as for an object and a +single mirror. There is thus an infinite trail of images +behind each mirror, the places of which any one can +assign.</p> + +<p>Every one may see the realisation of this arrangement +in a shop window, the two sides of which +are covered by parallel sheets of mirror-glass. An +infinite succession of +the objects in the +window is apparently +seen on both sides. +When the objects displayed +are glittering +new bicycles in a row +the effect is very striking; +but what we +are concerned with +here is a single small object like the little ball, and its +two trails of images. The electric force at any point +between the two sheets of tinfoil is exactly the same +as if the sheets were removed and charges alternately +negative and positive were placed at the image-points, +negative at the first images, positive at the second +images, and so on, each charge being the same in +amount as that on the ball. We have an "electric +kaleidoscope" with parallel mirrors. When the angle +between the conducting planes is an aliquot part of +360°, let us say 60°, the electrified point and the +images are situated, just as are the object and its image<span class='pagenum'><a name="Page_42" id="Page_42">42</a></span> +in Brewster's kaleidoscope, namely at the angular points +of a hexagon, the sides of which are alternately (as +shown in Fig. <a href="#f3">3</a>) of lengths twice the distance of the +electrified point from <i>A</i> and from <i>B</i>.</p> + +<div class="figcenter" style="width: 400px;"> +<a name="f4" id="f4"></a> +<img src="images/fig04.png" width="400" height="195" alt="Fig. 4." title="" /> +<p class="caption"><span class="smcap">Fig. 4.</span></p> +</div> + +<p>Now consider the spherical surface referred to at +p. <a href="#Page_37">37</a>, which is kept at uniform potential by a charge +at the external point <i>P</i>, and a charge <i>q'</i> at the inverse +point <i>P'</i> within the sphere. If <i>E</i> (Fig. <a href="#f4">4</a>) be any +point whatever on the surface, and <i>r</i>, <i>r'</i> be its distances +from <i>P</i> and <i>P'</i>, it is easy to prove by geometry that +the two triangles <i>CPE</i> and <i>CEP'</i> are similar, and +therefore <i>r'</i> = <i>ra</i> ⁄ <i>f</i>. [Here <i>a</i> ⁄ <i>f</i> is used to mean <i>a</i> +divided by <i>f</i>. The mark ⁄ is adopted instead of +the usual bar of the fraction, for convenience of +printing.] Now, by the explanation given above, the +potential produced at any point by a charge <i>q</i> at +another point, is equal to the ratio of the charge <i>q</i> to +the distance between the points. Thus the potential +at <i>E</i> due to the charge <i>q</i> at <i>P</i> is +<i>q</i> ⁄ <i>r</i>, +and that at <i>E</i> due to a charge <i>q'</i> at <i>P'</i> is +<i>q'</i> ⁄ <i>r'</i>. +Thus if <i>q'</i> = − <i>qa</i> ⁄ <i>f</i>, +<i>q'</i> at <i>P'</i> will produce a potential at +<i>E</i> = − <i>qa</i> ⁄ <i>fr'</i> = − <i>q</i> ⁄ <i>r</i>, +by the value of <i>r</i>. Hence <i>q</i> at <i>P</i> and − <i>qa</i> ⁄ <i>f</i> at <i>P'</i> +coexisting will give potential <i>q</i> ⁄ <i>r</i> + − <i>q</i> ⁄ <i>r</i> or zero, +at <i>E</i>. Thus the charge − <i>qa</i> ⁄ <i>f</i>, at the internal point<span class='pagenum'><a name="Page_43" id="Page_43">43</a></span> +<i>P'</i> will in presence of + <i>q</i> at <i>P</i> keep all points of the +spherical surface at zero potential. These two charges +represent the source and sink in the thermal analogue +of p. <a href="#Page_37">37</a> above.</p> + +<p>Now replace <i>S</i> by a spherical shell of metal connected +to the earth by a long fine wire, and imagine +all other conductors to be at a great distance from it. +If this be under the influence of the charge <i>q</i> at <i>P</i> +alone, a charge is induced upon it which, in presence +of <i>P</i>, maintains it at zero potential. The internal +charge − <i>qa</i> ⁄ <i>f</i>, and the induced distribution on the +shell are thus equivalent as regards the potential produced +by either at the spherical surface; for each +counteracts then the potential produced by <i>q</i> at <i>P</i>. +But it can be proved that if a distribution over an +equipotential surface can be made to produce the same +potential over that surface as a given internal distribution +does, they produce the same potentials at all +<i>external</i> points, or, as it is usually put, the external +fields are the same. This is part of the statement of +what has been called the "theorem of replacement" +discovered by Green, Gauss, Thomson, and Chasles +as described above.</p> + +<p>Another part of the statement of the theorem may +now be formulated. Coulomb showed long ago that +the surface-density of electricity at any point on a +conductor is proportional to the resultant field-intensity +just outside the surface at that point. Since the surface +is throughout at one potential this intensity is normal +to the surface. Let it be denoted by <i>N</i>, and <i>s</i> be the +surface-density: then according to the system of units +usually adopted 4π<i>s</i> = <i>N</i>.</p> + +<p>Let now the rate of diminution of potential per unit of<span class='pagenum'><a name="Page_44" id="Page_44">44</a></span> +distance outwards (or downward gradient of potential) +from the equipotential surface be determined for every +point of the surface, and let electricity be distributed +over the surface, so that the amount per unit area at +each point (the surface-density) is made numerically +equal to the gradient there divided by 4π. This, by +Coulomb's law, stated above, gives that field-intensity +just outside the surface which exists for the actual +distribution, and therefore, as can be proved, gives +the same field everywhere else outside the surface. +The external fields will therefore be equivalent, and +further, the amount of electricity on the surface will +be the same as that situated within it in the actual +distribution.</p> + +<p>Thus it is only necessary to find for − <i>qa</i> ⁄ <i>f</i> at <i>P'</i> +and <i>q</i> at <i>P</i>, the falling off gradient <i>N</i> of potential outside +the spherical surface at any point <i>E</i>, and to take +<i>N</i> ⁄ 4<i>π</i>, +to obtain <i>s</i> the surface-density at <i>E</i>. Calculation of this +gradient for the sphere gives +4π<i>s</i> = − <i>q</i> (<i>f</i><sup>2</sup> − <i>a</i><sup>2</sup>) ⁄ <i>ar</i><sup>3</sup>. +The surface-density is thus inversely as the cube of the +distance <i>PE</i>.</p> + +<p>If the influencing point <i>P</i> be situated within the +spherical shell, and the shell be connected to earth as +before, the induced distribution will be on its interior +surface. The corresponding point <i>P'</i> will now be outside, +but given by the same relation. And <i>a</i> will now +be greater than <i>f</i>, and the density will be given by +4π<i>s</i> = − <i>q</i> (<i>a</i><sup>2</sup> − <i>f</i><sup>2</sup>) ⁄ <i>ar</i><sup>3</sup>, where, <i>f</i> and <i>r</i> have the same +meanings with regard to <i>E</i> and <i>P</i> as before.</p> + +<p><i>P'</i> is in each case called the image of <i>P</i> in the +sphere <i>S</i>, and the charge − <i>qa</i> ⁄ <i>f</i> there supposed situated +is the <i>electric image</i> of the charge <i>q</i> at <i>P</i>. It will be +seen that an electric image is a charge, or system of<span class='pagenum'><a name="Page_45" id="Page_45">45</a></span> +charges, on one side of an electrified surface which +produces on the other side of that surface the same +electrical field as is produced by the actual electrification +of the surface.</p> + +<p>While by the theorem of replacement there is only +one distribution over a surface which produces at all +points on one side of a surface the same field as does +a distribution <i>D</i> on the other side of the surface, this +surface distribution may be equivalent to several different +arrangements of <i>D</i>. Thus the point-charge at <i>P'</i> +is only one of various image-distributions equivalent to +the surface-distribution in the sense explained. For +example, a uniform distribution over any spherical +surface with centre at <i>P'</i> (Fig. <a href="#f4">4</a>) would do as well, +provided this spherical surface were not large enough +to extend beyond the surface <i>S</i>.</p> + +<p>In order to find the potential of the sphere (Fig. <a href="#f4">4</a>) +when insulated with a charge <i>Q</i> upon it, in presence +of the influencing charge <i>q</i> at the external point <i>P</i>, it +is only necessary to imagine uniformly distributed over +the sphere, already electrified in the manner just +explained, the charge <i>Q</i> + <i>aq</i> ⁄ <i>f</i>. Then the whole +charge will be <i>Q</i>, and the uniformity of distribution +will be disturbed, as required by the action of the +influencing point-charge. The potential will be +<i>Q</i> ⁄ <i>a</i> + <i>q</i> ⁄ <i>f</i>. For a given potential <i>V</i> of the sphere, +the total charge is <i>aV</i> − <i>aq</i> ⁄ <i>f</i>, that is the charge is <i>aV</i> +over and above the induced charge.</p> + +<p>If instead of a single influencing point-charge at <i>P</i> +there be a system of influencing point-charges at +different external points, each of these has an image-charge +to be found in amount and situation by the +method just described, and the induced distribution is<span class='pagenum'><a name="Page_46" id="Page_46">46</a></span> +that obtained by superimposing all the surface distributions +found for the different influencing points.</p> + +<p>The force of repulsion between the point-charge <i>q</i> +and the sphere (with total charge <i>Q</i>) can be found at +once by calculating the sum of the forces between <i>q</i> at +<i>P</i> and the charges <i>Q</i> + <i>aq</i> ⁄ <i>f</i> at <i>C</i> and − <i>aq</i> ⁄ <i>f</i> at <i>P'</i>.</p> + +<p>This can be found also by calculating the energy of +the system, which will be found to consist of three +terms, one representing the energy of the sphere with +charge <i>Q</i> uninfluenced by an external charge, one +representing the energy on a small conductor (not a +point) at <i>P</i> existing alone, and a third representing the +mutual energy of the electrification on the sphere and +the charge <i>q</i> at <i>P</i> existing in presence of one another. +By a known theorem the energy of a system of conductors +is one half of the sum obtained by multiplying +the potential of each conductor by its charge and +adding the products together. It is only necessary +then to find the variation of the last term caused by +increasing <i>f</i> by a small amount <i>df</i>. This will be the +product <i>F . df</i> of the force <i>F</i> required and the displacement.</p> + +<p>Either method may be applied to find the forces of +attraction and repulsion for the systems of electrified +spheres described below.</p> + +<p>The problem of two mutually influencing non-intersecting +spheres, <i>S</i><sub>1</sub>, <i>S</i><sub>2</sub> (Fig. <a href="#f5">5</a>), insulated with +given charges, <i>q</i><sub>1</sub>, <i>q</i><sub>2</sub>, may now be dealt with +in the following manner. Let each be supposed at +first charged uniformly. By the known theorem referred +to above, the external field of each is the same +as if its whole charge were situated at the centre. +Now if the distribution on <i>S</i><sub>2</sub>, say, be kept unaltered,<span class='pagenum'><a name="Page_47" id="Page_47">47</a></span> +while that on <i>S</i><sub>1</sub> is allowed to change, the action of +<i>S</i><sub>2</sub> on <i>S</i><sub>1</sub> is the same as if the charge <i>q</i><sub>2</sub> were at the +centre <i>C</i><sub>2</sub> of <i>S</i><sub>2</sub>. Thus if <i>f</i> be the distance between +the centres <i>C</i><sub>1</sub>, <i>C</i><sub>2</sub>, and <i>a</i><sub>1</sub> be the radius of <i>S</i><sub>1</sub>, the +distribution will be that corresponding to <i>q</i><sub>1</sub> + <i>a</i><sub>1</sub><i>q</i><sub>2</sub> ⁄ <i>f</i> +uniformly distributed on <i>S</i><sub>1</sub> together with the induced +charge − <i>a</i><sub>1</sub><i>q</i><sub>2</sub> ⁄ <i>f</i>, which corresponds to the image-charge +at the point <i>I</i><sub>1</sub> (within <i>S</i><sub>1</sub>), the inverse of <i>C</i><sub>2</sub> +with respect to <i>S</i><sub>1</sub>. Now let the charge on <i>S</i><sub>1</sub> be +fixed in the state just supposed while that on <i>S</i><sub>2</sub> is +freed. The charge on <i>S</i><sub>2</sub> will rearrange itself under +the influence of <i>q</i><sub>1</sub> + <i>a</i><sub>1</sub><i>q</i><sub>2</sub> ⁄ <i>f</i> ( = <i>q'</i>) +and − <i>a</i><sub>1</sub><i>q</i><sub>2</sub> ⁄ <i>f</i>, considered +as at <i>C</i><sub>1</sub> and <i>I</i><sub>1</sub> respectively. The former of +these will give a distribution equivalent to <i>q</i><sub>2</sub> + <i>a</i><sub>2</sub><i>q'</i> ⁄ <i>f</i> +uniformly distributed over <i>S</i><sub>2</sub>, and an induced distribution +of amount − <i>a</i><sub>2</sub><i>q'</i> ⁄ <i>f</i> at <i>J</i><sub>1</sub>, the inverse point of <i>C</i><sub>1</sub> +with regard to <i>S</i><sub>2</sub>. The image-charge − <i>a</i><sub>1</sub><i>q</i><sub>2</sub> ⁄ <i>f</i> at <i>I</i><sub>1</sub> +in <i>S</i><sub>1</sub> will react on <i>S</i><sub>2</sub> and give an induced distribution +− <i>a</i><sub>2</sub> (− <i>a</i><sub>1</sub><i>q</i><sub>2</sub> ⁄ <i>f</i> ) <i>f'</i>, (<i>I</i><sub>1</sub><i>C</i><sub>2</sub> = <i>f'</i> ) corresponding to an +image-charge <i>a</i><sub>2</sub><i>a</i><sub>1</sub><i>q</i><sub>2</sub> ⁄ <i>ff'</i> at the inverse point <i>J</i><sub>2</sub> of <i>P</i><sub>1</sub> +with respect to <i>C</i><sub>2</sub><i>S</i><sub>2</sub>. Thus the distribution on <i>S</i><sub>2</sub> is +equivalent to <i>q</i><sub>2</sub> + <i>a</i><sub>2</sub><i>q'</i> ⁄ <i>f</i> − <i>a</i><sub>2</sub><i>a</i><sub>1</sub><i>q</i><sub>2</sub> ⁄ <i>ff'</i> +at the inverse point <i>J</i><sub>2</sub> of <i>P</i><sub>1</sub> +distributed<span class='pagenum'><a name="Page_48" id="Page_48">48</a></span> +uniformly over it, together with the two induced +distributions just described.</p> + +<div class="figcenter" style="width: 500px; position: relative;"><a name="f5" id="f5"></a><img src="images/fig05.png" width="500" height="245" alt="Fig. 5." title="" /> +<p class="caption"><span class="smcap">Fig. 5.</span></p></div> + +<p>In the same way these two induced distributions on +<i>S</i><sub>2</sub> may now be regarded as reacting on the distribution +on <i>S</i><sub>1</sub> as would point-charges − <i>a</i><sub>2</sub><i>q</i><sub>1</sub> ⁄ <i>f</i> +and <i>a</i><sub>2</sub><i>a</i><sub>1</sub><i>q</i><sub>2</sub> ⁄ <i>ff'</i>, +situated at <i>J</i><sub>1</sub> and <i>J</i><sub>2</sub> respectively, and would give two +induced distributions on <i>S</i><sub>1</sub> corresponding to their +images in <i>S</i><sub>1</sub>.</p> + +<p>Thus by partial influences in unending succession +the equilibrium state of the two spheres could be +approximated to as nearly as may be desired. An +infinite trail of electric images within each of the two +spheres is thus obtained, and the final state of each +conductor can be calculated by summation of the +effects of each set of images.</p> + +<p>If the final potentials, <i>V</i><sub>1</sub>, <i>V</i><sub>2</sub>, say, of the spheres are +given the process is somewhat simpler. Let first the +charges be supposed to exist uniformly distributed over +each sphere, and to be of amount <i>a</i><sub>1</sub><i>V</i><sub>1</sub>, <i>a</i><sub>2</sub><i>V</i><sub>2</sub> in the two +cases. The uniform distribution on <i>S</i><sub>1</sub> will raise the +potential of <i>S</i><sub>2</sub> above <i>V</i><sub>2</sub>, and to bring the potential +down to <i>V</i><sub>2</sub> in presence of this distribution we must +place an induced distribution over <i>S</i><sub>2</sub>, represented as +regards the external field by the image-charge +− <i>a</i><sub>2</sub><i>a</i><sub>1</sub><i>V</i><sub>1</sub> ⁄ <i>f</i> (at the image of <i>C</i><sub>1</sub> in <i>S</i><sub>2</sub>) where <i>f</i> is the +distance between the centres. The charge <i>a</i><sub>2</sub><i>V</i><sub>2</sub> on <i>S</i><sub>2</sub> +will similarly have an action on <i>S</i><sub>1</sub> to be compensated +in the same way by an image-charge − <i>a</i><sub>1</sub><i>a</i><sub>2</sub><i>V</i><sub>2</sub> ⁄ <i>f</i> at +the image of <i>C</i><sub>2</sub> in <i>S</i><sub>1</sub>. Now these two image-charges +will react on the spheres <i>S</i><sub>1</sub> and <i>S</i><sub>2</sub> respectively, and +will have to be balanced by induced distributions +represented by second image-charges, to be found in +the manner just exemplified. These will again react<span class='pagenum'><a name="Page_49" id="Page_49">49</a></span> +on the spheres and will have to be compensated as +before, and so on indefinitely. The charges diminish +in amount, and their positions approximate more and +more, according to definite laws, and the final state is +to be found by summation as before.</p> + +<p>The force of repulsion is to be found by summing +the forces between all the different pairs of charges +which can be formed by taking one charge of each +system at its proper point: or it can be obtained by +calculating the energy of the system.</p> + +<p>The method of successive influences was given +originally by Murphy, but the mode of representing +the effects of the successive induced charges by image-charges +is due to Thomson. Quite another solution +of this problem is, however, possible by Thomson's +method of electrical inversion.</p> + +<p>A similar process to that just explained for two +charged and mutually influencing spheres will give the +distribution on two concentric conducting spheres, +under the influence of a point-charge <i>q</i> at <i>P</i> between +the inner surface of the outer and the outer surface +of the inner, as shown in Fig. <a href="#f7">7</a>. There the influence +of <i>q</i> at <i>P</i>, and of the induced distributions on +one another, is represented by two series of images, +one within the inner sphere and one outside the outer. +These charges and positions can be calculated from the +result for a single sphere and point-charge.</p> + +<p>Thomson's method of electrical inversion, referred +to above, enabled the solutions of unsolved problems +to be inferred from known solutions of simpler cases +of distribution. We give here a brief account of the +method, and some of its results. First we have to +recall the meaning of geometrical inversion. In Fig. <a href="#f6">6</a> +<span class='pagenum'><a name="Page_50" id="Page_50">50</a></span> +the distances <i>OP</i>, <i>OP'</i>, <i>OQ</i>, <i>OQ'</i> fulfil the relation +<i>OP</i>.<i>OP'</i> = <i>OQ</i>.<i>OQ'</i> = <i>a</i><sup>2</sup>. Thus <i>P'</i> is (see p. <a href="#Page_37">37</a>) +the inverse of the point <i>P</i> with respect to a sphere of +radius <i>a</i> and centre <i>O</i> (indicated by the dotted line in +Fig. <a href="#f6">6</a>), and similarly <i>Q'</i> is the inverse of <i>Q</i> with +respect to the same sphere and centre. <i>O</i> is called the +centre of inversion, and the sphere of radius <i>a</i> is called +the sphere of inversion. Thus the sphere of Figs. <a href="#f1">1</a> +and <a href="#f4">4</a> is the sphere of inversion for the points <i>P</i> and +<i>P'</i>, which are inverse points of one another. For any +system of points <i>P</i>, <i>Q</i>, ..., another system <i>P'</i>, <i>Q'</i>, ... +of inverse points can be found, and if the first system +form a definite locus, the second will form a derived +locus, which is called the inverse of the former. Also +if <i>P'</i>, <i>Q'</i>, ... be regarded as the direct system, +<i>P</i>, <i>Q</i>, ... will be the corresponding inverse system +with regard to the same sphere and centre. <i>P'</i> is the +image of <i>P</i>, and <i>P</i> is the image of <i>P'</i>, and so on, with +regard to the same sphere and centre of inversion.</p> + +<div class="figcenter" style="width: 500px; position: relative;"><a name="f6" id="f6"></a><img src="images/fig06.png" width="500" height="236" alt="Fig. 6." title="" /> +<p class="caption"><span class="smcap">Fig. 6.</span></p></div> + +<p>The inverse of a circle is another circle, and therefore +the inverse of a sphere is another sphere, and the +inverse of a straight line is a circle passing through the +centre of inversion, and of an infinite plane a sphere<span class='pagenum'><a name="Page_51" id="Page_51">51</a></span> +passing through the centre of inversion. Obviously +the inverse of a sphere concentric with the sphere of +inversion is a concentric sphere.</p> + +<p>The line <i>P'Q'</i> is of course not the inverse of the line +<i>PQ</i>, which has for its inverse the circle passing through +the three points <i>O</i>, <i>P'</i>, <i>Q'</i>, as indicated in Fig. <a href="#f6">6</a>.</p> + +<p>The following results are easily proved.</p> + +<p>A locus and its inverse cut any line <i>OP</i> at the +same angle.</p> + +<p>To a system of point-charges <i>q</i><sub>1</sub>, <i>q</i><sub>2</sub>, ... at points <i>P</i><sub>1</sub>, +<i>P</i><sub>2</sub>, ... on one side of the surface of the sphere of inversion +there is a system of charges <i>aq</i><sub>1</sub> ⁄ <i>f</i><sub>1</sub>, <i>aq</i><sub>2</sub> ⁄ <i>f</i><sub>2</sub>, ... on the +other side of the spherical surface [<i>OP</i><sub>1</sub> = <i>f</i><sub>1</sub>, <i>OP</i><sub>2</sub> = <i>f</i><sub>2</sub>]. +This inverse system, as we shall call it, produces the +same potential at any point of the sphere of inversion, +as does the direct system from which it is derived.</p> + +<p>If <i>V</i>, <i>V'</i> be the potentials produced by the whole +direct system at <i>Q</i>, and by the whole inverse system +at <i>Q'</i>, <i>V'</i> ⁄ <i>V</i> = <i>r</i> ⁄ <i>a</i> = <i>a</i> ⁄ <i>r'</i>, where <i>OQ</i> = <i>r</i>, <i>OQ'</i> = <i>r'</i>.</p> + +<p>Thus if <i>V</i> is constant over any surface <i>S'</i>, <i>V'</i> is not +a constant over the inverse surface <i>S'</i>, unless <i>r</i> is a +constant, that is, unless the surface <i>S'</i> is a sphere concentric +with the sphere of inversion, in which case +the inverse surface is concentric with it and is an +equipotential surface of the inverse distribution.</p> + +<p>Further, if <i>q</i> be distributed over an element <i>dS</i> of +a surface, the inverse charge <i>aq</i> ⁄ <i>f</i> will be distributed +over the corresponding element <i>dS'</i> of the inverse +surface. But <i>dS'</i> ⁄ <i>dS</i> = <i>a</i><sup>4</sup> ⁄ <i>f</i><sup>4</sup> = <i>f'</i><sup>4</sup> ⁄ <i>a</i><sup>4</sup> where <i>f</i>, <i>f'</i> +are the distances of <i>O</i> from <i>dS</i> and <i>dS'</i>. Thus if <i>s</i> be +the density on <i>dS</i> and <i>s'</i> the inverse density on <i>dS'</i> +we have <i>s'</i> ⁄ <i>s</i> = <i>a</i><sup>3</sup> ⁄ <i>f'</i><sup>3</sup> = <i>f</i><sup>3</sup> ⁄ <i>a</i><sup>3</sup>.</p> + +<p>When <i>V</i> is constant over the direct surface, while<span class='pagenum'><a name="Page_52" id="Page_52">52</a></span> +<i>r</i> has different values for different directions of <i>OQ</i>, +the different points of the inverse surface may be +brought to zero potential by placing at <i>O</i> a charge +− <i>aV</i>. For this will produce at <i>Q'</i> a potential − <i>aV</i> ⁄ <i>r'</i> +which with <i>V'</i> will give at <i>Q'</i> a potential zero. This +shows that <i>V'</i> is the potential of the induced distribution +on <i>S'</i> due to a charge − <i>aV</i> at <i>O</i>, or that − <i>V'</i> is the +potential due to the induced charge on <i>S'</i> produced +by the charge <i>aV</i> at <i>O</i>.</p> + +<div class="figleft" style="width: 300px; position: relative;"><a name="f7" id="f7"></a><img src="images/fig07.png" width="300" height="283" alt="Fig. 7." title="" /> +<p class="caption"><span class="smcap">Fig. 7.</span></p> +</div> + +<p>Thus we have the conclusion that by the process of +inversion we get from a distribution in equilibrium, on +a conductor of any form, +an induced distribution on +the inverse surface supposed +insulated and conducting; +and conversely +we obtain from a given +induced distribution on an +insulated conducting surface, +a natural equilibrium +distribution on the inverse +surface. In each case the +inducing charge is situated at the centre of inversion. +The charges on the conductor (or conductors) after +inversion are always obtainable at once from the fact +that they are the inverses of the charges on the conductor +(or conductors) in the direct case, and the +surface-densities or volume-densities can be found +from the relations stated above.</p> + +<p>Now take the case of two concentric spheres +insulated and influenced by a point-charge <i>q</i> placed +at a point <i>P</i> between them as shown in Fig. <a href="#f7">7</a>. We +have seen at p. <a href="#Page_49">49</a> how the induced distribution, and the<span class='pagenum'><a name="Page_53" id="Page_53">53</a></span> +amount of the charge, on each sphere is obtained from +the two convergent series of images, one outside the +outer sphere, the other inside the inner sphere. We +do not here calculate the density of distribution at any +point, as our object is only to explain the method; but +the quantities on the spheres <i>S</i><sub>1</sub> and <i>S</i><sub>2</sub>, are respectively +− <i>q.OA.PB</i> ⁄ (<i>OP.AB</i>), − <i>q.OB.AP</i> ⁄ (<i>OP.AB</i>).</p> + +<p>It may be noticed that the sum of the induced +charges is − <i>q</i>, and that as the radii of the spheres are +both made indefinitely great, while the distance <i>AB</i> is +kept finite, the ratios <i>OA</i> ⁄ <i>OP</i>, <i>OB</i> ⁄ <i>OP</i> approximate +to unity, and the charges to − <i>q.PB</i> ⁄ <i>AB</i>, − <i>q.AP</i> ⁄ <i>AB</i>, +that is, the charges are inversely as the distances of +<i>P</i> from the nearest points of the two surfaces. But +when the radii are made indefinitely great we have +the case of two infinite plane conducting surfaces with +a point-charge between them, which we have described +above.</p> + +<p>Now let this induced distribution, on the two +concentric spheres, be inverted from <i>P</i> as centre of +inversion. We obtain two non-intersecting spheres, +as in Fig. <a href="#f5">5</a>, for the inverse geometrical system, and +for the inverse electrical system an equilibrium distribution +on these two spheres in presence of one +another, and charged with the charges which are the +inverses of the induced charges. These maintain the +system of two spheres at one potential. From this +inversion it is possible to proceed as shown by +Maxwell in his <i>Electricity and Magnetism</i>, vol. i, +§ 173, to the distribution on two spheres at two +different potentials; but we have shown above how +the problem may be dealt with directly by the method +of images.<span class='pagenum'><a name="Page_54" id="Page_54">54</a></span></p> + +<div class="figcenter" style="width: 250px; position: relative;"><a name="f8" id="f8"></a><img src="images/fig08.png" width="250" height="239" alt="Fig. 8." title="" /> +<p class="caption"><span class="smcap">Fig. 8.</span></p></div> + +<p>Again take the case of two parallel infinite planes +under the influence of a point-charge between them. +This system inverted from <i>P</i> as centre gives the +equilibrium distribution on two charged insulated +spheres in contact (Fig. <a href="#f8">8</a>); for this system is the +inverse of the planes and the charges upon them. +Another interesting case is that of the "electric +kaleidoscope" referred to above. Here the two infinite +conducting planes are inclined at an angle 360° ⁄ <i>n</i>, +where <i>n</i> is a whole number, and are therefore bounded +in one direction by the straight line which is their +intersection. The image points <i>I</i><sub>1</sub>, <i>J</i><sub>1</sub>, ..., of <i>P</i> +placed in the angle between the planes are situated as +shown in Fig. <a href="#f3">3</a>, and are <i>n</i> − 1 in number. This system +inverted from <i>P</i> as centre gives two spherical surfaces +which cut one another at the same angle as do the +planes. This system is one of electrical equilibrium +in free space, and therefore the problem of the +distribution on two intersecting spheres is solved, +for the case at least in which the angle of intersection +is an aliquot part of 360°. When the planes +are at right angles the result is that for two<span class='pagenum'><a name="Page_55" id="Page_55">55</a></span> +perpendicularly intersecting planes, for which Fig. <a href="#f9">9</a> +gives a diagram.</p> + +<div class="figcenter" style="width: 350px; position: relative;"><a name="f9" id="f9"></a><img src="images/fig09.png" width="350" height="419" alt="Fig. 9." title="" /> +<p class="caption"><span class="smcap">Fig. 9.</span></p></div> + +<p>But the greatest achievement of the method was +the determination of the distribution on a segment of +a thin spherical shell with edge in one plane. The +solution of this problem was communicated to M. +Liouville in the letter of date September 16, 1846, +referred to above, but without proof, which Thomson +stated he had not time to write out owing to preparation +for the commencement of his duties as Professor +of Natural Philosophy at Glasgow on November 1, +1846. It was not supplied until December 1868 and +January 1869; and in the meantime the problem had +not been solved by any other mathematician.</p> + +<p>As a starting point for this investigation the distribution +on a thin plane circular disk of radius <i>a</i> is +required. This can be obtained by considering the disk<span class='pagenum'><a name="Page_56" id="Page_56">56</a></span> +as a limiting case of an oblate ellipsoid of revolution, +charged to potential <i>V</i>, say. If Fig. <a href="#f10">10</a> represent the disk +and <i>P</i> the point at which the density is sought, so that +<i>CP</i> = <i>r</i>, and <i>CA</i> = <i>a</i>, +the density is <i>V</i> ⁄ {2π<sup>2</sup>√(<i>a</i><sup>2</sup> − <i>r</i><sup>2</sup>)}.</p> + +<p>The ratio <i>q</i> ⁄ <i>V</i>, of charge to potential, which is +called the electrostatic capacity of the conductor, is +thus 2<i>a</i> ⁄ π, that is <i>a</i> ⁄ 1.571. It is, as Thomson notes +in his paper, very remarkable that the Hon. Henry +Cavendish should have found long ago by experiment +with the rudest apparatus the electrostatic capacity of +a disk to be 1 ⁄ 1.57 of that of a sphere of the same +radius.</p> + +<table width="100%" summary="figs. 10, 11" border="0"> +<tr> +<td class="t50"><div class="figcenter" style="width: 250px; position: relative;"><a name="f10" id="f10"></a><img src="images/fig10.png" width="250" height="258" alt="Fig. 10." title="" /> +<p class="caption"><span class="smcap">Fig. 10.</span></p></div> +</td> +<td class="t50"><div class="figcenter" style="width: 250px; position: relative;"><a name="f11" id="f11"></a><img src="images/fig11.png" width="250" height="274" alt="Fig. 11." title="" /> +<p class="caption"><span class="smcap">Fig. 11.</span></p></div> +</td> +</tr> +</table> + +<p>Now invert this disk distribution with any point <i>Q</i> +as centre of inversion, and with radius of inversion <i>a</i>. +The geometrical inverse is a segment of a spherical surface +which passes through <i>Q</i>. The inverse distribution +is the induced distribution on a conducting shell uninsulated +and coincident with the segment, and under +the influence of a charge − <i>aV</i> situated at <i>Q</i> (Fig. <a href="#f11">11</a>). +Call this conducting shell the "bowl." If the surface-densities +at corresponding points on the disk and on +the inverse, say points <i>P</i> and <i>P'</i>, be <i>s</i> and <i>s'</i>, then, as<span class='pagenum'><a name="Page_57" id="Page_57">57</a></span> +on page <a href="#Page_51">51</a>, <i>s'</i> = <i>sa</i><sup>3</sup> ⁄ <i>QP'</i><sup>3</sup>. If we put in the value of +<i>s</i> given above, that of <i>s'</i> can be put in a form given +by Thomson, which it is important to remark is +independent of the radius of the spherical surface. +This expression is applicable to the other side of the +bowl, inasmuch as the densities at near points on +opposite sides of the plane disk are equal.</p> + +<p>If <i>v</i>, <i>v'</i> be the potentials at any point <i>R</i> of space, +due to the disk and to its image respectively, +− <i>v'</i> = <i>av</i> ⁄ <i>QR</i>. If then <i>R</i> be coincident with a point +<i>P'</i> on the spherical segment we have (since then +<i>v</i> = <i>V</i>) <i>V'</i> = <i>aV</i> ⁄ <i>QP'</i>, which is the potential due to +the induced distribution caused by the charge − <i>aV</i> +at <i>Q</i> as already stated.</p> + +<p>The fact that the value of <i>s'</i> does not involve the +radius makes it possible to suppose the radius infinite, +in which case we have the solution for a circular disk +uninsulated and under the influence of a charge of +electricity at a point <i>Q</i> in the same plane but outside +the bounding circle.</p> + +<p>Now consider the two parts of the spherical surface, +the bowl <i>B</i>, and the remainder <i>S</i> of the spherical +surface. <i>Q</i> with the charge − <i>aV</i> +may be regarded +as situated on the latter part of the surface. Any +other influencing charges situated on <i>S</i> will give distributions +on the bowl to be found as described above, +and the resulting induced electrification can be found +from these by summation. If <i>S</i> be uniformly electrified +to density <i>s</i>, and held so electrified, the inducing +distribution will be one given by <i>integration</i> over the +whole of <i>S</i>, and the bowl <i>B</i> will be at zero potential +under the influence of this electrification of <i>S</i>, just as +if <i>B</i> were replaced by a shell of metal connected to<span class='pagenum'><a name="Page_58" id="Page_58">58</a></span> +the earth by a long fine wire. The densities are equal +at infinitely near points on the two sides of <i>B</i>.</p> + +<p>Let the bowl be a thin metal shell connected with +the earth by a long thin wire and be surrounded by +a concentric and complete shell of diameter <i>f</i> greater +than that of the spherical surface, and let this shell be +rigidly electrified with surface density − <i>s</i>. +There will be no force within this shell due to its own +electrification, and hence it will produce no change +of the distribution in the interior. But the potential +within will be − 2π<i>fs</i>, +for the charge is − π<i>f</i><sup>2</sup><i>s</i>, and +the capacity of the shell is ½<i>f</i>. The potential of the +bowl will now be zero, and its electrification will just +neutralise the potential − 2π<i>fs</i>, that is, will be exactly +the free electrification required to produce potential 2π<i>fs</i>.</p> + +<p>To find this electrification let the value of <i>f</i> be only +infinitesimally greater than the diameter of the +spherical surface of which <i>B</i> is a part; then the +bowl is under the influence (1) of a uniform electrification +of density − <i>s</i> infinitely close to its outer surface, +and (2) of a uniform electrification of the same density, +which may be regarded as upon the surface which has +been called <i>S</i> above. It is obvious that by (1) a density +<i>s</i> is produced on the outer surface of the bowl, and no +other effect; by (2) an equal density at infinitely near +points on the opposite sides of the bowl is produced +which we have seen how to calculate. Thus the +distribution on the bowl freely electrified is completely +determined and the density can easily be calculated. +The value will be found in Thomson's paper.</p> + +<p>Interesting results are obtained by diminishing <i>S</i> +more and more until the shell is a complete sphere +with a circular hole in it. Tabulated results for<span class='pagenum'><a name="Page_59" id="Page_59">59</a></span> +different relative dimensions of <i>S</i> will be found in +Thomson's paper, "Reprint of Papers," Articles V, +XIV, XV. Also the reader will there find full particulars +of the mathematical calculations indicated in +this chapter, and an extension of the method to the +case of an influencing point not on the spherical surface +of which the shell forms part. Further developments +of the problem have been worked out by other writers, +and further information with references will be found +in Maxwell's <i>Electricity and Magnetism</i>, loc. cit.</p> + +<p>It is not quite clear whether Thomson discovered +<i>geometrical</i> inversion independently or not: very likely +he did. His letter to Liouville of date October 8, 1845, +certainly reads as if he claimed the geometrical transformation +as well as the application to electricity. +Liouville, however, in his Note in which he dwells on +the analytical theory of the transformation says, "La +transformation dont il s'agit est bien connue, du reste, +et des plus simples; c'est celle que M. Thomson lui-même +a jadis employée sous le nom de principe des +<i>images</i>." In Thomson and Tail's <i>Natural Philosophy</i>, +§ 513, the reference to the method is as follows: +"Irrespectively of the special electric application, the +method of images gives a remarkable kind of transformation +which is often useful. It suggests for mere +geometry what has been called the transformation by +reciprocal radius-rectors, that is to say...." Then +Maxwell, in his review of the "Reprint of Papers" +(<i>Nature</i>, vol. vii), after referring to the fact that the +solution of the problem of the spherical bowl remained +undemonstrated from 1846 to 1869, says that the +geometrical idea of inversion had probably been discovered +and rediscovered repeatedly, but that in his<span class='pagenum'><a name="Page_60" id="Page_60">60</a></span> +opinion most of these discoveries were later than 1845, +the date of Thomson's first paper.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">10</a></p> + +<p>A very general method of finding the potential at +any point of a region of space enclosed by a given +boundary was stated by Green in his 'Essay' for the +case in which the potential is known for every point of +the boundary. The success of the method depends on +finding a certain function, now called Green's function. +When this is known the potential at any point is at +once obtained by an integration over the surface. +Thomson's method of images amounts to finding for +the case of a region bounded by one spherical surface +or more the proper value of Green's function. Green's +method has been successfully employed in more complicated +cases, and is now a powerful method of attack +for a large range of problems in other departments +of physical mathematics. Thomson only obtained a +copy of Green's paper in January 1845, and probably +worked out his solutions quite independently of any +ideas derived from Green's general theory.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_61" id="Page_61">61</a></span></p> + +<h3>CHAPTER V</h3> + +<h4>THE CHAIR OF NATURAL PHILOSOPHY AT GLASGOW. +ESTABLISHMENT OF THE FIRST PHYSICAL +LABORATORY</h4> + +<p><span class="smcap">The</span> incumbent of the Chair of Natural Philosophy +in the University of Glasgow, Professor Meikleham, +had been in failing health for several years, and from +1842 to 1845 his duties had been discharged by another +member of the Thomson <i>gens</i>, Mr. David Thomson, +B.A., of Trinity College, Cambridge, afterwards +Professor of Natural Philosophy at Aberdeen. Dr. +Meikleham died in May 1846, and the Faculty thereafter +proceeded on the invitation of Dr. J. P. Nichol, +the Professor of Astronomy, to consider whether in +consequence of the great advances of physical science +during the preceding quarter of a century it was not +urgently necessary to remodel the arrangements for the +teaching of natural philosophy in the University. The +advance of science had indeed been very great. Oersted +and Ampère, Henry and Faraday and Regnault, Gauss +and Weber, had made discoveries and introduced +quantitative ideas, which had changed the whole aspect +of experimental and mathematical physics. The +electrical discoveries of the time reacted on the other +branches of natural philosophy, and in no small degree +on mathematics itself. As a result the progress of +that period has continued and has increased in rapidity,<span class='pagenum'><a name="Page_62" id="Page_62">62</a></span> +until now the accumulated results, for the most part +already united in the grasp of rational theory, have +gone far beyond the power of any single man to +follow, much less to master.</p> + +<p>It is interesting to look into a course of lectures +such as were usually delivered in the universities a +hundred years ago by the Professor of Natural Philosophy. +We find a little discussion of mechanics, +hydrostatics and pneumatics, a little heat, and a very +little optics. Electricity and magnetism, which in our +day have a literature far exceeding that of the whole +of physics only sixty years ago, could hardly be said to +exist. The professor of the beginning of the nineteenth +century, when Lord Kelvin's predecessor was +appointed, apparently found himself quite free to +devote a considerable part of each lecture to reflections +on the beauties of nature, and to rhetorical flights +fitter for the pulpit than for the physics lecture-table.</p> + +<p>In the intervening time the form and fashion of +scientific lectures has entirely changed, and the change +is a testimony to the progress of science. It is visible +even in the design of the apparatus. Microscopes, for +example, have a perfection and a power undreamed of +by our great-grandfathers, and they are supported on +stands which lack the ornamentation of that bygone +time, but possess stability and convenience. Everything +and everybody—even the professor, if that be +possible—must be business-like; and each moment of +time must be utilised in experiments for demonstration, +not for applause, and in brief and cogent statements +of theory and fact. To waste time in talk that is not +to the point is criminal. But withal there is need of +grace of expression and vividness of description, of<span class='pagenum'><a name="Page_63" id="Page_63">63</a></span> +clearness of exposition, of imagination, even of poetical +intuition: but the stern beauty of modern science is +only disfigured by the old artificial adornments and +irrelevancies.</p> + +<p>This is the tone and temper of science at the +present day: the task is immense, the time is short. +And sixty years since some tinge of the same cast of +thought was visible in scientific workers and teachers. +The Faculty agreed with Dr. Nichol that there was +need to bring physical teaching and equipment into +line with the state of science at the time; but they +wisely decided to do nothing until they had appointed +a Professor of Natural Philosophy who would be able +to advise them fully and in detail. They determined, +however, to make the appointment subject to such +alterations in the arrangements of the department as +they might afterwards find desirable.</p> + +<p>On September 11, 1846, the Faculty met, and having +considered the resolutions which had been proposed +by Dr. Nichol, resolved to the effect that the appointment +about to be made should not prejudice the right +of the Faculty to originate or support, during the incumbency +of the new professor, such changes in the +arrangements for conducting instruction in physical +science as it might be expedient to adopt, and that this +resolution should be communicated to the candidate +elected. The minute then runs: "The Faculty having +deliberated on the respective qualifications of the +gentlemen who have announced themselves candidates +for this chair, and the vote having been taken, it +carried unanimously in favour of Mr. William Thomson, +B.A., Fellow of St. Peter's College, Cambridge, and +formerly a student of this University, who is accordingly<span class='pagenum'><a name="Page_64" id="Page_64">64</a></span> +declared to be duly elected: and Mr. Thomson being +within call appeared in Faculty, and the whole of +this minute having been read to him he agreed to +the resolution of Faculty above recorded and accepted +the office." It was also resolved as follows: "The +Faculty hereby prescribe Mr. Thomson an essay on +the subject, <i>De caloris distributione per terræ corpus</i>, and +resolve that his admission be on Tuesday the 13th +October, provided that he shall be found qualified by +the Meeting and shall have taken the oath and made +the subscriptions which are required by law."</p> + +<p>At that time, and down to within the last fifteen +years, every professor, before his induction to his chair, +had to submit a Latin essay on some prescribed subject. +This was almost the last relic of the customs of the +days when university lectures were delivered in Latin, +a practice which appears to have been first broken +through by Adam Smith when Professor of Moral +Philosophy. Whatever it may have been in the +eighteenth century, the Latin essay at the end of the +nineteenth was perhaps hardly an infallible criterion of +the professor-elect's Latinity, and it was just as well to +discard it. But fifty years before, and for long after, +classical languages bulked largely in the curriculum of +every student of the Scottish Universities, and it is +undoubtedly the case that most of those who afterwards +came to eminence in other departments of +learning had in their time acquitted themselves well in +the old <i>Litteræ Humaniores</i>. This was true, as we have +seen, of Thomson, and it is unlikely that the form of +his inaugural dissertation cost him much more effort +than its matter.</p> + +<div class="figcenter" style="width: 550px; position: relative;"><a name="thomson" id="thomson"></a><img src="images/thomson.jpg" width="550" height="694" alt="Professor WILLIAM THOMSON" title="" /> +<p class="caption"><span class="smcap">Professor WILLIAM THOMSON</span>, 1846</p></div> + +<p>The subject chosen had reference no doubt to the<span class='pagenum'><a name="Page_65" id="Page_65">65</a></span> +papers on the theory of heat which Mr. Thomson had +already published. The thesis was presented to the +Faculty on the day appointed, and approved, and +Mr. Thomson having produced a certificate of his +having taken the oaths to government, and promised +to subscribe the formula of the Church of Scotland as +required by law, on the first convenient opportunity, +"the following oath was then administered to him, +which he took and subscribed: <i>Ego, Gulielmus +Thomson, B.A., physicus professor in hac Academia designatus, +promitto sancteque polliceor me in munere mihi demandato +studiose fideliterque versaturum.</i>" Professor +Thomson was then "solemnly admitted and received +by all the Members present, and took his seat as a +Member of Faculty."</p> + +<p>No translation of this essay was ever published, +but its substance was contained in various papers which +appeared later. The following reference to it is made +in an introduction attached to Article XI of his +<i>Mathematical and Physical Papers</i> (vol. i, 1882).</p> + +<p>"An application to Terrestrial Temperature, of the +principle set forth in the first part of this paper relating +to the age of thermal distributions, was made the +subject of the author's Inaugural Dissertation on the +occasion of his induction to the professorship of Natural +Philosophy in the University of Glasgow, in October +1846, '<i>De Motu Caloris per Terræ Corpus</i>'<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">11</a>: which, +more fully developed afterwards, gave a very decisive +limitation to the possible age of the earth as a habitation +for living creatures; and proved the untenability +of the enormous claims for TIME which, uncurbed +<span class='pagenum'><a name="Page_66" id="Page_66">66</a></span> +by physical science, geologists and biologists had begun +to make and to regard as unchallengeable. See 'Secular +Cooling of the Earth, Geological Time,' and several +other Articles below." Some statement of the argument +for this limitation will be given later. [See Chap. <a href="#Page_254">XIV.</a>]</p> + +<p>Thomson thus entered at the age of twenty-five on +what was to be his life work as a teacher, investigator, +and inventor. For he continued in office fifty-three +years, so that the united tenures of his predecessor and +himself amounted to only four years less than a +century! He took up his duties at the opening of +the college session in November, and promptly called +the attention of the Faculty to the deficiencies of the +equipment of apparatus, which had been allowed to +fall behind the times, and required to have added to it +many new instruments. A committee was appointed +to consider the question and report, and as a result of +the representations of this committee a sum of £100 +was placed at Professor Thomson's disposal to supply +his most pressing needs. In the following years repeated +applications for further grants were made and +various sums were voted—not amounting to more than +£500 or £600 in all—which were apparently regarded +as (and no doubt were, considering the times and the +funds at the disposal of the Faculty) a liberal provision +for the teaching of physical science. A minute of the +Faculty, of date Nov. 26, 1847, is interesting.</p> + +<p>After "emphatically deprecating" all idea that such +large annual expenditure for any one department was to +be regularly contemplated, the committee refer in their +report to the "inadequate condition of the department in +question," and express their satisfaction "with the +reasonable manner in which the Professor of Natural<span class='pagenum'><a name="Page_67" id="Page_67">67</a></span> +Philosophy has on all occasions readily modified his +demands in accordance with the economical suggestions +of the committee." They conclude by saying that they +"view his ardour and anxiety in the prosecution of +his profession with the greatest pleasure," and "heartily +concur in those anticipations of his future celebrity +which Monsr. Serville,<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">12</a> the French mathematician, has +recently thought fit to publish to the scientific world."</p> + +<p>Again, in April 1852, the Faculty agree to pay a sum +of £137 6<i>s.</i> 1½<i>d.</i> as the price of purchases of philosophical +apparatus already made, and approve of a +suggestion of the committee that the expenditure on +this behalf during the next year should not exceed +£50, and "they desire that the purchases shall be made +so far as is possible with the previously obtained concurrence +of the committee." It is easy to imagine +that the ardent young Professor of Natural Philosophy +found the leisurely methods of his older colleagues +much too slow, and in his enthusiasm anticipated consent +to his demands by ordering his new instruments +without waiting for committees and meetings and +reports.</p> + +<p>In an address at the opening of the Physical and +Chemical Laboratories of the University College of +North Wales, on February 2, 1885, Sir William +Thomson (as he was then) referred to his early +equipment and work as follows: "When I entered +upon the professorship of Natural Philosophy at +Glasgow, I found apparatus of a very old-fashioned +kind. Much of it was more than a hundred years +old, little of it less than fifty years old, and most of +it was worm-eaten. Still, with such appliances, year +<span class='pagenum'><a name="Page_68" id="Page_68">68</a></span> +after year, students of natural philosophy had been +brought together and taught as well as possible. The +principles of dynamics and electricity had been well +illustrated and well taught, as well taught as lectures +and so imperfect apparatus—but apparatus merely of +the lecture-illustration kind—could teach. But there +was absolutely no provision of any kind for experimental +investigation, still less idea, even, for anything +like students' practical work. Students' laboratories +for physical science were not then thought of."<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">13</a></p> + +<p>It appears that the class of Natural Philosophy +(there was then as a rule only one class in any subject, +though supplementary work was done in various ways) +met for systematic lectures at 9 a.m., which is the +hour still adhered to, and for what was called "Experimental +Physics" at 8 p.m.!</p> + +<p>The <i>University Calendar</i> for 1863-4 states that +"the Natural Philosophy Class meets two hours daily, +9 a.m. and 11 a.m. The first hour is chiefly spent in +statements of Principles, description of Results of +Observation, and Experimental Illustrations. The +second hour is devoted to Mathematical Demonstrations +and Exercises, and Examinations on all parts of the +Course.</p> + +<p>"The Text Books to be used are: 'Elements of +Dynamics' (first part now ready), Printed by George +Richardson, University Printer. 'Elements of Natural +Philosophy,' by Professors W. Thomson and P. G. +Tait (Two Treatises to be published before November. +Macmillan.<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">14</a>)</p> + +<p><span class='pagenum'><a name="Page_69" id="Page_69">69</a></span></p><p>"The shorter of the last mentioned Treatises will +be used for the work required of all students of +Natural Philosophy in the regular curriculum. The +whole or specified parts of the larger Treatise will be +prescribed in connection with voluntary examinations +and exercises in the Class, and for candidates for the +degree of M.A. with honours. Students who desire +to undertake these higher parts of the business of the +class, ought to be well prepared on all the subjects of +the Senior Mathematical Class.</p> + +<p>"The Laboratory in connection with the class is +open daily from 9 a.m. to 4 p.m. for Experimental +Exercises and Investigations, under the direction of +the Professor and his official assistant."</p> + +<p>In 1847 the meetings for experimental physics were +changed to 11 a.m. The hour 9 a.m. is still (1908) retained +for the regular meetings of the ordinary class, and +11 a.m. for meetings held twice a week for exercises +and tutorial work, attendance at which is optional.</p> + +<p>[A second graduating class has now been instituted +and is very largely attended. Each student attends three +lectures and spends four hours in the laboratory each +week. A higher class, in two divisions, is also held.]</p> + +<p>At an early date in his career as a professor Thomson +called in the aid of his students for experimental research. +In many directions the properties of matter +still lay unexplored, and it was necessary to obtain +exact data for the perfecting of the theories of elasticity, +electricity and heat, which had been based on the +researches of the first half of the nineteenth century. +To the authors of these theories—Gauss, Green, +Cauchy and others—he was a fit successor. Not +knowing all that had been done by these men of genius,<span class='pagenum'><a name="Page_70" id="Page_70">70</a></span> +he reinvented, as we have seen, some of their great +theorems, and in somewhat later work, notably in +electricity and magnetism, set the theories on a new +basis cleared of all extraneous and unnecessary matter, +and reduced the hypotheses and assumptions to the +smallest possible number, stated with the most careful +precautions against misunderstanding. As this work +was gradually accomplished the need for further experiment +became more and more clearly apparent. +Accordingly he established at the old College in the +High Street, what he has justly claimed was the first +physical laboratory for students.<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">15</a> An old wine-cellar +in the basement adjoining the Natural Philosophy +Class-room was first annexed, and was the scene of +early researches, which were to lead to much of +the best work of the present time. To this was +added a little later the Blackstone Examination-room, +which, disused and "left unprotected," was added to +the wine-cellar, and gave space for the increasing +corps of enthusiastic workers who came under the +influence of the new teacher, and were eager to be +associated with his work. A good many of the +researches which were carried out in this meagre +accommodation in the old College will be mentioned +in what follows.</p> + +<div class="figcenter" style="width: 550px; position: relative;"><a name="college" id="college"></a><img src="images/old_college.jpg" width="500" height="735" alt="INNER COURT OF THE OLD COLLEGE" title="" /> +<p class="caption"><span class="smcap">INNER COURT OF THE OLD COLLEGE</span><br /> +Showing Natural Philosophy Rooms</p></div> + +<p>[In the view of the inner court of the Old College +given opposite, the windows on the ground-floor to +<span class='pagenum'><a name="Page_71" id="Page_71">71</a></span>the right of the turret in front, are those of the Blackstone +Examination-room, which formed a large part +of the new Physical Laboratory. The windows above +these, on the second floor, are those of the Apparatus-room +of the Natural Philosophy Department. Between +the turret on the right of the picture and the angle of +the court are the windows of the Natural Philosophy +Class-room. The attic above the Apparatus-room +was at a later time occupied by the Engineering +Department, under Professor Macquorn Rankine.]</p> + +<p>Here again we may quote from the Bangor address:</p> + +<p>"Soon after I entered my present chair in the +University of Glasgow in 1846 I had occasion to +undertake some investigations of electrodynamic +qualities of matter, to answer questions suggested by +the results of mathematical theory, questions which +could only be answered by direct experiment. The +labour of observing proved too heavy, much of it +could scarcely be carried on without two or more +persons, working together. I therefore invited students +to aid in the work. They willingly accepted the +invitation, and lent me most cheerful and able help. +Soon after, other students, hearing that their class-fellows +had got experimental work to do, came to me +and volunteered to assist in the investigation. I could +not give them all work in the particular investigation +with which I had commenced—'the electric convection +of heat'—for want of means and time and +possibilities of arrangement, but I did all in my power +to find work for them on allied subjects (Electrodynamic +Properties of Metals, Moduluses of Elasticity +of Metals, Elastic Fatigue, Atmospheric Electricity, +etc.). I then had an ordinary class of a hundred<span class='pagenum'><a name="Page_72" id="Page_72">72</a></span> +students, of whom some attended lectures in natural +philosophy two hours a day, and had nothing more to +do from morning till night. These were the balmy +days of natural philosophy in the University of Glasgow—the +pre-Commissional days. But the majority +of the class really had very hard work, and many of +them worked after class-hours for self-support. Some +were engaged in teaching, some were city-missionaries, +intending to go into the Established Church of Scotland +or some other religious denomination of Scotland, +or some of the denominations of Wales, for I always +had many Welsh students. In those days, as now, +in the Scottish Universities all intending theological +students took a 'philosophical curriculum'—'zuerst +collegium logicum,' then moral philosophy, and (generally +last) natural philosophy. Three-fourths of my +volunteer experimentalists used to be students who +entered the theological classes immediately after the +completion of the philosophical curriculum. I well +remember the surprise of a great German professor +when he heard of this rule and usage: 'What! do +the theologians learn physics?' I said, 'Yes, they all +do; and many of them have made capital experiments. +I believe they do not find that their theology suffers +at all from (their) having learned something of +mathematics and dynamics and experimental physics +before they enter upon it.'"</p> + +<p>This statement, besides throwing an interesting light +on the conditions of university work sixty years ago, gives +an illustration of the wide interpretation in Scotland of +the term <i>Arts</i>. Here it has meant, since the Chair of +Natural Philosophy was founded in 1577, and held by +one of the Regents of the University, <i>Artes Liberales</i> in<span class='pagenum'><a name="Page_73" id="Page_73">73</a></span> +the widest sense, that is, the study of <i>Litteræ Humaniores</i> +(including mental and moral philosophy) and physical +and mathematical science. These were all deemed +necessary for a liberal education at that time: in the +scientific age in which we live it is more imperative +than ever that neither should be excluded from the +Arts curriculum of our Universities. The common +distinction between Arts and Science is a false one, +and the product of a narrow idea which is alien to the +traditions of our northern Universities.</p> + +<p>It is to be noted, however, that the laboratory thus +founded was essentially a research laboratory; it was +not designed for the systematic instruction of students +in methods of experimenting. Laboratories for this +purpose came later, and as a natural consequence. +But for the best students, ill prepared as, no doubt, +some of them were for the work of research, the +experience gained in such a laboratory was very valuable. +They learned—and, indeed, had to learn—in +an incidental manner how to determine physical constants, +such as specific gravities, thermal capacities, +electric resistances, and so forth. For, apart from the +<i>Relations des Expériences</i> of Regnault, and the magnetic +and electric work of Gauss and Weber, there was no +systematised body of information available for the +guidance of students. Good students could branch +out from the main line of inquiry, so as to acquire +skill in subsidiary determinations of this kind; to the +more easily daunted student such difficulties proved +formidable, and often absolutely deterrent.</p> + +<p>It is not easy for a physicist of the present day to +realise the state of knowledge of the time, and so +he often fails to recognise the full importance of<span class='pagenum'><a name="Page_74" id="Page_74">74</a></span> +Thomson's work. The want of precise knowledge +of physical constants was to a considerable extent +a consequence of the want of exact definitions of +quantities to be determined, and in a much greater +degree of the lack of any system of units of measurement. +The study of phenomena was in the main +merely qualitative; where an attempt had been made +to obtain quantitative determinations, the units employed +were arbitrary and dependent on apparatus in +the possession of the experimenter, and therefore +unavailable to others. In the department of heat, as +has been said, a great beginning had been made by +Regnault, in whose hands the exact determination of +physical constants had become a fine art.</p> + +<p>In electricity and magnetism there were already the +rudiments of quantitative measurement. But it was only +long after, when the actions of magnets and of electric +currents had been much further studied, that the +British Association entered on its great work of setting +up a system of absolute units for the measurement of +such actions. Up till then the resistance, for example, +of a piece of wire, to the passage of an electric current +along it, was expressed by some such specification as +that it was equal to the resistance of a certain piece of +copper wire in the experimenter's possession. It was +therefore practically impossible for experimenters elsewhere +to profit by the information. And so in other +cases. An example from Thomson's papers on the +"Dynamical Theory of Heat" may be cited here, +though it refers to a time (1851) when some progress +towards obtaining a system of absolute units had been +made. In § 118 (Art. XLVIII) he states that the +electromotive force of a thermoelectric couple of copper<span class='pagenum'><a name="Page_75" id="Page_75">75</a></span> +and bismuth, at temperatures 0° C. and 100° C. of its +functions, might be estimated from a comparison made +by Pouillet of the strength of the current sent by this +electromotive force through a copper wire 20 metres +long and 1 millimetre in diameter, with the strength +of a current decomposing water at a certain rate, were +it not that the specific resistances of different specimens +of copper are found to differ considerably from one +another. Hence, though an estimate is made, it is +stated that, without experiments on the actual wire +used by Pouillet, it was impossible to arrive at an +accurate result. Now if it had been in Pouillet's +power to determine accurately the resistance of his +circuit in absolute units, there would have been no +difficulty in the matter, and his result would have +been immediately available for the estimate required.</p> + +<p>When submarine cables came to be manufactured +and laid all this had to be changed. For they were +expensive; an Atlantic cable, for example, cost half a +million sterling. The state of the cable had to be +ascertained at short intervals during manufacture; a +similar watch had to be kept upon it during the process +of laying, and afterwards during its life of telegraphic +use. The observations made by one observer had +therefore to be made available to all, so that, with +other instruments and at another place, equivalent +observations could be made and their results quantitatively +compared with those of the former. To set up +a system of measurement for such purposes as these +involved much theoretical discussion and an enormous +amount of experimental investigation. This was +undertaken by a special committee of the Association, +and a principal part in furnishing discussions of theory<span class='pagenum'><a name="Page_76" id="Page_76">76</a></span> +and in devising experimental methods was taken by +Thomson. The committee's investigations took place +at a date somewhat later in Thomson's career than +that with which we are here dealing, and some account +of them will be given in a later chapter; but much +work, preparatory for and leading up to the determination +of electrical standards, was done by the +volunteer laboratory corps in the transformed wine-cellar +of the old College.</p> + +<p>The selection and realisation of electrical standards +was a work of extraordinary importance to the world +from every point of view—political, commercial, and +social. It not only rendered applications of electricity +possible in the arts and industries, but by relieving +experimental results from the vagueness of the specifications +formerly in use, made the further progress of +pure electrical science a matter in which every step +forward, taken by an individual worker, facilitated the +advance of all. But like other toilsome services, the +nature of which is not clear to the general public, +it has never received proper acknowledgment from +those who have profited by it. If Thomson had done +nothing more than the work he did in this connection, +first with his students and later with the British +Association Committee, he would have deserved well +of his fellow-countrymen.</p> + +<p>When Professor Thomson was entering on the +duties of his chair, and calling his students to his aid, +the discoveries of Faraday on the induction of currents +by the motion of magnets in the neighbourhood of +closed circuits of wire, or, what comes to the same +thing, the motion of such circuits in the "fields" of +magnets, had not been long given to the world, and<span class='pagenum'><a name="Page_77" id="Page_77">77</a></span> +were being pondered deeply by natural philosophers. +The time was ripe for a quantitative investigation of +current induction, like that furnished by the genius of +Ampère after the discovery by Oersted of the deflection +of a magnet by an electric current. Such an +investigation was immensely facilitated by Faraday's +conception of lines of magnetic force, the cutting of +which by the wire of the circuit gave rise to the +induced current. Indeed, the mathematical ideas +involved were indicated, and not obscurely, by Faraday +himself. But to render the mathematical theory +explicit, and to investigate and test its consequences, +required the highest genius. This work was accomplished +in great measure by Thomson, whose presentation +of electrodynamic theory helped Maxwell to the +view that light was an affair of the propagation of +electric and magnetic vibrations in an insulating +medium, the light-carrying ether.</p> + +<p>Another investigation on which he had already +entered in 1847 was of great importance, not only for +pure science but for the development and proper +economy of all industrial operations. The foundations +on which a dynamical theory of heat was to be raised +had been partly laid by Carnot and were being completed +on the experimental side by James Prescott +Joule, whom Thomson met in 1847 at the meeting +of the British Association at Oxford. The meeting +at Oxford in 1860 is memorable to the public at large, +mainly on account of the discussion which took place +on the Darwinian theory, and the famous dialectic +encounter between Bishop Wilberforce and Professor +Huxley; the Oxford meeting of 1894 will always be +associated with the announcement of the discovery of<span class='pagenum'><a name="Page_78" id="Page_78">78</a></span> +argon by Lord Rayleigh and Sir William Ramsay: +the meeting of 1847 might quite as worthily be +remembered as that at which Joule laid down, with +numerical exactitude, the first law of thermodynamics. +Joule brought his experimental results before the +Mathematical and Physical Section at that meeting; +and it appears probable that they would have received +scant attention had not their importance been forcibly +pointed out by Thomson. Communications thereafter +passed frequently between the two young physicists, +and there soon began a collaboration of great value to +science, and a friendship which lasted till the death +of Joule in 1884. [See p. <a href="#Page_88">88</a> below.]</p> + +<p>We shall devote the next few chapters to an account, +as free from technicalities as possible, of these great +divisions of Thomson's earlier original work as professor +at Glasgow.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_79" id="Page_79">79</a></span></p> + +<h3>CHAPTER VI</h3> + +<h4>FRIENDSHIP WITH STOKES AND JOULE. +EARLY WORK AT GLASGOW</h4> + +<p><span class="smcap">During</span> his residence at Cambridge Thomson gained +the friendship of George Gabriel Stokes, who had +graduated as Senior Wrangler and First Smith's +Prizeman in 1841. They discussed mathematical +questions together and contributed articles on various +topics to the <i>Cambridge Mathematical Journal</i>. In +1846 "Cambridge and Dublin" was substituted for +"Cambridge" in the title of the <i>Journal</i>, and a new +series was begun under the editorship of Thomson. +A feature of the earlier volumes of the new issue was a +series of Notes on Hydrodynamics written by agreement +between Thomson and Stokes, and printed in +vols. ii, iii, and v. The first, second, and fifth of +the series were written by Thomson, the others by +Stokes. The matter of these Notes was not altogether +novel; but many points were put in a new and more +truly physical light, and the series was no doubt of +much service to students, for whose use the articles +were intended. Some account of these Notes will be +given in a later chapter on Thomson's hydrodynamical +papers.</p> + +<p>For the mathematical power and sure physical +instinct of Stokes Thomson had always the greatest +admiration. When asked on one occasion who was<span class='pagenum'><a name="Page_80" id="Page_80">80</a></span> +the most outstanding worker in physical science on the +continent, he replied, "I do not know, but whoever he +is, I am certain that Stokes is a match for him." In a +report of an address which he delivered in June 1897, +at the celebration of the Jubilee of Sir George Stokes +as Lucasian Professor of Mathematics, Lord Kelvin referred +to their early intercourse at Cambridge in terms +which were reported as follows: "When he reflected on +his own early progress, he was led to recall the great +kindness shown to himself, and the great value which +his intercourse with Sir George Stokes had been to +him through life. Whenever a mathematical difficulty +occurred he used to say to himself, 'Ask Stokes what +he thinks of it.' He got an answer if answer was +possible; he was told, at all events, if it was unanswerable. +He felt that in his undergraduate days, and he +felt it more now."</p> + +<p>After the death of Stokes in February 1902, Lord +Kelvin again referred, in an enthusiastic tribute in +<i>Nature</i> for February 12, to these early discussions. +"Stokes's scientific work and scientific thought is but +partially represented by his published writings. He +gave generously and freely of his treasures to all who +were fortunate enough to have an opportunity of +receiving from him. His teaching me the principles +of solar and stellar chemistry when we were walking +about among the colleges sometime prior to 1852 +(when I vacated my Peterhouse Fellowship to be +no more in Cambridge for many years) is but one +example."</p> + +<p>The interchange of ideas between Stokes and +Thomson which began in those early days went on +constantly and seems to have been stimulating to both.<span class='pagenum'><a name="Page_81" id="Page_81">81</a></span> +The two men were in a sense complementary in nature +and temperament. Both had great power and great +insight, but while Stokes was uniformly calm, reflective, +and judicial, Thomson's enthusiasm was more outspokenly +fervid, and he was apt to be at times vehement +and impetuous in his eagerness to push on an investigation; +and though, as became his nationality, he was +cautious in committing himself to conclusions, he +exercised perhaps less reserve in placing his results +before the public of science.</p> + +<p>A characteristic instance of Thomson's vehement +pursuit of experimental results may be given here, +although the incidents occurred at a much later date in +his career than that with which we are at present +concerned. In 1880 the invention of the Faure +Secondary Battery attracted his attention. M. Faure +brought from Paris some cells made up and ready +charged, and showed in the Physical Laboratory at +Glasgow the very powerful currents which, in consequence +of their very low internal resistance, they +were capable of producing in a thick piece of copper +wire. The cells were of the original form, constructed +by coating strips of sheet lead on both sides with a paste +of minium moistened with dilute sulphuric acid, swathing +them in woollen cloth sewed round them, and then +rolling two together to form the pair of plates for one +cell.</p> + +<p>A supply of sheet lead, minium, and woollen cloth +was at once obtained, and the whole laboratory corps of +students and staff was set to work to manufacture +secondary batteries. A small Siemens-Halske dynamo +was telegraphed for to charge the cells, and the ventilating +steam-engine of the University was requisitioned<span class='pagenum'><a name="Page_82" id="Page_82">82</a></span> +to drive the dynamo during the night. Thus the +University stokers and engineer were put on double +shifts; the cells were charged during the night and the +charging current and battery-potential measured at +intervals.</p> + +<p>Then the cells were run down during the day, and +their output measured in the same way. Just as this +began, Thomson was laid up with an ailment which +confined him to bed for a couple of weeks or so; but +this led to no cessation of the laboratory activity. On +the contrary, the laboratory corps was divided into two +squads, one for the night, the other for the day, and the +work of charging and discharging, and of measurement +of expenditure and return of energy went on without +intermission. The results obtained during the day +were taken to Thomson's bedside in the evening, and +early in the morning he was ready to review those +which had been obtained during the night, and to suggest +further questions to be answered without delay. +This mode of working could not go on indefinitely, but +it continued until his assistants (some of whom had to +take both shifts!), to say nothing of the stokers and +students, were fairly well exhausted.</p> + +<p>On other occasions, when he was from home, he +found the post too slow to convey his directions to his +laboratory workers, and telegraphed from day to day +questions and instructions regarding the work on hand. +Thus one important result (anticipated, however, by +Villari) of the series of researches on the effects of +stress on magnetisation which forms Part VII of his +<i>Electrodynamic Qualities of Metals</i>—the fact that up +to a certain magnetising force the effect of pull, +applied to a wire of soft iron, is to increase the<span class='pagenum'><a name="Page_83" id="Page_83">83</a></span> +magnetisation produced, and for higher magnetising +forces to diminish it—was telegraphed to him on the +night on which the paper was read to the Royal +Society.</p> + +<p>It will thus be seen that Thomson, whether confined +to his room or on holiday, kept his mind fixed upon his +scientific or practical work, and was almost impatient +for its progress. Stokes worked mainly by himself; +but even if he had had a corps of workers and assistants, +it is improbable that such disturbances of hours of +attendance and laboratory and workshop routine would +have occurred, as were not infrequent at Glasgow when +Thomson's work was, in the 'sixties and 'seventies, at +its intensest.</p> + +<p>Stokes and Thomson were in succession presidents +of the Royal Society, Stokes from 1885 to 1890, and +Thomson (from 1892 as Lord Kelvin) from 1890 to +1895. This is the highest distinction which any +scientific man in this country can achieve, and it is +very remarkable that there should have been in recent +times two presidents in succession whose modes of +thought and mathematical power are so directly comparable +with those of the great founder of modern +natural philosophy. Stokes had the additional distinction +of being the lineal successor of Newton as +Lucasian Professor of Mathematics at Cambridge. But +it was reserved for Thomson to do much by the +publication of Thomson and Tait's <i>Natural Philosophy</i> +to bring back the current of teaching and thought in +dynamical science to the ideas of the <i>Principia</i>, and to +show how completely the fundamental laws, as laid +down in that great classic, avail for the inclusion of the +modern theory of energy, in all its transformations,<span class='pagenum'><a name="Page_84" id="Page_84">84</a></span> +within the category of dynamical action between +material systems.</p> + +<p>An exceedingly eminent politician, now deceased, +said some years ago that the present age was singularly +deficient in minds of the first quality. So far as +scientific genius is concerned, the dictum was singularly +false: we have here a striking proof of the contrary. +But then few politicians know anything of science; +indeed some of those who guide, or aspire to guide, +the destinies of the most scientific and industrial +empire the world has ever seen are almost boastful of +their ignorance. There are, of course, honourable +exceptions.</p> + +<p>It is convenient to refer here to the share which +Stokes and Thomson took in the physical explanation +of the dark lines of the solar spectrum, and to their +prediction of the possibility of determining the constitution +of the stars and of terrestrial substances by +what is now known as spectrum analysis. Thomson +used to give the physical theory of these lines in his +lectures, and say that he obtained the idea from Stokes +in a conversation which they had in the garden of +Pembroke at Cambridge, "some time prior to 1852" +(see the quotation from his <i>Nature</i> article quoted above, +p. <a href="#Page_80">80</a>, and the <i>Baltimore Lectures</i>, p. <a href="#Page_101">101</a>). This is +confirmed by a student's note-book, of date 1854, +which is now in the Natural Philosophy Department. +The statements therein recorded are perfectly definite +and clear, and show that at that early date the whole +affair of spectrum analysis was in his hands, and only +required confirmation by experiments on the reversal +of the lines of terrestrial substances by an atmosphere +of the substance which produced the lines, and a<span class='pagenum'><a name="Page_85" id="Page_85">85</a></span> +comparison of the positions of the bright lines of +terrestrial substances with those of the dark lines of +the solar spectrum. Why Thomson did not carry out +all these experiments it would be difficult to say. +Some of them he did make, for Professor John +Ferguson, who was a student of Natural Philosophy +in 1859-60, has recently told how he witnessed +Thomson make the experiment of reversing the lines +of sodium by passing the light from the salted flame +of a spirit lamp through vapour of sodium produced by +heating the metal in an iron spoon. A few days later, +says Professor Ferguson, Thomson read a letter to his +class announcing Bunsen and Kirchhoff's discovery.</p> + +<p>A letter of Stokes to Sir John Lubbock, printed in +the <i>Scientific Correspondence of Sir George Gabriel Stokes</i>, +states his recollection of the matter, and gives Thomson +the credit of having inferred the method of spectrum analysis, +a method to which Stokes himself makes no claim. +He says, "I know, I think, what Sir William Thomson +was alluding to. I knew well, what was generally +known, and is mentioned by Herschel in his treatise +on Light, that the bright D seen in flames is specially +produced when a salt of soda is introduced. I connected +it in my own mind with the presence of sodium, +and I suppose others did so too. The coincidence in +position of the bright and dark D is too striking to +allow us to regard it as fortuitous. In conversation +with Thomson I explained the connection of the dark +and bright line by the analogy of a set of piano strings +tuned to the same note, which, if struck, would give +out that note, and also would be ready to sound it, to +take it up, in fact, if it were sounded in air. This +would imply absorption of the aërial vibrations, as<span class='pagenum'><a name="Page_86" id="Page_86">86</a></span> +otherwise there would be a creation of energy. +Accordingly I accounted for the presence of the dark +D in the solar spectrum by supposing that there was +sodium in the atmosphere, capable of absorbing light of +that particular refrangibility. He asked me if there +were any other instances of such coincidences of bright +and dark lines, and I said I thought there was one +mentioned by Brewster. He was much struck with +this, and jumped to the conclusion that to find out +what substances were in the stars we must compare +the positions of the dark lines seen in their spectra +with the spectra of metals, etc....</p> + +<p>"I should have said that I thought Thomson was +going too fast ahead, for my notion at the time was +that, though a few of the dark lines might be traced +to elementary substances, sodium for one, probably +potassium for another, yet the great bulk of them were +probably due to compound vapours, which, like +peroxide of nitrogen and some other known compound +gases, have the character of selective absorption."</p> + +<p>It will be remembered that the experimental establishment +of the method of spectrum analysis was +published towards the end of 1859 by Bunsen and +Kirchhoff, to whom, therefore, the full credit of +discoverers must be given.</p> + +<p>Lord Kelvin in the later years of his life used to tell +the story of his first meeting with Joule at Oxford, +and of their second meeting a fortnight later in +Switzerland. He did so also in his address delivered on +the occasion of the unveiling of a statue of Joule, in +Manchester Town Hall, on December 7, 1893, and +we quote the narrative on account of its scientific and +personal interest. "I can never forget the British<span class='pagenum'><a name="Page_87" id="Page_87">87</a></span> +Association at Oxford in 1847, when in one of the +sections I heard a paper read by a very unassuming +young man, who betrayed no consciousness in his +manner that he had a great idea to unfold. I was +tremendously struck with the paper. I at first thought +it could not be true, because it was different from +Carnot's theory, and immediately after the reading of +the paper I had a few words with the author, James +Joule, which was the beginning of our forty years' +acquaintance and friendship. On the evening of the +same day, that very valuable institution of the British +Association, its conversazione, gave us opportunity for +a good hour's talk and discussion over all that either of +us knew of thermodynamics. I gained ideas which +had never entered my mind before, and I thought I, +too, suggested something worthy of Joule's consideration +when I told him of Carnot's theory. Then and +there in the Radcliffe Library, Oxford, we parted, both +of us, I am sure, feeling that we had much more to say +to one another and much matter for reflection in what +we had talked over that evening. But ... a fortnight +later, when walking down the valley of +Chamounix, I saw in the distance a young man +walking up the road towards me, and carrying in +his hand something which looked like a stick, but +which he was using neither as an alpenstock nor as a +walking-stick. It was Joule with a long thermometer +in his hand, which he would not trust by itself in the +<i>char-à-banc</i>, coming slowly up the hill behind him, +lest it should get broken. But there, comfortably and +safely seated in the <i>char-à-banc</i>, was his bride—the +sympathetic companion and sharer in his work of after +years. He had not told me in Section A, or in the<span class='pagenum'><a name="Page_88" id="Page_88">88</a></span> +Radcliffe Library, that he was going to be married in +three days, but now in the valley of Chamounix he +introduced me to his young wife. We appointed to +meet again a fortnight later at Martigny to make +experiments on the heat of a waterfall (Sallanches) with +that thermometer: and afterwards we met again and +again, and from that time, indeed, remained close friends +till the end of Joule's life. I had the great pleasure +and satisfaction for many years, beginning just forty +years ago, of making experiments along with Joule +which led to some important results in respect to the +theory of thermodynamics. This is one of the most +valuable recollections of my life, and is indeed as +valuable a recollection as I can conceive in the possession +of any man interested in science."</p> + +<p>At the beginning of his course of lectures each +session, Professor Thomson read, or rather attempted +to read, an introductory address on the scope and +methods of physical science, which he had prepared +for his first session in 1846. It set forth the fact that +in science there were two stages of progress—a natural +history stage and a natural philosophy stage. In the +first the discoverer or teacher is occupied with the +collection of facts, and their arrangement in classes +according to their nature; in the second he is concerned +with the relations of facts already discovered and +classified, and endeavours to bring them within the +scope of general principles or causes. Once the +philosophical stage is reached, its methods and results +are connected and enlarged by continued research after +facts, controlled and directed by the conclusions of +general theory. Thus the method is at first purely +inductive, but becomes in the second stage both<span class='pagenum'><a name="Page_89" id="Page_89">89</a></span> +inductive and deductive; the general theory predicts by +its deductions, and the verification of these by experiment +and observation give a validity to the theory +which no mere induction could afford. These stages +of scientific investigation are well illustrated by the +laws of Kepler arrived at by mere comparison of the +motions of the planets, and the deduction of these +laws, with the remarkable correction of the third law, +given by the theory of universal gravitation. The +prediction of the existence and place of the planet +Neptune from the perturbations of Uranus is an +excellent example of the predictive quality of a true +philosophical theory.</p> + +<p>The lecture then proceeded to state the province of +dynamics, to define its different parts, and to insist on +the importance of kinematics, which was described as +a purely geometrical subject, the geometry of motion, +considerations from which entered into every dynamical +problem. This distinction between dynamical and +kinematical considerations—between those in which +force is concerned and those into which enter only +the idea of displacement in space and in time—is +emphasised in Thomson and Tait's <i>Natural Philosophy</i>, +which commences with a long chapter devoted entirely +to kinematics.</p> + +<p>Whether Professor Thomson read the whole of the +Introductory Lecture on the first occasion is uncertain—Clerk +Maxwell is said to have asserted that it was +closely adhered to, for that one time only, and finished +in much less than the hour allotted to it. In later +years he had never read more than a couple of pages +when some new illustration, or new fact of science, +which bore on his subject, led him to digress from the<span class='pagenum'><a name="Page_90" id="Page_90">90</a></span> +manuscript, which was hardly ever returned to, and +after a few minutes was mechanically laid aside and +forgotten. Once on beginning the session he humorously +informed the assembled class that he did not +think he had ever succeeded in reading the lecture +through before, and added that he had determined that +they should hear the whole of it! But again occurred +the inevitable digression, in the professor's absorption +in the new topic the promise was forgotten, and the +written lecture fared as before! These digressions +were exceedingly interesting to the best students: +whether they compensated for the want of a carefully +prepared presentation of the elements of the subject, +suited to the wants of the mass of the members of the +class, is a matter which need not here be discussed. +All through his elementary lectures—introductory or +not—new ideas and new problems continually presented +themselves. An eminent physicist once remarked +that Thomson was perhaps the only living +man who made discoveries while lecturing. That was +hardly true; in the glow of action and stress of +expression the mind of every intense thinker often sees +new relations, and finds new points of view, which +amount to discoveries. But fecundity of mind has, +of course, its disadvantages: the unexpected cannot +happen without causing distractions to all concerned. +A mind which can see a theory of the physical +universe in a smoke-ring is likely, unless kept under +extraordinary and hampering restraint, to be tempted +to digress from what is strictly the subject in hand, +to the world of matters which that subject suggests. +Professor Thomson was, it must be admitted, too discursive +for the ordinary student, and perhaps did not<span class='pagenum'><a name="Page_91" id="Page_91">91</a></span> +study the art of boiling down physical theories to the +form most easily digestible. His eagerness of mind +and width of mental outlook gave his lectures a special +value to the advanced student, so that there was a +compensating advantage.</p> + +<p>The teacher of natural philosophy is really placed +in a position of extraordinary difficulty. The fabric +of nature is woven without seam, and to take it to +pieces is in a manner to destroy it. It must, after +examination in detail, be reconstructed and considered +as a whole, or its meaning escapes us. And here lies +the difficulty: every bit of matter stands in relation to +everything else, and both sides of every relation must +be considered. In other words, in the explanation of +any one phenomenon the explanation of all others is +more or less involved. This does not mean that investigation +or exposition is impossible, or that we +cannot proceed step by step; but it shows the foolishness +of that criticism of science and scientific method +which asks for complete or ultimate knowledge, and +of the popular demand for a simple form of words to +express what is in reality infinitely complex.</p> + +<p>In the earlier years of his professorship Professor +Thomson taught his class entirely himself, and gathered +round him, as he has told us in the Bangor address, +an enthusiastic band of workers who aided him in the +researches which he began on the electrodynamic +qualities of metals, the elastic properties of substances, +the thermal and electrical conductivities of metals, and +at a later date in the electric and magnetic work which +he undertook as a member of the British Association +Committee on Electrical Standards. The class met, +as has been stated, twice a day, first for lectures, then<span class='pagenum'><a name="Page_92" id="Page_92">92</a></span> +for exercises and oral examination. The changes +which took place later in the curriculum, and especially +the introduction of honours classes in the different +subjects, rendered it difficult, if not impossible, for two +hours' attendance to be given daily on all subjects, and +students were at first excused attendance at the second +hour, and finally such attendance became practically +optional. But so long as the old traditional curriculum +in Arts—of Humanity, Greek, Logic, Mathematics, +Moral Philosophy and Natural Philosophy—endured, +a large number of students found it profitable to attend +at both hours, and it was possible to give a large +amount of excellent tutorial instruction by the working +of examples and oral examination.</p> + +<p>Thomson always held that his commission included +the subject of physical astronomy, and though his +lectures on that subject were, as a rule, confined to a +statement of Kepler's laws and Newton's deductions +from them, he took care that the written and oral +examinations included astronomical questions, for which +the students were enjoined to prepare by reading +Herschel's <i>Outlines</i>, or some similar text-book. This +injunction not infrequently was disregarded, and discomfiture +of the student followed as a matter of course, +if he was called on to answer. Nor were the questions +always easy to prepare for by reading. A man might +have a fair knowledge of elementary astronomy, and +be unable to answer offhand such a question as, "Why +is the ecliptic called the ecliptic?" or to say, when the +lectures on Kepler had been omitted, short and tersely +just what was Newton's deduction from the third law +of the planetary motions.</p> + +<p>Home exercises were not prescribed as part of the<span class='pagenum'><a name="Page_93" id="Page_93">93</a></span> +regular work except from time to time in the "Higher +Mathematical Class" which for thirty years or more +of Thomson's tenure of office was held in the department. +But the whole ordinary class met every +Monday morning and spent the usual lecture hour in +answering a paper of dynamical and physical questions. +As many as ten, and sometimes eleven, questions were +set in these papers, some of them fairly difficult and +involving novel ideas, and by this weekly paper of +problems the best students, a dozen or more perhaps, +were helped to acquire a faculty of prompt and brief +expression. It was not uncommon for a good man to +score 80 or 90 or even 100 per cent. in the paper, no +small feat to accomplish in a single hour. But to +a considerable majority of the class, it is doubtful +whether the weekly examination was of much advantage: +they attempted one or two of the more +descriptive questions perhaps, but a good many did +next to nothing. The examinations came every week, +and so the preparation for one after another was neglected, +and as much procrastination of work ensued as +there would have been if only four or five papers a +session had been prescribed. Then the work of looking +over so many papers was a heavy task to the professor's +assistant, a task which became impossible when, for a +few years in the early 'eighties, the students in the +ordinary class numbered about 250.</p> + +<p>The subject of natural philosophy had become so +extensive in 1846 that Professor J. P. Nichol called +attention to the necessity for special arrangements for +its adequate teaching. What would he say if he could +survey its dimensions at the present time! To give +even a brief outline of the principal topics in dynamics,<span class='pagenum'><a name="Page_94" id="Page_94">94</a></span> +heat, acoustics, light, magnetism, and electricity is more +than can be accomplished in any course of university +lectures; and the only way to teach well and economically +the large numbers of students<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">16</a> who now throng +the physics classes is to give each week, say, three +lectures as well considered and arranged as possible, +without any interruption from oral examination, and +assemble the students in smaller classes two or three +times a week for exercises and oral examination.</p> + +<p>Thomson stated his views as to examinations and +lectures in the Bangor address. "The object of a +university is teaching, not testing, ... in respect to +the teaching of a university the object of examination +is to promote the teaching. The examination +should be, in the first place, daily. No professor should +meet his class without talking to them. He should +talk to them and they to him. The French call a +lecture a <i>conférence</i>, and I admire that idea. Every +lecture should be a conference of teachers and students. +It is the true ideal of a professorial lecture. I have +found that many students are afflicted when they +come up to college with the disease called 'aphasia.' +They will not answer when questioned, even when +the very words of the answer are put in their mouths, +or when the answer is simply 'yes' or 'no.' That +disease wears off in a few weeks, but the great cure +for it is in repeated and careful and very free interchange +of question and answer between teacher and +student.... Written examinations are very important, +as training the student to express with +<span class='pagenum'><a name="Page_95" id="Page_95">95</a></span>clearness and accuracy the knowledge he has gained, +but they should be once a week to be beneficial."</p> + +<p>The great difficulty now, when both classes and +subject have grown enormously, is to have free conversation +between professor and student, and yet give +an adequate account of the subject. To examine orally +in a thorough way two students in each class-hour is +about as much as can be done if there is to be any +systematic exposition by lecture at all; and thus the +conference between teacher and individual student can +occur only twice a year at most. Nevertheless Lord +Kelvin was undoubtedly right: oral examination and +the training of individual students in the art of clear +and ready expression are very desirable. The real +difficulties of the subject are those which occur to the +best students, and a discussion of them in the presence +of others is good for all. This is difficult nowadays, +for large classes cannot afford to wait while two +or three backward students grope after answers to +questions—which in many cases must be on points +which are sufficiently plain to the majority—to say +nothing of the temptation to disorder which the display +of personal peculiarities or oddities of expression +generally affords to an assembly of students. But time +will be economised and many advantages added, if +large classes are split up into sections for tutorial work, +to supplement the careful presentation of the subject +made in the systematic lectures delivered to the whole +class in each case. The introduction of a tutorial +system will, however, do far more harm than good, +unless the method of instruction is such as to foster the +self-reliance of the student, who must not be, so to +speak, spoon-fed: such a method, and the advantages<span class='pagenum'><a name="Page_96" id="Page_96">96</a></span> +of the weekly examination on paper may be secured, by +setting the tutorial class to work out on the spot exercises +prescribed by the lecturer. But the danger, which is +a very real one, can only be fully avoided by the +precautions of a skilful teacher, who in those small +classes will draw out and direct the ideas of his +students, rather than impart knowledge directly.</p> + +<p>After a few years Thomson found it necessary to +appoint an assistant, and Mr. Donald McFarlane, who +had distinguished himself in the Mathematics and +Natural Philosophy classes, was chosen. Mr. McFarlane +was originally a block-printer, and seems to have +been an apprentice at Alexandria in the Vale of +Leven, at the time of the passing of the first Reform +Bill. After some time spent in the cotton industry +of the district, he became a teacher in a village school +in the Vale of Leven, and afterwards entered the +University as a student. He discharged his duties in +the most faithful and self-abnegating manner until his +retirement in 1880, when he had become advanced in +years. He had charge of the instruments of the department, +got ready the lecture illustrations and attended +during lecture to assist in the experiments and supply +numerical data when required, prepared the weekly +class examination paper and read the answers handed +in, and assisted in the original investigations which +the professor was always enthusiastically pursuing. A +kind of universal physical genius was McFarlane; +an expert calculator and an exact and careful experimentalist. +Many a long and involved arithmetical +research he carried out, much apparatus he made in +a homely way, and much he repaired and adjusted. +Then, always when the professor was out of the way<span class='pagenum'><a name="Page_97" id="Page_97">97</a></span> +and calm had descended on the apparatus-room, if not +on the laboratory, McFarlane sat down to reduce his +pile of examination papers, lest Monday should arrive +with a new deluge of crude answers and queer mistakes, +ere the former had disappeared. On Friday +afternoons at 3 o'clock he gave solutions of the previous +Monday's questions to any members of the class who +cared to attend; and his clear and deliberate explanations +were much appreciated. An unfailing tribute +was rendered to him every year by the students, and +often took the form of a valuable gift for which one +and all had subscribed. A recluse he was in his way, +hardly anybody knew where he lived—the professor +certainly did not—and a man of the highest ability +and of the most absolute unselfishness. An hour in +the evening with one or two special friends, and +the study of German, were the only recreations of +McFarlane's solitary life. He was full of humour, and +told with keen enjoyment stories of the University +worthies of a bygone age. For thirty years he worked +on for a meagre salary, for during the earlier part of +that time no provision for assistants was made in the +Government grant to the Scottish Universities. By an +ordinance issued in 1861 by the University Commissioners, +appointed under the Act of 1858, a grant +of £100 a year was made from the Consolidated +Fund for an assistant in each of the departments +of Humanity, Greek, Mathematics, and Natural +Philosophy, and for two in the department of Chemistry; +and McFarlane's position was somewhat improved. +His veneration for Thomson was such as few students +or assistants have had for a master: his devotion resembled +that of the old <i>famulus</i> rather than the much<span class='pagenum'><a name="Page_98" id="Page_98">98</a></span> +more measured respect paid by modern assistants to +their chiefs.</p> + +<p>After his retirement McFarlane lived on in Glasgow, +and amused himself reading out-of-the-way Latin +literature and with the calculation of eclipses! He +finally returned to Alexandria, where he died in +February 1897. "Old McFarlane" will be held in +affectionate remembrance so long as students of the +Natural Philosophy Class in the 'fifties and 'sixties and +'seventies, now, alas! a fast vanishing band, survive.</p> + +<p>Soon after taking his degree of B.A. at Cambridge +in 1845, Thomson had been elected a Fellow of St. +Peter's College. In 1852 he vacated his Fellowship +on his marriage to Miss Margaret Crum, daughter of +Mr. Walter Crum of Thornliebank, near Glasgow, +but was re-elected in 1871, and remained thereafter a +Fellow of Peterhouse throughout his life.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_99" id="Page_99">99</a></span></p> + +<h3>CHAPTER VII</h3> + +<h4>THE "ACCOUNT OF CARNOT'S THEORY OF THE +MOTIVE POWER OF HEAT"—TRANSITION TO THE +DYNAMICAL THEORY OF HEAT</h4> + +<p><span class="smcap">The</span> meeting of Thomson and Joule at Oxford in 1847 +was fraught with important results to the theory of +heat. Thomson had previously become acquainted with +Carnot's essay, most probably through Clapeyron's +account of it in the <i>Journal de l'École Polytechnique</i>, 1834, +and had adopted Carnot's view that when work was +done by a heat engine heat was merely let down from a +body at one temperature to a body at a lower temperature. +Joule apparently knew nothing of Carnot's +theory, and had therefore come to the consideration of +the subject without any preconceived opinions. He +had thus been led to form a clear notion of heat as +something which could be transformed into work, and +<i>vice versa</i>. This was the root idea of his attempt to +find the dynamical equivalent of heat. It was obvious +that a heat engine took heat from a source and gave +heat to a refrigerator, and Joule naturally concluded +that the appearance of the work done by the engine +must be accompanied by the disappearance of a +quantity of heat of which the work done was the +equivalent. He carried this idea consistently through +all his work upon energy-changes, not merely in heat +engines but in what might be called electric engines.<span class='pagenum'><a name="Page_100" id="Page_100">100</a></span> +For he pointed out that the heat produced in the +circuit of a voltaic battery was the equivalent of the +energy-changes within the battery, and that, moreover, +when an electromagnetic engine was driven by the +current, or when electrochemical decomposition was +effected in a voltameter in the circuit, the heat evolved +in the circuit for a given expenditure of the materials +of the battery was less than it would otherwise have +been, by the equivalent of the work done by the engine, +or of the chemical changes effected in the voltameter. +Thus Joule was in possession at an earlier date than +Thomson of the fundamental notion upon which the +true dynamical theory of heat engines is founded. +Thomson, on the other hand, as soon as he had received +this idea, was able to add to it the conception, derived +from Carnot, of a reversible engine as the engine of +greatest efficiency, and to deduce in a highly original +manner all the consequences of these doctrines which +go to make up the ordinary thermodynamics even of +the present time. Though Clausius was the first, as +we shall see, to deduce various important theorems, yet +Thomson's discussion of the question had a quality +peculiarly its own. It was marked by that freedom +from unstated assumptions, from extraneous considerations, +from vagueness of statement and of thought, +which characterises all his applications of mathematics +to physics. The physical ideas are always set forth +clearly and in such a manner that their quantitative +representation is immediate: we shall have an example +of this in the doctrine of absolute temperature. In +most of the thermodynamical discussions which take +the great memoir of Clausius as their starting point, +temperature is supposed to be given by a hypothetical<span class='pagenum'><a name="Page_101" id="Page_101">101</a></span> +something which is called a perfect gas, and it is very +difficult, if not impossible, to gather a precise notion of +the properties of such a gas and of the temperature +scale thereon founded. Thomson's scale enables a +perfect gas to be defined, and the deviations of the +properties of ordinary gases from those of such a gas +to be observed and measured.</p> + +<p>The idea, then, which Joule had communicated to +Section A, when Thomson interposed to call attention +to its importance, was that work spent in overcoming +friction had its equivalent in the heat produced, that, in +fact, the amount of heat generated in such a case was +proportional to the work spent, quite irrespective of the +materials used in the process, provided no change of +the internal energy of any of them took place so as to +affect the resulting quantity of heat. This forced upon +physicists the view pointed to by the doctrine of the +immateriality of heat, established by the experiments of +Rumford and Davy, that heat itself was a form of +energy; and thus the principle of conservation of +energy was freed from its one defect, its apparent +failure when work was done against friction.</p> + +<p>Rumford had noted the very great evolution of heat +when gun-metal was rubbed by a blunt borer, and had +come to the reasonable conclusion that what was +evolved in apparently unlimited quantity by the abrasion +or cutting down of a negligible quantity of materials +could not be a material substance. He had also made +a rough estimate of the relation between the work +spent in driving the borer by horse-power and the heat +generated. Joule's method of determining the work-equivalent +of heat was a refinement of Rumford's, but +differed in the all-important respect that accurate<span class='pagenum'><a name="Page_102" id="Page_102">102</a></span> +means were employed for measuring the expenditure +of work and the gain of heat. He stirred a liquid, +such as water or mercury, in a kind of churn driven +by a falling weight. The range of descent of the +weight enabled the work consumed to be exactly +estimated, and a sensitive thermometer in the liquid +measured the rise of temperature; thus the heat +produced was accurately determined. The rise of +temperature was very slight, and the change of state of +the liquid, and therefore any possible change in its +internal energy, was infinitesimal. The experiments +were carried out with great care, and included very +exact measurements of the various corrections—for +example, the amount of work spent at pulleys and +pivots without affecting the liquid, and the loss of heat +by radiation. The experiments proved that the work +spent on the liquid and the heat produced were in +direct proportion to one another. He found, finally, in +1850, that 772 foot-pounds of work at Manchester +generated one British thermal unit, that is, as much +heat as sufficed to raise a pound of water from 60° F. +to 61° F. An approximation to this conclusion was +contained in the paper which he communicated to the +British Association at Oxford in 1847.</p> + +<p>The results of a later determination made with an +improved apparatus, and completed in 1878, gave a +very slightly higher result. When corrected to the +corresponding Fahrenheit degree on the air thermometer +it must be increased by somewhat less than one per +cent. The exact relation has been the subject during +the last twenty years of much refined experimental +work, but without any serious alteration of the number +indicated above.<span class='pagenum'><a name="Page_103" id="Page_103">103</a></span></p> + +<p>It is probable that in consequence of the conference +which he had with Joule at Oxford Thomson had +his thoughts turned for some time almost exclusively to +the dynamical theory of heat engines. He worked at +the subject almost continuously for a long time, sending +paper after paper to the Edinburgh Royal Society. +As we have seen, he had given Joule a description +of Carnot's essay on the Motive Power of Heat and +the conclusions, or some of them, therein contained. +Joule's result, and the thermodynamic law which it +established, gave the key to the correction of Carnot's +theory necessary to bring it into line with a complete +doctrine of energy, which should take account of work +done against frictional resistances.</p> + +<p>Mayer of Heilbronn had endeavoured to determine +the dynamical equivalent of heat in 1842, by calculating +from the knowledge available at the time of +the two specific heats of air—the specific heat at +constant pressure and the specific heat at constant +volume—the heat value of the work spent in compressing +air from a given volume to a smaller one. +The principle of this determination is easily understood, +but it involves an assumption that is not always +clearly perceived. Let the air be imagined confined +in a cylinder closed by a frictionless piston, which is +kept from moving out under the air pressure by force +applied from without. Let heat be given to the air so +as to raise its temperature, while the piston moves out +so as to keep the pressure constant. If the pressure be +<i>p</i> and the increase of volume be <i>dv</i>, the work done +against the external force is <i>pdv</i>. Let the rise of temperature +be one degree of the Centigrade scale, and the +mass of air be one gramme, the heat given to the gas<span class='pagenum'><a name="Page_104" id="Page_104">104</a></span> +is the specific heat <i>C<sub>p</sub></i> of the gas at constant pressure, +for there is only slight variation of specific heat with +temperature. But if the piston had been fixed the heat +required for the same rise of temperature would have +been <i>C<sub>v</sub></i>, the specific heat at constant volume. Now +Mayer assumed that the excess of the specific heat <i>C<sub>p</sub></i> +above <i>C<sub>v</sub></i> was the thermal equivalent of the work <i>pdv</i> +done in the former case. Thus he obtained the equation +<i>J</i> (<i>C<sub>p</sub></i> − <i>C<sub>v</sub></i>) = <i>pdv</i>, where <i>J</i> denotes the dynamical +equivalent of heat and <i>C<sub>p</sub></i>, <i>C<sub>v</sub></i> are taken in thermal +units. But if a be the coefficient of expansion of the air +under constant pressure (that is 1 ⁄ 273), and <i>v</i><sub>0</sub> be the +volume of the air at 0° C., we have dv = av<sub>0</sub>, so that +<i>J</i> (<i>C<sub>p</sub></i> − <i>C<sub>v</sub></i>) = <i>apv</i><sub>0</sub>. Now if <i>p</i> be one atmosphere, say +1.014 × 10<sup>6</sup> dynes per square centimetre, and the +temperature be the freezing point of water, the volume +of a gramme of air is 1 ⁄ .001293 in cubic centimetres. +Hence</p> + +<div class="center"><img class="floatInsert22" src="images/f104.png" alt="" title="" /> +</div> + +<p>from which, if <i>C<sub>p</sub></i> − <i>C<sub>v</sub></i> is known, the value of <i>J</i> can +be found.</p> + +<p>In Mayer's time the difference of the specific heats +of air was imperfectly known, and so <i>J</i> could not be +found with anything like accuracy. From Regnault's +experiments on the specific heat at constant pressure, and +from the known ratio of the specific heats as deduced +from the velocity of sound combined with Regnault's +result, the value of <i>C<sub>p</sub></i> − <i>C<sub>v</sub></i> may be taken as .0686. +Thus <i>J</i> works out to 42.2 × 10<sup>6</sup>, in ergs per calorie, +which is not far from the true value. Mayer obtained +a result equivalent to 36.5 × 10<sup>6</sup> ergs per calorie.</p> + +<p><span class='pagenum'><a name="Page_105" id="Page_105">105</a></span>The assumption on which this calculation is +founded is that there is no alteration of the internal +energy of the gas in consequence of expansion. If the +air when raised in temperature, and at the same time +increased in volume, contained less internal energy +than when simply heated without alteration of volume, +the energy evolved would be available to aid the +performance of the work done against external forces, +and less heat would be required, or, in the contrary +case, more heat would be required, than would be +necessary if the internal energy remained unaltered. +Thus putting <i>dW</i> for <i>pdv</i>, the work done, <i>e</i> for the +internal energy before expansion, and <i>dH</i> for the heat +given to the gas, we have obviously the equation</p> + +<div class="center"> +<i>JdH</i> = <i>de</i> + <i>dW</i> +</div> + +<p>where <i>de</i> is the change of internal energy due to +the alteration of volume, together with the alteration +of temperature. If now the temperature be altered +without expansion, no external work is done and <i>dW</i> +for that case is zero. Let <i>∂</i><i>e</i> and <i>∂</i><i>H</i> be the energy +change and the heat supplied, then in this case</p> + +<div class="center"> +<i>J</i><i>∂</i><i>H</i> = <i>∂</i><i>e</i> + <i>O</i> +</div> + +<p>Thus</p> + +<div class="center"> +<i>J</i> (<i>dH</i> − <i>∂</i><i>H</i>) = <i>de</i> − <i>∂</i><i>e</i> + <i>dW</i><br /> +</div> + +<p>and the assumption is that <i>de</i> = <i>∂</i><i>e</i>, so that +<i>dW</i> = <i>J</i> (<i>dH</i> − <i>∂</i><i>H</i>); that is, <i>dW</i> = <i>J</i> (<i>C<sub>p</sub></i> − <i>C<sub>v</sub></i>), when +the rise of temperature is 1° C. and the mass of air +is one gramme. This assumption requires justification, +and by an experiment of Joule's, which was +repeated in a more sensitive form devised by Thomson, +it was shown to be a very close approximation to the<span class='pagenum'><a name="Page_106" id="Page_106">106</a></span> +truth. Joule's experiment is well known: the explanation +given above may serve to make clear the nature +of the research undertaken later by Thomson and +Joule conjointly.</p> + +<p>The inverse process, the conversion of heat into +work, required investigation, and it is this that constitutes +the science of thermodynamics. It was the +subject of the celebrated <i>Réflexions sur la Puissance +Motrice du Feu, et sur les Machines Propres à +Développer cette Puissance</i>, published in 1824 by +Sadi Carnot, an uncle of the late President of the +French Republic. Only a few copies of this essay +were issued, and its text was known to very few +persons twenty-four years later, when it was reprinted +by the Academy of Sciences. Its methods and +conclusions were set forth by Thomson in 1849 in a +memoir which he entitled, "An Account of Carnot's +Theory of the Motive Power of Heat." Numerical +results deduced from Regnault's experiments on steam +were included; and the memoir as a whole led +naturally in Thomson's hands to a corrected theory of +heat engines, which he published in 1852. Carnot's +view of the working of a heat engine was founded on +the analogy of the performance of work by a stream of +water descending from a higher level to a lower. +The same quantity of water flows away in a given +time from a water wheel in the tail-race as is received +in that time by the wheel from the supply stream. +Now a heat engine receives heat from a supplying +body, or source, at one temperature and parts with +heat to another body (for example, the condenser of a +steam engine) at a lower temperature. This body is +usually called the refrigerator. According to Carnot<span class='pagenum'><a name="Page_107" id="Page_107">107</a></span> +these temperatures corresponded to the two levels in +the case of the water wheel; the heat was what +flowed through the engine. Thus in his theory as +much heat was given up by a heat engine to the body +at the lower temperature as was received by it from +the source. The heat was simply transferred from the +body at the higher temperature to the body at the +lower; and this transference was supposed to be the +source of the work.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">17</a></p> + +<p>The first law of thermodynamics based on Joule's +proportionality of heat produced to work expended, +and the converse assumed and verified <i>a posteriori</i>, +showed that this view is erroneous, and that the heat +delivered to the refrigerator must be less in amount +than that received from the source, by exactly the +amount which is converted into work, together with +the heat which, in an imperfect engine, is lost by conduction, +etc., from the cylinder or other working +chamber. This change was made by Thomson in +his second paper: but he found the ideas of Carnot of +direct and fruitful application in the new theory. These +were the cycle of operations and the ideal reversible +engine.</p> + +<p>In the Carnot cycle the working substance—which +might be a gas or a vapour, or a liquid, or a vapour and +its liquid in contact: it did not matter what for the +result—was supposed to be put through a succession of +changes in which the final state coincided with the +initial. Thus the substance having been brought +<span class='pagenum'><a name="Page_108" id="Page_108">108</a></span> +back to the same physical condition as it had when the +cycle began, has the same internal energy as it had at +the beginning, and in the reckoning of the work done +by or against external forces, nothing requires to be +set to the account of the working substance. This is +the first great advantage of the method of reasoning +which Carnot introduced.</p> + +<p>The ideal engine was a very simple affair: but the +notion of reversibility is difficult to express in a form +sufficiently definite and precise. Carnot does not +attempt this; he merely contents himself with describing +certain cycles of operations which obviously can be +carried through in the reverse order. Nor does +Thomson go further in his "Account of Carnot's +Theory," though he states the criterion of a perfect +engine in the words, "A perfect thermodynamic +engine is such that, whatever amount of mechanical +effect it can derive from a certain thermal agency, if an +equal amount be spent in working it backwards, an +equal reverse thermal effect will be produced." This +proposition was proved by Carnot: and the following +formal statement in the essay is made: "La puissance +motrice de la chaleur est independante des agents +mis en œuvre pour la réaliser: sa quantité est fixée +uniquement par les temperatures des corps entre +lesquels se fait, en dernier résultat, le transport du +calorique." The result involved in each, that the work +done in a cycle by an ideal engine depends on the +temperatures between which it works and not at all on +the working substance, is, as we shall see, of the greatest +importance. The proof of the proposition, by supposing +a more efficient engine than the ideal one to exist, +and to be coupled with the latter, so that the more<span class='pagenum'><a name="Page_109" id="Page_109">109</a></span> +efficient would perform the cycle forwards and the ideal +engine the same cycle backwards, is well known. In +Carnot's view the former would do more work by +letting down a given quantity of heat from the higher +to the lower temperature than was spent on the latter +in transferring the same quantity of heat from the +lower to the higher temperature, so that no heat would +be taken from or given to source or refrigerator, while +there would be a gain of work on the whole. This +would be equivalent to admitting that useful work +could be continually performed without any resulting +thermal or other change in the agents performing the +work. Even at that time this could not be admitted +as possible, and hence the supposition that a more +efficient engine than the reversible one could exist was +untenable.</p> + +<p>Carnot showed that the work done by an ideal +engine, in transferring heat from one temperature to +another, was to be found by means of a certain function +of the temperature, hence called "Carnot's function." +The corresponding function in the true dynamical +theory is always called Carnot's. A certain assignment +of value to it gave, as we shall see, Thomson's famous +absolute thermodynamic scale of temperature.</p> + +<p>In the light of the facts and theories which now +exist, and are almost the commonplaces of physical text +books, it is very interesting to review the ideas and +difficulties which occurred to the founders of the +science of heat sixty years ago. For example, Thomson +asks, in his "Account of Carnot's Theory," +what becomes of the mechanical effect which might +be produced by heat which is transferred from one body +to another by conduction. The heat leaves one body<span class='pagenum'><a name="Page_110" id="Page_110">110</a></span> +and enters another and no mechanical effect results: +if it passed from one to the other through a heat +engine, mechanical effect would be produced: what is +produced in place of the mechanical effect which is +lost? This he calls a very "perplexing question," and +hopes that it will, before long, be cleared up. He +states, further, that the difficulty would be entirely +avoided by abandoning Carnot's principle that mechanical +effect is obtained by "the transference of heat from +one body to another at a lower temperate." Joule urges +precisely this solution of the difficulty in his paper, +"On the Changes of Temperature produced by the +Rarefaction and Condensation of Air" (<i>Phil. Mag.</i>, May +1845). Thomson notes this, but adds, "If we do so, +however, we meet with innumerable other difficulties—insuperable +without further experimental investigation, +and an entire reconstruction of the theory of heat from +its foundation. It is in reality to experiment that we +must look, either for a verification of Carnot's axiom, +and an explanation of the difficulty we have been considering, +or for an entirely new basis of the Theory of +Heat."</p> + +<p>The experiments here asked for had already, as was +soon after perceived by Thomson, been made by Joule, +not merely in his determinations of the dynamical +equivalent of heat, but in his exceedingly important +investigation of the energy changes in the circuit of a +voltaic cell, or of a magneto-electric machine. Moreover, +the answer to this "very perplexing question" +was afterwards to be given by Thomson himself in his +paper, "On a Universal Tendency in Nature to the +Dissipation of Mechanical Energy," published in the +Edinburgh <i>Proceedings</i> in 1852.<span class='pagenum'><a name="Page_111" id="Page_111">111</a></span></p> + +<p>Again, we find, a page or two earlier in the "Account +of Carnot's Theory," the question asked with respect +to the heat evolved in the circuit of a magneto-electric +machine, "Is the heat which is evolved in one part of +the closed conductor merely transferred from those +parts which are subject to the inducing influence?" +and the statement made that Joule had examined this +question, and decided that it must be answered in the +negative. But Thomson goes on to say, "Before we +can finally conclude that heat is absolutely generated +in such operations, it would be necessary to prove +that the inducing magnet does not become lower in +temperature and thus compensate for the heat evolved +in the conductor."</p> + +<p>Here, apparently, the idea of work done in moving +the magnet, or the conductor in the magnetic field, +is not present to Thomson's mind; for if it had been, +the idea that the work thus spent might have its +equivalent, in part, at least, in heat generated in the +circuit, would no doubt have occurred to him and been +stated. This idea had been used just a year before by +Helmholtz, in his essay "Die Erhaltung der Kraft," +to account for the heat produced in the circuit by the +induced current, that is, to answer the first question +put above in the sense in which Joule answered it. +The subject, however, was fully worked out by +Thomson in a paper published in the <i>Philosophical +Magazine</i> for December 1851, to which we shall refer +later.</p> + +<p>Tables of the work performed by various steam +engines working between different stated temperatures +were given at the close of the "Account," and compared +with the theoretical "duty" as calculated for<span class='pagenum'><a name="Page_112" id="Page_112">112</a></span> +Carnot's ideal perfect engine. Of course the theoretical +duty was calculated from the temperatures of the boiler +and condenser; the much greater fall of temperature +from the furnace to the boiler was neglected as +inevitable, so that the loss involved in that fall is not +taken account of. Carnot's theory gave for the +theoretical duty of one heat unit (equivalent to 1390 +foot-pounds of work) 440 foot-pounds for boiler at +140° C. and condenser at 30° C.; and the best performance +recorded was 253 foot-pounds, giving a percentage +of 57.5 per cent. The worst was that of common engines +consuming 12 lb. of coal per horse-power per hour, and +gave 38.1 foot-pounds, or a percentage of 8.6 per cent. +These percentages become on the dynamical theory +68 and 10.3, since the true theoretical duty for the +heat unit is only 371 foot-pounds.</p> + +<p>It is worthy of notice that the indicator-diagram +method of graphically representing the changes in a +cycle of operations is adopted in Thomson's "Account," +but does not occur in Carnot's essay. The cycles +consist of two isothermal changes and two adiabatic +changes; that is, two changes at the temperatures of +the source and refrigerator respectively, and two +changes—from the higher to the lower temperature, +and from the lower to the higher. These changes are +made subject to the condition in each case that the +substance neither gains nor loses energy in the form of +heat, but is cooled in the one case by expansion and +heated in the other by compression. The indicator +diagram was due not to Thomson but to Clapeyron +(see p. <a href="#Page_99">99</a> above), who used it to illustrate an account +of Carnot's theory.</p> + +<p>There appeared in the issue of the Edinburgh<span class='pagenum'><a name="Page_113" id="Page_113">113</a></span> +<i>Philosophical Transactions</i> for January 2, 1849, along +with the "Account of Carnot's Theory," a paper by +James Thomson, entitled, "Theoretical Considerations +on the Effect of Pressure in Lowering the Freezing +Point of Water." The author predicted that, unless +the principle of conservation of energy was at fault, +the effect of increase of pressure on water in the act of +freezing would be to lower the freezing point; and he +calculated from Carnot's theory the amount of lowering +which would be produced by a given increment of +pressure. The prediction thus made was tested by +experiments carried out in the Physical Laboratory by +Thomson, and the results obtained completely confirmed +the conclusions arrived at by theory. This +prediction and its verification have been justly regarded +as of great importance in the history of the dynamical +theory of heat; and they afford an excellent example +of the predictive character of a true scientific theory. +The theory of the matter will be referred to in the +next chapter.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_114" id="Page_114">114</a></span></p> + +<h3>CHAPTER VIII</h3> + +<h4>THERMODYNAMICS AND ABSOLUTE THERMOMETRY</h4> + +<p><span class="smcap">The</span> first statement of the true dynamical theory of +heat, based on the fundamental idea that the work +done in a Carnot cycle is to be accounted for by an +excess of the heat received from the source over the +heat delivered to the refrigerator, was given by Clausius +in a paper which appeared in <i>Poggendorff's Annalen</i> in +March and April 1850, and in the <i>Philosophical +Magazine</i> for July 1850, under a title which is a +German translation of that of Carnot's essay. In that +paper the First Law of Thermodynamics is explicitly +stated as follows: "In all cases in which work is +produced by the agency of heat, a quantity of heat +proportional to the amount of work produced is expended, +and, inversely, by the expenditure of that +amount of work exactly the same amount of heat is +generated." Modern thermodynamics is based on this +principle and on the so-called Second Law of Thermodynamics; +which is, however, variously stated by +different authors. According to Clausius, who used in +his paper an argument like that of Carnot based on the +transference of heat from the source to the refrigerator, +the foundation of the second law was the fact that heat +tends to pass from hotter to colder bodies. In 1854 +(<i>Pogg. Ann.</i>, Dec. 1854) he stated his fundamental +principle explicitly in the form: "Heat can never<span class='pagenum'><a name="Page_115" id="Page_115">115</a></span> +pass from a colder to a hotter body, unless some other +change, connected therewith, take place at the same +time," and gives in a note the shorter statement, which +he regards as equivalent: "Heat cannot of itself pass +from a colder to a hotter body."</p> + +<p>We shall not here discuss the manner in which +Clausius applied this principle: but he arrived at and +described in his paper many important results, of which +he must therefore be regarded as the primary discoverer. +His theory as originally set forth was lacking in +clearness and simplicity, and was much improved by +additions made to it on its republication, in 1864, with +other memoirs on the Theory of Heat.</p> + +<p>In the <i>Transactions R.S.E.</i>, for March 1851, +Thomson published his great paper, "On the Dynamical +Theory of Heat." The object of the paper was stated +to be threefold: (1) To show what modifications of +Carnot's conclusions are required, when the dynamical +theory is adopted: (2) To indicate the significance +in this theory of the numerical results deduced from +Regnault's observations on steam: (3) To point out +certain remarkable relations connecting the physical +properties of all substances established by reasoning +analogous to that of Carnot, but founded on the +dynamical theory.</p> + +<p>This paper, though subsequent to that of Clausius, is +very different in character. Many of the results are +identical with those previously obtained by Clausius, +but they are reached by a process which is preceded +by a clear statement of fundamental principles. These +principles have since been the subject of discussion, and +are not free from difficulty even now; but a great step +in advance was made by their careful formulation in<span class='pagenum'><a name="Page_116" id="Page_116">116</a></span> +Thomson's paper, as a preliminary to the erection of the +theory and the deduction of its consequences. Two +propositions are stated which may be taken as the First +and Second Laws of Thermodynamics. One is +equivalent to the First Law as stated in p. <a href="#Page_116">116</a>, the +other enunciates the principle of Reversibility as a +criterion of "perfection" of a heat engine. We quote +these propositions.</p> + +<p>"Prop. I (Joule).—When equal quantities of mechanical +effect are produced by any means whatever +from purely thermal sources, or lost in purely thermal +effects, equal quantities of heat are put out of existence +or are generated."</p> + +<p>"Prop. II (Carnot and Clausius).—If an engine be +such that when worked backwards, the physical and +mechanical agencies in every part of its motions are all +reversed, it produces as much mechanical effect as can +be produced by any thermodynamic engine, with the +same temperatures of source and refrigerator, from a +given quantity of heat."</p> + +<p>Prop. I was proved by assuming that heat is a form +of energy and considering always the work effected by +causing a working substance to pass through a closed +cycle of changes, so that there was no change of +internal energy to be reckoned with.</p> + +<p>Prop. II was proved by the following "axiom": +"It is impossible, by means of inanimate material +agency, to derive mechanical effect from any portion +of matter by cooling it below the temperature of the +coldest of the surrounding objects." This is rather a +postulate than an axiom; for it can hardly be contended +that it commands assent as soon as it is stated, even from +a mind which is conversant with thermal phenomena.<span class='pagenum'><a name="Page_117" id="Page_117">117</a></span> +It sets forth clearly, however, and with sufficient +guardedness of statement, a principle which, when the +process by which work is done is always a cyclical one, +is not found contradicted by experience, and one, +moreover, which can be at once explicitly applied to +demonstrate that no engine can be more efficient than +a reversible one, and that therefore the efficiency of a +reversible engine is independent of the nature of the +working substance.</p> + +<p>It has been suggested by Clerk Maxwell that this +"axiom" is contradicted by the behaviour of a gas. +According to the kinetic theory of gases an elevation +of temperature consists in an increase of the kinetic +energy of the translatory motion of the gaseous +particles; and no doubt there actually is, from time +to time, a passage of some more quickly moving +particles from a portion of a gas in which the average +kinetic energy is low, to a region in which the +average kinetic energy is high, and thus a transference +of heat from a region of low temperature to one of +higher temperature. Maxwell imagined a space filled +with gas to be divided into two compartments A and B +by a partition in which were small massless trapdoors, +to open and shut which required no expenditure of +energy. At each of these doors was stationed a "sorting +demon," whose duty it was to allow every particle +having a velocity greater than the average to pass through +from A to B, and to stop all those of smaller velocity +than the average. Similarly, the demons were to +prevent all quickly moving particles from going across +from B to A, and to pass all slowly moving particles. +In this way, without the expenditure of work, all the +quickly moving particles could be assembled in one<span class='pagenum'><a name="Page_118" id="Page_118">118</a></span> +compartment, and all the slowly moving particles in the +other; and thus a difference of temperatures between +the two compartments could be brought about, or a +previously existing one increased by transference of +heat from a colder to a hotter mass of gas.</p> + +<p>Contrary to a not uncommon belief, this process does +not invalidate Thomson's axiom as he intended it to be +understood. For the gas referred to here is what he +would have regarded as the working substance of the +engine, by the cycles of which all the mechanical effect +was derived; and it is not, at the end of the process, +in the state as regards average kinetic energy of the +particles in which it was at first. That this was his +answer to the implied criticism of his axiom contained +in Maxwell's illustration, those who have heard +him refer to the matter in his lectures are well aware. +But of course it is to be understood that the substance +returns to the same state only in a statistical sense.</p> + +<p>Thomson's demonstration that a reversible engine is +the most efficient is well known, and need not here +be repeated in detail. The reversible engine may be +worked backwards, and the working substance will +take in heat where in the direct action it gave it out, +and <i>vice versa</i>: the substance will do work against +external forces where in the direct action it had work +done upon it, and <i>vice versa</i>: in short, all the physical +and mechanical changes will be of the same amount, +but merely reversed, at every stage of the backward +process. Thus if an engine A be more efficient than +a reversible one B, it will convert a larger percentage +of an amount of heat <i>H</i> taken in at the source into +work than would the reversible one working between +the same temperatures. Thus if <i>h</i> be the heat given<span class='pagenum'><a name="Page_119" id="Page_119">119</a></span> +to the refrigerator by A, and <i>h'</i> that given by B when +both work directly and take in <i>H</i>; <i>h</i> must be less than +<i>h'</i>. Then couple the engines together so that B works +backwards while A works directly. A will take in <i>H</i> +and deliver <i>h</i>, and do work equivalent to <i>H-h</i>. B +will take <i>h'</i> from the refrigerator and deliver <i>H</i> to the +source, and have work equivalent to <i>H-h'</i> spent upon +it. There will be no heat on the whole given to or +taken from the source; but heat <i>h'-h</i> will be taken +from the refrigerator, and work equivalent to this will +be done. Thus <i>by a cyclical process</i>, which leaves the +working substance as it was, work is done at the +expense of heat taken from the refrigerator, which +Thomson's postulate affirms to be impossible. Therefore +the assumption that an engine more efficient than +the reversible engine exists must be abandoned; and +we have the conclusion that all reversible engines are +equally efficient.</p> + +<p>Thomson acknowledged in his paper the priority of +Clausius in his proof of this proposition, but stated that +this demonstration had occurred to him before he was +aware that Clausius had dealt with the matter. He +now cited, as examples of the First Law of Thermodynamics, +the results of Joule's experiments regarding +the heat produced in the circuits of magneto-electric +machines, and the fact that when an electric current +produced by a thermal agency or by a battery drives +a motor, the heat evolved in the circuit by the passage +of the current is lessened by the equivalent of the +work done on the motor.</p> + +<div class="figcenter" style="width: 450px; position: relative;"><a name="f12" id="f12"></a><img src="images/fig12.png" width="450" height="417" alt="Fig. 12." title="" /> +<p class="caption"><span class="smcap">Fig. 12.</span></p></div> + +<p>In the Carnot cycle, the first operation is an isothermal +expansion (<i>AB</i> in Fig. <a href="#f12">12</a>), in which the substance +increases in volume by <i>dv</i>, and takes in from<span class='pagenum'><a name="Page_120" id="Page_120">120</a></span> +the source heat of amount <i>Mdv</i>. The second +operation is an adiabatic expansion, <i>BC</i>, in which +the volume is further increased and the temperature +sinks by <i>dt</i> to the temperature of the refrigerator. +The third operation is an isothermal compression, +<i>CD</i>, until the volume and pressure are such that +an adiabatic compression <i>DA</i> will just bring the +substance back to the original state. If ∂<i>p</i> ⁄ ∂<i>t</i> be the +rate of increase of pressure with temperature when +the volume is constant, the step of pressure from one +isothermal to the other is <i>∂p ⁄ ∂t . dt</i>; and thus the area +of the closed cycle in the diagram which measures the +external work done in the succession of changes is +<i>∂p ⁄ ∂t . dtdv</i>. Now, by the second law, the work done +must be a certain fraction of the work-equivalent of +the heat, <i>Mdv</i>, taken in from the source. This +fraction is independent of the nature of the working +substance, but varies with the temperature, and is<span class='pagenum'><a name="Page_121" id="Page_121">121</a></span> +therefore a function of the temperature. Its ratio to +the difference of temperature <i>dt</i> between source and +refrigerator was called "Carnot's function," and the +determination of this function by experiment was at +first perhaps the most important problem of thermodynamics. +Denoting it by <i>μ</i>, we have the equation</p> + +<div class="center"><img class="floatInsert22" src="images/f121.png" alt="" title="" /> +</div> + +<p>which may be taken as expressing in mathematical +language the second law of thermodynamics. <i>M</i> is here +so chosen that <i>Mdv</i> is the heat expressed in units of +work, so that μ does not involve Joule's equivalent of +heat. This equation was given by Carnot: it is here +obtained by the dynamical theory which regards the +work done as accounted for by disappearance, not +transference merely, of heat.</p> + +<p>The work done in the cycle becomes now <i>μMdtdv</i>, +or if <i>H</i> denote <i>Mdv</i>, it is <i>μHdt</i>. The fraction of the +heat utilised is thus <i>μdt</i>. This is called the <i>efficiency</i> of +the engine for the cycle.</p> + +<p>From the first law Thomson obtained another fundamental +equation. For every substance there is a relation +connecting the pressure <i>p</i> (or more generally +the stress of some type), the volume <i>v</i> (or the configuration +according to the specified stress), and the +temperature. We may therefore take arbitrary changes +of any two of these quantities: the relation referred +to will give the corresponding change of the third. +Thomson chose <i>v</i> and <i>t</i> as the quantities to be varied, +and supposed them to sustain arbitrary small changes +<i>dv</i> and <i>dt</i> in consequence of the passage of heat to the<span class='pagenum'><a name="Page_122" id="Page_122">122</a></span> +substance from without. The amount of heat taken +in is <i>Mdv</i> + <i>Ndt</i>, where <i>Mdv</i> and <i>Ndt</i> are heats +required for the changes taken separately. But the +substance expanding through <i>dv</i> does external work +pdv. Thus the net amount of energy given to the +substance from without is <i>Mdv</i> + <i>Ndt</i> − <i>pdv</i> or +(<i>M</i> − <i>p</i>) <i>dv</i> + <i>Ndt</i>; and if the substance is made to +pass through a cycle of changes so that it returns to +the physical state from which it started, the whole +energy received in the cycle must be zero. From this +it follows that the rate of variation of <i>M</i> − <i>p</i> when the +temperature but not the volume varies, is equal to +the rate of variation of <i>N</i> when the volume but not +the temperature varies. To see that this relation +holds, the reader unacquainted with the properties +of perfect differentials may proceed thus. Let the +substance be subjected to the infinitesimal closed cycle +of changes defined by (1) a variation consisting +of the simultaneous changes <i>dv</i>, <i>dt</i> of volume and +temperature, (2) a variation − <i>dv</i> of volume only, +(3) a variation − <i>dt</i> of temperature only. <i>M</i> − <i>p</i> and +<i>N</i> vary so as to have definite values for the beginning +and end of each step, and the proper mean values can +be written down for each step at once, and therefore +the value of (<i>M</i> − <i>p</i>) <i>dv</i> + <i>Ndt</i> obtained. Adding +together these values for the three steps we get the +integral for the cycle. The condition that this should +vanish is at once seen to be the relation stated above.</p> + +<p>This result combined with the equation <i>A</i> derived +from the second law, gives an important expression +for Carnot's function.</p> + +<p>We shall not pursue this discussion further: so +much is given to make clear how certain results as to<span class='pagenum'><a name="Page_123" id="Page_123">123</a></span> +the physical properties of substances were obtained, +and to explain Thomson's scale of absolute thermodynamic +temperature, which is by far the most important +discovery within the range of theoretical thermodynamics.</p> + +<p>There are several scales of temperature: in point of +fact the scale of a mercury-in-glass thermometer is +defined by the process of graduation, and therefore +there are as many such scales as there are thermometers, +since no two specimens of glass expand in precisely the +same way. Equal differences of temperature do not +correspond to equal increments of volume of the mercury: +for the glass envelope expands also and in its +own way. On the scale of a constant pressure gas +thermometer changes of temperature are measured by +variations of volume of the gas, while the pressure is +maintained constant; on a constant volume gas thermometer +changes of temperature are measured by +alterations of pressure while the volume of the gas is +kept constant. Each scale has its own independent +definition, thus if the pressure of the gas be kept +constant, and the volume at temperature 0° C. be <i>v</i><sub>0</sub> +and that at any other temperature be <i>v</i><sub>1</sub> we define the +numerical value <i>t</i>, this latter temperature, by the equation +<i>v</i> = <i>v</i><sub>0</sub> (1 + <i>Et</i>), where <i>E</i> is 1 ⁄ 100 of the increase +of volume sustained by the gas in being raised from +0° C. to 100° C. These are the temperatures of +reference on an ordinary centigrade thermometer, that +is, the temperature of melting ice and of saturated +steam under standard atmospheric pressure, respectively. +Thus <i>t</i> has the value (<i>v</i> ⁄ <i>v</i><sub>0</sub> − 1) ⁄ <i>E</i>, and is the temperature +(on the constant pressure scale of the gas thermometer) +corresponding to the volume <i>v</i>. Equal<span class='pagenum'><a name="Page_124" id="Page_124">124</a></span> +differences of temperature are such as correspond to +equal increments of the volume at 0° C.</p> + +<p>Similarly, on the constant volume scale we obtain a +definition of temperature from the pressure <i>p</i>, by the +equation <i>t</i> = (<i>p</i> ⁄ <i>p</i><sub>0</sub> − 1) ⁄ <i>E'</i>, where <i>p</i><sub>0</sub> is the pressure +at 0° C., and <i>E'</i> is 1 ⁄ 100 of the change of pressure +produced by raising the temperature from 0° C. to +100° C.</p> + +<p>For air <i>E</i> is approximately 1 ⁄ 273, and thus +<i>t</i> = 273 (<i>v</i> − <i>v</i><sub>0</sub>) ⁄ <i>v</i><sub>0</sub>. If we take the case of <i>v</i> = 0, +we get <i>t</i> = − 273. Now, although this temperature +may be inaccessible, we may take it as zero, and the +temperature denoted by <i>t</i> is, when reckoned from this +zero, 273 + <i>t</i>. This zero is called the absolute zero +on the constant pressure air thermometer. The value +of <i>E'</i> is very nearly the same as that of <i>E</i>; and we get +in a similar manner an absolute zero for the constant +volume scale. If the gas obeyed Boyle's law exactly +at all temperatures, <i>E</i> would not differ from <i>E'</i>.</p> + +<p>It was suggested to Thomson by Joule, in a letter +dated December 9, 1848, that the value of <i>μ</i> might be +given by the equation <i>μ</i> = <i>JE</i> ⁄ (1 + <i>Et</i>). Here we +take heat in dynamical units, and therefore the factor +<i>J</i> is not required. With these units Joule's suggestion +is that <i>μ</i> = <i>E</i> ⁄ (1 + <i>Et</i>), or with <i>E</i> = 1 ⁄ 273 +<i>μ</i> = 1 ⁄ (273 + <i>t</i>), that is, <i>μ</i> = 1 ⁄ <i>T</i> where <i>T</i> is the +temperature reckoned in centigrade degrees from the +absolute zero of the constant pressure air thermometer.</p> + +<p>The possibility of adopting this value of μ was shown +by Thomson to depend on whether or not the heat +absorbed by a given mass of gas in expanding without +alteration of temperature is the equivalent of the work +done by the expanding gas against external pressure. +<span class='pagenum'><a name="Page_125" id="Page_125">125</a></span> +The heat <i>H</i> absorbed by the air in expanding from +volume <i>V</i> to another volume <i>V'</i> at constant temperature +is the integral of <i>Mdv</i> taken from the former +volume to the latter. But by the value of <i>M</i> given +on p. <a href="#Page_121">121</a>, if <i>W</i> be the integral of <i>pdv</i>, that is the work +done by the air in the expansion, ∂<i>W</i> ⁄ ∂<i>t</i> = μ<i>H</i>. +The equation fulfilled by the gas at constant pressure +(the defining equation for <i>t</i>), <i>v</i> = <i>v</i><sub>0</sub> (1 + <i>Et</i>), +gives for the integral of <i>pdv</i>, that is <i>W</i>, the +equation <i>W</i> = <i>pv</i><sub>0</sub> (1 + <i>Et</i>) log (<i>V'</i> ⁄ <i>V</i>), so that +∂<i>W</i> ⁄ ∂<i>t</i> = <i>EW</i> ⁄ (1 + <i>Et</i>). Thus μ<i>H</i> = <i>EW</i> ⁄ (1 + <i>Et</i>).</p> + +<p>Hence it follows that if μ = <i>E</i> ⁄ (1 + <i>Et</i>), the value +of <i>H</i> will be simply <i>W</i>. Thus Joule's suggested value +of μ is only admissible if the work done by the gas in +expanding from a given volume to any other is the +equivalent of the heat absorbed; or, which is the same +thing, if the external work done in compressing the +gas from one volume to another is the equivalent of +the heat developed.</p> + +<p>This result naturally suggests the formation of a +new scale of thermometry by the adoption of the +defining relation <i>T</i> = 1 ⁄ μ, where <i>T</i> denotes temperature. +A scale of temperature thus defined is proposed +in the paper by Joule and Thomson, "On the Thermal +Effects of Fluids in Motion," Part II, which was +published in the <i>Philosophical Transactions</i> for June 1854, +and is what is now universally known as Thomson's +scale of absolute thermodynamic temperature. It can, +of course, be made to give 100 as the numerical value +of the temperature difference between 0° C. and 100° C. +by properly fixing the unit of <i>T</i>. This scale was the +natural successor, in the dynamical theory, of one +which Thomson had suggested in 1848, and which<span class='pagenum'><a name="Page_126" id="Page_126">126</a></span> +was founded, according to Carnot's idea, on the condition +that a unit of heat should do the same amount +of work in descending through each degree. This, as +he pointed out, might justly be called an <i>absolute</i> scale, +since it would be independent of the physical properties +of any substance. In the same sense the scale defined +by <i>T</i> = 1 ⁄ μ is truly an absolute scale.</p> + +<p>The new scale gives a simple expression for the +efficiency of a perfect engine working between two +physically given temperatures, and assigns the numerical +values of these temperatures; for the heat <i>H</i> taken +in from the source in the isothermal expansion which +forms the first operation of the cycle (p. <a href="#Page_120">120</a>) is <i>Mdv</i>, +and, as we have seen, the work done in the cycle is +<i>∂</i><i>p</i> ⁄ <i>∂</i><i>t</i> . <i>dtdv</i>, or μ<i>Hdt</i>. If we adopt the expression 1 ⁄ <i>T</i> +for μ, we may put <i>dT</i> for <i>dt</i>; and we obtain for the +work done the expression <i>HdT</i> ⁄ <i>T</i>. The work done +is thus the fraction <i>dT</i> ⁄ <i>T</i> of the heat taken in, and +this is what is properly called the efficiency of the +engine for the cycle.</p> + +<p>If we suppose the difference of temperatures between +source and refrigerator to be finite, <i>T</i> − <i>T'</i>, say, then +since <i>T</i> is the temperature of the source, we have for +the efficiency (<i>T</i> − <i>T'</i>) ⁄ <i>T</i>. If the heat taken in be +<i>H</i>, the heat rejected is <i>HT'</i> ⁄ <i>T</i>, so that the heat +received by the engine is to the heat rejected by it in +the ratio of <i>T'</i> to <i>T</i>. Thus, as was done by Thomson, +we may define the temperatures of the source and +refrigerator as proportional to the heat taken in from +the source and the heat rejected to the refrigerator by +a perfect engine, working between those temperatures. +The scale may be made to have 100 degrees between +the temperature of melting ice and the boiling point,<span class='pagenum'><a name="Page_127" id="Page_127">127</a></span> +as already explained. We shall return to the comparison +of this scale with that of the air thermometer. +At present we consider some of the thermodynamic +relations of the properties of bodies arrived at by +Thomson.</p> + +<p>First we take the working substance of the engine +as consisting of matter in two states or phases; for +example, ice and water, or water and saturated steam. +Let us apply equation (A) to this case. If <i>v</i><sub>1</sub>, <i>v</i><sub>2</sub> be +the volume of unit of mass in the first and second +states respectively, the isothermal expansion of the first +part of the cycle will take place in consequence of the +conversion of a mass <i>dm</i> from the first state to the +second. Thus <i>dv</i>, the change of volume, is <i>dm</i> (<i>v</i><sub>2</sub> − <i>v</i><sub>1</sub>). +Also if <i>L</i> be the latent heat of the substance in the +second state, <i>e.g.</i> the latent heat of water, <i>Mdv</i> = <i>Ldm</i>; +so that <i>M</i> (<i>v</i><sub>2</sub> − <i>v</i><sub>1</sub>) = <i>L</i>. If <i>dp</i> be the step of pressure +corresponding to the step <i>dT</i> of temperature, equation +(A) becomes</p> + +<div class="center"><img class="floatInsert22" src="images/f127.png" alt="" title="" /> +</div> + +<p>In the case of coexistence of the liquid and solid +phases, this gives us the very remarkable result that a +change of pressure <i>dp</i> will raise or lower the temperature +of coexistence of the two phases, that is, the melting +point of the solid, by the difference of temperature, <i>dT</i>, +according as <i>v</i><sub>2</sub> is greater or less than <i>v</i><sub>1</sub> Thus a +substance like water, which expands in freezing, so that +<i>v</i><sub>2</sub> − <i>v</i><sub>1</sub> is negative, has its freezing point lowered by +increase of pressure and raised by diminution of pressure. +This is the result predicted by Professor James Thomson +and verified experimentally by his brother (p. <a href="#Page_113">113</a> above). +On the other hand, a substance like paraffin wax,<span class='pagenum'><a name="Page_128" id="Page_128">128</a></span> +which contracts in solidifying, would have its melting +point raised by increase of pressure and lowered by a +diminution of pressure.</p> + +<p>The same conclusions would be applicable when the +phases are liquid and vapour of the same substance, if +there were any case in which <i>v</i><sub>2</sub> − <i>v</i><sub>1</sub> is negative. As +it is we see, what is well known to be the case, that the +temperature of equilibrium of a liquid with its vapour +is raised by increase of pressure.</p> + +<p>Another important result of equation (B), as applied +to the liquid and vapour phases of a substance, is the +information which it gives as to the density of the +saturated vapour. When the two phases coexist the +pressure is a function of the temperature only. Hence +if the relation of pressure to temperature is known, +<i>dp</i> ⁄ <i>dT</i> can be calculated, or obtained graphically from +a curve; and the volume <i>v</i><sub>2</sub> per unit mass of the +vapour will be given in terms of <i>dp</i> ⁄ <i>dT</i>, the temperature +<i>T</i>, and the volume <i>v</i> per unit mass of the liquid. +The density of saturated steam at different temperatures +is very difficult to measure experimentally with any +approach of accuracy: but so far as experiment goes +equation (B) is confirmed. The theory here given is +fully confirmed by other results, and equation (B) is +available for the calculation of <i>v</i><sub>2</sub> for any substance for +which the relation between <i>p</i> and <i>T</i> is known. It is +thus that the density of saturated steam can best be found.</p> + +<p>We can obtain another important result for the case +of the working substance in two phases from equation +(B). The relation is</p> + +<div class="center"><img class="floatInsert22" src="images/f128.png" alt="" title="" /> +</div> + +<p><span class='pagenum'><a name="Page_129" id="Page_129">129</a></span></p> + +<p>where <i>c</i> and <i>h</i> are the specific heats of the substance in +the two phases respectively, and <i>L</i> is the latent heat of +the second phase at absolute temperature <i>T</i>.</p> + +<p>We shall obtain the relation in another way, which +will illustrate another mode of dealing with a cycle of +operations which Thomson employed. Any small +step of change of a substance may be regarded as made +up of a step of volume, say, followed by a step of temperature, +that is, by an isothermal step followed by an +adiabatic step. In this way any cycle of operations +whatever may be regarded as made up of a series of +Carnot cycles. But without regarding any cycle of a +more general kind than Carnot's as thus compounded, +we can draw conclusions from it by the dynamical +theory provided only it is reversible. Suppose a +gramme, say, of the substance to be taken at a specified +temperature <i>T</i> in the lower phase, and to be changed +to the other phase at that temperature. The heat +taken in will be <i>L</i> and the expansion will be <i>v</i><sub>2</sub> − <i>v</i><sub>1</sub>. +Next, keeping the substance in the second phase, and +in equilibrium with the first phase (that is, for example, +if the second phase is saturated vapour, the saturation +is to continue in the further change), let the substance +be lowered in temperature by <i>dT</i>. The heat given +out by the substance will be <i>hdT</i>, where <i>h</i> is the +specific heat of the substance in the second phase. +Now at the new temperature <i>T</i> − <i>dT</i> let the substance +be wholly brought back to the second phase; the heat +given out will be <i>L</i> − <i>∂</i><i>L</i> ⁄ <i>∂</i><i>T</i> . <i>dT</i>. Finally, let the +substance, now again all in the first phase, be brought +to the original temperature: the heat taken in will be +<i>cdt</i>, where <i>c</i> is the specific heat in the first phase. +Thus the net excess of heat taken in over heat given +<span class='pagenum'><a name="Page_130" id="Page_130">130</a></span> +out in the cycle is (<i>∂</i><i>L</i> ⁄ <i>∂</i><i>T</i> + <i>c</i> − <i>h</i>) <i>dT</i>. This must, +in the indicator diagram for the changes specified, be +the area of the cycle or (<i>v</i><sub>2</sub> − <i>v</i><sub>1</sub>) <i>∂</i><i>p</i> ⁄ <i>∂</i><i>T</i> . <i>dT</i>. But by +equation (B) <i>L</i> ⁄ <i>T</i> (<i>v</i><sub>2</sub> − <i>v</i><sub>1</sub>) = <i>∂</i><i>p</i> ⁄ <i>∂</i><i>T</i>, and the area +of the cycle is (<i>L</i> ⁄ <i>T</i>) <i>dT</i>. Equating the two +expressions thus found for the area we get equation (C).</p> + +<p>This relation was arrived at by Clausius in his paper +referred to above, and the priority of publication is +his: it is here given in the form which it takes when +Thomson's scale of absolute temperature is used.</p> + +<p>Regnault's experimental results for the heat required +to raise unit mass of water from the temperature of +melting ice to any higher temperature and evaporate it +at that temperature enable the values of <i>L</i> ⁄ <i>T</i> and <i>∂</i><i>L</i> ⁄ <i>∂</i><i>T</i> +to be calculated, and therefore that of <i>h</i> to be found. +It appears that <i>h</i> is negative for all the temperatures to +which Regnault's experimental results can be held to +apply. This, as was pointed out by Thomson, means +that if a mass of saturated vapour is made to expand so +as at the same time to fall in temperature, it must have +heat given to it, otherwise it will be partly condensed +into liquid; and, on the other hand, if the vapour be +compressed and made to rise in temperature while at +the same time it is kept saturated, heat must be taken +from it, otherwise the vapour will become superheated +and so cease to be saturated.</p> + +<p>It is convenient to notice here the article on <i>Heat</i> +which Thomson wrote for the ninth edition of the +<i>Encyclopædia Britannica</i>. In that article he gave a +valuable discussion of ordinary thermometry, of thermometry +by means of the pressures of saturated vapour of +different substances—steam-pressure thermometers, he +called them—of absolute thermodynamic thermometry,<span class='pagenum'><a name="Page_131" id="Page_131">131</a></span> +all enriched with new experimental and theoretical +investigations, and appended to the whole a valuable +synopsis, with additions of his own, of the Fourier +mathematics of heat conduction.</p> + +<p>First dealing with temperature as measured by the +expansion of a liquid in a less expansible vessel, he +showed how it is in reality numerically reckoned. This +amounted to a discussion of the scale of an ordinary +mercury-in-glass thermometer, a subject concerning +which erroneous statements are not infrequently made +in text-books. A sketch of Thomson's treatment of it +is given here.</p> + +<p>Considering this thermometer as a vessel consisting +of a glass bulb and a long glass stem of fine and uniform +bore, hermetically sealed and containing only mercury +and mercury vapour, he explained the numerical +relation between the temperature as shown by the +instrument and the volumes of the mercury and vessel. +The scale is really defined by the method of graduation +adopted. Two points of reference are marked on the +stem at which the top of the mercury stands when the +vessel is immersed (1) in melting ice, (2) in saturated +steam under standard atmospheric pressure. The stem +is divided into parts of equal volume of bore between +these two points and beyond each of them. For a +centigrade thermometer the bore-space between the +two points is divided into 100 equal parts, and the +lower point of reference is marked 0 and the upper 100, +and the other dividing marks are numbered in accordance +with this along the stem. Each of these parts of +the bore may be called a degree-space.</p> + +<p>Now let the instrument contain in its bulb and +stem, up to the mark 0, <i>N</i> degree-spaces, and let <i>v</i> be<span class='pagenum'><a name="Page_132" id="Page_132">132</a></span> +the volume of a degree-space at that temperature. The +volume up to the mark 0 will be <i>Nv</i>, at that temperature; +and if the substance of the vessel be quite uniform +in quality and free from stress, <i>N</i> will be the same for +all temperatures. If <i>v</i><sub>0</sub> be the volume of a degree-space +at the temperature of melting ice the volume of the +mercury at that temperature will be <i>Nv</i><sub>0</sub>. If <i>G</i> be +the expansion of the glass when the volume of a +degree-space is increased from <i>v</i><sub>0</sub> to <i>v</i> by the rise +of temperature, then <i>v</i> = <i>v</i><sub>0</sub> (1 + <i>G</i>). The volume +of the mercury has been increased therefore to +(<i>N</i> + <i>n</i>) <i>v</i><sub>0</sub> (1 + <i>G</i>) by the same rise of temperature, if +the top of the column is thereby made to rise from the +mark 0 so as to occupy <i>n</i> degree-spaces more than before. +But if <i>E</i> be the expansion of the mercury between +the temperature of melting ice and that which has +now been attained, the volume of the mercury is also +<i>Nv</i><sub>0</sub> (1 + <i>E</i>). Hence <i>N</i> (1 + <i>E</i>) = (<i>N</i> + <i>n</i>) (1 + <i>G</i>). +This gives <i>n</i> = <i>N</i> (<i>E</i> − <i>G</i>) ⁄ (1 + <i>G</i>).</p> + +<p>If we take, as is usual, <i>n</i> as measuring the temperature, +and substitute for it the symbol <i>t</i>, we have, since +<i>N</i> = 100 (1 + <i>G</i><sub>100</sub>) ⁄ (<i>E</i><sub>100</sub> − <i>G</i><sub>100</sub>),</p> + +<div class="center"> +<img class="floatInsert22" src="images/f132.png" alt="" title="" /> +</div> + +<p>In this reckoning the definition of any temperature, let +us say 37° C., is the temperature of the vessel and its +contents when the top of the mercury column stands +at the mark 37 above 0, on the scale defined by the +graduation of the instrument; but the numerical +signification with relation to the volumes is given by +equation (D). This shows that the numerical measure<span class='pagenum'><a name="Page_133" id="Page_133">133</a></span> +of any temperature involves both the expansion of the +vessel and that of the glass vessel between the temperature +of melting ice and the temperature in question. +This result may be contrasted with the erroneous statement +frequently made that equal increments of temperature +correspond to equal increments of the volume +of the thermometric substance. It also shows that +different mercury-in-glass thermometers, however accurately +made and graduated, need not agree when +placed in a bath at any other temperature than 0° C. +or 100° C. This fact, and the results of the comparison +of thermometers made with different kinds of glass +with the normal air thermometer, which was carried out +by Regnault, were always insisted on by Thomson in +his teaching when he dealt with the subject of heat. +The scale of a mercury-in-glass thermometer is too +often in text-books, and even in Acts of Parliament +regarded as a perfectly definite thing, and the expansion +of a gas is not infrequently defined by this indefinite +scale, instead of being used as it ought to be, as the basis +of definition of the scale of the gas thermometer. The +whole treatment of the so-called gaseous laws is too +often, from a logical point of view, a mass of confusion.</p> + +<p>In his article on <i>Heat</i> Thomson gave two definitions +of the scale of absolute temperature. One is that +stated on p. <a href="#Page_126">126</a> above, namely, that the temperature of +the source and refrigerator are in the ratio of the heat +taken in from the source to the heat given to the +refrigerator, when the engine describes a Carnot cycle +consisting of two isothermal and two adiabatic +changes.</p> + +<p>The other definition is better adapted for general use, +as it applies to any cycle whatever which is reversible.<span class='pagenum'><a name="Page_134" id="Page_134">134</a></span> +Let the working substance expand under constant +pressure by an amount <i>dv</i> (<i>AB'</i> in Fig. <a href="#f12">12</a>), and let +heat <i>H</i> be given to the substance at the same time. +The external work done is <i>pdv</i>. Thomson called +<i>pdv</i> ⁄ <i>H</i> the work ratio. Now let the temperature be +raised by <i>dT</i> without giving heat to the substance or +taking heat from it, and let the corresponding pressure +rise be <i>dp</i>; and call <i>dp</i> ⁄ <i>p</i> the pressure ratio. The +temperature ratio <i>dT</i> ⁄ <i>T</i> is equal to the product of the +work ratio and the pressure ratio, that is,</p> + +<div class="center"><img class="floatInsert22" src="images/f134b.png" alt="" title="" /> +</div> + +<p>This is clearly true; for <i>dvdp</i> is the area of a cycle +like <i>AB'C'D</i>, represented in Fig. <a href="#f12">12</a>, for which an +amount of heat <i>H</i> is taken in, though not in this case +strictly at one temperature. And clearly, since in +Fig. <a href="#f12">12</a> the change from <i>B'</i> to <i>B</i> is adiabatic, <i>H</i> is the +heat which would have to be taken in for the isothermal +change <i>AB</i> in the Carnot cycle <i>ABCD</i>, which has the +same area as AB'C'D. Thus the efficiency of the +cycle is <i>dvdp</i> ⁄ <i>H</i>, and this by the former definition +is <i>dT</i> ⁄ <i>T</i>.</p> + +<p>Or we may regard the matter thus:—The amount +of heat <i>H</i> which corresponds to an infinitesimal expansion +<i>dv</i> may be used in equation (A) whether the +expansion is isothermal or not, if we take <i>T</i> as the +average temperature of the expansion. Hence we +have <i>dp</i> ⁄ <i>dT</i> = <i>H</i> ⁄ (<i>dv</i>.<i>T</i>), that is, <i>dT</i> ⁄ <i>T</i> = <i>dpdv</i> ⁄ <i>H</i>. +The theorem on p. <a href="#Page_128">128</a> is obtained by what is virtually +this process.<span class='pagenum'><a name="Page_135" id="Page_135">135</a></span></p> + +<h3><span class="smcap">Comparison of Absolute Scale with Scale of +Air Thermometer</span></h3> + +<p>The comparison which Joule and Thomson carried +out of the absolute thermodynamic scale with the scale +of the constant pressure gas thermometer has already +been referred to, and it has been shown that the two +scales would exactly agree, that is, absolute temperature +would be simply proportional to the volume of the gas +in a gas thermometer kept at the temperature to be +measured, if the internal energy of the gas were not +altered by an alteration of volume without alteration of +temperature, that is, if the <i>de</i> − <i>∂</i><i>e</i> of p. <a href="#Page_107">107</a> above +were zero. Joule tested whether this was the case +by immersing two vessels, connected by a tube which +could be opened or closed by a stopcock, in the water of +a calorimeter, ascertaining the temperature with a very +sensitive thermometer, and then allowing air which +had already been compressed into one of the vessels to +flow into the other, which was initially empty. It +was found that no alteration of temperature of the +water of the calorimeter that could be observed was +produced. But the volume of the air had been +doubled by the process, and if any sensible alteration +of internal energy had taken place it would have shown +itself by an elevation or a lowering of the temperature +of the water, according as the energy had been +diminished or increased.</p> + +<p>Thomson suggested that the gas to be examined +should be forced through a pipe ending in a fine nozzle, +or, preferably, through a plug of porous material placed +in a pipe along which the gas was forced by a pump, +and observations made of the temperature in the steady<span class='pagenum'><a name="Page_136" id="Page_136">136</a></span> +stream on both sides of the plug. The experiments +were carried out with a plug of compressed cotton-wool +held between two metal disks pierced with holes, in a +tube of boxwood surrounded also by cotton-wool, and +placed in a bath of water closely surrounding the supply +pipe. This was of metal, and formed the end of a long +spiral all immersed in the bath. Thus the temperature +of the gas approaching the plug was kept at a uniform +temperature determined by a delicate thermometer; +another thermometer gave the temperature in the +steady stream beyond the plug.</p> + +<p>In the case of hydrogen the experiments showed a +slight heating effect of passage through the plug; air, +oxygen, nitrogen and carbonic acid were cooled by the +passage.</p> + +<p>The theory of the matter is set forth in the original +papers, and in a very elegant manner in the article on +<i>Heat</i>. The result of the analysis shows that if <i>∂</i><i>w</i> be +the positive or negative work-value of the heat which +will convert one gramme of the gas after passage to its +original temperature; and <i>T</i> be absolute temperature, +and <i>v</i> volume of a gramme of the gas at pressure <i>p</i>, and +the difference of pressure on the two sides of the plug +be <i>dp</i>, the equation which holds is</p> + +<div class="center"> +<a name="FNanchor_18_18" id="FNanchor_18_18"></a> + <a href="#Footnote_18_18" class="fnanchor"> + <img class="floatInsert35" src="images/136.png" alt="" title="" /> + </a> +</div> + +<p class="after">It was found by Joule and Thomson that <i>∂</i><i>w</i> was +proportional to <i>dp</i> for values of <i>dp</i> up to five or six +atmospheres. At different temperatures, however, in +the case of hydrogen the heating effect was found to +diminish with rise of temperature, being .100 of a +degree centigrade at 4° or 5° centigrade, and .155 at +temperatures of from 89° to 93° centigrade for a +difference of pressure due to 100 inches of mercury.</p> + +<p>If there is neither heating nor cooling <i>∂</i><i>w</i> = 0, and +we obtain by integration <i>T</i> = <i>Cv</i>, where <i>C</i> is a +constant.</p> + +<p><span class='pagenum'><a name="Page_138" id="Page_138">138</a></span> +Elaborate discussions of the theory of this experiment +will be found in modern treatises on thermodynamics, +and in various recent memoirs, and the differential +equation has been modified in various ways, and integrated +on various suppositions, which it would be +out of place to discuss here.</p> + +<p>The cooling effect of passing a gas such as air or +oxygen through a narrow orifice has been used to +liquefy the gas. The stream of gas is pumped along a +pipe towards the opening, and that which has passed +the orifice and been slightly cooled is led on its way +back to the pump along the outside of the pipe by +which more gas is approaching the orifice, and so +cools slightly the advancing current. The gas which +emerges later is thus cooler than that which emerged +before, and the process goes on until the issuing gas is +liquefied and falls down into the lower part of the pipe +surrounding the orifice, whence it can be drawn off +into vessels constructed to receive and preserve it.</p> + +<p>It is possible thus to liquefy hydrogen, which shows +that at the low temperature at which the process is +usually started (an initial cooling is applied) the passage +through the orifice has a cooling effect as in the other +cases.</p> + +<p>Another idea, that of <i>thermodynamic motivity</i>, on +which Thomson suggested might be founded a fruitful +presentation of the subject of thermodynamics, may be +mentioned here. It was set forth in a letter written +to Professor Tait in May 1879. If a system of bodies +be given, all at different temperatures, it is possible to +reduce them to a common temperature, and by doing +so to extract a certain amount of mechanical energy +from them. The temperatures must for this purpose +<span class='pagenum'><a name="Page_139" id="Page_139">139</a></span> +be equalised by perfect thermodynamic engines working +between the final temperature <i>T</i><sub>0</sub>, say, and the +temperatures of the different parts of the system. This +process is one of the levelling up and the levelling +down of temperature; and the temperature <i>T</i><sub>0</sub> is such +that exactly the heat given out at <i>T</i><sub>0</sub> by certain engines, +receiving heat from bodies of higher temperature than +<i>T</i><sub>0</sub>, is supplied to the engines which work between <i>T</i><sub>0</sub> +and bodies at lower temperatures. The whole useful +work obtained in this way was called by Thomson the +<i>motivity</i> of the system. Of course equalisation of +temperature may be obtained by conduction, and in +this case the energy which might be utilised is lost. +With two equal and similar bodies at absolute temperatures +<i>T</i>, <i>T'</i> the temperature to which they are reduced +when their motivity is extracted is √(<i>TT'</i>). If the +temperatures are equalised by conduction the resulting +temperature is higher, being ½(<i>T</i> + <i>T'</i>). Thus, if only +the two bodies are available for engines to work +between, the motivity is the measure of the energy +lost when conduction brings about equalisation of +temperature.</p> + +<p>A very suggestive paper on the subject was published +by Lord Kelvin in the <i>Trans. R.S.E.</i>, vol. 28, 1877-8.</p> + +<h3><span class="smcap">Dissipation of Energy</span></h3> + +<p>In connection with the theory of heat must be +mentioned Thomson's great generalisation, the theory +of the dissipation of energy.<a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">19</a> Most people have some +<span class='pagenum'><a name="Page_140" id="Page_140">140</a></span> +notion of the meaning of the physical doctrine of conservation +of energy, though in popular discourses it is +usually misstated. What is meant is that in a finite +material system, which is isolated in the sense that +it is not acted on by force from without, the total +amount of energy—that is, energy of motion and energy +of relative position (including energy of chemical affinity) +of the parts—remains constant. The usual misstatement +is that the energy of the universe is constant. +This may be true if the <i>universe</i> is finite; if the +universe is infinite in extent the statement has no +meaning. In any case, we know nothing about the +universe as a whole, and therefore make no statements +regarding it.</p> + +<p>But while there is thus conservation or constancy of +amount of energy in an isolated and finite material +system, this energy may to residents on the system +become unavailable. For useful work within such a +system is done by conversion of energy from one form +to another and the total amount remains unchanged. +But if this conversion is prevented all processes which +involve such conversion must cease, and among these +are vital processes.</p> + +<p>The unavailable form which the energy of the +system with which we are directly and at present +concerned, whatever may become of us ultimately, is +taking, according to Thomson's theory, is universally +diffused heat. How this comes about may be seen as +follows. Even a perfect engine, if the refrigerator be +at the lowest available temperature, rejects a quantity +of heat which cannot be utilised for the performance of +the work. This heat is diffused by conduction and +radiation to surrounding bodies, and so to bodies more<span class='pagenum'><a name="Page_141" id="Page_141">141</a></span> +remote, and the general temperature of the system is +raised. Moreover, as heat engines are imperfect there +is heat rejected to the surroundings by conduction, and +produced by work done against friction, so that the +heat thrown on the unavailable or waste heap is still +further increased.</p> + +<p>Conduction of heat is the great agency by which +energy is more and more dispersed in this unavailable +form throughout the totality of material bodies. +As has been seen, available motivity is continually +wasted through its agency; and in the flow of heat +in the earth and in the sun and other unequally heated +bodies of our system the waste of energy is prodigious. +Aided by convection currents in the air and in the +ocean it continually equalises temperatures, but does so +at an immense cost of useful energy.</p> + +<p>Then in our insanely wasteful methods of heating +our houses by open fires, of half burning the coal used +in boiler furnaces, and allowing unconsumed carbon to +escape into the atmosphere in enormous quantities, while +a very large portion of the heat actually generated is +allowed to escape up chimneys with heated gases, the +store of unavailable heat is being added to at a rate +which will entail great distress, if not ruin, on humanity +at no indefinitely distant future. It will be the height +of imprudence to trust to the prospect, not infrequently +referred to at the present time, of drawing on the +energy locked up in the atomic structure of matter. +He would be a foolish man who would wastefully +squander the wealth he possesses, in the belief that he +can recoup himself from mines which all experience +so far shows require an expenditure to work them far +beyond any return that has as yet been obtained.</p> + +<p><span class='pagenum'><a name="Page_142" id="Page_142">142</a></span> +It is not apart from our present theme to urge that +it is high time the question of the national economy of +fuel, and the desirability of utilising by afforestation the +solar energy continually going to waste on the surface +of the earth, were dealt with by statesmen. If statesmen +would but make themselves acquainted with the +results of physical science in this magnificent region of +cosmic economics there would be some hope, but, alas! +as a rule their education is one which inevitably leads +to neglect, if not to disdain of physical teaching.</p> + +<p>From the causes which have been referred to, energy +is continually being dissipated, not destroyed, but +locked up in greater and greater quantity in the general +heat of bodies. There is always friction, always heat +conduction and convection, so that as our stores of +motional or positional energy, whether of chemical +substances uncombined, the earth's motion, or what +not, are drawn upon, the inevitable fraction, too often +a large proportion, is shed off and the general temperature +raised. After a large part of the whole existent +energy has gone thus to raise the dead level of things, +no difference of temperature adequate for heat engines +to work between will be possible, and the inevitable +death of all things will approach with headlong +rapidity.</p> + +<h3><span class="smcap">Thermoelasticity and Thermoelectricity</span></h3> + +<p>In the second definition of the scale of absolute +temperature just discussed, stress of any type may be +substituted for pressure, and the corresponding displacement +s for the change of volume. Thus for a piece of +elastic material put through a cycle of changes we<span class='pagenum'><a name="Page_143" id="Page_143">143</a></span> +may substitute <i>dS</i> for <i>dp</i> and <i>Ads</i> for <i>dv</i>; where <i>A</i> is +such a factor that <i>AdSds</i> is the work done in the displacement +<i>ds</i> by the stress <i>dS</i>. As an example consider a +wire subjected to simple longitudinal stress <i>S</i>. Longitudinal +extension is produced, but this is not the only +change; there is at the same time lateral contraction. +However, <i>s</i> within certain limits is proportional to <i>S</i>.</p> + +<p>Let heat <i>dH</i> in dynamical measure be given to the +wire while the stress <i>S</i> is maintained constant, and let +the extension increase from <i>s</i> to <i>s</i> + <i>ds</i>. The stress <i>S</i> +will do work <i>ASds</i> <i>on the wire</i>, and the work ratio will +be − <i>ASds</i> ⁄ <i>dH</i>. Now let the stress be increased to +<i>S</i> + <i>dS</i> while the extension is kept constant, and the +absolute temperature raised from <i>T</i> to <i>T</i> + <i>dT</i>. The +stress ratio (as we may call it) is <i>dS</i> ⁄ <i>S</i> and the temperature +ratio <i>dT</i> ⁄ <i>T</i>. Thus we obtain (p. <a href="#Page_134">134</a> above)</p> + +<div class="center"><img class="floatInsert22" src="images/f143a.png" alt="" title="" /> +</div> + +<p>In his <i>Heat</i> article Thomson used the alteration <i>e</i> +of strain under constant stress (that is <i>ds</i> ⁄ <i>l</i>, where <i>l</i> is +the length of the wire) corresponding to an amount of +heat sufficient to raise the temperature under constant +stress by 1°. Hence if <i>K</i> be the specific heat under +constant stress, and <i>le</i> be put for <i>ds</i> in the sense just +stated, we have</p> + +<div class="center"><img class="floatInsert25" src="images/f143b.png" alt="" title="" /> +</div> + +<p>where ρ is the density, since <i>dH</i> = <i>K</i>ρ<i>lA</i>.</p> + +<p>The ratio of <i>dH</i> to the increase <i>ds</i> of the extension +is positive or negative, that is, the substance absorbs +or evolves heat, when strained under the condition of<span class='pagenum'><a name="Page_144" id="Page_144">144</a></span> +constant stress, according as <i>dS</i> ⁄ <i>dT</i> is negative or positive. +Or we may put the same thing in another way +which is frequently useful. If a wire subjected to +constant stress has heat given to it, <i>ds</i> is negative or +positive, in other words the wire shortens or lengthens, +according as <i>dS</i> ⁄ <i>dT</i> is positive or negative, that is, +according as the stress for a given strain is increased +or diminished by increase of temperature.</p> + +<p>It is known from experiment that a metal wire +expands under constant stress when heat is given to it, +and thus we learn from the equation (F) that the +stress required for a given strain is diminished when +the temperature of the wire is raised. Again, a strip of +india-rubber stretched by a weight is shortened if its +temperature is raised, consequently the stress required +for a given strain is increased by rise of temperature.</p> + +<p>These results, from a qualitative point of view, are +self-evident. But from what has been set forth it will +be obvious that an equation exactly similar to (F) +holds whether the change <i>ds</i> of <i>s</i> is taken as before +under constant stress, or at uniform temperature, or +whether the change <i>dS</i> of <i>S</i> is effected adiabatically or +at constant strain.</p> + +<p>In all these cases the same equation</p> + +<div class="center"><img class="floatInsert25" src="images/f144.png" alt="" title="" /> +</div> + +<p>applies, with the change of meaning of <i>dT</i> involved.</p> + +<p>This equation differs from that of Thomson as +given in various places (<i>e.g.</i> in the <i>Encyclopædia +Britannica</i> article on <i>Elasticity</i> which he also wrote) +in the negative sign on the right-hand side, but the<span class='pagenum'><a name="Page_145" id="Page_145">145</a></span> +difference is only apparent. According to his specification +a <i>pressure</i> would be a positive stress, and an +<i>expansion</i> a positive displacement, and in applying the +equation to numerical examples this must be borne in +mind so that the proper signs may be given to each +numerical magnitude. As an example of adiabatic +change, a sudden extension of the wire already referred +to by an increase of stress <i>dS</i> may be considered. If +there is not time for the passage of heat from or to the +surroundings of the wire, the change of temperature +will be given by equation (G).</p> + +<p>This equation was applied by Thomson (article +<i>Elasticity</i>) to find the relation between what he called +the kinetic modulus of elasticity and the static modulus, +that is, between the modulus for adiabatic strain and +the modulus for isothermal strain.</p> + +<p>The augmentation of the strain produced by raising +the temperature 1° is <i>e</i>, and therefore <i>edT</i>, that is, +− <i>Te</i><sup>2</sup><i>dS</i> ⁄ <i>K</i>ρ, is the increase of strain due to the sudden +rise of temperature <i>dT</i>. This added to the isothermal +strain produced by <i>dS</i> will give the whole adiabatic +strain. Thus if <i>M</i> be the static or isothermal modulus, +the adiabatic strain is <i>dS</i> ⁄ <i>M</i> − <i>Te</i><sup>2</sup><i>dS</i> ⁄ <i>K</i>ρ. If <i>M'</i> +denote the kinetic or adiabatic modulus its value +is <i>dS</i> divided by the whole adiabatic strain, that +is, <i>M'</i> = <i>M</i> ⁄ (1 − <i>MTe</i><sup>2</sup> ⁄ <i>K</i>ρ) and the ratio +<i>M'</i> ⁄ <i>M</i> = 1 ⁄ (1 − <i>MTe</i><sup>2</sup> ⁄ <i>K</i>ρ).</p> + +<p>It is well known and easy to prove, without the use +of any theorem which can be properly called thermodynamic, +that this ratio of moduli is equal to the ratio of +the specific heat <i>K</i> of the substance, under the condition +of constant stress, to the specific heat <i>N</i> under +the condition of constant strain of the corresponding<span class='pagenum'><a name="Page_146" id="Page_146">146</a></span> +type. This, indeed, is self-evident if two changes of +stress, one isothermal the other adiabatic, <i>which produce +the same steps of displacement ds</i>, be considered, and it +be remembered that the step <i>∂</i><i>T</i> of temperature which +accompanies the adiabatic change may be regarded as +made up of a step − <i>dT</i> of temperature, accompanying +a displacement ds effected at constant stress, and then +two successive steps <i>dT</i> and <i>∂</i><i>T</i> effected, at constant +strain, along with the steps of stress <i>dS</i>. The ratio +<i>M'</i> ⁄ <i>M</i> is easily seen to have the value (<i>∂</i><i>T</i> + <i>dT</i>) ⁄ <i>dt</i>, +and since − <i>KdT</i> + <i>N</i> (<i>∂</i><i>T</i> + <i>dT</i>) = 0, by the adiabatic +condition, the theorem is proved.</p> + +<p>Laplace's celebrated result for air, according to +which the adiabatic bulk-modulus is equal to the +static bulk-modulus multiplied by the ratio of the +specific heat of air pressure constant to the specific +heat of air volume constant, is a particular example of +this theory.</p> + +<p>Thomson showed in the <i>Elasticity</i> article how, by +the value of <i>M'</i> ⁄ <i>M</i>, derived as above from thermodynamic +theory, the value of <i>K</i> ⁄ <i>N</i> could be obtained +for different substances and for different types of stress, +and gave very interesting tables of results for solids, +liquids, and gases subjected to pressure-stress (bulk-modulus) +and for solids subjected to longitudinal stress +(Young's modulus).</p> + +<p>The discussion as to the relation of the adiabatic +and isothermal moduli of elasticity is part of a very +important paper on "Thermoelastic, Thermomagnetic, +and Thermoelectric Properties of Matter," which he +published in the <i>Philosophical Magazine</i> for January +1878. This was in the main a reprint of an article +entitled, "On the Thermoelastic and Thermomagnetic<span class='pagenum'><a name="Page_147" id="Page_147">147</a></span> +Properties of Matter, Part I," which appeared in +April 1855 in the first number of the <i>Quarterly +Journal of Mathematics</i>. Only thermoelasticity was +considered in this article; the thermomagnetic results +had, however, been indicated in an article on "Thermomagnetism" +in the second edition of the <i>Cyclopædia of +Physical Science</i>, edited and in great part written by +Professor J. P. Nichol, and published in 1860. For +the same <i>Cyclopædia</i> Thomson also wrote an article entitled, +"Thermo-electric, Division I.—Pyro-Electricity, +or Thermo-Electricity of Non-conducting Crystals," +and the enlarged <i>Phil. Mag.</i> article also contained the +application of thermodynamics to this kind of thermoelectric +action.</p> + +<p>This great paper cannot be described without a +good deal of mathematical analysis; but the student +who has read the earlier thermodynamical papers of +Thomson will have little difficulty in mastering it. +It must suffice to say here that it may be regarded as +giving the keynote of much of the general thermodynamic +treatment of physical phenomena, which forms +so large a part of the physical mathematics of the +present day, and which we owe to Willard Gibbs +Duhem, and other contemporary writers.</p> + +<p>Thomson had, however, previous to the publication +of this paper, applied thermodynamic theory to thermoelectric +phenomena. A long series of papers containing +experimental investigations, and entitled, +"Electrodynamic Qualities of Metals," are placed in the +second volume of his <i>Mathematical and Physical Papers</i>. +This series begins with the Bakerian Lecture (published +in the <i>Transactions of the Royal Society</i> for +1856) which includes an account of the remarkable<span class='pagenum'><a name="Page_148" id="Page_148">148</a></span> +experimental work accomplished during the preceding +four or five years by the volunteer laboratory corps in the +newly-established physical laboratory in the old College. +The subjects dealt with are the Electric Convection +of Heat, Thermoelectric Inversions, the Effects of +Mechanical Strain and of Magnetisation on the Thermoelectric +Qualities of Metals, and the Effects of +Tension and Magnetisation on the Electric Conductivity +of Metals. It is only possible to give here a +very short indication of the thermodynamic treatment, +and of the nature of Thomson's remarkable discovery +of the electric convection of heat.</p> + +<p>It was found by Seebeck in 1822 that when a +circuit is formed of two different metals (without any +cell or battery) a current flows round the circuit if the +two junctions are not at the same temperature. For +example, if the two metals be rods of antimony and +bismuth, joined at their extremities so as to form a +complete circuit, and one junction be warmed while +the other is kept at the ordinary temperature, a current +flows across the hot junction in the direction from +bismuth to antimony. Similarly, if a circuit be made +of a copper wire and an iron wire, a current passes +across the warmer junction from copper to iron. The +current strength—other things being the same—depends +on the metals used; for example, bismuth and antimony +are more effective than other metals.</p> + +<p>It was found by Peltier that when a current, say +from a battery, is sent round such a circuit, that junction +is cooled and that junction is heated by the passage +of the current, which, being respectively heated and +cooled, would without the cell have caused a current to +flow in the same direction. Thus the current produced<span class='pagenum'><a name="Page_149" id="Page_149">149</a></span> +by the difference of temperature of the junctions +causes an absorption of heat from the warmer junction, +and an evolution of heat at the colder junction.</p> + +<p>This naturally suggested to Thomson the consideration +of a circuit of two metals, with the junctions at +different temperatures, as a heat engine, of which the +hot junction was the source and the cold junction the +refrigerator, while the heat generated in the circuit by +the current and other work performed, if there was +any, was the equivalent of the difference between the +heat absorbed and the heat evolved. Of course in such +an arrangement there is always irreversible loss of heat +by conduction; but when such losses are properly +allowed for the circuit is capable of being correctly +regarded as a reversible engine.</p> + +<p>Shortly after Seebeck's discovery it was found by +Cumming that when the hot junction was increased +in temperature the electromotive force increased more +and more slowly, at a certain temperature of the hot +junction took its maximum value, and then as the +temperature of the hot junction was further increased +began to diminish, and ultimately, at a sufficiently high +temperature, in most instances changed sign. The +temperature of maximum electromotive force was +found to be independent of the temperature of the +colder junction. It is called the temperature of the +neutral point, from the fact that if the two junctions +of a thermoelectric circuit be kept at a constant small +difference of temperature, and be both raised in temperature +until one is at a higher temperature than the +neutral point, and the other is at a lower, the electromotive +force will fall off, until finally, when this point +is reached, it has become zero.<span class='pagenum'><a name="Page_150" id="Page_150">150</a></span></p> + +<p>Thus it was found that for every pair of metals +there was at least one such temperature of the hot +junction, and it was assumed, with consequences in +agreement with experimental results, that when the +temperature was the neutral temperature there was +neither absorption nor evolution of heat at the junction. +But then the source provided by the thermodynamic +view just stated had ceased to exist. The +current still flowed, there was evolution of heat at +the cold junction, and likewise Joulean evolution of +heat in the wires of the circuit in consequence of their +resistance. Hence it was clear that energy must be +obtained elsewhere than at the junctions. Thomson +solved the problem by showing that (besides the +Joulean evolution of heat) there is absorption (or +evolution) of heat when a current flows in a conductor +along which there is a gradient of temperature. +For example, when an electric current flows along an +unequally heated copper wire, heat is evolved where +the current flows from the hot parts to the cold, and +heat is absorbed where the flow is from cold to hot. +When the hot junction is at the temperature of zero +absorption or evolution of heat—the so-called neutral +temperature—the heat absorbed in the flow of the circuit +along the unequally heated conductors is greater than +that evolved on the whole, by an amount which is the +equivalent of the energy electrically expended in the +circuit in the same time.</p> + +<p>It was found, moreover, that the amount of heat +absorbed by a given current in ascending or descending +through a given difference of temperature is different +in different metals. When the current was unit +current and the temperature difference also unity,<span class='pagenum'><a name="Page_151" id="Page_151">151</a></span> +Thomson called the heat absorbed or evolved in a +metal the specific heat of electricity in the metal, a +name which is convenient in some ways, but misleading +in others. The term rather conveys the notion +that electricity has a material existence. A substance +such as copper, lead, water, or mercury has a specific +heat in a perfectly understood sense; electricity is not +a substance, hence there cannot be in the same proper +sense a specific heat of electricity.</p> + +<p>However, this absorption and evolution of heat was +investigated experimentally and mathematically by +Thomson, and is generally now referred to in thermoelectric +discussions as the "Thomson effect."</p> + +<p>Part VI (<i>Trans. R.S.</i>, 1875) of the investigations of +the electrodynamic qualities of metals dealt with the +effects of stretching and compressing force, and of torsion, +on the magnetisation of iron and steel and of nickel and +cobalt.</p> + +<p>One of the principal results was the discovery that +the effect of longitudinal pull is to increase the inductive +magnetisation of soft iron, and of transverse thrust +to diminish it, so long as the magnetising field does +not exceed a certain value. When this value, which +depends on the specimen, is exceeded, the effect of +stress is reversed. The field-intensity at which the +effect is reversed is called the Villari critical intensity, +from the fact, afterwards ascertained, that the result +had previously been established by Villari in Italy. +No such critical value of the field was found to exist +for steel, or nickel, or cobalt.</p> + +<p>In some of the experiments the specimen was put +through a cycle of magnetic changes, and the results +recorded by curves. These proved that in going from<span class='pagenum'><a name="Page_152" id="Page_152">152</a></span> +one state to another and returning the material lagged +in its return path behind the corresponding states in +the outward path. This is the phenomenon called +later "hysteresis," and studied in minute detail by +Ewing and others. Thomson's magnetic work was +thus the starting point of many more recent researches.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_153" id="Page_153">153</a></span></p> + +<h3>CHAPTER IX</h3> + +<h4>HYDRODYNAMICS—DYNAMICAL THEOREM OF +MINIMUM ENERGY—VORTEX MOTION</h4> + +<p>Thomson devoted great attention from time to time +to the science of hydrodynamics. This is perhaps +the most abstruse subject in the domain of applied +mathematics, and when viscosity (the frictional resistance +to the relative motion of particles of the fluid) +is taken into account, passes beyond the resources of +mathematical science in its present state of development. +But leaving viscosity entirely aside, and dealing +only with so-called perfect fluids, the difficulties are +often overwhelming. For a long time the only kind +of fluid motion considered was, with the exception of +a few simple cases, that which is called irrotational +motion. This motion is characterised by the analytical +peculiarity, that the velocity of an element of the fluid +in any direction is the rate of variation per unit distance +in that direction of a function of the coordinates (the +distances which specify the position) of the particle. +This condition very much simplifies the analysis; but +when it does not hold we have much more serious +difficulties to overcome. Then the elements of the +fluid have what is generally, but quite improperly, +called molecular rotation. For we know little of the +<i>molecules</i> of a fluid; even when we deal with infinitesimal +elements, in the analysis of fluid motion, we are<span class='pagenum'><a name="Page_154" id="Page_154">154</a></span> +considering the fluid in mass. But what is meant +is elemental rotation, a rotation of the infinitesimal +elements as they move. We have an example of such +motion in the air when a ring of smoke escapes from +the funnel of a locomotive or the lips of a tobacco-smoker, +in the motion of part of the liquid when a cup +of tea is stirred by drawing the spoon from one side to +the other, or when the blade of an oar is moving +through the water. In these last two cases the depressions +seen in the surface are the ends of a vortex +which extends between them and terminates on the +surface. In all these examples what have been called +<i>vortices</i> are formed, and hence the name vortex motion +has been given to all those cases in which the condition +of irrotationality is not satisfied.</p> + +<p>The first great paper on vortex motion was published +by von Helmholtz in 1858, and ten years later a +memoir on the same subject by Thomson was published +in the <i>Transactions of the Royal Society of +Edinburgh</i>. In that memoir are given very much +simpler proofs of von Helmholtz's main theorems, and, +moreover, some new theorems of wide application to +the motion of fluids. One of these is so comprehensive +that it may be said with truth to contain the +whole of the dynamics of a perfect fluid. We go on +to indicate the contents of the principal papers, as far +as that can be done without the introduction of analysis +of a difficult description.</p> + +<p>In Chapter VI reference has been made to the +"Notes on Hydrodynamics" published by Thomson +in the <i>Cambridge and Dublin Mathematical Journal</i> +for 1848 and 1849. These Notes were not intended +to be entirely original, but were composed for the<span class='pagenum'><a name="Page_155" id="Page_155">155</a></span> +use of students, like Airy's <i>Tracts</i> of fifteen years +before.</p> + +<p>The first Note dealt with the equation of continuity, +that is to say, the mathematical expression of the +obvious fact that if any region of space in a moving +fluid be considered, the excess of rate of flow into the +space across the bounding surface, above the rate of +flow out, is equal to the rate of growth of the quantity +of fluid within the space. The proof given is that +now usually repeated in text-books of hydrodynamics.</p> + +<p>The second Note discussed the condition fulfilled at +the bounding surface of a moving fluid. The chief +mathematical result is the equation which expresses +the fact, also obvious without analysis, that there is +no flow of the fluid across the surface. In other +words, the component of the motion of a fluid particle +in the immediate neighbourhood of the surface at any +instant, taken in the direction perpendicular to the +surface, must be equal to the motion of the surface in +that direction at the same instant.</p> + +<p>The third Note, published a year later (February 1849), +is of considerable scientific importance. It is entitled, +"On the Vis Viva of a Liquid in Motion." What +used to be called the "vis viva" of a body is double +what is now called the energy of motion, or kinetic +energy, of the body. The term liquid is merely a +brief expression for a fluid, the mass of which per +unit volume is the same throughout, and suffers no +variation. The fluid, moreover, is supposed devoid of +friction, that is, the relative motions of its parts are +unresisted by tangential force between them. The +chief theorem proved and discussed may be described +as follows.</p> + +<p><span class='pagenum'><a name="Page_156" id="Page_156">156</a></span>The liquid is supposed to fill the space within a +closed envelope, which fulfils the condition of being +"simply continuous." The condition will be understood +by imagining any two points <i>A</i>, <i>B</i>, within the +space, to be joined by two lines <i>ACB</i>, <i>ADB</i> both lying +within the space. These two lines will form a circuit +<i>ACBDA</i>. If now this circuit, however it may be +drawn, can be contracted down to a point, without +any part of the circuit passing out of the space, the +condition is fulfilled. Clearly the space within the +surface of an anchor-ring, or a curtain-ring, would not +fulfil this condition, for one part of the circuit might +pass from <i>A</i> to <i>B</i> round the ring one way, and the +other from <i>A</i> to <i>B</i> the other way. The circuit could +not then be contracted towards a point without passing +out of the ring.</p> + +<p>Now let the liquid given at rest in such a space be +set in motion by any arbitrarily specified variation of +position of the envelope. The liquid within will be set +in motion in a manner depending entirely on the motion +of the envelope. It is possible to conceive of other +motions of the liquid than that taken, which all agree in +having the specified motion of the surface. Thomson's +theorem asserts that the motion actually taken has less +kinetic energy than that of any of the other motions +which have the same motion of the bounding surface.</p> + +<p>The motion produced has the property described by +the word "irrotational," that is, the elements of the +fluid have no spinning motion—they move without +rotation. A small portion of a fluid may describe any +path—may go round in a circle, for example—and yet +have no rotation. The reader may imagine a ball +carried round in a circle, but in such a way that no<span class='pagenum'><a name="Page_157" id="Page_157">157</a></span> +line in the body ever changes its direction. The body +has translation, but no spin.</p> + +<p>Irrotationality of a fluid is secured, as stated above, +when the velocity of each element in any direction is +the rate of variation per unit distance in that direction +of a certain function of the coordinates, the distances, +taken parallel to three lines perpendicular to one +another and drawn from a point, which specify the +position of the particle. In fact, what is called a +velocity-potential exists, similar to the potential described +in Chapter IV above, for an electric field. +This condition, together with the specified motion of +the surface, suffices to determine the motion of the +fluid.</p> + +<p>Two important particular consequences were pointed +out by Thomson: (1) that the motion of the fluid at +any instant depends solely on the form and motion of +the bounding surface, and is therefore independent of +the previous motion; and (2) that if the bounding +surface be instantaneously brought to rest, the liquid +throughout the vessel will also be instantly brought to +rest.</p> + +<p>This theorem was afterwards generalised by Thomson +(<i>Proc. R.S.E.</i>, 1863), and applied to any material +system of connected particles set into motion by +specified velocities simultaneously and suddenly imposed +at selected points of the system. It was already +known that the kinetic energy of a system of bodies +connected in any manner, and set in motion by +impulses applied at specified points, was either a +maximum or a minimum, as compared with that for +any other motion compatible with these impulses, and +with the connections of the system. This was proved<span class='pagenum'><a name="Page_158" id="Page_158">158</a></span> +by Lagrange in the <i>Mécanique Analytique</i> as a generalisation +of a theorem given by Euler for a rigid body +set into rotation by an impulse.</p> + +<p>Bertrand proved in 1842 that when the impulses +applied are given in amount, and are applied at specified +points, the system starts off with kinetic energy greater +than that of any other motion which is consistent with +the given impulses and the connections of the system. +This other motion must be such as could be produced +in the system by the given impulses, together with any +other set of impulses capable of doing no work on the +whole.</p> + +<p>Thomson's theorem is curiously complementary to +Bertrand's. Let the system be acted on by impulses +applied at certain specified points, and by no other +impulses of any kind; and let the impulses be such +as to start those selected points with any prescribed +velocities. The system will start off with kinetic +energy which is less than that of any other motion +which the system could have consistently with the +prescribed velocities, and which it could be constrained +to take by impulses which do no work on the whole. +In each case the difference of energies is the energy +of the motion which must be compounded with one +motion to give the other which is compared with it.</p> + +<p>A simple example, such as might be taken of the +particular case considered by Euler, may help to make +these theorems clear. Imagine a straight uniform rod +to lie on a horizontal table, between which and the +rod there is no friction. Let the rod be struck a blow +at one end in a horizontal direction at right angles to +the length of the rod. If no other impulse acts, the +end of the rod will move off with a certain definite<span class='pagenum'><a name="Page_159" id="Page_159">159</a></span> +velocity, and the other parts of the rod (which is +supposed perfectly unbending) will be started by the +connections of the system. It is obvious that any +number of other motions of the rod can be imagined, +all of which give the same motion of the extremity +struck. But the actual motion taken is one of turning +about that point of the rod which is two-thirds of the +length from the end struck. If the reader will consider +the kinetic energy for any other horizontal turning +motion consistent with the same motion of the end, he +will find that the kinetic energy is greater than that of +the motion just specified. This motion could be produced +by applying at the point about which the rod +turns the impulse required to keep that point at rest. +The impulse so applied would do no work. The +actual value is <small><sup>1</sup>⁄<sub>8</sub></small><i>mv</i><sup>2</sup>, where <i>m</i> denotes the mass of the +rod and <i>v</i> the velocity of the end. If the motion +taken were one of rotation about a point of the rod at +distance <i>x</i> from the end struck, the kinetic energy would +be <i>m</i> (4<i>l</i><sup>2</sup> − 6<i>lx</i> + 3<i>x</i><sup>2</sup>) <i>v</i><sup>2</sup> ⁄ 6<i>x</i><sup>2</sup>, +where 2<i>l</i> is the length of +the rod, and this has its least value <small><sup>1</sup>⁄<sub>8</sub></small><i>mv</i><sup>2</sup> for <i>x</i> = 4<i>l</i> ⁄ 3. +For example, <i>x</i> = 2<i>l</i> gives <small><sup>1</sup>⁄<sub>6</sub></small><i>mv</i><sup>2</sup>, which is greater than +the value just found.</p> + +<p>Bertrand's theorem applied to this case of motion +is not quite so easy, perhaps, to understand. The +motion which is said to have maximum energy is one +given by a specified impulse at the end struck, and +this, in the absence of any other impulses, would be a +motion of minimum energy. But let the alternative +motion, which is to be compared with that actually +taken, be one constrained by additional impulses such +as can together effect no work, and the existence of +the maximum is accounted for. The kinetic energy<span class='pagenum'><a name="Page_160" id="Page_160">160</a></span> +produced is one-half the product of the impulse into +the velocity of the point struck, that is ½<i>Iv</i>, and it +has just been seen that this is the product of <small><sup>1</sup>⁄<sub>6</sub></small><i>mv</i><sup>2</sup> +by the factor (4<i>l</i><sup>2</sup> − 6<i>lx</i> + 3<i>x</i><sup>2</sup>) ⁄ <i>x</i><sup>2</sup>. This factor is +3<i>I</i> ⁄ <i>mv</i>, and is a minimum when <i>x</i> = 4<i>l</i> ⁄ 3. Thus +for a given <i>I</i>, <i>v</i> will have its maximum value when +the factor referred to is least, and ½<i>Iv</i> will then be a +maximum.</p> + +<p>The bar can be constrained to turn about another +point by a fixed pivot there situated. An impulse +will be applied to the rod by the pivot, simultaneously +with the blow; and it is obvious that this impulse +does no work, since there is no displacement of the +point to which it is applied.</p> + +<p>The two theorems are consequences of one principle. +The constraint in each case increases what may be +called the effective inertia, which may be taken as +<i>I</i> ⁄ <i>v</i>. Thus when <i>v</i> is given, <i>I</i> is increased by any +constraint compelling the rod to rotate about a +particular axis, and so ½<i>Iv</i>, or the kinetic energy, is +increased. On the other hand, when <i>I</i> is given the +same constraint diminishes <i>v</i>, and so ½<i>Iv</i> is diminished.</p> + +<p>A short paper published in the B. A. Report for +1852 points out that the lines of force near a small +magnet, placed with its axis along the lines of force +in a uniform magnetic field, as it would rest under +the action of the field, are at corresponding points +similar to those of the field of an insulated spherical +conductor, under the inductive influence of a distant +electric change. Further, the fact is noted that, if the +magnet be oppositely directed to the field, the lines of +force are curved outwards, just as the lines of flow of +a uniform stream would be by a spherical obstacle, at<span class='pagenum'><a name="Page_161" id="Page_161">161</a></span> +the surface of which no eddies were caused. This is +one of those instructive analogies between the theory +of fluid motion and other theories involving perfectly +analogous fundamental ideas, which Thomson was +fond of pointing out, and which helped him in his +repeated attempts to imagine mechanical representations +of physical phenomena of different kinds.</p> + +<p>With these may be placed another, which in lectures +he frequently dwelt on—a simple doublet, as it is +called, consisting of a point-source of fluid and an +equal and closely adjacent point-sink. A short tube +in an infinite mass of liquid, which is continually +flowing in at one end and out at the other, may serve +as a realisation of this arrangement. The lines of +flow outside the tube are exactly analogous to the lines +of force of a small magnet; and if at the same time +there exist a uniform flow of the liquid in the direction +of the length of the tube, the field of flow will be an +exact picture of the field of force of the small magnet, +when it is placed with its length along the lines of a +previously existing uniform field. The flow in the +doublet will be with or against the general flow +according as the magnet is directed with or against the +field.</p> + +<p>The paper on vortex-motion has been referred to +above, and an indication given of the nature of the +fluid-motion described by this title. There are, however, +two cases of fluid-motion which are referred to +as vortices, though the fundamental criterion of vortex-motion—the +non-existence of a velocity-potential—is +satisfied in only one of them. The exhibition of +one of these was a favourite experiment in Thomson's +ordinary lectures, as his old students will remember.<span class='pagenum'><a name="Page_162" id="Page_162">162</a></span> +If water in a large bowl is stirred rapidly with a +teaspoon carried round and round in a circle about the +axis of the bowl, the surface will become concave, and +the form of the central part will be a paraboloid of +revolution about the vertical through the lowest point, +that is to say, any section of that part of the surface +made by a vertical plane containing the axis will be +a parabola symmetrical about the axis. The motion +can be better produced by mounting the vessel on a +whirling-table, and rotating it about the vertical axis +coinciding with its axis of figure; but the phenomenon +can be quite well seen without this machinery. In +this case the velocity of each particle of the water is +proportional to its distance from the axis, and the +whole mass, when relative equilibrium is set up, turns, +as if it were rigid, about the axis of the vessel. Each +element of the fluid in this "forced vortex," as it is +called, is in rotation, and, like the moon, makes one +turn in one revolution about the centre of its path. +This is, therefore, a true, though very simple, case of +vortex-motion.</p> + +<p>On the other hand, what may be called a "free +vortex" may exist, and is approximated to sometimes +when water in a vessel is allowed to run off through +an escape pipe at the bottom. The velocity of an +element in this "vortex" is inversely proportional to +its distance from the centre, and the form of the free +surface is quite different from that in the other case. +The name "free vortex" is often given to this case +of motion, but there is no vortex-motion about it +whatever.</p> + +<p>Thomson's great paper on vortex-motion was read +before the Royal Society of Edinburgh in 1867, and<span class='pagenum'><a name="Page_163" id="Page_163">163</a></span> +was recast and augmented in the following year. It +will be possible to give here only a sketch of its scope +and main results.</p> + +<p>The fluid is supposed contained in a closed fixed +vessel which is either simply or multiply continuous +(see p. <a href="#Page_156">156</a>), and may contain immersed in it simply or +multiply continuous solids. When these solids exist +their surfaces are part of the boundary of the liquid; +they are surrounded by the liquid unless they are anywhere +in contact with the containing vessel, and their +density is supposed to be the same as that of the liquid. +They may be acted on by forces from without, and +they act on the liquid with pressure-forces, and either +directly or through the liquid on one another.</p> + +<p>The first result obtained is fairly obvious. The +centre of mass of the whole system must remain at +rest whatever external forces act on the solids, since +the density is the same everywhere within the vessel, +and the vessel is fixed; that is to say, there is no +momentum of the contents of the vessel in any +direction. For whatever motion of the solids is set +up by the external forces, must be accompanied by a +motion of the liquid, equal and opposite in the sense +here indicated.</p> + +<p>After a discussion of what he calls the impulse of the +motion, which is the system of impulsive forces on the +movable solids which would generate the motion from +rest, Thomson proceeds to prove the important proposition +that the rotational motion of every portion +of the liquid mass, if it is zero at any one instant for +every portion of the mass, remains always zero. This +is done by considering the angular momentum of any +small spherical portion of the liquid relatively to an<span class='pagenum'><a name="Page_164" id="Page_164">164</a></span> +axis through the centre of the sphere, and proving that +in order that it may vanish, for every axis, the component +velocities of the fluid at the centre must be +derivable from a velocity-potential. The angular +momentum of a particle about an axis is the product +of the component of the particle's momentum, +at right angles to the plane through the particle and +the axis, by the distance of the particle from the axis. +The sum of all such products for the particles making +up the body (when proper account is taken of the +signs according to the direction of turning round the +axis) is the angular momentum. The proof of this +result adopted is due to Stokes. The angular velocities +of an element of fluid at a point <i>x</i>, <i>y</i>, <i>z</i>, about the axes +of <i>x</i>, <i>y</i>, <i>z</i> are shown to be ½ (<i>∂</i><i>w</i> ⁄ <i>∂</i><i>y</i> − <i>∂</i><i>v</i> ⁄ <i>∂</i><i>z</i>), etc.</p> + +<p>The condition was therefore shown to be necessary; +it remained to prove that it was sufficient. This is +obvious at once from the definition of the velocity-potential, +which must now be supposed to exist in +order that its sufficiency may be proved. If any +diameter of the spherical portion be taken as the axis, +and any plane through that axis be considered, the +velocity of a particle at right angles to that plane can +be at once expressed as the rate at which the velocity-potential +varies per unit distance along the circle, +symmetrical about the axis, on which the particle lies. +The integral of the velocity-potential round this circle +vanishes, and so the angular momentum for any thin +uniform ring of particles about the axis also vanishes, +and as the sphere is made up of such rings, the whole +angular momentum is zero. Thus the condition is +sufficient.</p> + +<p>Thomson then proves that if the angular momentum<span class='pagenum'><a name="Page_165" id="Page_165">165</a></span> +thus considered be zero for every portion of the liquid +at any one instant, it remains zero at every subsequent +instant; that is, no physical action whatsoever could +set up angular momentum within the fluid, which, it is +to be remembered, is supposed to be frictionless. The +proof here given cannot be sketched because it depends +on the differential equation of continuity satisfied by +the velocity-potential throughout the fluid (the same +differential equation, in fact, that is satisfied by the +distribution of temperature in a uniform conducting +medium in the stationary state), and the consequent +expression of this function for any spherical space in +the fluid as a series of spherical harmonic functions. +To a reader to whom the properties of these functions +are known the process can present no difficulty.</p> + +<p>An entirely different proof of this proposition is +given subsequently in the paper, and depends on a new +and very general theorem, which has been described as +containing almost the whole theory of the motion of a +fluid. This depends on what Thomson called the flow +along any path joining any two points <i>P</i>, <i>Q</i> in the +fluid. Let <i>q</i> be the velocity of the fluid at any element +of length <i>ds</i> of such a path, and <i>θ</i> be the angle between +the direction of <i>ds</i> (taken positive in the sense from <i>P</i> to +<i>Q</i>) and the direction of <i>q</i>: <i>q</i> cos <i>θ</i> . <i>ds</i> is the flow along <i>ds</i>. +If <i>u</i>, <i>v</i>, <i>w</i> be the components of <i>q</i> at <i>ds</i>, parallel to the +axes, and <i>dx</i>, <i>dy</i>, <i>dz</i> be the projections of <i>ds</i> on the axes, +<i>udx</i> + <i>vdy</i> + <i>wdz</i> is the same thing as <i>q</i> cos <i>θ</i> . <i>ds</i>. +The sum of the values of either of these expressions for +all the elements of the path between <i>P</i> and <i>Q</i> is the +flow along the path. The statement that <i>u</i>, <i>v</i>, <i>w</i> are +the space-rates of variation of a function <i>φ</i> (of <i>x</i>, <i>y</i>, <i>z</i>) +parallel to the axes, or that <i>q</i> cos <i>θ</i> is the space-rate of<span class='pagenum'><a name="Page_166" id="Page_166">166</a></span> +variation of <i>φ</i> along <i>ds</i>, merely means that this sum is +the same for whatever path may be drawn from <i>P</i> to <i>Q</i>. +This, however, is only the case when the paths are +so taken that in each case the value of <i>φ</i> returns after +variation along a closed path to the value which it had +at the starting point, that is, the closed path must be +capable of being contracted to a point without passing +out of space occupied by irrotationally moving fluid.</p> + +<p>Since the flow from <i>P</i> to <i>Q</i> is the same for any two +paths which fulfil this condition, the flow from <i>P</i> to <i>Q</i> +by any one path and from <i>Q</i> to <i>P</i> by any other must +be zero. The flow round such a closed path is not +zero if the condition is not fulfilled, and its value was +called by Thomson the circulation round the path.</p> + +<p>The general theorem which he established may now +be stated. Consider any path joining <i>PQ</i>, and moving +with the fluid, so that the line contains always the +same fluid particles. Let <i>u</i>̇, <i>v</i>̇, <i>w</i>̇ be the time-rates of +change of <i>u</i>, <i>v</i>, <i>w</i> at an element <i>ds</i> of the path, at any +instant, and <i>du</i>, <i>dv</i>, <i>dw</i> the excesses of the values of <i>u</i>, <i>v</i>, +<i>w</i> at the terminal extremity of <i>ds</i> above the values at the +other extremity; then the time-rate of variation of +<i>udx</i> + <i>vdy</i> + <i>wdz</i> is <i>u</i>̇<i>dx</i> + <i>v</i>̇<i>dy</i> + <i>w</i>̇<i>dz</i> + <i>udu</i> + <i>vdv</i> + <i>wdw</i> +or <i>u</i>̇<i>dx</i> + <i>v</i>̇<i>dy</i> + <i>w</i>̇<i>dz</i> + <i>qdq</i>, where <i>q</i> has the +meaning specified above. Thus if <i>S</i> be the flow for +the whole path <i>PQ</i>, and <i>Ṡ</i> its time-rate of variation, <i>S'</i> +denote the sum of <i>u</i>̇<i>dx</i> + <i>v</i>̇<i>dy</i> + <i>w</i>̇<i>dz</i> along the path +from <i>P</i> to <i>Q</i>, and <i>q</i><sub>1</sub>, <i>q</i><sub>0</sub> the resultant fluid velocities at +<i>Q</i> and <i>P</i>, we get <i>Ṡ</i> = <i>S'</i> + ½(<i>q</i><sub>1</sub><sup>2</sup> − <i>q</i><sub>0</sub><sup>2</sup>). This is +Thomson's theorem. If the curve be closed, that is, if +<i>P</i> and <i>Q</i> be coincident, <i>q</i><sub>1</sub> = <i>q</i><sub>0</sub> and <i>Ṡ</i> = <i>S'</i>. But in +certain circumstances <i>S'</i> is zero, and so therefore is +also <i>Ṡ</i>. Thus in the circumstances referred to, as the<span class='pagenum'><a name="Page_167" id="Page_167">167</a></span> +closed path moves with the fluid <i>Ṡ</i> is continually zero, +and it follows that if <i>Ṡ</i> is zero at any instant it remains +zero ever after. But <i>Ṡ</i> is only zero if <i>u</i>, <i>v</i>, <i>w</i> are derivable +from a potential, single valued in the space in +which the closed path is drawn, so that the path could +be shrunk down to a point without ever passing out of +such space. In a perfect fluid if this condition is once +fulfilled for a closed curve moving with the fluid, it is +fulfilled for this curve ever after.</p> + +<p>The circumstances in which <i>S'</i> is zero are these:—the +external force, per unit mass, acting on the fluid at +any point is to be derivable from a potential-function, +and the density of the fluid is to be a function of the +pressure (also a function of the coordinates); and these +functions must be such as to render <i>S'</i> always zero for +the closed path. This condition is manifestly fulfilled +in many important cases; for example, the forces are +derivable from a potential due to actions, such as +gravity, the origin of which is external to the fluid; +and the density is a function of the pressure (in the +present case it is a constant), such that the part of <i>S'</i> +which depends on pressure and density vanishes for the +circuit.</p> + +<p>It is to be clearly understood that the motion of a +fluid may be irrotational although the value of <i>S</i> does +not vanish for <i>every</i> closed path that can be drawn in +it. The fluid may occupy multiply continuous space, +and the path may or may not be drawn so that <i>S</i> shall +be zero; but what is necessary for irrotational motion +within any space is that <i>S</i> should vanish for all paths +which are capable of being shrunk down to zero without +passing out of that space. <i>S</i> need not vanish for a +path which cannot be so shrunk down, but it must, if the<span class='pagenum'><a name="Page_168" id="Page_168">168</a></span> +condition just stated is fulfilled, have the same value +for any two paths, one of which can be made to pass +into the other by change of position without ever passing +in whole or in part out of the space. The potential +is always single valued in fluid filling a singly continuous +space such as that within a spherical shell, or between +two concentric shells; within a hollow anchor-ring +the potential, though it exist, and the motion be irrotational, +is not single valued. In the latter case the +motion is said to be <i>cyclic</i>, in the former <i>acyclic</i>.</p> + +<p>A number of consequences are deduced from this +theorem; and from these the properties of vortices, +which had previously been discovered by von Helmholtz, +immediately follow. First take any surface whatever +which has for bounding edge a closed curve drawn in +the fluid, and draw from any element of this surface, +of area <i>dS</i>, a line perpendicular to the surface towards +the side chosen as the positive side, and calculate the +angular velocity <i>ω</i>, say, of the fluid about that normal +from the components of angular velocity determined in +the manner explained at p. <a href="#Page_164">164</a>. This Thomson +called the <i>rotation</i> of the element. Now take the product +<i>ωdS</i> for the surface element. It is easy to see that +this is equal to half the circulation round the bounding +edge of the element. As the fluid composing the +element moves the area <i>dS</i> may change, but the circulation +round its edge by Thomson's theorem remains +unaltered. Thus <i>ω</i> alters in the inverse ratio of <i>dS</i>, +and the line drawn at right angles to the surface at <i>dS</i>, +if kept of length proportional to <i>ω</i>, will lengthen or +shorten as <i>dS</i> contracts or expands.</p> + +<p>Now sum the values of <i>ωdS</i> for the finite surface +enclosed by the bounding curve. It follows from the<span class='pagenum'><a name="Page_169" id="Page_169">169</a></span> +fact that <i>ωdS</i> is equal to half the circulation round the +edge of <i>dS</i>, that this sum, which is usually denoted by +Σ<i>ωdS</i>, is equal to half the circulation round the closed +curve which forms the edge of the surface. Also as +the fluid moves the circulation round the edge remains +unaltered, and therefore so does also Σ<i>ωdS</i> for the +elements enclosed by it. It is important to notice that +this sum being determined by the circulation in the +bounding curve is the same for all surfaces which have +the same boundary.</p> + +<p>The equality of 2Σ<i>ωdS</i> for the surface to the circulation +round its edge was expressed by Thomson as an +analytical theorem of integration, which was first given +by Stokes in a Smith's Prize paper set in 1854. It is +here stated, apparently by an oversight, that it was first +given in Thomson and Tait's <i>Natural Philosophy</i>, § 190. +In the second edition of the <i>Natural Philosophy</i> the +theorem is attributed to Stokes. It is now well known +as Stokes's theorem connecting a certain surface integral +with a line integral, and has many applications both in +physics and in geometry.</p> + +<p>Now consider the resultant angular velocity at any +point of the fluid, and draw a short line through that +point in the direction of the axis of rotation. That +line may be continued from point to point, and will +coincide at every one of its points with the direction of +the axis of rotation there. Such an axial curve, as it +may be called, it is clear moves with the fluid. For +take any infinitesimal area containing an element of +the line; the circulation round the edge of this area is +zero, since there is no rotation about a line perpendicular +to the area. Hence the circulation along the axial +curve is zero, and the axial curves move with the fluid.</p> + +<p><span class='pagenum'><a name="Page_170" id="Page_170">170</a></span>Take now any small plane area <i>dS</i> moving with the +fluid, and draw axial lines through every point of its +boundary. These will form an axial tube enclosing +<i>dS</i>. If <i>θ</i> be the angle between the direction of resultant +rotation and a perpendicular to <i>dS</i>, the cross-section +of the tube at right angles to the normal, and to the +axial lines which bound it, is <i>dS</i>.cos<i>θ</i>. Let these +axial lines be continued in both directions from the +element <i>dS</i>. They will enclose a tube of varying +normal cross-section; but the product of rotation and +area of normal cross-section has everywhere the same +value. A vortex-tube with the fluid within it is called +a vortex-filament.</p> + +<p>It will be seen that this vortex-tube must be endless, +that is, it must either return into itself, or be infinitely +long in one or both directions. For if it were terminated +anywhere within the fluid, it would be possible to +form a surface, starting from a closed circuit round the +tube, continued along the surface of the tube to the +termination, and then closed by a cap situated beyond +the termination. At no part of this surface would +there be any rotation, and Σ<i>ωdS</i>, which is equal to +the circulation, would be zero for it; and of course +this cannot be the case. Thus the tube cannot terminate +within the fluid. It can, however, have both of its +ends on the surface, or one on the bounding surface +and the other at infinity, if the fluid is infinitely +extended in one direction, but in that case the termination +is only apparent. The section is widened out at +the surface; some of the bounding lines pass across +to the other apparent termination, when it also +lies on the surface, while the other lines pass off to +infinity along the surface, and correspond to other lines<span class='pagenum'><a name="Page_171" id="Page_171">171</a></span> +coming in from infinity to the other termination. +Whether the surface is infinite or not, the vortex is +spread out into what is called a vortex-sheet, that is, in +a surface on the two sides of which the fluid moves +with different tangential velocities.</p> + +<p>Through a vortex-ring or tube, the fluid circulates +in closed lines of flow, each one of which is laced +through the tube. The circulation along every line of +flow which encloses the same system of vortex-tubes +has the same value.</p> + +<p>If any surface be drawn cutting a vortex-tube, it is +clear from the definition of the tube that the value of +Σ<i>ωdS</i> for every such surface must be the same. This +Thomson calls the "rotation of the tube."</p> + +<p>As was pointed out first by von Helmholtz, vortex-filaments +correspond to circuits carrying currents and +the velocity in the surrounding fluid to magnetic field-intensity. +The "rotation of the tube" corresponds to +the strength of the current, and sources and sinks to +positive and negative magnetic poles. Thomson made +great use of this analogy in his papers on electromagnetism.</p> + +<p>Examples of vortex-tubes are indicated on p. <a href="#Page_154">154</a>; +and the reader may experiment with vortices in liquids +with water in a tea-cup, or in a river or pond, at +pleasure. Air vortices may be experimentally studied +by means of a simple apparatus devised by Professor +Tait, which may be constructed by anyone.</p> + +<div class="figleft" style="width: 150px; position: relative;"><a name="f13" id="f13"></a><img src="images/fig13.png" width="150" height="165" alt="Fig. 13." title="" /> +<p class="caption"><span class="smcap">Fig. 13.</span></p></div> + +<p>In one end of a packing-box, about 2ft. long by +18in. wide and 18in. deep, a circular hole is cut, +and the edges of the hole are thinned down to a blunt +edge. This can be closed at pleasure by a piece of +board. The opposite end is removed, and a sheet of<span class='pagenum'><a name="Page_172" id="Page_172">172</a></span> +canvas stretched tightly in its place, and tacked to the +ends of the sides. Through two holes bored in one of +the sides the mouths of two flasks with bent necks +protrude into the box. One of these flasks contains +ammonia, the other hydrochloric acid. When the hole +at one end is closed up by a slip of tinplate, and the +liquids are heated with a spirit-lamp, the vapours form +a cloud of sal-ammoniac within the box, which is +retained during its formation. The hole is then +opened, and the canvas struck smartly with the palm +of the open hand. Immediately a beautiful ring of +smoke emerges, clear-cut and definite +as a solid, and moves across the room. +(See Fig. <a href="#f13">13</a>.) Of course, it is a ring +of air, made visible by the smoke carried +with it. By varying the shape of the +aperture—for example, by using instead +of the hole cut in the wood, a slide of +tinplate with an elliptic hole cut in it—the vortex-rings +can be set in vibration as they are created, and the +vibrations studied as the vortex moves.</p> + +<p>Still more beautiful vortices can be formed in water +by using a long tank of clear water to replace the air +in which the vortex moves, and a compartment at one +end filled with water coloured with aniline, instead of +the smoke-box. A hole in the dividing partition +enables the vortex to be formed, and a piston arrangement +fitted to the opposite side enables the impulse to +the water to be given from without.</p> + +<p>From the account of the nature of vortex-motion +given above, it will be clear that vortices in a perfect +fluid once existent must be ever existent. To create a +vortex within a mass of irrotationally moving perfect<span class='pagenum'><a name="Page_173" id="Page_173">173</a></span> +fluid is physically impossible. It occurred to Thomson, +therefore, that ordinary matter might be portions of a +perfect fluid, filling all space, differentiated from the +surrounding fluid by the rotation which they possess. +Such matter would fulfil the law of conservation, as it +could neither be created nor destroyed by any physical +act.</p> + +<p>The results of such experiments led Thomson to +frame his famous vortex-atom theory of matter, a +theory, however, which he felt ultimately was beset +with so many difficulties as to be unworkable.</p> + +<p>The paper on vortex-motion also deals with the +modification of Green's celebrated theorem of analysis, +which, it was pointed out by Helmholtz, was necessary +to adapt it to a space which is multiply continuous. +The theorem connects a certain volume-integral taken +throughout a closed space with an integral taken over +the bounding surface of the space. This arises from +the fact noticed above that in multiply continuous +space (for example, the space within an endless tube) the +functions which are the subject of integration may +not be single valued. Such a function would be the +velocity-potential for fluid circulating round the tube—cyclic +motion, as it was called by Thomson. If a +closed path of any form be drawn in such a tube, starting +from a point <i>P</i>, and doubling back so as to return +to <i>P</i> without making the circuit of the tube, the +velocity-potential will vary along the tube, but will +finally return to its original value when the starting +point is reached. And the circulation round this +circuit will be zero. But if the closed path make the +circuit of the tube, the velocity-potential will continuously +vary along the path, until finally, when <i>P</i> is<span class='pagenum'><a name="Page_174" id="Page_174">174</a></span> +reached again, the value of the function is greater (or +less) than the value assumed for the starting point, by +a certain definite amount which is the same for every +circuit of the space. If the path be carried twice +round in the same direction, the change of the function +will be twice this amount, and so on. The space +within a single endless tube such as an anchor-ring is +doubly continuous; but much more complicated cases +can be imagined. For example, an anchor-ring with a +cross-connecting tube from one side to the other would +be triply continuous.</p> + +<p>Thomson showed that the proper modification of the +theorem is obtained by imagining diaphragms placed +across the space, which are not to be crossed by any +closed path drawn within the space, and the two +surfaces of each of which are to be reckoned as part +of the bounding surface of the space. One such +diaphragm is sufficient to convert a hollow anchor-ring +into a singly continuous space, two would be required +for the hollow anchor-ring with cross-connection, and +so on. The number of diaphragms required is always +one less than the degree of multiplicity of the +continuity.</p> + +<p>The paper also deals with the motion of solids in +the fluid and the analogous motions of vortex-rings and +their attraction by ordinary matter. These can be +studied with vortex-rings in air produced by the +apparatus described above. Such a ring made to pass +the re-entrant corner of a wall—the edge of a window +recess, for example—will appear to be attracted. A +large sphere such as a large terrestrial globe serves also +very well as an attracting body.</p> + +<p>Two vortex-rings projected one after the other also<span class='pagenum'><a name="Page_175" id="Page_175">175</a></span> +act on one another in a very curious manner. Their +planes are perpendicular to the direction of motion, +and the fluid is moving round the circular core of the +ring. There is irrotational cyclic motion of the fluid +through the ring in one direction and back outside, as +shown in Fig. <a href="#f13">13</a>, which can be detected by placing a +candle flame in the path of the centre. The first ring, +in consequence of the existence of that which follows +it, moves more slowly, and opens out more widely, the +following ring hastens its motion and diminishes in +diameter, until finally it overtakes the former and +penetrates it. As soon as it has passed through it +moves ahead more and more slowly, until the one +which has been left behind begins to catch it up, and +the changes which took place before are repeated. +The one penetrating becomes in its turn the penetrated, +and so on in alternation. Great care and skill are, +however, necessary to make this interesting experiment +succeed.</p> + +<p>We have not space to deal here with other hydrodynamical +investigations, such as the contributions +which Thomson made to the discussion of the many +difficult problems of the motion of solids through a +liquid, or to his very numerous and important contributions +to the theory of waves. The number and +importance of his hydrodynamical papers may be +judged from the fact that there are no less than +fifty-two references to his papers, and thirty-five +to Thomson and Tait's <i>Natural Philosophy</i> in the +latest edition of Lamb's <i>Hydrodynamics</i>, and that +many of these are concerned with general theorems +and results of great value.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_176" id="Page_176">176</a></span></p> + +<h3>CHAPTER X</h3> + +<h4>THE ENERGY THEORY OF ELECTROLYSIS—ELECTRICAL +UNITS—ELECTRICAL OSCILLATIONS</h4> + +<h3><span class="smcap">Electrolysis and Electrical Units</span></h3> + +<p><span class="smcap">In</span> December 1851 Thomson communicated an +important paper to the <i>Philosophical Magazine</i> on +"The Mechanical Theory of Electrolysis," and +"Applications of Mechanical Effect to the Measurement +of Electromotive Forces, and of Galvanic +Resistances, in Absolute Units."</p> + +<p>In the first of these he supposed a machine of the +kind imagined by Faraday, consisting of a metal disk, +rotating uniformly with its plane at right angles to the +lines of force of a uniform magnetic field, and touched +at its centre and its circumference by fixed wires, to +send a current through an electrochemical apparatus, +to which the wires are connected. A certain amount +of work <i>W</i> was supposed to be spent in a given time, +during which a quantity of heat <i>H</i> was evolved in the +circuit, and a certain amount of work <i>M</i> spent in the +chemical apparatus in effecting chemical change. If +<i>H</i> be taken in dynamical units, <i>W</i> = <i>H</i> + <i>M</i>.</p> + +<p>The work done in driving the disk, if the intensity +of the field is <i>I</i>, the current produced <i>c</i>, the radius of the +disc <i>r</i>, and the angular velocity of turning <i>w</i>, is ½<i>Ir</i><sup>2</sup><i>cw</i>.</p> + +<p>Thomson assumed that the work done in the electrochemical +apparatus was equal to the heat of chemical<span class='pagenum'><a name="Page_177" id="Page_177">177</a></span> +combination of the substance or substances which +underwent the chemical action, taken with the proper +sign according to the change, if more compound substances +than one were acted on. Hence <i>M</i> represented +this resultant heat of combination.</p> + +<p>The electrochemical apparatus was a voltameter +containing a definite compound to be electrolysed, or a +voltaic cell or battery. And by Faraday's experiments +on electrolysis it was known that the amount of +chemical action was proportional to the whole quantity +of electricity passed through the cell in a given time, so +that the rate at which energy was being spent in the +cell was at any instant proportional to the current at +that instant.</p> + +<p>The chemical change could be measured by considering +only one of the elements set free, or made to +combine, by the passage of the current, and considering +the quantity of heat <i>θ</i>, say, for the whole chemical +change in the cell corresponding to the action on unit +mass of that element. Thus if <i>E</i> denote the whole +quantity of that element operated on the heat of combination +in the vessel was <i>θ</i><i>E</i>. If <i>E</i> be taken for +unit of time, and ε denote the quantity set free by the +passage of unit quantity of electricity, then <i>E</i> = <i>εc</i>, +since a current conveys <i>c</i> units of electricity in one +second. The number <i>ε</i> is a definite quantity of the +element, and is called its electrochemical equivalent. +Again, from Joule's experiments, <i>H</i> = <i>Rc</i><sup>2</sup>, if <i>R</i> denote +the resistance of the current, and so</p> + +<div class="center"><img class="floatInsert13" src="images/f177a.png" alt="" title="" /> +</div> + +<p>and</p> + +<div class="center"><img class="floatInsert25" src="images/f177b.png" alt="" title="" /> +</div> + +<p><span class='pagenum'><a name="Page_178" id="Page_178">178</a></span></p> + +<p>The quantity ½<i>Ir</i><sup>2</sup><i>w</i> is the electromotive force due to +the disk.</p> + +<p>Thus <i>c</i> was positive or negative according as ½<i>Ir</i><sup>2</sup><i>w</i> +was greater or less than <i>θε</i>, and was zero when +½<i>Ir</i><sup>2</sup><i>w</i> = <i>θε</i>. Thus the electromotive force of the disk +was opposed by a back electromotive force <i>θε</i> due to +the chemical action in the voltameter or battery, to +which the wires from the disk were connected.</p> + +<p>The conclusion arrived at therefore was that the +electromotive force (or, as it was then termed, the +intensity) of the electrochemical action was equal to +the dynamical value of the whole chemical change +effected by a current of unit strength in unit of time.</p> + +<p>From this result Thomson proceeded to calculate +the electromotive forces required to effect chemical +changes of different kinds, and those of various types of +voltaic cell. Supposing a unit of electricity to be +carried by the current through the cell, he considered +the chemical changes which accompanied its passage, +and from the known values of heats of combination +calculated their energy values. In some parts the +change was one of chemical combination, in others +one of decomposition of the materials, and regard had +to be paid to the sign of the heat-equivalent. By +properly summing up the whole heat-equivalents a net +total was obtained which, according to Thomson, was +the energy consumed in the passage of unit current, +and was therefore the electromotive force. The +theory was incomplete, and required to be supplemented +by thermodynamic theory, which shows that +besides the electromotive force there must be included +in the quantity set against the sum of heats a term +represented by the product of the absolute temperature<span class='pagenum'><a name="Page_179" id="Page_179">179</a></span> +multiplied by the rate of variation of electromotive +force with alteration of temperature. Thus the theory +is only applicable when the electromotive force is not +affected by variation of temperature. The necessary +addition here indicated was made by Helmholtz.</p> + +<p>In the next paper, which appeared in the same +number (December 1851) of the <i>Philosophical Magazine</i>, +the principle of work is applied to the measurement +of electromotive forces and resistances in absolute +units. The advantages of such units are obvious. +Nearly the whole of the quantitative work of the older +experimenters was useless except for those who had +actually made the observations: it was hardly possible +for one man to advance his researches by employing +data obtained by others. For the results were expressed +by reference to apparatus and materials in the possession +of the observers, and to these others could obtain access +only with great difficulty and at great expense—to say +nothing of the uncertainty of comparisons made to +enable the results of one man to be linked on to those +made elsewhere, and with other apparatus, by another. +It was imperative, therefore, to obtain absolute units—units +independent of accidents of place and apparatus—for +the expression of currents, electromotive forces, and +resistances, so as to enable the results of the work of +experiments all over the world to be made available to +every one who read the published record. (See Chap. +<a href="#Page_244">XIII</a>.)</p> + +<p>The magneto-electric machine imagined in the +former paper gave a means of estimating the electromotive +force of a cell or battery in absolute units. The +same kind of machine is used here, in the simpler form +of a sliding conductor connecting a pair of insulated<span class='pagenum'><a name="Page_180" id="Page_180">180</a></span> +rails laid with their plane perpendicular to the lines of +force of a uniform magnetic field. If the rails be +connected by a wire, and the slider be moved so as to +cut across the lines of force, a current will be produced +in the circuit. The current can be measured in terms +of the already known unit of current, that current +which flowing in a circle of radius unity produces a +magnetic field at the centre of 2π units. This current, +<i>c</i>, say, in strength, flowing in the circuit, renders a +dynamical force <i>cIl</i> necessary to move the slider of +length <i>l</i> across the lines of force of the field of intensity +<i>I</i>, and if the speed of the slider required for the current +<i>c</i> be <i>v</i>, the rate at which work is done in moving the +slider is <i>cIlv</i>. This must be the rate at which work is +done in the circuit by the current, and if the only +work done be in the heating of the conductor, we have +<i>cIlv</i> = <i>Rc</i><sup>2</sup>, or <i>Ilv</i> = <i>Rc</i>, so that <i>Ilv</i> is the electromotive +force. Any electromotive force otherwise +produced, which gave rise to the same current, must +obviously be equal to <i>Ilv</i>, so that the unit of electromotive +force can thus be properly defined.</p> + +<p>Thomson used a foot-grain-second system of units; +but from this arrangement are now obtained the C.G.S. +units of electromotive force and resistance. If <i>I</i> is one +C.G.S. unit, <i>l</i> one centimetre, and <i>v</i> one centimetre +per second, we have unit electromotive force in the +C.G.S. system. Also in one C.G.S. unit of resistance +if <i>c</i> be unity as well as <i>Ilv</i>.</p> + +<p>The idea of the determination of a resistance in +absolute units on correct principles was due to W. +Weber, who also gave methods of carrying out the +measurement; and the first determination was made +by Kirchhoff in 1849. Thomson appears, however,<span class='pagenum'><a name="Page_181" id="Page_181">181</a></span> +to have been the first to discuss the subject of units +from the point of view of energy. This mode of regarding +the matter is important, as the absolute units are so +chosen as to enable work done by electric and magnetic +forces to be reckoned in the ordinary dynamical units. +A vast amount of experimental resource and skill has +been spent since that time on the determination of +resistance, though not more than the importance of +the subject warranted. We shall have to return to +the subject in dealing with the work of the British +Association on Electrical Standards, of which Thomson +was for long an active member.</p> + +<h3><span class="smcap">Electrical Oscillations</span></h3> + +<p>In his famous tract on the conservation of energy, +published in 1847, von Helmholtz discussed some +puzzling results obtained by Riess in the magnetisation +of iron wires by the current of a Leyden jar discharge +flowing in a coil surrounding them, and by the fact, +observed by Wollaston, that when water was decomposed +by Leyden jar discharges a mixture of oxygen +and hydrogen appeared at each electrode, and suggested +that possibly the discharge was oscillatory in character.</p> + +<p>In 1853 the subject was discussed mathematically +by Thomson, in a paper which was to prove fruitful +in our own time in a manner then little anticipated. +The jar is given, let us say, with the interior coating +charged positively, and the exterior coating charged +negatively. A coil or helix of wire has its ends +connected to the two coatings, and a current immediately +begins in the wire, and gradually (not slowly) +increases in strength. Accompanying the creation of +the current is the production of a magnetic field, that<span class='pagenum'><a name="Page_182" id="Page_182">182</a></span> +is, the surrounding space is made the seat of magnetic +action. The magnetic field, as we shall see from +another investigation of Thomson's, almost certainly +involves motion in or of a medium—the ether—filling +the space where the magnetic action is found to exist. +The charge of the jar consists of a state of intense and +peculiar strain in the glass plate between the coatings. +When the plates are connected by the coil, this state +of strain breaks down and motion in the medium +ensues, not merely between the plates, but also in the +surrounding space—in fact, in the whole field. This +motion—which is not to be confused with bodily displacement +of finite parts of the medium—is opposed +by something akin to inertia of the medium (the +property that confers energy on matter when in +motion), so that when the motion is started it persists, +until it is finally wiped out by resistance of the nature +of friction. The inertia here referred to depends on +the mode in which the coil is wound, or whether it +contains or not an iron core.</p> + +<p>If the work done in charging a Leyden jar or electric +condenser, by bringing the charge to the condenser in +successive small portions, is considered, it is at once +clear that it must be proportional to the square of the +whole quantity of electricity brought up. For whatever +the charge may be, let it be brought up from a +great distance in a large number <i>N</i> of equal instalments. +The larger the whole amount the larger must +each instalment be, and therefore the greater the +amount accumulated on the condenser when any +given number of instalments have been deposited. +But the greater any charge that is being brought up, +and also the greater the charge that has already arrived,<span class='pagenum'><a name="Page_183" id="Page_183">183</a></span> +the greater is the repulsion that must be overcome in +bringing up that instalment, in simple proportion in +each case, and therefore the greater the work done. +Thus the whole work done in bringing up the charge +must be proportional to <i>Q</i><sup>2</sup>. We suppose it to be +½<i>Q</i><sup>2</sup> ⁄ <i>C</i>, where <i>C</i> is a constant depending on the +condenser and called its capacity.</p> + +<p>The idea of the charge as a quantity of some kind of +matter, brought up and placed on the insulated plate +of the condenser, has only a correspondence to the +fact, which is that the medium between the plates is +the seat, when the condenser is charged, of a store of +energy, which can only be made available by connecting +the plates of the condenser by a wire or other +conductor. The charge is only a surface aspect of the +state of the medium, apparently a state of strain, to +which the energy belongs.</p> + +<p>When a wire is used to connect the plates the state +of strain disappears; the energy comes out from the +medium between the plates by motion sideways of the +tubes of strain (so that the insulating medium is under +longitudinal tension and lateral pressure) which, according +to Faraday's conception of lines of electric force +connecting the charge on a body with the opposite +charges on other bodies, run from plate to plate, when +the condenser is in equilibrium in the changed state. +These tubes move out with their ends on the wire, +carrying the energy with them, and the ends run +towards one another along the wire; the tube shortens +in the process, and energy is lost in the wire. The +ends of a tube thus moving represent portions of the +charges which were on the plates, and the oppositely-directed +motions of the opposite charges represent a<span class='pagenum'><a name="Page_184" id="Page_184">184</a></span> +current along the wire from one conductor to the +other. The motion of the tubes is accompanied by +the development of a magnetic field, the lines of force +of which are endless, and the direction of which at +every point is perpendicular at once to the length of +the tube and to the direction in which it is there +moving. In certain circumstances the tube, by the +time its ends have met, will have wholly disappeared in +the wire, and the whole energy will have gone to heat +the wire: in other circumstances the ends will meet +before the tube has disappeared, the ends will cross, +and the tube will be carried back to the condenser and +reinserted in the opposite direction. At a certain +time this will have happened to all the tubes, though +they will have lost some of their energy in the process; +and the condenser will again be charged, though in the +opposite way to that in which it was at first. Then +the tubes will move out again, and the same process +will be repeated: once more the condenser will be +charged, but in the same direction as at first, and once +more with a certain loss of energy. Again the process +of discharge and charge will take place, and so on, +again and again, until the whole energy has disappeared. +This process represents, according to the modern theory +of the flow of energy in the electromagnetic field, +with more or less accuracy, what takes place in the +oscillatory discharge of a condenser.</p> + +<p>The motion of the tubes with their ends on the wire +represents a certain amount of energy, commonly regarded +as kinetic, and styled electrokinetic energy. If +<i>c</i> denote the current, that is, the rate, − <i>dQ</i>/<i>dt</i>, at +which the charge of the condenser is being changed, +and <i>L</i> a quantity called self-inductance, depending<span class='pagenum'><a name="Page_185" id="Page_185">185</a></span> +mainly on the arrangement of the connecting wire—whether +it is wound in a coil or helix, with or without +an iron core, or not—the electrokinetic energy will be +½<i>Lc</i><sup>2</sup>. This is analogous to the kinetic energy ½<i>mv</i><sup>2</sup> +of a body (say a pendulum bob) of mass <i>m</i> and velocity +<i>v</i>, so that <i>L</i> represents a quantity for the conducting +arrangement analogous to inertia, and <i>c</i> is the analogue +of the velocity of the body. The whole energy at +any instant is thus</p> + +<div class="center"> +½<i>Q</i><sup>2</sup> ⁄ <i>C</i> + ½<i>Lc</i><sup>2</sup>, or ½<i>Q</i><sup>2</sup> ⁄ <i>C</i> + ½<i>L</i> (<i>dQ</i> ⁄ <i>dt</i>)<sup>2</sup>.<br /> +</div> + +<p>The loss of energy due to heating of the conducting +connection is not completely understood, though its +quantitative laws have been quite fully ascertained and +expressed in terms of magnitudes that are capable of +measurement. It was found by Joule to be proportional +to the second power, or square, of the current, +and to a quantity <i>R</i> depending on the conductor, and +called its resistance. The generation of heat in the +conductor seems to be due to some kind of frictional +action of particles of the conductor set up by the penetration +of the Faraday tubes into it. A conductor is +unable to bear any tangential action exerted upon it by +Faraday tubes, which, however, when they exist, begin +and end at material particles, except when they are +endless, as they may be in the radiation of energy. +When the Faraday tubes are moving with any ordinary +speed they are not at their ends perpendicular to the +conducting surface from which they start or at which +they terminate, but are there more or less inclined to +the surface, and consequently there is tangential action +which appears to displace the particles (not merely +at the surface, unless the alternation is very rapid) +<span class='pagenum'><a name="Page_186" id="Page_186">186</a></span> +relatively to one another and so cause frictional +generation of heat.</p> + +<p>The time rate of generation of heat is thus <i>Rc</i><sup>2</sup>, or +<i>R</i> (<i>dQ</i> ⁄ <i>dt</i>)<sup>2</sup>, when the units in which <i>R</i> and <i>c</i> are +expressed are such as to make this quantity a rate of +doing work in the true dynamical sense. This is the +rate at which the sum of energy already found is being +diminished, and so the equation</p> + +<div class="center"><img class="floatInsert30" src="images/f186a.png" alt="" title="" /> +</div> + +<p>holds, or leaving out the common factor <i>dQ</i> ⁄ <i>dt</i>, the +equation</p> + +<div class="center"><img class="floatInsert25" src="images/f186b.png" alt="" title="" /> +</div> + +<p>This last equation was established by Thomson, and +is precisely that which would be obtained for a pendulum +bob of mass <i>L</i>, pulled back towards the position +of equilibrium with a force <i>Q</i> ⁄ <i>C</i>, where <i>Q</i> is the displacement +from the middle position, and having its +motion damped out by resisting force of amount <i>R</i> per +unit of the velocity.</p> + +<p>It is more instructive perhaps to take the oscillatory +motion of a spiral spring hung vertically with a weight +on its lower end, as that which has a differential +equation equivalent to the equation just found. When +the stretch is of a certain amount, there is equilibrium—the +action of the spring just balances the weight,—and +if the spring be stretched further there will be a +balance of pull developed tending to bring the system +back towards the equilibrium position. If left to itself +the system gets into motion, which, if the resistance is<span class='pagenum'><a name="Page_187" id="Page_187">187</a></span> +not too great, is added to until the equilibrium position +is reached; and the motion, which is continued by the +inertia of the mass, only begins to fall off as that +position is passed, and the pull of the spring becomes +insufficient to balance the weight. Thus the mass +oscillates about the position of equilibrium, and the +oscillations are successively smaller and smaller in +extent, and die out as their energy is expended finally +in doing work against friction.</p> + +<p>If the resisting force for finite motion is very +great, as for example when the vibrating mass of the +pendulum or spring is immersed in a very viscous fluid, +like treacle, oscillation will not take place at all. After +displacement the mass will move at first fairly quickly, +then more and more slowly back to the position +of equilibrium, which it will, strictly speaking, only +exactly reach after an infinite time. The resisting +force is here indefinitely small for an indefinitely small +speed, but it becomes so great when any motion ensues, +that as the restoring force falls off with the displacement, +no work is finally done by it, except to move the +body through the resisting medium.</p> + +<p>The differential equation is applicable to the spring +if <i>Q</i> is again taken as displacement from the equilibrium +position, <i>L</i> as the inertia of the vibrating body, 1 ⁄ <i>C</i> +as the pull exerted by the spring per unit of its +extension (that is, the stiffness of the spring), and <i>R</i> has +the same meaning as before.</p> + +<p>In this case of motion, as well as in that of the +pendulum, energy is carried off by the production of +waves in the medium in which the vibrator is immersed. +These are propagated out from the vibrator as their +source, but no account of them is taken in the differential<span class='pagenum'><a name="Page_188" id="Page_188">188</a></span> +equation, which in that respect is imperfect. There +is no difficulty, only the addition of a little complication, +in supplying the omission.</p> + +<p>The formation of such waves by the spiral spring +vibrator can be well shown by immersing the vibrating +body in a trough of water, and the much greater rate +of damping out of the motion in that case can then be +compared with the rate of damping in air.</p> + +<p>It has been indicated that the differential equation +does not represent oscillatory motion if the value of +<i>R</i> is too great. The exact condition depends on the +roots of the quadratic equation <i>Lx</i><sup>2</sup> + <i>Rx</i> + 1 ⁄ <i>C</i> = 0, +obtained by writing 1 for <i>Q</i>, and <i>x</i> for <i>d</i> ⁄ <i>dt</i>, and +then treating <i>x</i> as a quantity. These roots are +− <i>R</i> ⁄ 2<i>L</i> ± √(<i>R</i><sup>2</sup> ⁄ 4<i>L</i><sup>2</sup> − 1 ⁄ <i>CL</i>), and are therefore real +or imaginary according as 4<i>L</i> ⁄ <i>C</i> is less or greater +than <i>R</i><sup>2</sup>. If the roots are real, that is, if <i>R</i><sup>2</sup> be greater +than 4<i>L</i> ⁄ <i>C</i>, the discharge will not be oscillatory; the +Faraday tubes referred to above will be absorbed in +the wire without any return to the condenser. The +corresponding result happens with the vibrator when +<i>R</i> is sufficiently great, or <i>L</i> ⁄ <i>C</i> sufficiently small (a +weak spring and a small mass, or both), to enable the +condition to be fulfilled.</p> + +<p>If, however, the roots of the quadratic are imaginary, +that is, if 4<i>L</i> ⁄ <i>C</i> be greater than <i>R</i><sup>2</sup> (a condition which +will be fulfilled in the spring analogue, by making the +spring sufficiently stiff and the mass large enough to +prevent the friction from controlling the motion) the +motion is one in which <i>Q</i> disappears by oscillations +about zero, of continually diminishing amplitude. A +complete discussion gives for the period of oscillation +<span class='pagenum'><a name="Page_189" id="Page_189">189</a></span> +4π<i>L</i> ⁄ √(4<i>L</i> ⁄ <i>C</i> − <i>R</i><sup>2</sup>), or if <i>R</i> be comparatively small, +2π√(<i>LC</i>). The charge <i>Q</i> falls off by the fraction +<i>e</i><sup> − <i>RT</i>⁄2<i>L</i></sup> (where <i>e</i> is the number 2.71828...) in each +period <i>T</i>, and so gradually disappears.</p> + +<p>Thus electric oscillations are produced, that is to +say, the charged state of the condenser subsides by +oscillations, in which the charged state undergoes +successive reversals, with dissipation of energy in the +wire; and both the period and the rate of dissipation +can be calculated if <i>L</i>, <i>C</i>, and <i>R</i> are known, or can +be found, for the system. These quantities can be +calculated and adjusted in certain definite cases, and as +the electric oscillations can be experimentally observed, +the theory can be verified. This has been done by +various experimenters.</p> + +<p>Returning to the pendulum illustration, it will be +seen that the pendulum held deflected is analogous to +the charged jar, letting the pendulum go corresponds +to connecting the discharging coil to the coatings, the +motion of the pendulum is the analogue of that motion +of the medium in which consists the magnetic field, +the friction of the air answers to the resistance of the +wire which finally damps out the current. The inertia +or mass of the bob is the analogue of what Thomson +called the electromagnetic inertia of the coil and +connections; what is now generally called the self-inductance +of the conducting system. The component +of gravity along the path towards the lowest point, +answers to the reciprocal, 1 ⁄ <i>C</i>, of the capacity of the +condenser.</p> + +<p>It appears from the analogy that just as the oscillations +of a pendulum can be prevented by immersing +the bob in a more resisting medium, such as treacle or +oil, so that when released the pendulum slips down to<span class='pagenum'><a name="Page_190" id="Page_190">190</a></span> +the vertical without passing it, so by properly proportioning +the resistance in the circuit to the electromagnetic +inertia of the coil, oscillatory discharge of the +Leyden jar may also be rendered impossible.</p> + +<p>All this was worked out in an exceedingly instructive +manner in Thomson's paper; the account of the +matter by the motion of Faraday tubes is more recent, +and is valuable as suggesting how the inertia effect of +the coil arises. The analogy of the pendulum is a +true one, and enables the facts to be described; but it +is to be remembered that it becomes evident only as a +consequence of the mathematical treatment of the +electrical problem. The paper was of great importance +for the investigation of the electric waves used in +wireless telegraphy in our own time. It enabled the +period of oscillation of different systems to be calculated, +and so the rates of exciters and receivers of +electric waves to be found. For such vibrators are +really Leyden jars, or condensers, caused to discharge +in an oscillatory manner.</p> + +<p>This application was not foreseen by Thomson, and, +indeed, could hardly be, as the idea of electric waves in +an insulating medium came a good deal later in the +work of Maxwell. Yet the analogy of the pendulum, +if it had then been examined, might have suggested +such waves. As the bob oscillates backwards and +forwards the air in which it is immersed is periodically +disturbed, and waves radiate outwards from it through +the surrounding atmosphere. The energy of these +waves is exceedingly small, otherwise, as pointed out +above, a term would have to be included in the theory +of the resisted motion of the pendulum to account for +this energy of radiation. So likewise when the electric<span class='pagenum'><a name="Page_191" id="Page_191">191</a></span> +vibrations proceed, and the insulating medium is the +seat of a periodically varying magnetic field, electromagnetic +waves are propagated outwards through the +surrounding medium—the ether—and the energy +carried away by the waves is derived from the initial +energy of the charged condenser. In strictness also +Thomson's theory of electric oscillations requires an +addition to account for the energy lost by radiation. +This is wanting, and the whole decay of the amount +of energy present at the oscillator is put down to the +action of resistance—that is, to something of the nature +of frictional retardation. Notwithstanding this defect +of the theory, which is after all not so serious as certain +difficulties of exact calculation of the self-inductance +of the discharging conductor, the periods of vibrators +can be very accurately found. When these are known +it is only necessary to measure the length of an +electrical wave to find its velocity of propagation. +When electromagnetic waves were discovered experimentally +in 1888 by Heinrich Hertz, it was thus that +he was able to demonstrate that they travelled with the +velocity of light.</p> + +<p>Thomson suggested that double, triple and quadruple +flashes of lightning might be successive flashes of an +oscillatory discharge. He also pointed out that if a +spark-gap were included in a properly arranged condenser +and discharging wire, it might be possible, by means of +Wheatstone's revolving mirror, to see the sparks produced +in the successive oscillations, as "points or short +lines of light separated by dark intervals, instead of a +single point of light, or of an unbroken line of light, +as it would be if the discharge were instantaneous, or +were continuous, or of appreciable duration."</p> + +<p><span class='pagenum'><a name="Page_192" id="Page_192">192</a></span>This anticipation was verified by experiments made +by Feddersen, and published in 1859 (<i>Pogg. Ann.</i>, +108, 1859). The subject was also investigated in +Helmholtz's laboratory at Berlin, by N. Schiller, who, +determining the period for condensers with different +substances between the plates, was able to deduce the +inductive capacities of these substances (<i>Pogg. Ann.</i>, +152, 1874). [The specific inductive capacity of an +insulator is the ratio of the capacity of a condenser +with the substance between the plates to the capacity +of an exactly similar condenser with air between the +plates.]</p> + +<p>The particular case of non-oscillatory discharge +obtained by supposing <i>C</i> and <i>Q</i> both infinitely great +and to have a finite ratio <i>V</i> (which will be the potential, +p. <a href="#Page_34">34</a>, of the charged plate), is considered in the paper. +The discharging conductor is thus subjected to a +difference of potential suddenly applied and maintained +at one end, while the other end is kept at potential +zero. The solution of the differential equation for +this case will show how the current rises from zero in +the wire to its final steady value. If <i>c</i> be put as before +for the current − <i>dQ</i> ⁄ <i>dt</i>, and the constant value <i>V</i> for +<i>Q</i> ⁄ <i>C</i>, the equation is</p> + +<div class="center"><img class="floatInsert22" src="images/f192a.png" alt="" title="" /> +</div> + +<p>which gives, since <i>c</i> = 0 when <i>t</i> = 0,</p> + +<div class="center"><img class="floatInsert22" src="images/f192b.png" alt="" title="" /> +</div> + +<p>Thus, when an infinite time has elapsed the current +has become <i>V</i> ⁄ <i>R</i>, the steady value.</p> + +<p><span class='pagenum'><a name="Page_193" id="Page_193">193</a></span>Thomson concludes by showing how, by measuring +the non-oscillatory discharge of a condenser (the +capacity of which can be calculated) by means of an +electrodynamometer and an ordinary galvanometer +arranged in series, what W. Weber called the duration +of the discharging current may be determined. From +this Thomson deduced a value for the ratio of the +electromagnetic unit of electricity to the electrostatic +unit, and indicated methods of determining this ratio +experimentally. This ratio is of fundamental importance +in electromagnetic theory, and is essentially of the +nature of a speed. According to Maxwell it is the +speed of propagation of electromagnetic waves in an +insulating medium for which the units are defined. +It was first determined in the Glasgow laboratory by +Mr. Dugald McKichan, and has been determined +many times since. It is practically identical with the +speed of light as ascertained by the best experiments.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_194" id="Page_194">194</a></span></p> + +<h3>CHAPTER XI</h3> + +<h4>THOMSON AND TAIT'S 'NATURAL PHILOSOPHY'—GYROSTATIC +ACTION—'ELECTROSTATICS AND MAGNETISM'</h4> + +<h3><span class="smcap">The 'Natural Philosophy'</span></h3> + +<p><span class="smcap">Professor Tait</span> was appointed to the Chair of +Natural Philosophy in the University of Edinburgh +in 1860, and came almost immediately into frequent +contact with Thomson. Both were Peterhouse +men, trained by the same private tutor—William +Hopkins—both were enthusiastic investigators in +mathematical as well as in experimental physics, they +taught in the sister universities of Edinburgh and +Glasgow, and had much the same kind of classes to +deal with and the same educational problems to solve. +Tait was an Edinburgh man—an old school-fellow of +Clerk Maxwell at the Edinburgh Academy—and had +therefore been exposed to that contact, in play and in +work, with compeers of like age and capabilities, which +is one of the best preparations for the larger school and +more serious struggles of life. Thomson's early education, +under his father's anxious care, had no doubt +certain advantages, and his early entrance into college +classes gave him to a great extent that intercourse +with others for which such advantages are never +complete compensation. The two men had much<span class='pagenum'><a name="Page_195" id="Page_195">195</a></span> +community of thought and experience, and the literary +partnership into which they entered was hailed as one +likely to do much for the progress of science.</p> + +<p>In some ways, however, Thomson and Tait were +very different personalities. Thomson troubled himself +little with metaphysical subtleties, his conceptions were +like those of Newton, absolutely clear so far as they +went; he never, in his teaching at least, showed any +disposition to discuss the "foundations of dynamics," +or the conception of motion in a straight line. These +were taken for granted like the fundamental ideas in a +book on geometry; and the student was left to do +what every true dynamical student must do for himself +sooner or later—to compare the abstractions of +dynamics with the products of his experience in the +world of matter and force. Perhaps a little guidance +now and then in the difficulties about conceptions, +which beset every beginner, might not have been +amiss: but Thomson was so intent on the concrete +example in hand—pendulum or gyrostat, or what not—that +he left each man to form or correct his own +ideas by the lessons which such examples afford to +every one who carefully examines them.</p> + +<p>Tait, on the other hand, though he continually +denounced metaphysical discussion, was in reality much +more metaphysical than Thomson, and seemed to take +pleasure in the somewhat transcendental arguments +with regard to matters of analysis which were put +forward, especially in the <i>Elements of Quaternions</i>, by +Sir William Rowan Hamilton, of Dublin, a master +whom he much revered. But there is metaphysics +and metaphysics! and the pronouncements +of professed metaphysicians were often characterised<span class='pagenum'><a name="Page_196" id="Page_196">196</a></span> +as non-scientific and fruitless, which no doubt they +were from the physical point of view.</p> + +<p>Then Tait was strongly convinced of the importance +for physics of the quaternion analysis: Thomson +was not, to say the least; and this was probably the +main reason why the vectorial treatment of displacement, +velocities, and other directed quantities, has no +place in the joint writings of the two Scottish professors. +In controversy Tait was a formidable antagonist: +when war was declared he gave no quarter and +asked for none, though he never fought an unchivalric +battle. He admired foreign investigators—and especially +von Helmholtz—but he was always ready to put +on his armour and place lance in rest for the cause +of British science. Thomson was much less of a combatant, +though he also could bravely splinter a spear +with an opponent on occasion, as in the memorable +discussion with Huxley on the Age of the Earth.</p> + +<p>Tait's professorial lectures were always models of +clear and logical arrangement. Every statement bore +on the business in hand; the experimental illustrations, +always carefully prepared beforehand, were called for +at the proper time and were invariably successful. +With Thomson it was otherwise: his digressions, +though sometimes inspired and inspiring, were fatal to +the success of the utmost efforts of his assistants to +make his lectures successful systematic expositions of +the facts and principles of elementary physics.</p> + +<p>As has been stated in Chapter IV, two books were +announced in 1863 as in course of preparation for the +ensuing session of College. These were not published +until 1867 and 1873; the first issued was the famous +<i>Treatise on Natural Philosophy</i>, the second was entitled<span class='pagenum'><a name="Page_197" id="Page_197">197</a></span> +<i>Elements of Natural Philosophy</i>, and consisted in the +main of part of the non-mathematical or large type +portions of the <i>Treatise</i>. The scheme of the latter +was that of an articulated skeleton of statements of +principles and results, printed in ordinary type, with +the mathematical deductions and proofs in smaller +type. As was to be expected, the <i>Elements</i>, to a student +whose mathematical reading was wide enough to +tackle the <i>Treatise</i>, was the more difficult book of +the two to completely master. But the continued +large print narrative, as it may be called, is extremely +valuable. It is a memorial of a habit of mind which +was characteristic of both authors. They kept before +them always the idea or thing rather than its symbol; +and thus the edifice which they built up seemed never +obscured by the scaffolding and machinery used in its +erection. And as far as possible in processes of deduction +the ideas are emphasised throughout; there is no +mere putting in at one end and taking out at the +other; the result is examined and described at every +stage. As in all else of Thomson's work, physical +interpretation is kept in view at every step, and made +available for correction and avoidance of errors, and +the suggestion of new inquiries.</p> + +<p>The book as it stands consists of "Division I, +Preliminary" and part of "Division II, Abstract +Dynamics." Division I includes the chapter on Kinematics +already referred to, a chapter on Dynamical +Laws and Principles, chapters on Experience and +Measures and Instruments. Division II is represented +only by Chapter V, Introductory; Chapter VI, Statics +of a Particle and Attractions; and Chapter VII, Statics +of Solids and Fluids. Thus Abstract Dynamics is<span class='pagenum'><a name="Page_198" id="Page_198">198</a></span> +without the more complete treatment of Kinetics to +which, as well as to Statics, the discussion of Dynamical +Laws and Principles was intended to be an introduction. +But to a considerable extent, as we shall see, Kinetics +is treated in this introductory chapter: indeed, the discussion +of the general theorems of dynamics and their +applications to kinetics is remarkably complete.</p> + +<p>In Volume II it was intended to include chapters on +the kinetics of a particle and of solid and fluid bodies, +on the vibrations of solid bodies, and on wave-motion +in general. It was expected also to contain a chapter +much referred to in Volume I, on "Properties of +Matter." That the work was not completed is a +matter of keen regret to all physicists, regret, however, +now tempered by the fact that many of the subjects of +the unfulfilled programme are represented by such +works as Lord Rayleigh's <i>Theory of Sound</i>, Lamb's +<i>Hydrodynamics</i>, and Routh's <i>Dynamics of a System of +Rigid Bodies</i>. But all deeply lament the loss of the +"Properties of Matter." No one can ever write it as +Thomson would have written it. His students obtained +in his lectures glimpses of the things it might have +contained, and it was most eagerly looked for. If that +chapter only had been given, the loss caused by the +discontinuance of the book would not have been so +irreparable.</p> + +<p>The first edition of the book was published by the +Clarendon Press, Oxford. It was printed by Messrs. +Constable, of Edinburgh, and is a beautiful specimen of +mathematical typography. In some ways the first +edition is exceedingly interesting, for it is not too much +to say that its issue had an influence on dynamical +science, and its exposition in this country, only second<span class='pagenum'><a name="Page_199" id="Page_199">199</a></span> +to that due to Newton's <i>Principia</i>. Three other works, +perhaps, have had the same degree and kind of influence +on mathematical thought—Laplace's <i>Mécanique Céleste</i>, +Lagrange's <i>Mécanique Analytique</i>, and Fourier's <i>Théorie +Analytique de la Chaleur</i>.</p> + +<p>The second edition was issued by the Cambridge +University Press as Parts I and II in 1878 and 1883. +Various younger mathematicians now of eminence—Professor +Chrystal, of Edinburgh, and Professor +Burnside, of Greenwich, may be mentioned—read the +proofs, and it is on the whole remarkably free from +typographical and other errors. With the issue of +Part II, the continuation was definitely abandoned.</p> + +<p>In the second edition many topics are more fully +discussed, and the contents include a very valuable +account of cycloidal motion (or oscillatory motion, as +it is more usually called), and of a revised version of the +chapter on Statics which forms the concluding portion +of the book, and which discusses some of the great +problems of terrestrial and cosmical physics.</p> + +<p>Various speculations have been indulged in, from +time to time, as to the respective parts contributed to +the work by the two authors, but these are generally +very wide of the mark. The mode of composition of +the sections on cycloidal (oscillatory) motion gives some +idea of Thomson's method of working. His proofs +(of "T and T-<i>dash</i>" as the authors called the book) +were carried with him by rail and steamer, and he +worked incessantly (without, however, altogether withdrawing +his attention from what was going on around +him!) at corrections and additions. He corrected +heavily on the proofs, and then overflowed into +additional manuscript. Thus, when he came to the<span class='pagenum'><a name="Page_200" id="Page_200">200</a></span> +short original § 343, he greatly extended that in the +first instance, and proceeded from section to section +until additions numbered from § 343<i>a</i> to § 343<i>p</i>, +amounting in all to some ten pages of small print, +had been interpolated. Similarly § 345 was extended +by the addition of §§ 345 (i) to 345 (xxviii), mainly +on gyrostatic domination. The method had the disadvantage +of interrupting the printers and keeping type +long standing, but the matter was often all the more +inspiring through having been produced under pressure +from the printing office. Indeed, much was no doubt +written in this way which, to the great loss of dynamical +science, would otherwise never have been written at all.</p> + +<p>The kinematical discussion begins with the consideration +of motion along a continuous line, curved +or straight. This naturally suggests the ideas of +curvature and tortuosity, which are fully dealt with +mathematically, before the notion of velocity is introduced. +When that is done, the directional quality of +velocity is not so much insisted on as is now the case: +for example, a point is spoken of as moving in a +curve with a uniform velocity; and of course in the +language of the present time, which has been rendered +more precise by vector ideas, if not by vector-analysis, +the velocity of a point which is continually changing +the direction of its motion, cannot be uniform. The +same remark may be made regarding the treatment of +acceleration: in both cases the reference of the quantity +to three Cartesian axes is immediate, and the changes +of the components, thus fixed in direction, are alone +considered.</p> + +<p>There can be no doubt that greater clearness is +obtained by the process afterwards insisted on by Tait,<span class='pagenum'><a name="Page_201" id="Page_201">201</a></span> +of considering by a hodographic diagram the changes of +velocity in successive intervals of time, and from these +discovering the direction and magnitude of the rate +of change at each instant. This method is indeed +indicated at § 37, but no diagram is given, and the +properties of the hodograph are investigated by means +of Cartesians. The subject is, however, treated in the +<i>Elements</i> by the method here indicated.</p> + +<p>Remarkable features of this chapter are the very +complete discussion of simple harmonic or vibratory +motion, the sections on rotation, and the geometry of +rolling and precessional motion, and on the curvature +of surfaces as investigated by kinematical methods. +A remark made in § 96 should be borne in mind by +all who essay to solve gyrostatic problems. It is that +just as acceleration, which is always at right angles to +the motion of a point, produces a change in the direction +of the motion but none in the <i>speed</i> of the point (it +does influence the <i>velocity</i>), so an action, tending always +to produce rotation about an axis at right angles to +that about which a rigid body is already rotating, will +change the direction of the axis about which the body +revolves, but will produce no change in the rate of +turning.<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">20</a></p> + +<p><span class='pagenum'><a name="Page_202" id="Page_202">202</a></span>A very full and clear account of the analysis of +strains is given in this chapter, in preparation for the +treatment of elasticity which comes later in the book; +and a long appendix is added on Spherical Harmonics, +which are defined as homogeneous functions of the +coordinates which satisfy the differential equation of +the distribution of temperature in a medium in which +there is steady flow of heat, or of distribution of +potential in an electrical field. This appendix is +within its scope one of the most masterly discussions +of this subject ever written, though, from the point +of view of rigidity of proof, required by modern +function-theory, it may be open to objection.</p> + +<p>In the next chapter, which is entitled "Dynamical +Laws and Principles," the authors at the outset declare +their intention of following the <i>Principia</i> closely in the +discussion of the general foundations of the subject. +Accordingly, after some definitions the laws of motion +are stated, and the opportunity is taken to adopt and +enforce the Gaussian system of absolute units for +dynamical quantities. As has been indicated above, +the various difficulties more or less metaphysical which +must occur to every thoughtful student in considering +Newton's laws of motion are not discussed, and probably +such a discussion was beyond the scheme which the +authors had in view. But metaphysics is not altogether +excluded. It is stated that "matter has an innate +power of resisting external influences, so that every +body, as far as it can, remains at rest, or moves +uniformly in a straight line," and it is stated that this +property—inertia—is proportional to the quantity of +matter in the body. This statement is criticised by +Maxwell in his review of the <i>Natural Philosophy</i> in<span class='pagenum'><a name="Page_203" id="Page_203">203</a></span> +<i>Nature</i> in 1879 (one of the last papers that Maxwell +wrote). He asks, "Is it a fact that 'matter has any +power, either innate or acquired, of resisting external +influences'? Does not every force which acts on a +body always produce that change in the motion of the +body by which its value, as a force, is reckoned? Is +a cup of tea to be accused of resisting the sweetening +influence of sugar, because it persistently refuses to +turn sweet unless the sugar is put into it?"</p> + +<p>This innate power of resisting is merely the <i>materiæ +vis insita</i> of Newton's "Definitio III," given in the +<i>Principia</i>, and the statement to which Maxwell objects +is only a free translation of that definition. Moreover, +when a body is drawn or pushed by other bodies, it +reacts on those bodies with an equal force, and this +reaction is just as real as the action: its existence is +due to the inertia of the body. The definition, from +one point of view, is only a statement of the fact that +the acceleration produced in a body in certain circumstances +depends upon the body itself, as well as on the +other bodies concerned, but from another it may be +regarded as accounting for the reaction. The mass +or inertia of the body is only such a number that, for +different bodies in the same circumstances as to the +action of other bodies in giving them acceleration, the +product of the mass and the acceleration is the same +for all. It is, however, a very important property of +the body, for it is one factor of the quantum of kinetic +energy which the body contributes to the energy of +the system, in consequence of its motion relatively to +the chosen axes of reference, which are taken as at +rest.</p> + +<p>The relativity of motion is not emphasised so greatly<span class='pagenum'><a name="Page_204" id="Page_204">204</a></span> +in the <i>Natural Philosophy</i> as in some more modern +treatises, but it is not overlooked; and whatever may +be the view taken as to the importance of dwelling on +such considerations in a treatise on dynamics, there +can be no doubt that the return to Newton was on +the whole a salutary change of the manner of teaching +the subject.</p> + +<p>The treatment of force in the first and second laws +of motion is frankly causal. Force is there the <i>cause</i> +of rate of change of momentum; and this view Professor +Tait in his own writings has always combated, +it must be admitted, in a very cogent manner. According +to him, force is merely rate of change of momentum. +Hence the forces in equations of motion are only +expressions, the values of which as rates of change of +momentum, are to be made explicit by the solution of +such equations in terms of known quantities. And +there does not seem to be any logical escape from +this conclusion, though, except as a way of speaking, +the reference to cause disappears.</p> + +<p>The discussion of the third law of motion is particularly +valuable, for, as is well known, attention was +therein called to the fact that in the last sentences of +the <i>Scholium</i> which Newton appended to his remarks +on the third law, the rates of working of the acting +and reacting forces between the bodies are equal and +opposite. Thus the whole work done in any time +by the parts of a system on one another is zero, +and the doctrine of conservation of energy is virtually +contained in Newton's statement. The only point in +which the theory was not complete so far as ordinary +dynamical actions are concerned, was in regard to +work done against friction, for which, when heat was<span class='pagenum'><a name="Page_205" id="Page_205">205</a></span> +left out of account, there was no visible equivalent. +Newton's statement of the equality of what Thomson +and Tait called "activity" and "counter-activity" is, +however, perfectly absolute. In the completion of the +theory of energy on the side of the conversion of heat +into work, Thomson, as we have seen, took a very +prominent part.</p> + +<p>After the introduction of the dynamical laws the +most interesting part of this chapter is the elaborate +discussion which it contains of the Lagrangian equations +of motion, of the principle of Least Action, with the +large number of extremely important applications of +these theories. The originality and suggestiveness of +this part of the book, taken alone, would entitle it to +rank with the great classics—the <i>Mécanique Céleste</i>, +the <i>Mécanique Analytique</i>, and the memoirs of Jacobi +and Hamilton—all of which were an outcome of the +<i>Principia</i>, and from which, with the <i>Principia</i>, the +authors of the <i>Natural Philosophy</i> drew their inspiration.</p> + +<p>It is perhaps the case, as Professor Tait himself +suggested, that no one has yet arisen who can bend to +the fullest extent the bow which Hamilton fashioned; +but when this Ulysses appears it will be found that +his strength and skill have been nurtured by the study +of the <i>Natural Philosophy</i>. Lagrange's equations are +now, thanks to the physical reality which the expositions +and examples of Thomson and Tait have given to +generalised forces, coordinates, and velocities, applied +to all kinds of systems which formerly seemed to be +outside the range of dynamical treatment. As Maxwell +put it, "The credit of breaking up the monopoly of +the great masters of the spell, and making all their +charms familiar in our ears as household words, belongs<span class='pagenum'><a name="Page_206" id="Page_206">206</a></span> +in great measure to Thomson and Tait. The two +northern wizards were the first who, without compunction +or dread, uttered in their mother tongue the +true and proper names of those dynamical concepts, +which the magicians of old were wont to invoke only +by the aid of muttered symbols and inarticulate equations. +And now the feeblest among us can repeat the +words of power, and take part in dynamical discussions +which a few years ago we should have left to our +betters."</p> + +<p>A very remarkable feature in this discussion is the +use made of the idea of "ignoration of coordinates." +The variables made use of in the Lagrangian equations +must be such as to enable the positions of the parts of +the system which determine the motion to be expressed +for any instant of time. These parts, by their displacements, +control those of the other parts, through +the connections of the system. They are called the +independent coordinates, and sometimes the "degrees +of freedom," of the system. Into the expressions of the +kinetic and potential energies, from which by a formal +process the equations of motion, as many in number +as there are degrees of freedom, are derived, the value +of these variables and of the corresponding velocities +enter in the general case. But in certain cases some +of the variables are represented by the corresponding +velocities only, and the variables themselves do not +appear in the equations of motion. For example, when +fly-wheels form part of the system, and are connected +with the rest of the system only by their +bearings, the angle through which the wheel has +turned from any epoch of time is of no consequence, +the only thing which affects the energy of the system<span class='pagenum'><a name="Page_207" id="Page_207">207</a></span> +is the angular velocity or angular momentum of the +wheel. The system is said by Thomson and Tait in +such a case to be under gyrostatic domination. (See +"Gyrostatic Action," p. <a href="#Page_214">214</a> below.)</p> + +<p>Moreover, since the force which is the rate of +growth of the momentum corresponding to any coordinate +is numerically the rate of variation with that +coordinate of the difference of the kinetic and potential +energies, every force is zero for which the coordinate +does not appear; and therefore the corresponding momentum +is constant. But that momentum is expressed +by means of the values of other coordinates +which do appear and their velocities, with the velocities +for the absent coordinates; and as many equations are +furnished by the constant values of such momenta +as there are coordinates absent. The corresponding +velocities can be determined from these equations in +terms of the constant momenta and the coordinates +which appear and their velocities. The values so found, +substituted in the expressions for the kinetic and potential +energies, remove from these expressions every +reference to the absent coordinates. Then from the +new expression for the kinetic energy (in which a +function of the constant momenta now appears, and +is taken as an addition to the potential energy) the +equations of motion are formed for the coordinates +actually present, and these are sufficient to determine +the motion. The other coordinates are thus in a +certain sense ignored, and the method is called that of +"ignoration of coordinates."</p> + +<p>Theorems of action of great importance for a +general theory of optics conclude this chapter; but of +these it is impossible to give here any account, without<span class='pagenum'><a name="Page_208" id="Page_208">208</a></span> +a discussion of technicalities beyond the reading of +ordinary students of dynamics.</p> + +<p>In an Appendix to Part I an account is given of Continuous +Calculating Machines. Ordinary calculating +machines, such as the "arithmometer" of Thomas of +Colmar, carry out calculations and exhibit the result +as a row of figures. But the machines here described +are of a different character: they exhibit their results +by values of a continuously varying quantity. The +first is one for predicting the height of the tides for +future time, at any port for which data have been +already obtained regarding tidal heights, by means of +a self-registering tide-gauge. Two of these were +made according to the ideas set forth in this Appendix; +one is in the South Kensington Museum, the other is +at the National Physical Laboratory at Bushy House, +where it is used mainly for drawing on paper curves of +future tidal heights, for ports in the Indian Ocean. +From these curves tide-tables are compiled, and issued +for the use of mariners and others.</p> + +<p>Another machine described in this Appendix was +designed for the mechanical solution of simultaneous +linear equations. It is impossible to explain here the +interesting arrangement of six frames, carrying as many +pulleys, adjustable along slides (for the solution of +equations involving six unknown quantities), which +Thomson constructed, and which is now in the +Natural Philosophy Department at Glasgow. The +idea of arranging the first practical machine for this +number of variables, was that it might be used for the +calculation of the corrections on values already found +for the six elements of a comet or asteroid. The +machine was made, but some mechanical difficulties<span class='pagenum'><a name="Page_209" id="Page_209">209</a></span> +arose in applying it, and the experiments with it +were not at the time persevered with. Very possibly, +however, it may yet be brought into use.</p> + +<div class="figcenter" style="width: 550px; position: relative;"><a name="f14" id="f14"></a><img src="images/fig14.png" width="550" height="225" alt="Fig.14." title="" /> +<p class="caption"><span class="smcap">Fig. 14.</span></p></div> + +<p>But the most wonderful of these mechanical arrangements +is the machine for analysing the curves +drawn by a self-registering tide gauge, so as to exhibit +the constants of the harmonic curves, and thus enable +the prediction of tidal heights to be carried out either +by the tide-predicting machine, or by calculation. +One day in 1876, Thomson remarked to his brother, +James Thomson, then Professor of Engineering at +Glasgow, that all he required for the construction of a +tidal analyser was a form of integrating machine more +satisfactory for his purpose than the usual type of integrator +employed by surveyors and naval architects. +James Thomson at once replied that he had invented, +a long time before, what he called a disk-globe-cylinder-integrator. +This consisted of a brass disk, with its +plane inclined to the horizontal, which could be turned +about its axis by a wheel gearing in teeth on the edge +of the disk, and driven by the operator in a manner +which will presently appear. Parallel and close to the +disk, but not touching it, was placed a horizontal cylinder<span class='pagenum'><a name="Page_210" id="Page_210">210</a></span> +of brass, about 2 inches in diameter (called the registering +cylinder), and between the disk and this cylinder +was laid a metal ball about 2½ inches in diameter. +When the disk was kept at rest, and the ball was +rolled along between the cylinder and disk, the trace +of its rolling on the latter was a straight horizontal +line passing through the centre. Supposing then that the +point of contact of the ball with the disk was on one +side, at a distance from the centre, and that the disk +was then turned, the ball was by the friction between +it and the disk made to roll, and so to turn the +cylinder. The angular velocity of rolling, and therefore +the angular velocity of the cylinder, was proportional +to the speed of the part of the disk in contact +with it, that is, to <i>y</i>. It was also proportional to +the speed of turning of the disk.</p> + +<p>The mode by which this machine effects an integration +will now be evident. Imagine the area to be found +to lie between a curve and a straight datum line, drawn +on a band of paper. This is stretched on a large cylinder, +with the datum line round the cylinder. We call this +the paper-cylinder. The distances of the different points +of the curve from the datum line are values of <i>y</i>. A horizontal +bar parallel to the cylinder carries a fork at one end +and a projecting style at the other. The globe just +fits between the prongs of the fork, and when the bar +is moved in the direction of its length carries the ball +along the disk and cylinder. When the style at the +other end is on the datum line, the centre of the ball +is at the centre of the disk, and the turning of the disk +does not turn the cylinder. When the bar is displaced +in the line of its own length to bring the style from the +datum line to a point on the curve, the ball is displaced<span class='pagenum'><a name="Page_211" id="Page_211">211</a></span> +a distance <i>y</i>, and there is a corresponding turning of the +cylinder by the action of the ball. In the use of the +instrument the paper-cylinder is turned by the operator +while the style is kept on the curve, and the disk is +turned by the gearing already referred to, which is +driven by a shaft geared with that of the paper-cylinder. +Thus the displacement of the ball is always <i>y</i>, the +ordinate of the curve, and for any displacement <i>dx</i> +along the datum line, the registering cylinder is turned +through an angle proportional to <i>ydx</i>. Thus any finite +angle turned through is proportional to the integral of +<i>ydx</i> for the corresponding part of the curve: a scale +round one end of the registering cylinder gives that +angle. Thomson immediately perceived that this +extremely ingenious integrating machine was just +what he required for his purpose. The curve of +tidal heights drawn (on a reduced scale, of course) by a +tide-gauge, is really the resultant of a large number of +simple curves, represented by a series of harmonic +terms, the coefficients of which are certain integrals. +The problem is the evaluation of these integrals; and +the method usually employed is to obtain them by +measurement of ordinates of the curve and an elaborate +process of calculation. But one of them is simply the +integral area between the curve and the datum line +corresponding to the mean water level, and the others +are the integrals of quantities of the type <i>y</i> sin <i>nx</i> . <i>dx</i>, +where <i>y</i> is the ordinate of the curve, and <i>n</i> a number +inversely proportional to the period of the tidal constituent +represented by the term.</p> + +<p>All that was necessary, in order to give the integral +of a term <i>y</i> sin <i>nx</i> . <i>dx</i>, was to make the disk oscillate +about its axis as the paper-cylinder was turned through<span class='pagenum'><a name="Page_212" id="Page_212">212</a></span> +an angle proportional to <i>x</i>. Thus one disk, globe, and +cylinder was arranged exactly as has been described for +the integral of <i>ydx</i>, and with this as many others as +there were harmonic terms to be evaluated from the +curve were combined as follows. The disks were +placed all in one plane with their centres all on one +horizontal line, and the cylinders with their axes also in +line, and a single sliding bar, with a fork for each globe, +gave in each case the displacement <i>y</i> from the centre +of the disk.</p> + +<p>The requisite different speeds of oscillation were given +to the disks by shafts geared with the paper-cylinder, +by trains of wheels cut with the proper number of +teeth for the speed required.</p> + +<p>Thus the angles turned through by the registering +cylinders when a curve on the paper-cylinder was +passed under the style were proportional to the integrals +required, and it was only necessary to calibrate the +graduation of the scales of these cylinders by means +of known curves to obtain the integrals in proper +units.</p> + +<p>One of these machines, which analyses four harmonic +constituents, is in the Natural Philosophy Department +at Glasgow; a much larger machine, to analyse a +tidal curve containing five pairs of harmonic terms, or +eleven constituents in all, was made for the British +Association Committee on Tidal Observations, and is +probably now in the South Kensington Museum.</p> + +<p>But still more remarkable applications which +Thomson made of his brother's integrating machine +were to the mechanical integration of linear differential +equations, with variable coefficients, to the integration +of the general linear differential equation of any order,<span class='pagenum'><a name="Page_213" id="Page_213">213</a></span> +and, finally, to the integration of any differential +equation of any order.</p> + +<p>These applications were all made in a few days, +almost in a few hours, after James Thomson first +described the elementary machine, and papers containing +descriptions of the combinations required were at +once dictated by Thomson to his secretary, and +despatched for publication. Very possibly he had +thought out the applications to some extent before; +but it is unlikely that he had done so in detail. But, +even if it were so, the connection of a series of machines +by the single controlling bar, and the production of +the oscillations of the disks, all controlled, as they were, +by the motion of a simple point along the curve, so as +to give the required Fourier coefficients, were almost +instantaneous, and afford an example of invention +amounting to inspiration.</p> + +<p class="tb">There should be noticed here also the geometrical +slide for use in safety-valves, cathetometers and other +instruments, and the hole-slot-and-plane mode of so +supporting an instrument now used in all laboratories. +These were Thomson's inventions, and their importance +is insisted on in the <i>Natural Philosophy</i>.</p> + +<p>In Part II, the principal subjects treated are attractions, +elasticity, such great hydrostatical examples as the +equilibrium theory of the tides and the equilibrium +of rotating liquid spheroids, and such problems of +astronomical and terrestrial dynamics as the distribution +of matter in the earth, with the bearing on this +subject of the precession of the equinoxes, tidal friction, +the earth's rigidity, the effects of elastic tides, the +secular cooling of the earth, the age of the earth, and<span class='pagenum'><a name="Page_214" id="Page_214">214</a></span> +the "age of the sun's heat." Of these, with the exception +of the age of the earth, we shall not attempt to +give any account. The importance of the original +contributions to elasticity contained in the book is +indicated by the large space devoted to the <i>Natural +Philosophy</i> in Professor Karl Pearson's continuation of +Todhunter's <i>History of Elasticity</i>. The heavy task of +editing Part II was performed mainly by Sir George +Darwin, who made many notable additions from his +own researches to the matter contained in the first +edition.</p> + +<p>In the next chapter an attempt will be made to +present Thomson's views on the subject of the age +of the earth. These, when they were published, +attracted much attention, and received a good deal of +hostile criticism from geologists and biologists, whose +processes they were deemed to restrict to an entirely +inadequate period of time.</p> + +<h3><span class="smcap">Gyrostatic Action</span></h3> + +<p>Thomson in his lectures and otherwise gave a great +deal of attention to the motion of gyrostats, and to the +effect of the inclusion of gyrostats in a system on its +properties. Reference has been made to the treatment +of "gyrostatic domination" in "Thomson and Tait." +A gyrostat consists of a disk or wheel with a massive +rim, which revolves within a case or framework, by +which the whole arrangement can be moved about, or +supported, without interfering with the wheel. The +ordinary toy consisting of wheel with a massive rim, and +a light frame, is an example. But much larger and +more carefully made instruments, in which the wheel<span class='pagenum'><a name="Page_215" id="Page_215">215</a></span> +is entirely enclosed, give the most interesting experiments. +The body seems to have its properties entirely +altered by the rotation of the wheel, and of course the +case prevents any outward change from being visible.</p> + +<div class="figcenter" style="width: 550px; position: relative;"><a name="f15" id="f15"></a><img src="images/fig15.png" width="550" height="283" alt="Fig. 15." title="" /> +<p class="caption"><span class="smcap">Fig. 15.</span></p></div> + +<p>Figure 15 shows one form of gyrostat mounted +on a horizontal frame, held in the hands of an experimenter. +The axis of the fly-wheel is vertical within +the tubular part of the case; the fly-wheel is within +the part on which is engraved an arrow-head to show +the direction of rotation. Round the case in the +plane of the wheel is a projecting rim sharpened to +an edge, on which the gyrostat can be supported in +other experiments. To the rim are screwed two projecting +pivots, which can turn in bearings on the two +sides of the frame as shown. The centre of mass of +the wheel is on the level of these pivots, so that the +instrument will remain with either end of the axis up.</p> + +<p>If the fly-wheel be not in rotation, the experimenter +can carry the arrangement about, and the fly-wheel +and case move with it as if the gyrostat were merely<span class='pagenum'><a name="Page_216" id="Page_216">216</a></span> +an ordinary rigid body. But now remove the gyrostat +from the frame, and set the wheel in rotation. This +is done by an endless cord wrapped round a small +pulley fast on the axle (to which access is obtained by +a hole just opposite in the case) and passed also round +a larger pulley on the shaft of a motor. When the +motor is started the cord must be tightened only very +gently at first, so that it slips on the pulley, otherwise +the motor would be retarded, and possibly burned +by the current. The fly-wheel gradually gets up +speed, and then the cord can be brought quite tight +so that no slipping occurs. When the speed is great +enough the cord is cut with a stroke from a sharp +knife and runs out.</p> + +<p>The gyrostat is now replaced on its pivots in the +frame, with its axis vertical, and moved about as it +was before. If the experimenter, holding the frame +as shown, turns round in the direction of the arrow, +which is that of rotation, nothing happens. If, however, +he turns round the other way, the gyrostat +immediately turns on its pivots so as to point the other +end of the axis up. If the experimenter continues +his turning motion, the gyrostat is now quiescent: +for it is being carried round now in the direction of +rotation. Thus, with no gravitational stability at all +(since the centre is on a level with the pivots) the +gyrostat is in stable equilibrium when carried round +in the direction of rotation, but is in unstable +equilibrium when carried round the opposite way.</p> + +<p>Thus, if the observer knew nothing of the rotation +of the fly-wheel, and could see and feel only the +outside of the case, the behaviour of the instrument +might well appear very astonishing.<span class='pagenum'><a name="Page_217" id="Page_217">217</a></span></p> + +<p>This is a case of what Thomson and Tait call +"gyrostatic domination," which is treated very fully +in their Sections 345 (vi) to 345 (xxviii) of Part I. +It may be remarked here that this case of motion +may be easily treated mathematically in an exceedingly +elementary manner, and the instability of the one +case, and the stability of the other, made clear to the +beginner who has only a notion of the composition +of angular momenta about different axes.</p> + +<p>A year or two ago it was suggested by Professor +Pickering, of Harvard, that the fact that the outermost +satellite of Saturn revolves in the direction opposite +to the planet's rotation, may be due to the fact that +originally Saturn rotated in the direction of the motion +of this moon, but inasmuch as his motion round the +sun was opposite in direction to his rotation, he was +turned, so to speak, upside down, like the gyrostat! +The other satellites, it is suggested, were thrown off +later, as their revolution is direct. Professor Pickering +refers to an experiment (similar to that described above) +which he gives as new. Thomson had shown this +experiment for many years, as an example of the +general discussion in "Thomson and Tait," and its +theory had already been explicitly published.<a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">21</a></p> + +<p>Many other experiments with gyrostats used to be +shown by Thomson to visitors. Many of these are +indicated in "Thomson and Tait." The earth's +precessional motion is a gyrostatic effect due to the +differential attraction of the sun, which tends to bring +the plane of the equator into coincidence with the +ecliptic, and so alters the direction of the axis of +rotation. Old students will remember the balanced +<span class='pagenum'><a name="Page_218" id="Page_218">218</a></span>globe—with inclined material axis rolling round a +horizontal ring—by which the kinematics of the +motion could be studied, and the displacement of the +equinoxes on the ecliptic traced.</p> + +<div class="figcenter" style="width: 300px; position: relative;"><a name="f16" id="f16"></a><img src="images/fig16.png" width="300" height="455" alt="Fig. 16." title="" /> +<p class="caption"><span class="smcap">Fig. 16.</span></p></div> + +<p>Another example of the gyrostatic domination discussed +in "Thomson and Tait" is given in the very +remarkable address entitled "A Kinetic Theory of +Matter," which Sir William Thomson delivered to +Section A of the British Association at Montreal, in +1884. Figure 16 shows an ordinary double "coach +spring," the upper and lower members of which +carry two hooked rods as shown. If the upper hook +is attached to a fixed support, and a weight is hung +on the lower, the spring will be drawn out, and the +arrangement will be in equilibrium under a certain +elongation. If the weight be pulled down further<span class='pagenum'><a name="Page_219" id="Page_219">219</a></span> +and then left to itself, it will vibrate up and down +in a period depending upon the equilibrium elongation +produced by the weight. The same thing will happen +if a spiral spring be substituted for the coach spring. +A spherical case, through which the hooked rods pass +freely, hides the internal parts from view.</p> + +<div class="figcenter" style="width: 300px; position: relative;"><a name="f17" id="f17"></a><img src="images/fig17.png" width="300" height="445" alt="Fig. 17." title="" /> +<p class="caption"><span class="smcap">Fig. 17.</span></p></div> + +<p>Figure 17 shows two hooked rods, as in the former +case, attached by swivels to two opposite corners of +a frame formed of four rods jointed together at their +ends. Each of these is divided in the middle for the +insertion of a gyrostat, the axis of which is pivoted +on the adjacent ends of the two halves of the rod. +A spherical case, indicated by the circle, again hides +the internal arrangement from inspection, but permits +the hooked rods to move freely up and down. The +swivels allow the frame, gyrostats and all, to be turned +about the line of the hooks.<span class='pagenum'><a name="Page_220" id="Page_220">220</a></span></p> + +<p>If now the gyrostats be not in rotation, the frame +will be perfectly limp, and will not in the least resist +pull applied by a weight. But if the gyrostats be rotated +in the directions shown by the circles, with arrowheads +drawn round the rods, there will be angular +momentum of the whole system about the line joining +the hooks, and if a weight or a force be applied to +pull out the frame along that line, the pull will be +resisted just as it was in the other case by the spring. +Moreover, equilibrium will be obtained with an +elongation proportional to the weight hung on, and +small oscillations will be performed just as if there +were a spring in the interior instead of the gyrostats.</p> + +<p>According as the frame is pulled out, or shortened, +the angular momentum of the gyrostats about the line +joining the hooks is increased or diminished, and the +frame, carrying the gyrostats with it, turns about +the swivels in one direction or the other, at the rate +necessary to maintain the angular momentum at a +constant value. But this will not be perceived from +without.</p> + +<p>The rotation of the fly-wheels thus gives to the otherwise +limp frame the elasticity which the spring possesses; +without dissection of the model the difference cannot +be perceived. This illustrates Thomson's idea that the +elasticity of matter may be due to motion of molecules +or groups of molecules of the body, imbedded in a +connecting framework, deformed by applied forces as +in this model, and producing displacements which are +resisted in consequence of the motion.</p> + +<p>And here may be mentioned also Thomson's explanation +of the phenomenon, discovered by Faraday, +of the rotation of the plane of a beam of polarised light<span class='pagenum'><a name="Page_221" id="Page_221">221</a></span> +which is passed along the lines of force of a magnetic +field. This rotation is distinct altogether from that +which is produced when polarised light is passed along +a tube filled with a solution of sugar or tartaric acid. +If the ray be reflected after passage, and made to +retraverse the medium, the rotation is annulled in +the latter case, it is doubled in the former. This +led Thomson to the view that in sugar, tartaric acid, +quartz, etc., the turning is due to the structure of +the substance, and in the magnetic field to rotation +already existing in the medium. He used to say that +a very large number of minute spiral cavities all in +the same direction, and all right-handed or all left-handed, +in the sugar or quartz, would give the effect; +on the other hand, the magnetic phenomenon could +only be produced by some arrangement analogous to +a very large number of tops, or gyrostats, imbedded in +the medium with their axes all in one direction (or +preponderatingly so) and all turning the same way. +The rotation of these tops or gyrostats Thomson +supposed to be caused by the magnetic field, and to +be essentially that which constitutes the magnetisation +of the medium.</p> + +<p>Let the frame of the gyrostatic spring-balance +described above, turn round the line joining the hooks +so as to exactly compensate, by turning in the opposite +direction, the angular momentum about that line given +by the fly-wheels; then the arrangement will have no +angular momentum on the whole; and a large number +of such balances, all very minute and hooked together, +will form a substance without angular momentum in any +part. But now by the equivalent of a magnetic force +along the lines of the hooks, let a different angular<span class='pagenum'><a name="Page_222" id="Page_222">222</a></span> +turning of the frames be produced; the medium will +possess a specific angular momentum in every part. +If a wave of transverse vibrations which are parallel +to one direction (that is, if the wave be plane-polarised) +enter the medium in the direction of the axes of the +frames, the direction of vibration will be turned as the +wave proceeds, that is, the plane of polarisation will +be turned round.</p> + +<p>More recent research has shown an effect of a +magnetic field on the spectrum of light produced in +the field, and viewed with a spectroscope in a direction +at right angles to the field—the Zeeman effect, +as it is called—and the explanation of this effect by +equations of moving electric charges, which are essentially +gyrostatic equations, is suggestive of an analogy +or correspondence between the systems of moving +electrons which constitute these charges, and some +such gyrostatic molecules as Thomson imagined. It +has been pointed out that the Zeeman effect, in its +simple forms at least, can be exactly imitated by the +motion of an ordinary pendulum having a gyrostat in +its bob, with its axis directed along the suspension +rod.<a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">22</a></p> + +<h3><span class="smcap">Electrostatics and Magnetism</span></h3> + +<p>In the ten years from 1863 to 1873 Thomson was +extremely busy with literary work. In 1872, five +years after the publication of the treatise on <i>Natural +Philosophy</i>, and just before the appearance of the +<i>Elements</i>, Messrs. Macmillan & Co. published for him +a collection of memoirs entitled <i>Reprint of Papers on</i> +<span class='pagenum'><a name="Page_223" id="Page_223">223</a></span> +<i>Electrostatics and Magnetism</i>. The volume contains +596 pages, and the subjects dealt with range from the +"Uniform Motion of Heat and its Connection with +the Mathematical Theory of Electricity" (the paper +already described in Chapter II above) and the discussion +of Electrometers and Electrostatic Measuring +Instruments, to a complete mathematical theory of +magnetism. The subject of electrostatics led naturally +to the consideration of electrical measuring instruments +as they existed forty years ago (about 1867), and their +replacement by others, the indications of which from +day to day should be directly comparable, and capable +of being interpreted in absolute units. Down to that +time people had been obliged to content themselves +with gold-leaf electroscopes, and indeed it was impossible +for accurate <i>measuring</i> instruments to be invented +until a system of absolute units had been completely +worked out. The task of fixing upon definitions of +units and of realising them in suitable standards had +been begun by the British Association, and it was as +part of the Report of that Committee to the Dundee +Meeting in 1867 that Thomson's paper on Electrometers +first appeared.</p> + +<p>It was there pointed out that an electrometer is +essentially an instrument for measuring differences of +electric potential between conductors, by means of +effects of electrostatic force. Such a difference is what +a gold-leaf electroscope indicates for its gold leaves and +the walls surrounding the air-space in which they are +suspended. As electroscopes used to be constructed, +these walls were made of glass imperfectly covered, if +at all, by conducting material, and the electroscope +was quite indefinite and uncertain in its action. The<span class='pagenum'><a name="Page_224" id="Page_224">224</a></span> +instrument was also, as made, quite insensitive. Recently, +however, it has been rehabilitated in reputation, +and brought into use as a very sensitive indicator of +effects of radio-activity.</p> + +<p>Thomson described in this paper six species of +electrometers of his own devising. The best known +of these are his quadrant electrometer and his attracted-disk +electrometers. The former is to be found in +some form or other in every laboratory nowadays, +and need not be described in detail. The action is of +two conductors—the two pairs of opposite quadrants +of a shallow, horizontal, cylindrical box, made by +dividing the box into four by two slits at right angles—upon +an electrified slip of aluminium suspended by a +two-thread suspension within the box, with its length +along one of the slits. The two pairs of opposite +quadrants are at the potential difference to be measured, +and the slip of aluminium, or "needle," has each end +urged round from a quadrant at higher potential towards +one at a lower, and these actions conspire to turn the +slip against its tendency to return to the position in +which the two threads are in one plane. Thus the +deflection (measured by the displacement of a reflected +ray of light used as index) gives an indication of the +amount of the potential difference.</p> + +<p>The electrification of the "needle" was kept up by +enclosing the quadrantal box within an electrified +Leyden jar, to the interior coating of which contact +is made by a platinum wire, depending from the needle +to sulphuric acid contained in the jar. The whole +apparatus was enclosed in a conducting case connected +to earth. This made its action perfectly definite. +Variations of this electrification of the jar were shown<span class='pagenum'><a name="Page_225" id="Page_225">225</a></span> +by an attached attracted-disk electrometer, the principle +of which we shall merely indicate.</p> + +<p>The quadrant electrometer has now been vastly +increased in sensibility by the use of a single quartz +fibre as suspension. By the invention of this fibre, +which is exceedingly strong and is, moreover, so +definite in its elastic properties that it comes back +at once exactly to its former zero state after twist, +Mr. C. V. Boys has increased the delicacy of all kinds +of suspended indicators many fold. But it ought to be +remembered that a Dolezalek electrometer, with some +hundred or more times the sensibility of the bifilar +instrument, was only made possible by its predecessor.</p> + +<p>Attracted-disk-electrometers simply measure, either +by weighing or by the deflection of a spring, the +attractive force between two parallel disks at different +potentials. From the determination of this force, and +the measurement of the distance between the disks (or +better, of an alteration of the distance) a difference of +potentials can be determined, and a unit for it obtained, +which is in direct and known relation to ordinary +dynamical units. Thomson's "Absolute Electrometer" +was designed specially for accurate determinations of +this kind. Another form, called the Long Range +Electrometer, was devised for the measurement of the +potentials of the charged conductors in electric machines +and Leyden jars.</p> + +<p>Accurate determinations of the sparking resistance +between parallel plates charged to different potentials +in air were made by means of attracted-disk-electrometers +in the course of some important experiments +described in the <i>Electrostatics and Magnetism</i>. These +results have been much referred to in later researches.</p> + +<p><span class='pagenum'><a name="Page_226" id="Page_226">226</a></span>A small attracted-disk-electrometer was used as indicated +above to keep a watch on the electrification of +the Leyden jar of the quadrant instrument, and a +small induction machine was added, by turning which +the operator could make good any loss of charge of +the jar.</p> + +<p>This electrical machine was an example of an apparatus +on precisely the same principle as the Voss or +Wimshurst machines of the present day. In it by a set +of moving carriers, influenced by conductors, the charges +of the latter were increased according to a compound +interest principle only interfered with by leakage to +the air or by the supports. Several forms of this +machine, on the same principle, were constructed by +Thomson, and described in 1868; but he afterwards +found that he had been anticipated by C. F. Varley in +1860. Still later it was discovered that a similar +instrument had been made a century before by +Nicholson, and called by him the "Revolving +Doubler."</p> + +<p>The experiments which Thomson made on atmospheric +electricity at the old College tower, and by +means of portable electrometers in Arran and elsewhere, +can only be mentioned. They led no doubt +to some improvements on electrometers which he made, +the method of bringing the nozzle of a water-dropper, +or a point on a portable electrometer to the potential +of the air, by the inductive action on a stream of water-drops +in the one case, or the particles of smoke from +a burning match in the other. He invented a self-acting +machine, worked by a stream of water-drops, +for accumulating electric charges, on the principle of +the revolving doubler. It was this apparently that<span class='pagenum'><a name="Page_227" id="Page_227">227</a></span> +led to the machines with revolving carriers, to which +reference has been made above.</p> + +<p>The mathematical theory of magnetism which +Thomson gave in 1849, in the <i>Phil. Trans. R.S.</i>, was, +when completed by various later papers, a systematic +discussion of the whole subject, including electromagnetism +and diamagnetism. To a large extent the +ground covered by the 1849 paper had been traversed +before by Poisson, and partially by Murphy and Green; +but Thomson stated that one chief object of his +memoir was to formally construct the theory without +reference to the two magnetic fluids, by means of +which the facts of experiment and conclusions of theory +had so far been expressed. He found it, however, convenient +to introduce the idea of positive and negative +magnetic matter (attracting and repelling as do charges +of positive and negative electricity), which are to be +regarded as always present in equal amounts, not only +in a magnet as a whole, but in every portion of a +magnet; and at first sight this might appear like a +return to the magnetic fluids. But it amounts on the +whole rather to a conception of a magnet as a conglomeration +of doublets of magnetic matter (that is, +very close, equal and inseparable charges of the two +kinds of matter), the arrangement of which can be +changed by the action of magnetic force. This idea +is set forth now in all the books on magnetism and +electricity. There can be no doubt that the systematic +presentment of the subject by Thomson, and the +theorems and ideas of magnetic force and magnetic +permeability by which he rendered the clear, and +therefore mathematical, notions of Faraday explicitly +quantitative, had much influence in furthering the<span class='pagenum'><a name="Page_228" id="Page_228">228</a></span> +progress of electrical science, and so leading on the one +hand to the electromagnetic theories of Maxwell, and +on the other to modern research on the magnetic +properties of iron, and to the correct ideas which now +prevail as to construction of dynamo-electric machines +and motors.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_229" id="Page_229">229</a></span></p> + +<h3>CHAPTER XII</h3> + +<h4>THE AGE OF THE EARTH</h4> + +<p><span class="smcap">From</span> his student days throughout his life, Lord +Kelvin took a keen interest in geological questions. +He was always an active member of the Geological +Society of Glasgow, and was its president for twenty-one +years (1872-1893). The distribution of heat in +the substance of the earth was the subject of his +inaugural dissertation as Professor of Natural Philosophy; +and previously, as a student, he had written an +essay on "The Figure of the Earth," for which he +had been awarded a University Gold Medal. He +never ceased to ponder over the problems of terrestrial +physics, and he wrote much on the subject. His +papers are to be found as Appendices to Thomson and +Tait's <i>Natural Philosophy</i>, and in vol. ii of his <i>Popular +Lectures and Addresses</i>, which is devoted to geology +and general physics.</p> + +<p>His conclusions regarding the age of the earth have +been referred to in the last chapter. The first allusion +to the subject was contained (see p. <a href="#Page_65">65</a> above) in his +inaugural dissertation "<i>De Caloris distributione in Terræ +Corpus</i>"; but he returned to it again in a communication +made to the Royal Society of Edinburgh in +December, 1865, and entitled "The Doctrine of +Uniformity in Geology briefly refuted." On February +27, 1868, he delivered to the Geological Society of<span class='pagenum'><a name="Page_230" id="Page_230">230</a></span> +Glasgow an address entitled "On Geological Time," +in which the necessity for limiting geological and +other changes to an almost infinitesimal fraction of the +vast periods at that time demanded was insisted on, +and which gave rise to much discussion.</p> + +<p>The address began with a protest against the old +uniformitarian view of geological changes as expressed +by Playfair in his <i>Illustrations of the Huttonian +Theory</i>. The first objection taken to the idea that +"in the continuation of the different species of animals +and vegetables that inhabit the earth, we discern +neither a beginning nor an end; in the planetary +motions where geometry has carried the eye so far, +both into the future and the past, we discover no +mark either of the commencement or the termination +of the present order" is, that the stability of the motions +of the heavenly bodies, to which reference is made in +this statement, is founded upon what is essentially an +approximate calculation, which leaves out, by intention, +the consideration of frictional resistance.</p> + +<p>He points out, for example, that the friction which +accompanies the relative motion of the waters of the +earth and the land is attended by the production of +heat, and that, by the doctrine of the conservation of +energy, heat cannot be produced without a disappearance +of an equivalent quantity of energy, either of +motion or of position. The chief source of this +energy is the earth's rotation. Since the earth turns +under the moon and the tidal spheroid—that is, the +earth's shape as distorted by the heaping up of the +waters in the tides—remains on the whole stationary +with respect to the moon, the solid matter of the +earth turns under the distribution of the water, held<span class='pagenum'><a name="Page_231" id="Page_231">231</a></span> +more or less fixed by the moon, as does a fly-wheel +under a stationary friction band round its rim. Then +just as the band held fixed retards the fly-wheel, so +the earth must be retarded in its rotation by this +water-brake. In the earth's rotation there is a store +of kinetic energy which, roughly estimated, would not +be exhausted in less than ten million million years, +although drawn upon continuously by friction, or +other actions, at the rate of one million horse-power; +so that, no immediate catastrophe, such as that we +should be involved in by the stoppage or considerable +retardation of the spinning motion of the earth, is +possible. But it was pointed out by Thomson that +the best results of astronomical observation show that +the earth would in one hundred years fall behind a +perfect time-keeper, with which its rotation kept pace +at the beginning of the time, by about twenty seconds. +The tendency is to make the earth turn slower, and +the moon to increase its distance and move more slowly +in its orbit, but with a resultant effect towards coincidence +of the period of the earth's rotation with that of +revolution of the moon round the earth. After this +coincidence has been attained, however, the solar tides +will tend to make the moon fall in towards the earth.</p> + +<p>If then the earth be rotating more and more slowly, +as time goes on, at present, it must have been rotating +more rapidly in past time. A thousand million years +ago, at the present rate of retardation, the earth must +have been rotating one seventh part of its speed faster +than it is rotating at present, and this would give for +centrifugal force at the surface one thousand million +years ago, greater than the centrifugal force at present, +in the ratio of 64 to 49. Apparently therefore the<span class='pagenum'><a name="Page_232" id="Page_232">232</a></span> +earth must have solidified at a much later date than +that epoch, a date when it was rotating much more +nearly with the angular speed which it has now; +otherwise the figure of the earth would have deviated +much more from the spherical form than it actually +does. On the other hand, one hundred million years +ago centrifugal force would be only three per cent. +greater than it is at present, and consolidation of the +earth at that less remote period would give a shape to +the earth not very different from that which it now +possesses. The argument therefore from tidal retardation +would cut down the time available for geological +and biological changes to something not much more +than one hundred million years, perhaps to less.</p> + +<p>A second argument for limitation of the time available +for such processes is derived from the sun's heat. +The sun cannot be regarded as a miraculous body +producing its light and heat from nothing. Changes +of the constitution of the sun must be continually +proceeding, to account for its enormous radiation of +energy into space, a radiation of which only an infinitesimal +part is received by the bodies of the solar +system, and a still more minute portion by the earth. +The effects of the sun's light and heat on the earth +show how enormous must be the quantity of energy +lost from the sun in a year. How is this loss of energy +to be accounted for? What is the physical change which +gives rise to it? In 1854 Thomson put forward the +theory that the sun's heat is kept up by the falling in +of meteors on the sun's surface, but he afterwards saw +reason to abandon that view. Helmholtz had advocated +the theory that the sun was a body heated by the +coming together of the matter composing it by its<span class='pagenum'><a name="Page_233" id="Page_233">233</a></span> +mutual attraction, a process which, although the sun +is now a continuous mass, is to be regarded as still +going on. It is easy to calculate the exhaustion of +potential energy caused by the coming together of the +matter of the sun from universal dispersion through +infinite space to a sphere of uniform density of the +present size of the sun. The result is about as much +energy as would be generated by burning seven million +million million million million tons of coal. The +amount radiated in each hour is about as much as +would be generated by burning something like nine +tons of coal every hour on every square yard of the +sun's surface. It is certain that the sun must be still +contracting, and if it contracts sufficiently to just make +good this expenditure by the further exhaustion of +potential energy involved in the closer aggregation of +the matter, it must diminish in radius in each year by +as much as 130 feet.</p> + +<p>The amount of energy generated by the falling +together of the matter of the sun from universal diffusion +to the dimensions which the sun has at present, is +only about 13,000,000 times the amount now radiated +per annum. In Thomson's paper Pouillet's estimate +of the energy radiated per second is used, and this +number is raised to 20,000,000. Taking the latter +estimate, the whole potential energy exhausted by the +condensation of the sun's mass to uniform density +would suffice for only 20,000,000 years' supply. But +the sun is undoubtedly of much greater density in the +central parts than near the surface, and so the energy +exhausted must be much greater than that stated above. +This will raise the number of years provided for. On +the other hand, a considerable amount of energy would<span class='pagenum'><a name="Page_234" id="Page_234">234</a></span> +be dissipated during the process of condensation, and +this would reduce the period of radiation estimated. +Thomson suggests that 50,000,000, or 100,000,000, +years is a possible estimate.</p> + +<p>It is not unlikely that the rate of radiation in past +time, when the sun had not nearly condensed to its +present size, was so much less than it is at present +that the period suggested above may have to be considerably +augmented. Another source of radiation, +which seems to be regarded by some authorities as a +probable, if not a certain, one, has been suggested +in recent years—the presence of radio-active substances +in the sun. So far as we know, Lord +Kelvin did not admit that this source of radiation +was worthy of consideration; but of course, granted +its existence to an extent comparable with the energy +derivable from condensation of the sun's mass, the +"age of the sun's heat" would have to be very greatly +extended. These are matters, however, on which further +light may be thrown as research in radio-activity +progresses. Lord Kelvin was engaged when seized with +his last illness in discussing the changes of energy in a +gaseous, or partially gaseous, globe, slowly cooling and +shrinking in doing so; and a posthumous paper on the +subject will shortly be published which may possibly +contain further information on this question of solar +physics.</p> + +<p>But Thomson put forward a third argument in the +paper on Geological Time, which has always been +regarded as the most important. It is derived from +the fact, established by abundant observations, that the +temperature in the earth's crust increases from the surface +inwards; and that therefore the earth must be<span class='pagenum'><a name="Page_235" id="Page_235">235</a></span> +continually losing heat by conduction from within. If +the earth be supposed to have been of uniform temperature +at some period of past time and in a molten +state, and certain assumptions as to the conductive +power and melting point of its material be made, the +time of cooling until the gradient of temperature at the +surface acquired its present value can be calculated. +This was done by Thomson in a paper published in the +<i>Transactions, R.S.E.</i>, in 1862. We propose to give +here a short sketch of his argument, which has excited +much interest, and been the cause of some controversy.</p> + +<p>In order to understand this argument, the reader +must bear in mind some fundamental facts of the flow +of heat in a solid. Let him imagine a slab of any +uniform material, say sandstone or marble, the two +parallel faces of which are continually maintained at +two different temperatures, uniform over each face. +For example, steam may be continually blown against +one face, while ice-cold water is made to flow over the +other. Heat will flow across the slab from the hotter +face to the colder. It will be found that the rate of +flow of heat per unit area of face, that is per square +centimetre, or per square inch, is proportional to the +difference of the temperatures in the slab at the two +faces, and inversely proportional to the thickness of the +slab. In other words, it is proportional to the fall of +temperature from one face to the other taken per unit +of the thickness, that is, to the "gradient of temperature" +from one face to the other. Moreover, comparing +the flow in one substance with the flow in +another, we find it different in different substances for +the same gradient of temperature. Thus we get +finally a flow of heat across unit area of the slab which<span class='pagenum'><a name="Page_236" id="Page_236">236</a></span> +is equal to the gradient of temperature multiplied by a +number which depends on the material: that number +is called the "conductivity" of the substance.</p> + +<p>Now, borings made in the earth show that the temperature +increases inwards, and the same thing is +shown by the higher temperatures found in deeper +coal mines. By means of thermometers sunk to +different depths, the rate of increase of temperature +with depth has been determined. Similar observations +show that the daily and annual variations of temperature +caused by the succession of day and night, and +summer and winter, penetrate to only a comparatively +small depth below the surface—three or four feet in +the former case, sixty or seventy in the latter. Leaving +these variations out of account, since the average of +their effects over a considerable interval of time must +be nothing, we have in the earth a body at every point +of the crust of which there is a gradient of increasing +temperature inwards. The amount of this may be +taken as one degree of Fahrenheit's scale for every +50 feet of descent. This gradient is not uniform, but +diminishes at greater depths. Supposing the material +of uniform quality as regards heat-conducting power, +the mathematical theory of a cooling globe of solid +material (or of a straight bar which does not lose heat +from its sides) gives on certain suppositions the +gradients at different depths. The surface gradient +of 1° F. in 50 feet may be taken as holding for 5000 +feet or 6000 feet or more.</p> + +<p>This gradient of diminution of temperature outwards +leads inevitably to the conclusion that heat must be +constantly flowing from the interior of the earth +towards the surface. This is as certain as that heat<span class='pagenum'><a name="Page_237" id="Page_237">237</a></span> +flows along a poker, one end of which is in the fire, +from the heated end to the other. The heat which +arrives at the surface of the earth is radiated to the +atmosphere or carried off by convection currents; +there is no doubt that it is lost from the earth. Thus +the earth must be cooling at a rate which can be +calculated on certain assumptions, and it is possible on +these assumptions to calculate backwards, and determine +the interval of time which must have elapsed since +the earth was just beginning to cool from a molten +condition, when of course life cannot have existed on +its surface, and those geological changes which have +effected so much can hardly have began.</p> + +<p>Considering a globe of uniform material, and of +great radius, which was initially at one temperature, +and at a certain instant had its surface suddenly brought +to, let us say, the temperature of melting ice, at which +the surface was kept ever after, we can find, by +Fourier's mathematical theory of the flow of heat, the +gradient of temperature at any subsequent time for a +point on the surface, or at any specified distance within +it. For a point on the surface this gradient is simply +proportional to the initial uniform temperature, and +inversely proportional to the square root of the product +of the "diffusivity" of the material (the ratio of the +conductivity to the specific heat) by the interval of +time which has elapsed since the cooling was started. +Taking a foot as the unit of length, and a year as the +unit of time, we find the diffusivity of the surface strata +to be 400. If we take the initial temperature as +7000 degrees F.—which is high enough for melting +rock—and take the interval of time which has elapsed +as 100,000,000 years, we obtain at the surface a<span class='pagenum'><a name="Page_238" id="Page_238">238</a></span> +gradient approximately equal to that which now exists. +A greater interval of time would give a lower gradient, +a smaller interval would give a higher gradient than +that which exists at present. A lower initial temperature +would require a smaller interval of time, a higher +initial temperature a longer interval for the present +gradient.</p> + +<p>With the initial temperature of 7,000 degrees F., +an interval of 4,000,000 years would give a surface +gradient of 1° F. in 10 ft. Thus, on the assumption +made, the surface gradient of temperature has diminished +from <sup>1</sup>⁄<sub>10</sub> to <sup>1</sup>⁄<sub>50</sub> in about 96,000,000 years. +After 10,000 years from the beginning of the cooling +the gradient of temperature would be 2° F. per foot. +But, as Thomson showed, such a large gradient +would not lead to any sensible augmentation of the +surface temperature, for "the radiation from earth and +atmosphere into space would almost certainly be so +rapid" as to prevent this. Hence he inferred that +conducted heat, even at that early period, could not +sensibly affect the general climate.</p> + +<p>Two objections (apart from the assumptions already +indicated) will readily occur to any one considering +this theory, and these Thomson answered by anticipation. +The first is, that no natural action could +possibly bring the surface of a uniformly heated globe +instantaneously to a temperature 7000° lower, and +keep it so ever after. In reply to this Thomson +urged "that a large mass of melted rock, exposed freely +to our earth and sky, will, after it once becomes +crusted over, present in a few hours, or a few days, or +at most a few weeks, a surface so cool that it can be +walked over with impunity. Hence, after 10,000<span class='pagenum'><a name="Page_239" id="Page_239">239</a></span> +years, or indeed, I may say, after a single year, its +condition will be sensibly the same as if the actual +lowering of temperature experienced by the surface had +been produced in an instant, and maintained constant +ever after." The other objection was, that the earth +was probably never a uniformly heated solid 7000° F. +above the present surface temperature as assumed for +the purpose of calculation. This Thomson answers +by giving reasons for believing that "the earth, although +once all melted, or melted all round its surface, did, in +all probability, really become a solid at its melting +temperature all through, or all through the outer layer +which has been melted; and not until the solidification +was thus complete, or nearly so, did the surface begin +to cool."</p> + +<p>Thomson was inclined to believe that a temperature +of 7000° F. was probably too high, and results of +experiments on the melting of basalt and other rocks +led him to prefer a much reduced temperature. This, +as has already been pointed out, would give a smaller +value for the age of the earth. In a letter on the +subject published in <i>Nature</i> (vol. 51, 1895) he states +that he "is not led to differ much" from an estimate +of 24,000,000 years founded by Mr. Clarence King +(<i>American Journal of Science</i>, January 1893) on experiments +on the physical properties of rocks at high +temperatures.</p> + +<p>It is to be observed that the assumptions made above +that the physical constants of the material are constant +throughout the earth, and at all temperatures, are +confessedly far from the truth. Nevertheless Thomson +strongly held that the uncertainty of the data +can at most extend the earth's age to some value<span class='pagenum'><a name="Page_240" id="Page_240">240</a></span> +between 20,000,000 and 200,000,000 of years, and +that the enormously long periods which were wont to +be asked for by geologists and biologists for the changes +of the earth's surface and the development of its flora +and fauna, cannot possibly be conceded.</p> + +<p>In <i>Nature</i> for January 3, 1895, Professor John +Perry suggested that very possibly the conductivity of +the material composing the interior of the earth was +considerably higher than that of the surface strata. If +this were so, then, as can be shown without difficulty, +the attainment of the present gradient would be very +greatly retarded, and therefore the age of the earth +correspondingly increased. The question then arose, +and was discussed, as to whether the rocks and other +materials at high temperatures were more or less +conducting than at low temperatures, and experiments +on the subject were instituted and carried out. On +the whole, the evidence seemed to show that the conductivity +of most substances is diminished, not increased, +by the rise of temperature, and so far as it went, +therefore, the evidence was against Professor Perry's +suggestion. On the other hand, he contended that +the inside of the earth may be a mass of great rigidity, +partly solid and partly fluid, possessing a "quasi-conductivity" +which might greatly increase the period +of cooling. The subject is a difficult one both from a +mathematical and from the physical point of view, and +further investigation is necessary, especially of the +behaviour of materials under the enormous stresses +which they undoubtedly sustain in the interior of the +earth.</p> + +<p>After the publication of the paper on Geological +Time a reply to it was made by Professor Huxley, in<span class='pagenum'><a name="Page_241" id="Page_241">241</a></span> +an address to the Geological Society of London, +delivered on February 19, 1869. He adopted the <i>rôle</i> +of an advocate retained for the defence of geology +against what seems to have been regarded as an unwarranted +attack, made by one who had no right to +offer an opinion on a geological question. For, after +a long and eloquent "pleading," he concludes his +address with the words: "My functions, as your +advocate, are at an end. I speak with more than the +sincerity of a mere advocate when I express the belief +that the case against us has entirely broken down. +The cry for reform which has been raised from without +is superfluous, inasmuch as we have long been +reforming from within with all needful speed; and the +critical examination of the grounds upon which the +very grave charge of opposition to the principles of +Natural Philosophy has been brought against us, rather +shows that we have exercised a wise discrimination in +declining to meddle with our foundations at the bidding +of the first passer-by who fancies our house is not so +well built as it might be." To this Thomson rejoined +in an address entitled "Of Geological Dynamics," +also delivered to the Geological Society of Glasgow +on April 5, 1869; and to this, with Professor Huxley's +address, the reader must be referred for the objection, +brought against Thomson's arguments, and the replies +which were immediately forthcoming. This is not +the place to discuss the question, but reference may be +made to an interesting paper on the subject in the +<i>Glasgow Herald</i> for February 22, 1908, by Professor +J. W. Gregory, in which the suggestion of Professor +Perry, of a nearer approach to uniformity of temperature +in the interior of the earth than Thomson had<span class='pagenum'><a name="Page_242" id="Page_242">242</a></span> +thought possible, is welcomed as possibly extending the +interval of time available to a period sufficient for all +purposes. In Professor Gregory's opinion, "Lord +Kelvin in one respect showed a keener insight than +Huxley, who, referring to possible changes in the rate +of rotation of the earth, or in the heat given forth +from the sun or in the cooling of the earth, declared +that geologists are Gallios, 'who care for none of these +things.' An ever-increasing school of geologists now +cares greatly for these questions, and reveres Lord +Kelvin as one of the founders of the geology of the +inner earth."</p> + +<p>After all, the problem is not one to be dealt with by +the geologist or biologist alone, but to be solved, so far +as it can be solved at all, by a consideration of all +relevant evidence, from whatsoever quarter it may +come. It will not do in these days for scientific men +to shut themselves up within their special departments +and to say, with regard to branches of science which +deal with other aspects of nature and other problems +of the past, present and future of that same earth on +which all dwell and work, that they "care for none of +these things." This is an echo of an old spirit, not +yet dead, that has done much harm to the progress of +science. The division of science into departments is +unavoidable, for specialisation is imperative; but it is +all the more necessary to remember that the divisions +set up are more or less arbitrary, and that there are +absolutely no frontiers to be guarded and enforced. +Chemistry, physiology, and physics cannot be walled +off from one another without loss to all; and geology +has suffered immensely through its having been regarded +as essentially a branch of natural history, the<span class='pagenum'><a name="Page_243" id="Page_243">243</a></span> +devotees of which have no concern with considerations +of natural philosophy. Lord Kelvin's dignified questions +were unanswerable. "Who are the occupants of +'our house,' and who is the 'passer-by'? Is geology +not a branch of physical science? Are investigations, +experimental and mathematical, of underground temperature +not to be regarded as an integral part of +geology?... For myself, I am anxious to be regarded +by geologists not as a mere passer-by, but as +one constantly interested in their grand subject, and +anxious in any way, however slight, to assist them in +their search for truth."</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_244" id="Page_244">244</a></span></p> + +<h3>CHAPTER XIII</h3> + +<h4>BRITISH ASSOCIATION COMMITTEE ON ELECTRICAL +STANDARDS</h4> + +<p><span class="smcap">When</span> Professor Thomson began his work as a teacher +in the University of Glasgow, there was, as has already +been noticed, great vagueness of specification of physical +quantities. Few of the formal definitions of units of +measurement, now to be found in the pages of every +elementary text book, had been framed, and there +was much confusion of quantities essentially distinct, +a confusion which is now, to some extent at least, +guarded against by the adoption of a definite unit, +with a distinctive name for each magnitude to be +measured. Thus rate of working, or activity, was +confused with work done; the condition for maximum +activity in the circuit of a battery or dynamo was often +quoted as the condition of greatest efficiency, that is of +greatest economy of energy, although it was exactly +that in which half the available energy was wasted.</p> + +<p>Partly as a consequence of this vagueness of specification, +there was a great want of knowledge of the values +of physical constants; for without exact definitions of +quantities to be determined, such definitions as would +indicate units for their measurement, related to ordinary +dynamical units according to a consistent scheme, it was +impossible to devise satisfactory experimental methods<span class='pagenum'><a name="Page_245" id="Page_245">245</a></span> +to do for electricity and magnetism what had been +done by Regnault and others for heat.</p> + +<p>The first steps towards the construction of a complete +system of units for the quantitative measurement +of magnetic and electric quantities were taken by +Gauss, in his celebrated paper entitled <i>Intensitas vis +magneticæ terrestris ad mensuram absolutam revocata</i>, +published in 1832. In this he showed how magnetic +forces could be expressed in absolute units, and thus be +connected with the absolute dynamical units which +Gauss, in the same paper, based on chosen fundamental +units of length, mass, and time. Thus the modern +system of absolute units of dynamical quantities, and +its extension to magnetism, are due to the practical +insight of a great mathematician, not to the experimentalists +or "practicians" of the time.</p> + +<p>Methods of measuring electric quantities in absolute +units were described by W. Weber, in Parts II and +III of his <i>Elecktrodynamische Maassbestimmungen</i>, published +in 1852. These were great steps in advance, +and rendered further progress in the science of absolute +measurement comparatively easy. But they remained +the only steps taken until the British Association +Committee began their work. We have already +(pp. <a href="#Page_74">74-76</a>) referred to the great importance of that +work, not only for practical applications but also for the +advancement of science. But it was not a task which +struck the imagination or excited the wonder of the +multitude. For the realisation of standards of resistance, +for example, involved long and tedious investigations +of the effects of impurities on the resistance of +metals, and the variation of resistance caused by change +of temperature and lapse of time. Then alloys had to<span class='pagenum'><a name="Page_246" id="Page_246">246</a></span> +be sought which would have a temperature effect of +small amount, and which were stable and durable in all +their properties.</p> + +<p>The discoveries of the experimentalist who finds +a new element of hitherto undreamed-of properties +attract world-wide attention, and the glory of the +achievement is deservedly great. But the patient, +plodding work which gives a universal system of units +and related standards, and which enables a great +physical subject like electricity and magnetism to +rise from a mere enumeration of qualitative results to +a science of the most delicate and exact measurement, +and to find its practical applications in all the affairs +of daily life and commerce, is equally deserving of the +admiration and gratitude of mankind. Yet it receives +little or no recognition.</p> + +<p>The construction of a standard of resistance was the +first task undertaken by the committee; but other +units, for example of quantity of electricity, intensity +of electric field and difference of potential, had also to +be defined, and methods of employing them in experimental +work devised. It would be out of place to +endeavour to discuss these units here, but some idea of +the manner in which their definitions are founded on +dynamical conceptions may be obtained from one or +two examples. Therefore we shall describe two simple +experiments, which will illustrate this dynamical +foundation. An account has been given in Chapter XI +of the series of electrometers which Thomson invented +for the measurement of differences of electric potential. +These all act by the evaluation in terms of ordinary +dynamical units of the force urging an electrified body +from a place of higher towards a place of lower potential.<span class='pagenum'><a name="Page_247" id="Page_247">247</a></span></p> + +<p>Some indication of the meaning of electrical +quantities has been given in Chapter IV. Difference +of electric potential between two points in an electric +field was there defined as the dynamical work done +in carrying a unit of positive electricity against the +forces of the field from the point of lower to the point +of higher potential. Now by the definition of unit +quantity of electricity given in electrical theory—that +quantity which, concentrated at a point at unit distance +from an equal quantity also concentrated at a point, is +repelled with unit force—we can find, by the simple +experiment of hanging two pith balls (or, better, two +hollow, gilded beads of equal size) by two fine fibres +of quartz, a metre long, say, electrifying the two balls +as they hang in contact, and observing the distance at +which they then hang, the numerical magnitude in +absolute units of a charge of electricity, and apply that +to finding the charge on a large spherical conductor +and the potential at points in its field also in absolute +units. If <i>m</i> be the mass of a ball, <i>g</i> gravity in cm. +sec. units, <i>d</i> the distance in cms. of the centres of +the balls apart, and <i>l</i> the length in cms. of a thread, +the charge <i>q</i>, say, on each ball is easily found to be +<img class="floatInsert18" src="images/f247.png" alt="" title="" /> +Thus the charge is got in +absolute centimetre-gramme-second units in terms of +the mass <i>m</i> obtained by ordinary weighing, and <i>l</i> and <i>d</i> +obtained by easy and exact measurements.</p> + +<p>If one of the balls be now taken away without discharging +the other, and the latter be placed in the field +of a large electrified spherical conductor, the fibre will +be deflected from the vertical by the force on the ball. +Let the two centres be now on the same level. That +force is got at once from the angle of deflection (which is<span class='pagenum'><a name="Page_248" id="Page_248">248</a></span> +easily observed), the charge on the ball, and the value +of <i>m</i>. The electric field-intensity is obtained by +dividing the value of the force by <i>q</i>. The field intensity +multiplied by <i>D</i>, the distance apart in cms. of the +centres of the ball and the conductor, gives the potential +at the centre of the ball in C.G.S. units. Multiplication +again by <i>D</i> gives the charge on the conductor.</p> + +<p>When it made its first Report in 1862 (to the meeting +at Cambridge) the committee consisted of Professors +A. Williamson, C. Wheatstone, W. Thomson, W. H. +Miller, Dr. A. Matthiessen, and Mr. F. Jenkin. At +the next meeting, at Newcastle, it had been augmented +by the addition of Messrs. Balfour Stewart, C. W. +Siemens, Professor Clerk Maxwell, Dr. Joule, Dr. +Esselbach, and Sir Charles Bright. The duty with +which the committee had been charged was that of +constructing a suitable standard of resistance. A reference +to the account given in Chapter X above, of the +derivation of what came to be called the electromagnetic +unit of difference of potential, or electromotive force, +by means of a simple magneto-electric machine—a +disk turning on a uniform magnetic field, or the simple +rails and slider and magnetic field arrangement there +described—will show how from this unit and the +electromagnetic unit of current (there also defined) the +unit of resistance is defined. It is the resistance of +the circuit of slider, rails, and connecting wire, when +with this electromagnetic unit of electromotive force +the unit of current is made to flow.</p> + +<p>This was one clear and definite way of defining the +unit of current, and of attaining the important object +of connecting the units in such a way that the rate of +working in a circuit, or the energy expended in any +time, should be expressed at once in ordinary dynamical<span class='pagenum'><a name="Page_249" id="Page_249">249</a></span> +units of activity or energy. A considerable number of +proposals were discussed by the committee; but it was +finally determined to take the basis here indicated, and +to realise a standard of resistance in material of constant +and durable properties, which should have some simple +multiple of the unit of resistance, in the system of +dynamical units based on the centimetre as unit of +length, the gramme as unit of mass, and the second +as unit of time—the so-called C.G.S. system. The +comparison of the different metals and alloys available +was a most important but exceedingly laborious +series of investigations, carried out mainly by Dr. +Matthiessen and Professor Williamson.</p> + +<p>Professor Thomson suggested to the committee the +celebrated method of determining the resistance of a +circuit by revolving a coil, which formed the main +part of the circuit about a vertical axis in the earth's +magnetic field. An account of the experiments made +with this method is contained in the Report of 1863. +They were carried out at King's College, London, +where Maxwell was then Professor of Experimental +Physics, by Maxwell, Balfour Stewart, and Fleeming +Jenkin. The theoretical discussion and the description +of the experiments was written by Maxwell, the details +of the apparatus were described by Jenkin.</p> + +<p>The principle of the method is essentially the same +as that of the simple magneto-electric machine, to +which reference has just been made. Two parallel +coils of wire were wound in channels cut round rings +of brass, which, however, were cut across by slots +filled with vulcanite, to prevent induced currents from +circulating in the brass. These coils were mounted +in a vertical position and could be driven as a rigid +system, at a constant measured speed, about a vertical<span class='pagenum'><a name="Page_250" id="Page_250">250</a></span> +axis passing through the centre of the system. Between +the coils at this centre was hung, from a steady support, +a small magnetic needle by a single fibre of silk; and a +surrounding screen prevented the needle and suspension +from being affected by currents of air.</p> + +<p>The ends of the coil were connected together so +that the whole revolved as a closed circuit about the +vertical axis. When the coil system was at right +angles to the magnetic meridian there was a magnetic +induction through it of amount <i>AH</i>, where <i>A</i> denotes +the effective area of the coils, and <i>H</i> the horizontal +component of the earth's magnetic field. By one +half-turn the coil was reversed with reference to this +magnetic induction, and as the coil turned an induced +current was generated, which depended at any instant +on the rate at which the magnetic induction was varying +at the instant, on the inductive electromotive force +due to the varying of the current in the coil itself, and +on the resistance of the circuit. A periodic current +thus flowed in one direction <i>relatively to the coil</i> in one +half-turn from a position perpendicular to the magnetic +meridian, and in the opposite direction in the next +half-turn. But as the position of the coil was reversed +in every half-turn as well as the current in it, the current +flowed on the whole in the same average direction +relatively to the needle, and but for self-induction +would have had its maximum value always when the +plane of the coil was in the magnetic meridian.</p> + +<p>The needle was deflected as it would have been +by a certain average current, and the deflection was +opposed by the action of the earth's horizontal magnetic +field <i>H</i>. But this was the field cut by the coil +as it turned, and therefore (except for a small term<span class='pagenum'><a name="Page_251" id="Page_251">251</a></span> +depending on the turning of the coil in the field of +the needle) the value of <i>H</i> did not appear in the result, +and did not require to be known.</p> + +<p>Full details of the theory of this method and of the +experiments carried out to test it will be found in +various memoirs and treatises<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">23</a>; but it must suffice +here to state that the resistance of the coil was determined +in this way, by a large series of experiments, +before and after every one of which the resistance was +compared with that of a German-silver standard. The +resistance of this standard therefore became known +in absolute units, and copies of it, or multiples or +sub-multiples of it, could be made.</p> + +<p>A unit called the B.A. unit, which was intended to +contain 10<sup>9</sup> C.G.S. electromagnetic units of resistance, +was constructed from these experiments, and copies of +it were soon after to be found in nearly all the physical +laboratories of the world. Resistance boxes were +constructed by various makers, in which the coils were +various multiples of the B.A. unit, so that any resistance +within a certain range could be obtained by +connecting these coils in series (which was easily done +by removing short circuiting plugs), and thus the +absolute units of current electromotive force and +resistance came into general use.</p> + +<p>In 1881 Lord Rayleigh and Professor Schuster +carried out a very careful repetition of the British +Association experiments with the same apparatus at +the Cavendish Laboratory, and obtained a somewhat +different result. They found that the former result +<span class='pagenum'><a name="Page_252" id="Page_252">252</a></span>was about 1.17 per cent. too small. Lord Rayleigh next +carried out an independent set of experiments by the +same method with improved apparatus, and found that +this percentage error must be increased to about 1.35.</p> + +<p>It may be noticed here that the simple disk machine, +of Thomson's illustration of the absolute unit of +electromotive force, has been used by Lorenz to give +a method of determining resistance which is now +recognised as the best of all. It is sketched here that +the reader may obtain some idea of later work on this +very important subject; work which is a continuation +of that of the original British Association Committee by +their successors. A circuit is made up of a standard +coil of wire, the ends of which are made to touch at the +circumference and near the centre of the disk, which is +placed symmetrically with respect to a cylindrical coil, +and within it. A current is sent round this coil from +a battery, and produces a magnetic field within the +coil, the lines of magnetic force of which pass across +the plane of the disk. This current, or a measured +fraction of it, is also made to flow through the standard +coil. The disk is now turned at a measured speed +about its axis, so that the electromotive force due to +the cutting of the field tends to produce a current in +the standard coil of wire. The electromotive force +of the disk is made to oppose the potential difference +between the ends of this coil due to the current, so +that no current flows along the disk or the wires connecting +it with the standard coil. The magnetic field +within the coil can be calculated from the form and +dimensions of the coil and the current in it (supposed +for the moment to be known), and the electromotive +force of the disk is obtained in terms of its dimensions<span class='pagenum'><a name="Page_253" id="Page_253">253</a></span> +and its speed and the field intensity. But this electromotive +force, which is proportional to the current in +the coil, is equal to the product of the resistance of +the wire and the same current, or a known fraction of +it. Thus the current appears on both sides of the +equation and goes out, and the value of the resistance +is found in absolute units.</p> + +<p>Lord Rayleigh obtained, by this method, a result +which showed that the B.A. unit was 1.323 per cent. +too small; and exact experiments have been made by +others with concordant results. Values of the units +have been agreed on by International Congresses as +exact enough for general work, and with these units +all electrical researches, wherever made, are available +for use by other experimenters.</p> + +<p>A vast amount of work has been done on this +subject during the last forty years, and though the +value of the practical unit of resistance—10<sup>9</sup> C.G.S. +units, now called the "ohm"—is taken as settled, and +copies can now be had in resistance boxes, or separately, +adjusted with all needful accuracy, at the National +Physical Laboratory and at the Bureau of Standards +at Washington, and elsewhere, experiments are being +made on the exact measurement of currents; while a +careful watch is kept on the standards laid up at these +places to see whether any perceptible variation of their +resistance takes place with lapse of time.</p> + +<p>The British Association Committee also worked out +a complete system of units for all electrical and magnetic +quantities, and gave the first systematic statement +of their relations, that is, of the so-called dimensional +equations of the quantities. This will be found in the +works to which reference has already been made (p. <a href="#Page_251">251</a>).</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_254" id="Page_254">254</a></span></p> + +<h3>CHAPTER XIV</h3> + +<h4>THE BALTIMORE LECTURES</h4> + +<p><span class="smcap">The</span> Baltimore Lectures were delivered in 1884 at +Johns Hopkins University, soon after the Montreal +meeting of the British Association. The subject +chosen was the Wave Theory of Light; and the idea +underlying the course was to discuss the difficulties of +this theory to "Professorial fellow-students in physical +science." A stenographic report of the course was +taken by Mr. A. S. Hathaway, and was published soon +after. The lectures were revised by Lord Kelvin, and +the book now known as <i>The Baltimore Lectures</i> was +published just twenty years later (in 1904) at the +Cambridge University Press. It is absolutely impossible +in such a memoir as the present to give any account of +the discussions contained in the lectures as now published. +The difficulties dealt with can for the most +part only be understood by those who are acquainted +with the wave theory of light in its details, and such +readers will naturally go direct to the book itself.</p> + +<p>Some of the difficulties, however, were frequently +alluded to in Lord Kelvin's ordinary lectures, and all +his old students will remember the animation with +which he discussed the apparent anomaly of a medium +like the luminiferous ether, which is of such enormous +rigidity that (on the elastic solid theory) a wave of transverse +oscillation is propagated through it with a speed of<span class='pagenum'><a name="Page_255" id="Page_255">255</a></span> +3 × 10<sup>10</sup> centimetres (186,000 miles) per second, and +yet appears to offer no impediment to the slow motion +of the heavenly bodies. For Lord Kelvin adopted the +elastic solid theory of propagation of light as "the only +tenable foundation for the wave theory of light in the +present state of our knowledge," and dismissed the +electromagnetic theory (his words were spoken in 1884, +it is to be remembered) with the statement of his +strong view that an electric displacement perpendicular +to the line of propagation, accompanied by a magnetic +disturbance at right angles to both, is inadmissible.</p> + +<p>And he goes on to say that "when we have an +electromagnetic theory of light," electric displacement +will be seen as in the direction of propagation, with +Fresnelian vibrations perpendicular to that direction. +In the preface, of date January 1904, the insufficiency of +the elastic solid theory is admitted, and the question of +the electromagnetic theory again referred to. He says +there that the object of the Baltimore Lectures was to +ascertain how far the phenomena of light could be +explained within the limits of the elastic solid theory. +And the answer is "everything <i>non-magnetic; nothing +magnetic</i>." But he adds, "The so-called electromagnetic +theory of light has not helped us hitherto," and that +the problem is now fully before physicists of constructing +a "comprehensive dynamics of ether, electricity, +and ponderable matter which shall include electrostatic +force, magnetostatic force, electromagnetism, electrochemistry, +and the wave theory of light."</p> + +<p>All this is exceedingly interesting, for it seems to +make clear Lord Kelvin's attitude with respect to the +electromagnetic theory of Maxwell, which is now +regarded by most physicists as affording on the whole<span class='pagenum'><a name="Page_256" id="Page_256">256</a></span> +a satisfactory account, if not a dynamical theory +in the sense understood by Lord Kelvin, of light-propagation. +That there is an electric displacement +perpendicular to the direction of propagation and a +magnetic displacement (or motion) perpendicular to +both seems proved by the experiments of Hertz, and +the velocity of propagation of these disturbances has +been found to be that of light. Of course it remains +to be found out in what the electric and magnetic +changes consist, and whether the ether has or has not +an atomic structure. Towards the answer to this +question on electromagnetic presuppositions some +progress has already been made, principally by Larmor. +And, after all, while we may imagine that we know +something more definite of dynamical actions on +ponderable matter, it is not quite certain that we do: +we are more familiar with them, that is almost all. +We know, for example, that at every point in the +gravitational field of the earth we may set up a +gravitation vector, or field-intensity; for a particle of +matter there is subjected to acceleration along that +direction. But of the <i>rationale</i> of the action we know +nothing, or next to nothing. So we set up electric and +magnetic vectors in an insulating medium, corresponding +to electric and magnetic effects which we can +observe; and it is not too much to say that we know +hardly less in this case than we do in the other, of the +inner mechanism of the action of which we see the +effects.</p> + +<p>Returning to the difficulty of the elastic solid theory, +that while its rigidity is enormous, it offers no obstacle +to the planets and other heavenly bodies which move +through it, it may be interesting to recall how Lord<span class='pagenum'><a name="Page_257" id="Page_257">257</a></span> +Kelvin used to deal with it in his elementary lectures. +The same discussion was given in the Introductory +Lecture at Baltimore. The difficulty is not got over +by an explanation of what takes place: it is turned by +showing that a similar difficulty exists in reconciling +phenomena which can be observed every day with such +ordinary materials as pitch or shoemakers' wax. A +piece of such wax can be moulded into a tuning-fork +or a bell, and will then, if struck, sound a musical note +of definite pitch. This indicates, for rapidly alternating +deformations started by a force of short duration, +the existence of internal forces of the kind called elastic, +that is, depending on the amount of deformation caused, +not on the rate at which the deformation is increasing +or diminishing, as is the case for the so-called "viscous +forces" which are usually displayed by such material. +But the tuning-fork or bell, if left lying on the table, +will gradually flatten down into a thin sheet under +only its own weight. Here the deformation is opposed +only by viscous forces, which, as the change is very +slow, are exceedingly small.</p> + +<p>But let a large slab of it, three or four inches thick, +be placed in a glass jar ten or twelve inches in diameter, +already partly filled with water, and let some ordinary +corks be imprisoned beneath, while some lead bullets +are laid on the upper surface. After a month or two +it will be found that the corks have disappeared from +the water into the wax, and that the orifices which +they made in entering it have healed up completely; +similarly the bullets have sunk down into the slab, +leaving no trace behind. After two or three months +more, the corks will be seen to be bursting their way +out through the upper surface of the slab, and the<span class='pagenum'><a name="Page_258" id="Page_258">258</a></span> +bullets will be found in the water below. The very +thing has taken place that would have happened if +water had been used instead of pitch, only it has taken +a very much longer time to bring it about. The corks +have floated up through the wax in consequence of +hydrostatic upward force exerted by the wax acting as a +fluid; and the bullets have sunk down in consequence +of the excess of their weights above the upward +hydrostatic force exerted on them as on the corks. +The motion in both cases has been opposed by the +viscous forces called into play.</p> + +<p>The application of this to the luminiferous ether is +immediate. Let the ether be regarded as a substance +which can perform vibrations only "when times and +forces are suitable," that is, when the forces producing +distortion act for only an infinitesimal time (as in the +starting of the tuning-fork by a small blow), and are +not too great. Vibrations may be set up locally, and +the medium may have a true rigidity by which they +are propagated to more remote parts; that is to say, +waves travel out from the centre of disturbance. On +the other hand, if the forces are long continued, even if +they be small, they produce continuously increasing +change of shape. Thus the planets move seemingly +without resistance.</p> + +<p>The conclusion is that the apparently contradictory +properties of the ether are no more mysterious than the +properties of pitch or shoemakers' wax. And, after all, +matter is still a profound mystery.</p> + +<p>Dynamical illustrations, which old Glasgow students +will recognise, appear continually in the lectures. +They will remember, almost with affection, the system +of three <i>particles</i> (7 lb. or 14 lb. weights!) joined<span class='pagenum'><a name="Page_259" id="Page_259">259</a></span> +together in a vertical row by stout spiral springs of +steel, which were always to be taken as massless, and +will recall Lord Kelvin's experiments with them, +demonstrating the three modes of vibration of a system +of three masses, each of which influenced those next it +on the two sides. Here they will find the problem +solved for any number of particles and intervening +springs, and the solution applied to an extension of the +massive molecule which von Helmholtz imbedded in the +elastic ether, and used to explain anomalous dispersion. +A highly complex molecule is suggested, consisting of +an outer shell embedded in the ether as in the simpler +case, a second shell within that connected to the outer +by a sufficient number of equal radial springs, a third +within and similarly connected to the second by radial +springs, and so on. This molecule will have as many +modes of vibration as there are sets of springs, and can +therefore impart, if it is set into motion, a complex +disturbance to the ether in which it is imbedded.</p> + +<p>The modification of this arrangement by which +Lord Kelvin explained the phosphorescence of such +substances as luminous paint is also described, and +will be recognised by some as an old friend. A +number, two dozen or so, of straight rods of wood +eighteen inches long are attached to a steel wire four +or five inches apart, like steps on a ladder made with a +single rope along the centres of the steps. The wire +is so attached to each rod that the rod must turn with +the wire if the latter is twisted round. Each rod is +loaded with a piece of lead at each end to give it more +moment of inertia about the wire. The wire, with +this "ladder" attached to it, is rigidly attached to the +centre of a cross-bar at the top, which can be made to<span class='pagenum'><a name="Page_260" id="Page_260">260</a></span> +swing about the wire as an axis and so impart twisting +vibrations to the wire in a period depending on this +driver. Sliding weights attached to the bar enable its +moment of inertia to be changed at pleasure. The +lower end of the wire carries a cross-bar with two +vanes, immersed in treacle in a vessel below. When +the period of the exciter was very long the waves of +torsion did not travel down the "ladder," but when +the period was made sufficiently short the waves +travelled down and were absorbed in the treacle below. +In the former case the vibrations persisted; the case +was analogous to that of phosphorescence.</p> + +<div class="figcenter" style="width: 300px; position: relative;"><a name="f18" id="f18"></a><img src="images/fig18.png" width="300" height="473" alt="Fig. 18." title="" /> +<p class="caption"><span class="smcap">Fig. 18.</span></p></div> + +<p>Incidentally a full and very attractive account of the +elastic solid theory is given in these lectures, accompanied +as it is by characteristic digressions on points of<span class='pagenum'><a name="Page_261" id="Page_261">261</a></span> +interest which suggest themselves, and on topics on +which the lecturer held strong opinions, such, for +example, as the absurd British system of weights and +measures. The book reads in many places like a +report of some of the higher mathematical lectures +which were given every session at Glasgow; and on +that account, if on no other, it will be read by the +old students of the higher class with affectionate +interest. But the discussions of the great fundamental +difficulty presented at once by dispersion—the fact, +that is, that light of different wave lengths has different +velocities in ordinary transparent matter—the +discussions of the various theories of dispersion that +have been put forward, the construction of the molecules, +gyrostatic and non-gyrostatic, with all their +remarkable properties, which Lord Kelvin invents in +order to frame a dynamical mechanism which will +imitate the action of matter as displayed in the complex +manifestations of the optical phenomena, not only of +isotropic matter, but of crystals, will ever afford instruction +to every mathematician who has the courage +to attack this subject, and remain as a monument to +the extraordinary genius of their author.</p> + +<p>A subject is touched on in these lectures which has +not been dealt with in the present review of Lord +Kelvin's work. By four lines of argument—by the heat +of combination of copper and zinc, together with the +difference of electric potential developed when these +metals are put in contact, from the thickness of a capillary +film of soap and water (measured by Rücker and +Reinold) just before it gives way, and the work spent in +stretching it, from the kinetic theory of gases and the +estimated length of free path of a particle (given also by<span class='pagenum'><a name="Page_262" id="Page_262">262</a></span> +Loschmidt and by Johnstone Stoney), and from the +undulatory theory of light—Lord Kelvin estimated +superior and inferior limits to the "size of the atoms" +of bodies, or, more properly speaking, of the molecular +structure of the matter. We cannot discuss these arguments—and +they can be read at leisure by any one who +will consult Volume I (Constitution of Matter) of Lord +Kelvin's <i>Popular Lectures and Addresses</i>, for his Royal +Institution Lecture on the subject, there given in full—but +we may state his conclusion. Let a drop of water, +a rain drop, for example, be magnified to the size of the +earth, that is, from a sphere a quarter of an inch, or +less, in diameter to a sphere 8000 miles in diameter, +and let the dimensions of the molecular structure be +magnified in the same proportion. "The magnified +structure would be more coarse-grained than a heap of +small shot, but probably less coarse-grained than a heap +of cricket-balls."</p> + +<p>Of course, it is not intended here to convey the idea +that the molecules are spheres like shot or cricket-balls; +they undoubtedly have a structure of their own. And +no pronouncement is made as to the divisibility or +non-divisibility of the molecules. All that is alleged is +that if the division be carried to a minuteness near to +or beyond that of the dimensions of the structure, +portions of the substance will be obtained which have +not the physical properties of the substance in bulk.</p> + +<p>The recent interesting researches of chemists and +physicists into phenomena which seem to demonstrate +the disintegration, not merely of molecules, but even of +the atomic structure of matter, attracted Lord Kelvin's +attention in his last years, and <i>suo more</i> he endeavoured +to frame dynamical explanations of electronic (or, as he<span class='pagenum'><a name="Page_263" id="Page_263">263</a></span> +preferred to call it, "electrionic") action. But though +keenly interested in all kinds of research, he turned +again and again to the older theories of light, and his +dynamical representations of the ether and of crystals, +with renewed vigour and enthusiasm.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_264" id="Page_264">264</a></span></p> + +<h3>CHAPTER XV</h3> + +<h4>SPEED OF TELEGRAPH SIGNALLING—LAYING OF SUBMARINE +CABLES—TELEGRAPH INSTRUMENTS—NAVIGATIONAL +INSTRUMENTS, COMPASS AND +SOUNDING MACHINE</h4> + +<h3><span class="smcap">Theory of Signalling</span></h3> + +<p><span class="smcap">When</span> the question of laying an Atlantic cable began +to be debated in the middle of the nineteenth century, +Professor Thomson undertook the discussion of the +theory of signalling through such a cable. It was not +generally understood by practical telegraphists that the +conditions of working would be very different from +those to which they were accustomed on land lines, +and that the instruments employed on such lines would +be useless for a cable. Such a cable consists of a +copper conductor separated from the sea-water by a +coating of gutta-percha; it forms an elongated Leyden +jar of very great capacity, which, when a battery is +connected to one end of the conducting core, is +gradually charged up, first at that end, and later and +later at greater distances from it, and then is gradually +discharged again when the battery is withdrawn and +the end of the conductor connected to earth. Here, +again, an application of Fourier's analysis solved the +problem, which, with certain modifications, and on +the supposition that the working is slow, is essentially +the same problem as the diffusion of heat along a<span class='pagenum'><a name="Page_265" id="Page_265">265</a></span> +conducting bar, or the diffusion of a salt solution +along a column of water. The signals are retarded +(and this was one of the results of the investigation) +in such a manner "that the time required to reach a +stated fraction of the maximum strength of current at +the remote end," when a given potential difference is +applied at the other, or home end, is proportional to +the product of the capacity and resistance of the cable, +each taken per unit of the length, and also proportional +to the square of the length of cable. In other words, +the retardation is proportional to the product of the +resistance of the copper conductor and the total +capacity of the cable. This gave a practical rule of +great importance for guidance in the manufacture of +submarine cables. The conductor should have the +highest conductivity obtainable, and should therefore +be of pure copper; the insulating covering should, +while forming a nearly absolutely non-conducting +sheath, have as low a specific inductive capacity as +possible. The first of these conditions ran counter to +some views that had been put forward, to the effect +that it was only necessary to have the internal conductor +highly conducting on its surface; and some +controversy on the subject ensued. The inverse square +law, as it was called, was vehemently called in question, +from a mistaken interpretation of some experiments +that were made to test it. For if the potential at the +home end be regularly altered, according to the simple +harmonic law, so that the number of periods of oscillation +in a second is <i>n</i>, the changes of potential are +propagated with velocity 2√(π<i>n</i>⁄<i>cr</i>), where <i>c</i> and <i>r</i> are +the capacity and resistance of the cable, each taken +per unit length. In this case, for a long cable, there<span class='pagenum'><a name="Page_266" id="Page_266">266</a></span> +is a velocity of propagation independent of the length; +and this fact seems to have misled the experimenters. +Thomson's view prevailed, and the result was the +establishment, first by Thomas Bolton & Sons, Stoke-on-Trent, +of mills for the manufacture of high +conductivity copper, which is now a great industry.</p> + +<p>The Fourier mathematics of the conduction of heat +along a bar suffices to solve the problem, so long as the +signalling is so slow as not to bring into play electromagnetic +induction to any serious extent. For rapid +signalling in which very quick changes of current are +concerned the electromotive forces due to the growth +or dying out of the current would be serious, and the +theory of diffusion would not apply. But ordinary +cable working is quite slow enough to enable such +electromotive forces to be disregarded.</p> + +<h3><span class="smcap">Laying of First American Cables</span></h3> + +<p>The first cable of 1858 was laid by the U.S. frigate +<i>Niagara</i> and H.M.S. <i>Agamemnon</i>, after having been +manufactured with all the precautions suggested by +Professor Thomson's researches. It is hard to realise +how difficult such an enterprise was at the time. The +manufacture of a huge cable, the stowage of it in cable +tanks on board the vessels, the invention of laying and +controlling and picking-up machinery had to be faced +with but little experience to guide the engineers. +Here again Thomson, by his knowledge of dynamics +and true engineering instinct, was of great assistance. +In 1865 he read a very valuable paper on the forces +concerned in the laying and lifting of deep-sea cables, +showing how the strains could be minimised in various<span class='pagenum'><a name="Page_267" id="Page_267">267</a></span> +practical cases of importance—for example, in the lifting +of a cable for repairs.</p> + +<p>A first Atlantic cable had been partly laid in 1857 +by the <i>Niagara</i>, when it broke in 2000 fathoms of +water, about 330 miles from Valentia, where the laying +had begun. An additional length of 900 miles was +made, and the enterprise was resumed. This time it +was decided that the two vessels, each with half of +the cable on board, should meet and splice the cable in +mid-ocean, and then steam in opposite directions, the +<i>Agamemnon</i> towards Valentia, the <i>Niagara</i> towards +Newfoundland. Professor Thomson was engineer in +charge of the electrical testing on board of the <i>Agamemnon</i>. +After various mishaps the cable was at last safely +laid on August 6, 1858, and congratulations were +shortly after exchanged between Great Britain and the +United States. On September 6 it was announced +that signals had ceased to pass, and an investigation of +the cause of the stoppage was undertaken by Professor +Thomson and the other engineers. The report stated +that the cable had been too hastily made, that, in fact, +it was not good enough, and that the strains in laying +it had been too great and unequal. It was found +impossible to repair it, so that there was no option but +to abandon it.</p> + +<p>This cable probably suffered seriously from the +violent means which seem to have been employed to +force signals through it. Now only a very moderate +difference of potential is applied to a cable at the sending +end, and speed of signalling is obtained by the use +of instruments, the moving parts of which have little +inertia, and readily respond to only an exceedingly +feeble current.<span class='pagenum'><a name="Page_268" id="Page_268">268</a></span></p> + +<p>A second cable was made and laid in 1865 by the +Great Eastern, which could take on board the whole +at once and steam from shore to shore. It was also +well adapted for cable work through having both screw +and paddles. As Thomson points out, "steerage +way" could be got on the vessel by driving the screw +ahead, so as to send a stream of water astern towards +the rudder, while the paddles were driven astern to +prevent the ship from going ahead. This was of great +advantage in manœuvring on many occasions.</p> + +<p>This cable also broke, but a third was laid successfully +in 1866 by the same vessel, and the second was +recovered and repaired, so that two good cables were +secured for commercial working. On both expeditions +Professor Thomson acted as electrical engineer, and +received the honour of knighthood and the thanks +of the Anglo-American Telegraph Company on his +return home, when he was also presented with the +freedom of the city of Glasgow.</p> + +<p>He afterwards acted as engineer for the French +Atlantic Cable, for the Brazilian and River Plate +Company, and for the Commercial Company, whose +two new Atlantic cables were laid in 1882-4.</p> + +<h3><span class="smcap">Mirror Galvanometer and Siphon Recorder</span></h3> + +<p>Since whatever the potential applied at the sending +end of the cable might be (and, of course, as has been +stated, this potential had to be kept to as low a value +as possible) the current at the receiving end only rose +gradually, it was necessary to have as delicate a receiving +instrument as possible, so that it would quickly +respond to the growing and still feeble current. For<span class='pagenum'><a name="Page_269" id="Page_269">269</a></span> +unless the cable could be worked at a rate which would +permit of charges per word transmitted which were +within the reach of commercial people, it was obvious +that the enterprise would fail of its object. And as a +cable could not cost less than half a million sterling, +the revenue to be aimed at was very considerable. +This problem Thomson also solved by the invention +of his mirror galvanometer. The suspended magnet +was made of small pieces of watch-spring cemented +to a small mirror, so that the whole moving part +weighed only a grain or two. Its inertia, or resistance +to being set into motion, was thus very small, and it +was hung by a single fibre of silk within a closed +chamber at the centre of the galvanometer coil. A +ray of light from a lamp was reflected to a white paper +scale in front of the mirror, which as it turned caused +a spot of illumination to move along the paper. A +motion of this long massless index to the left was +regarded as a dot, a motion to the right as a dash, and +the Morse alphabet could therefore be employed. This +instrument was used in the 1858 cable expedition, and +a special form of suspension was invented for it by +Thomson, to enable it to be used on board ship. The +suspension thread, instead of being held at one end only, +was stretched from top to bottom of the chamber in +which the needle hung, and kept tight by being secured +at both ends. Thus the minimum of disturbance was +caused to the mirror by the rolling or pitching of the +ship.</p> + +<p>The galvanometer was also enclosed in a thick iron +case to guard it against the magnetic field due to the +iron of the ship. The "iron-clad galvanometer" first +used in submarine telegraphy (on the 1858 expedition<span class='pagenum'><a name="Page_270" id="Page_270">270</a></span> +in the U.S. frigate <i>Niagara</i>) is in the collection of historical +apparatus in the Natural Philosophy Department +of the University of Glasgow.</p> + +<p>The mirror galvanometer then invented has become +one of the most useful instruments of the laboratory. +Mirror deflection is now used also for the indicators of +many kinds of instruments.</p> + +<p>The galvanometer was replaced later by another +invention of Professor Thomson—the siphon recorder. +Here a small and delicate pen was formed by a piece +of very fine glass tube (vaccination tubing, in fact) in +the form of a siphon, of which the shorter end dipped +into an ink-bottle, while the other end wrote the +message in little zig-zag notches on a ribbon of paper +drawn past it by machinery. The siphon was moved +to and fro by the signalling currents, which flowed in +a small coil hung between the poles of an electromagnet, +excited by a local battery, and the ink was +spirted in a succession of fine drops from the pen +to the paper. This was accomplished by electrifying +the ink-bottle and ink by a local electrical machine, +and keeping the paper in contact with an uninsulated +metal roller. Electric attraction between the electrified +ink and the unelectrified paper thus drew the ink-drops +out, and the pen, which never touched the paper, was +quite unretarded by friction. Both these instruments +had the inestimable advantage that the to and fro +motions of the spot of light or the pen took place +independently of ordinary earth-currents through the +cable.</p> + +<p>The arrangement of magnet and suspended coil in +this instrument has become widely known as that of +the "d'Arsonval galvanometer." This application was<span class='pagenum'><a name="Page_271" id="Page_271">271</a></span> +anticipated by Thomson, and is distinctly mentioned +in his recorder patent, long before such galvanometers +were ever used. It was later proposed by several +experimenters before M. d'Arsonval.</p> + +<p>It is not too much to say that, by his discussion of +the speed of signalling, his services as an electrical +engineer, and especially by his invention of instruments +capable of responding to very feeble currents, Thomson +made submarine telegraphy commercially possible. +Later he entered into partnership with Mr. C. F. +Varley and Professor Fleeming Jenkin. A combination +of inventions was made by the firm: Varley had +patented a method of signalling by condensers, and +Jenkin later suggested and patented an automatic key +for "curb-sending" on a cable—that is, signalling by +placing one pole of the battery for an interval a little +shorter than the usual one to the line, and then reversing +the battery for the remainder. This gave sharper +signals, as the reversal helped to discharge the cable +more rapidly than it would have been by the mere +connection to earth between two signals. The firm +of Thomson, Varley & Jenkin took a prominent part +in cable work; and Thomson and Jenkin acted as +engineers for many large undertakings. They employed +a staff of young electricians at the cable-works +at Millwall and elsewhere, keeping watch over the +cable during manufacture, and sent them to sea as +representatives and assistants to perform similar duties +during the process of cable-laying. On their staff +were many men who have come to eminence in +electrical and engineering pursuits in later life.<span class='pagenum'><a name="Page_272" id="Page_272">272</a></span></p> + +<h3><span class="smcap">Mariners' Compass and Sounding Machine</span></h3> + +<p>After the earlier Atlantic expeditions Sir William +Thomson turned his attention to the construction of +navigational instruments, and invented the mariner's +compass and wire-sounding apparatus which are now +so well known. He had come to the conclusion that +the compasses in use had much too large needles +(some of them bar-magnets seven or eight inches +long!) to respond quickly and certainly to changes of +course, and, what was still more serious, to admit of +the application of correcting magnets, and of masses of +soft-iron to annul the action of the magnetism of the +ship.</p> + +<p>The compass card consists of a paper ring, on which +the "points" and degrees are engraved in the ordinary +way, and is kept circular by a light ring of +aluminium. Threads of silk extend radially from the +rim to a central boss of aluminium in which is a cap of +aluminium. In the top of the cap is a sapphire bearing, +which rests on an iridium point projecting upward +from the compass bowl. Eight magnets of glass-hard +steel, from 3¼ inches to 2 inches long, and about the +thickness of a knitting-needle, which form the compass +needle, are strung like the steps of a rope ladder, +on two silk threads attached to four of the radial +threads.</p> + +<p>The weight of the card is extremely small—only +170½ grains; that is less than <sup>2</sup>⁄<sub>5</sub> of an ounce. But the +matter is not merely made small in amount; it is +distributed on the whole at a great distance from the +axis; consequently the period of free vibration is long, +and the card is very steady. The great lightness of<span class='pagenum'><a name="Page_273" id="Page_273">273</a></span> +the card also causes the error due to friction on the +point of support to be very small.</p> + +<p>The errors of the compass in an iron ship are mainly +the semicircular error and the quadrantal error. We +can only briefly indicate how these arise and how they +are corrected. The ship's magnetism may be considered +as partly permanent, and partly inductive. The former +changes only very slowly, the latter alters as the ship +changes course and position. For the ship is a combination +of longitudinal, transverse, and vertical girders +and beams. As a whole it is a great iron or steel +girder, but its structure gives it longitudinal, transverse, +and vertical magnetisation. This disturbs the compass, +which is also affected by the magnetisation of the iron +or steel masts and spars, or of iron or steel carried as +cargo.</p> + +<p>The semicircular error is due to a great extent to +permanent magnetism, but also in part to induced +magnetism. It is so called because when the ship's +head is turned through 360°, the error attains a +maximum on two courses 180° apart. It may amount +to over 20° in an ordinary iron vessel, and to 30° or +40° in an armour-clad. It is corrected by two sets +of steel magnets placed with their centres under the +needle in the binnacle. One set have their lengths +fore and aft, the others in the thwart-ship direction. +These magnets annul the error on the north and south +and on the east and west courses, due to the two horizontal +components of magnetic force produced mainly +by the permanent magnetism of the ship. A regular +routine of swinging the ship when marks on the shore +(the true bearings of which from the ship are known) +are available, is followed for the adjustment.<span class='pagenum'><a name="Page_274" id="Page_274">274</a></span></p> + +<p>The quadrantal error is so called because its maxima +are found on four compass courses successively a +quadrant, or 90°, from one another. It amounts in +general to from 5° to 10° at most. It is due to induced +magnetism, and is corrected by a pair of soft-iron +spheres, placed on the two sides of the compass with +their centres in a line transverse to the ship, through +the centre of the compass needle. There are, however, +exceptional cases in which they are placed in the fore +and aft line one afore, the other abaft, the needle. +When the quadrantal error has once been annulled it +is always zero, for as the induced magnetism changes, +so does that of the spheres, and the adjustment remains +good. In a new ship the permanent magnetism slowly +alters, and so the semicircular correction has to be +improved from time to time by changing the magnets.</p> + +<p>These adjustments are not quite all that have to be +made; but enough has been stated to show how the +process of compensation can be carried out with the +Thomson compass. The immensely-too-large magnets +used formerly as compass needles, through a mistaken +notion, apparently, that more directive force would be +got by their means, rendered the quadrantal adjustment +an impossibility. The card swinging round brought +the large needles into different positions relatively to +the iron balls, when these were used, and exerted an inductive +action on them which reacted on the needles, +producing more error, perhaps, than was corrected.</p> + +<p>Thomson invented also an instrument called a +"deflector," by which it is possible to adjust a compass +when sights of sun or stars, or bearings of terrestrial +objects, cannot be obtained. By means of it the +directive forces on the needles on different courses<span class='pagenum'><a name="Page_275" id="Page_275">275</a></span> +can be compared. Then the adjustment is made by +placing the correctors so that the directive force is as +nearly as may be the same on all courses. The +compass is then quite correct.</p> + +<p>The theory of deviations of the compass, it is right +to say, was discussed first partially by Poisson, but +afterwards very completely and elegantly by the late +Mr. Archibald Smith of Jordanhill, whose memoirs, +now incorporated in the <i>Admiralty Manual of Deviations +of the Compass</i>, led to Lord Kelvin's inventions.</p> + +<p>Lord Kelvin's compass is now almost universally in +use in the merchant service of this country, and in +most of the navies of the world. It has added greatly +to the certainty and safety of navigation.</p> + +<p>The sounding machine is also well known. At +first pianoforte wire was used for deep-sea sounding by +Commodore Belknap of the U.S. Navy, and by others, +on Sir William Thomson's recommendation. Finally, +a form of machine was made by which a sinker could +be lowered to the bottom of the sea and brought up +again in a few minutes; so that it was possible to take +a sounding without the long delay involved in the old +method with a reel of hemp-rope, which often tempted +shipmasters to run risks of going ashore rather than +stop the ship for the purpose. The wire offered little +resistance to motion through the water, and by a +proper winding machine, with brake to prevent the +wire from running out too fast and kinking, when it +was almost certain to break, one man could quickly +sound and heave up again, while another attended to +the wire and sinker. A gauge consisting of a long +quill-tube closed at the upper end, and coated inside +with chromate of silver, showed by the action of the<span class='pagenum'><a name="Page_276" id="Page_276">276</a></span> +sea-water on the coating how far the water had passed +up the tube, compressing the air above it; and from +this, by placing the tube along a wooden rule properly +graduated, the depth was read off at once. With the +improved machine a ship approaching the shore in +thick weather could take soundings at short intervals +without stopping, and discover at once any beginning +of shallowing of the water, and so avoid danger.</p> + +<p>The single wire is not now used, as a thin stranded +wire is found safer and quite as effective. The gauge +also has been improved. The apparatus can be seen in +any well-found sea-going vessel; though there are still, +or were until not very long ago, steam vessels without +this apparatus, though crossing the English Channel +with passengers. These depended for soundings on +the obsolete hemp-rope, wrapped round an iron spindle +held vertically on the deck by members of the ship's +company, while the cord was unwound by the descent +of the sinker.<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">24</a></p> + +<p>Sir William Thomson's electrical and other inventions +are too numerous to specify here, and they are +in constant use wherever precision of measurement is +aimed at or required. Long ago he invented electrometers +for absolute measurements of electrical potential +("electric pressure"); more recently his current-balances +have given the same precision to electrodynamic +measurement of currents. All his early instruments +were made by Mr. James White, Glasgow. The +<span class='pagenum'><a name="Page_277" id="Page_277">277</a></span>business founded by Mr. White, and latterly carried +on at Cambridge Street, has developed immensely, and +is now owned by a limited liability company—Messrs. +Kelvin and James White (Limited).</p> + +<p>For many years Sir William Thomson was a keen +yachtsman, and his schooner yacht, the <i>Lalla Rookh</i>, +was well known on the Clyde and in the Solent. An +expert navigator, he delighted to take deep-sea voyages +in his yacht, and went more than once as far as +Madeira. Many navigational and hydrodynamical +problems were worked out on these expeditions. For +a good many years, however, he had given up sea-faring +during his times of relaxation, and lived in +Glasgow and London and in Largs, Ayrshire, where +he built, in 1875, a large and comfortable house, looking +out towards the Firth and the Argyleshire lochs he +knew and loved so well.</p> + +<p>In the course of his deep-sea expeditions in his yacht +he became impressed with the utility of Sumner's +method of determining the position of a ship. Let us +suppose that at a given instant the altitude of the sun +is determined from the ship. The Greenwich meantime, +and therefore the longitude at which the sun is +vertical, is known by chronometer, and the declination +of the sun is known from the Nautical Almanac. +The point on the earth vertically under the sun can +be marked on the chart, and a circle (or rather, what +would be a circle on a terrestrial globe) drawn round +it from every point of which the sun would have the +observed altitude. The ship is at a point on this +circle. Some time after the altitude of the sun is +observed again, and a new "circle" is drawn. If the +first "circle" be bodily shifted on the chart along the<span class='pagenum'><a name="Page_278" id="Page_278">278</a></span> +distance run in the interval, it will intersect the second +in two points, one of which will be the position of the +ship, and it is generally possible to tell which, without +danger of mistake.</p> + +<p>Sir William Thomson printed tables for facilitating +the calculations in the use of Sumner's method, and continually +used them in his own voyages. He was well +versed in seamanship of all kinds, and used his experience +habitually to throw light on abstruse problems of +dynamics. Some of these will be found in "Thomson +and Tait"; for instance, in Part I, § 325, where a +number of nautical phenomena are cited in illustration +of an important principle of hydrodynamics. The fifth +example stated is as follows: "In a smooth sea, with +moderate wind blowing parallel to the shore, a sailing +ship heading towards the shore, with not enough of +sail set, can only be saved from creeping ashore by +setting more sail, and sailing rapidly towards the shore, +or the danger that is to be avoided, so as to allow her +to be steered away from it. The risk of going ashore +in fulfilment of Lagrange's equations is a frequent +incident of 'getting under way' while lifting anchor +or even after slipping from moorings." His seamanship +was well known to shipmasters, with whom he +had much intercourse, and whose intelligence and +practical skill he held in very high regard.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_279" id="Page_279">279</a></span></p> + +<h3>CHAPTER XVI</h3> + +<h4>LORD KELVIN IN HIS CLASS-ROOM AND LABORATORY</h4> + +<p><span class="smcap">It</span> is impossible to convey to those who never studied +at Glasgow any clear conception of Thomson as he +appeared to students whom he met daily during the +session. His appearance at meetings of the British +Association, and his vivacious questionings of the +various authors of papers, his absorption in his subject +and oblivion to the flight of time when he read a paper +himself, will long be remembered by scientific men: +but though they suffice to suggest what he was like in +his own lecture-room, the picture lacks the setting of +furniture, apparatus, assistants, and students, which +all contributed to the unique impression made by +his personality on his pupils. The lecture-table—with +long straight front and ends refracted inward, +flanked by higher small round tables supported on +cylindrical pillars—laden with instruments; the painted +diagrams of the solar spectrum and of the paths of +coloured rays through a prism, hung round the walls; +the long wire with the cylindrical vibrator attached, +for experiments on torsion, and the triple spiral spring +vibrator, which hung at the two ends of the long blackboard; +the pendulum thirty feet long, consisting of a +steel wire and a twelve-pound cannon-ball as bob, +suspended from the apex of the dome-roof above the +lecture-table; the large iron wheel in the beautiful<span class='pagenum'><a name="Page_280" id="Page_280">280</a></span> +oriel window on the right of the lecturer, and the +collection of optical instruments on the table in front +of the central window spaces, from which the small +iron-framed panes—dear to the heart of the architect—had +been removed; the clock on either side of the +room, one motionless, the other indicating the time, +and having attached to it the alarm which showed +when the "angry bell" outside had ceased to toll; the +ten benches of eager and merry students, which filled +the auditorium; all these combined to form a scene +which every student fondly recalls, and which cannot +be adequately described. A similar scene, with some +differences of arrangement and having its own particular +associations, will occur to every student who attended +in the Old College.</p> + +<p>The writer will never forget the lecture-room when +he first beheld it, from his place on Bench VIII, a few +days after the beginning of session 1874-5. Sir +William Thomson, with activity emphasised rather +than otherwise by his lameness, came in with the +students, passed behind the table, and, putting up +his eye-glass, surveyed the apparatus set out. Then, +as the students poured in, an increasing stream, the +alarm weight was released by the bell-ringer, and fell +slowly some four or five feet, from the top of the clock +to a platform below. By the time the weight had +descended the students were in their places, and then, +as Thomson advanced to the table, all rose to their +feet, and he recited the third Collect from the Morning +Service of the Church of England. It was the +custom then, and it is still one better honoured in +the observance than in the breach (which has become +rather common) to open all the first and second classes<span class='pagenum'><a name="Page_281" id="Page_281">281</a></span> +of the day with prayer; and the selection of the +prayers was left to the discretion of the professors. +Next came the roll-call by the assistant; each name +was called in its English, or Scottish (for the clans +were always well represented) form, and the answer +"adsum" was returned.</p> + +<p>Then the Professor began his lecture, generally with +the examination of one of the students, who rose +in his place when his name was called. Thomson, +as the quotation in Chapter VI from the Bangor +Address shows, was fond of oral examination, and +after the second hour had begun to decline as one of +regular attendance, habitually devoted ten or fifteen +minutes to asking questions and criticising the answers. +The names of the students to be questioned were +selected at random from the class register, or by a +kind of lottery, carried out by placing a small card for +each student in a box on the table, and drawing a +name whenever a member of the class was to be +examined. The interest in the drawing each day was +intense, for there was a glorious uncertainty as to +what might be the line of examination adopted. +Sometimes, in the midst of a criticism of an answer, +an idea would suddenly occur to the Professor, +and he would enlarge upon it, until the forgotten +examinee slipped quietly back into his seat, to be no +more disturbed at least for that day! And how great +the relief if the ordeal was well passed and the card +was placed in that receptacle of the blessed, the compartment +reserved for those who had been called and +duly passed the assize! But there was a third compartment +reserved for the cards of those unfortunates who +failed to satisfy the judge! The reader may have<span class='pagenum'><a name="Page_282" id="Page_282">282</a></span> +anticipated the fact that the three divisions of this +fateful box were commonly known to students by the +names of the three great habitations of spirits described +in the <i>Divina Commedia</i> of Dante.</p> + +<p>As has been stated, the oral examination with which +the lectures opened was the cause of a good deal of +excitement, which was added to by the element of +chance introduced by drawing the names from the +purgatorial compartment of the box. The ordeal was +dreaded by backward students, whom Thomson found, +as he said, aphasic, when called on to answer in +examination, but who certainly were anything but +aphasic in more congenial circumstances. Occasionally +they abstained from responding to their names, modestly +seeking the seclusion of the crowd, and some little +time would be spent in ascertaining whether the +examinee-designate was present. When at last he was +discovered, he generally rose with a fervent appeal to +his fellows on either side to help him in his need.</p> + +<p>McFarlane used to tell of an incident which illustrated +the ingenuity with which it was sometimes +attempted to evade the ordeal of the <i>viva voce</i> +examination. One afternoon, when he was busily +preparing the lecture-illustrations for next day, a student +came into the class-room, and engaging him in conversation +on some point of dynamics, regarding which +he professed to have a difficulty, hovered round the +box which contained the three compartments popularly +known as Purgatory, Heaven, and Hell! Always +when McFarlane left the room to bring something +from the adjoining cabinet of apparatus, he found, when +he returned, his inquiring friend hurriedly quitting the +immediate vicinity of the box. At last the student<span class='pagenum'><a name="Page_283" id="Page_283">283</a></span> +took leave, with many apologies for giving so much +trouble. As McFarlane suspected would be the case, +the ticket bearing the name of that student was no +longer to be found! He used to conclude the story as +follows: "I just made a new ticket for him, and +placed it on the top of the other tickets, and next day +Sir William called him, the very first time." What +were his feelings, who had fondly thought himself safe +for the session, and now found himself subjected to a +"heckling" which he probably expected would be +repeated indefinitely, may be imagined.</p> + +<p>The subject of the first lecture which the writer +attended was simple harmonic motion, and was illustrated +by means of pendulums, spiral springs with +weights, a long vertical rod of steel tipped with an ivory +ball and fastened to a heavy base, tuning-forks, etc.</p> + +<p>The motion was defined as that of a particle moving +along the diameter of a circle—the "auxiliary circle," +Thomson called it—so as always to keep pace, as +regards displacement in the direction along that +diameter, with a particle moving with uniform speed +in the circle. Then the velocity and acceleration +were found, and it was shown that the particle was +continually accelerated towards the centre in proportion +to the distance of the particle from that point. +The constant ratio of acceleration to displacement +was proved to be equal to the square of the angular +velocity in the auxiliary circle, and from this fact, +and the particular value of the acceleration when the +particle was at either end of its range of motion, an +expression for the period in terms of the speed and +radius of the auxiliary circle was deduced. Then +the ordinary simple pendulum formula was obtained.<span class='pagenum'><a name="Page_284" id="Page_284">284</a></span></p> + +<p>This mode of treatment of an elementary matter, so +entirely different from anything in the ordinary text-books, +arrested the attention at once, and conveyed, to +some at least of those present, an idea of simple harmonic +motion which was directly applicable to all kinds +of cases, such as the motion of the air in a sound wave, +or of the medium which conveys the waves of light.</p> + +<p>The subject of Kepler's laws was dealt with in the +early lectures of every course, and Newton's deductions +were insisted on as containing the philosophy of the +whole question, leading, as they did, to the single +principle from which the laws could be deduced, and +the third law corrected when the mass of the planet +was comparable with that of the sun. Sometimes +Thomson would read the remarkable passage in +Hegel's <i>Logik</i>, in which he refers to the Newtonian +theory of gravitation and says, "The planets are not +pulled this way and that, they move along in their +orbits like the blessed gods," and remark upon it. +On one occasion his remark was, "Well, gentlemen, +if these be his physics, what must his metaphysics be?" +And certainly that a <i>philosopher</i> should deny, as Hegel +seemed to do, all merit to the philosophical setting in +which Newton placed the empirical results of Kepler, +is a very remarkable phenomenon.</p> + +<p>The vivacity and enthusiasm of the Professor at that +time were very great. The animation of his countenance +as he looked at a gyrostat spinning, standing on a +knife-edge on the glass plate in front of him, and +leaning over so that its centre of gravity was on one +side of the point of support; the delight with which he +showed that hurrying of the precessional motion caused +the gyrostat to rise, and retarding the precessional<span class='pagenum'><a name="Page_285" id="Page_285">285</a></span> +motion caused the gyrostat to fall, so that the freedom +to "precess" was the secret of its not falling; the +immediate application of the study of the gyrostat to +the explanation of the precession of the equinoxes, and +illustration by a model of a terrestrial globe, arranged so +that the centre should be a fixed point, while its axis—a +material spike of brass—rolled round a horizontal +circle, the centre of which represented the pole of the +ecliptic, and the diameter of which subtended an angle +at the centre of the globe of twice the obliquity of the +ecliptic; the pleasure with which he pointed to the +motion of the equinoctial points along a circle surrounding +the globe on a level with its centre, and +representing the plane of the ecliptic, and the smile +with which he announced, when the axis had rolled +once round the circle, that 26,000 years had elapsed—all +these delighted his hearers, and made the lecture +memorable.</p> + +<p>Then the gyrostat, mounted with its axis vertical on +trunnions on a level with the fly-wheel, and resting on +a wooden frame carried about by the professor! The +delight of the students with the quiescence of the +gyrostat when the frame, gyrostat and all, was carried +round in the direction of the spin of the fly-wheel, and +its sudden turning upside down when the frame was +carried round the other way, was extreme, and when +he suggested that a gyrostat might be concealed on a +tray of glasses carried by a waiter, their appreciation +of what would happen was shown by laughter and a +tumult of applause.</p> + +<p>Some would have liked to follow the motions of +spinning bodies a little more closely, and to have made +out clearly why they behaved as they did. Apparently<span class='pagenum'><a name="Page_286" id="Page_286">286</a></span> +Thomson imagined the whole affair was self-evident, +for he never gave more than the simple parallelogram +diagram showing the composition, with the already +existing angular momentum about the axis of the top, +of that generated about another axis, in any short time, +by the action of gravity.</p> + +<p>As a matter of fact, the stability and instability of +the gyrostat on the tray give the best possible illustration +of the two different forms of solution of the differential equation, +Ӫ + μӨ = 0, according as μ is positive or +negative; though it is also possible to explain the +inversion very simply from first principles. All this +was no doubt regarded by Thomson as obvious; but it +was far from being self-evident to even good students +of the ordinary class, who, without exception, were +beginning the study of dynamics.</p> + +<p>Thomson's absorption in the work of the moment +was often very great, and on these occasions he much +disliked to be brought down to sublunary things by +any slight mischance or inconvenience. Examples +will occur to every old pupil of the great emphasis +with which he commanded that precautions should be +taken to prevent the like from happening again. Copies +of Thomson and Tait's <i>Natural Philosophy</i>—"T and +T'" was its familiar title—and of other books, including +Barlow's Tables and other collections of +numerical data, were always kept on the lecture-table. +But occasionally a laboratory student would stray in +after everything had been prepared for the morning +lecture, and carry off <i>Barlow</i> to make some calculation, +and of course forget to return it. Next morning some +number would be wanted from <i>Barlow</i> in a hurry, and +the book would be missing. Then Thomson would<span class='pagenum'><a name="Page_287" id="Page_287">287</a></span> +order that <i>Barlow</i> should be chained to the lecture-table, +and enjoin his assistant to see that that was +done without an hour's delay!</p> + +<p>On one occasion, after working out part of a calculation +on the long fixed blackboard on the wall +behind the table, his chalk gave out, and he dropped +his hand down to the long ledge which projected +from the bottom of the board to find another piece. +None was just there; and he had to walk a step or +two to obtain one. So he enjoined McFarlane, his +assistant, who was always in attendance, to have a +sufficient number of pieces on the ledge in future, to +enable him to find one handy wherever he might +need it. McFarlane forgot the injunction, or could +not obtain more chalk at the time, and the same thing +happened next day. So the command was issued, +"McFarlane, I told you to get plenty of chalk, and you +haven't done it. Now have a <i>hundred</i> pieces of chalk +on this ledge to-morrow; remember, a <i>hundred</i> pieces; +I will count them!" McFarlane, afraid to be caught +napping again, sent that afternoon for several boxes of +chalk, and carefully laid the new shining white sticks +on the shelf, all neatly parallel at an angle to the edge. +The shelf was about sixteen feet long, so that there +was one piece of chalk for every two inches, and the +effect was very fine. The class next morning was +delighted, and very appreciative of McFarlane's diligence. +Thomson came in, put up his eye-glass, looked +at the display, smiled sweetly, and, turning to the +applauding students, began his lecture.</p> + +<p>From time to time there were special experiments, +which excited the interest of the class to an extraordinary +degree. One was the determination of the<span class='pagenum'><a name="Page_288" id="Page_288">288</a></span> +velocity of a bullet fired from a rifle into a Robins +ballistic pendulum. The pendulum, consisting of a +massive bob of lead attached to a rigid frame of iron +bars turning about knife-edges, was set up behind the +lecture-table, and the bullet was fired by Thomson +from a Jacob rifle into the bob of the pendulum. +The velocity was deduced from the deflection of the +pendulum, its known moment of inertia about the +line of the knife-edges, the distance of the line of fire +from that line, and the mass of the bullet.</p> + +<p>In some of the notices of Lord Kelvin that have +appeared in the newspapers, the imagination of the +writers has converted the Jacob rifle into one which +Professor Thomson carried in the early years of the +volunteer movement, as a member of a Glasgow corps. +It is still used in the Natural Philosophy Department +for the same experiment, and is a muzzle-loading +rifle of large calibre, which throws an ounce bullet. +It was invented by the well-known Indian sportsman, +Colonel Jacob, for big-game shooting in India. +Thomson held a commission as captain in the K (or +University) Company of rifle volunteers, and so did +not shoulder a rifle, except when he may have indulged +in target practice.</p> + +<p>The front bench students were always in a state +of excitement, mingled in some cases perhaps with a +little trepidation. For the target was very near them, +and though danger was averted by placing a large +wooden screen in front of the bob, to prevent splinters +of the bullet from flying about in the event of its +missing the target and striking the iron casing of the +bob, there was a slight amount of nervousness as to +what might happen. The rifle, loaded by McFarlane,<span class='pagenum'><a name="Page_289" id="Page_289">289</a></span> +who had weighed out the charge of powder (so many +drams) from a prescription kept in a cavity of the +stock, was placed on the table, and two rests, provided +with <big>V</big> notches to receive the rifle, were placed in the +proper position to enable a bull's eye to be obtained. +Thomson generally produced a small box of cotton +wool, and inserted a little in each of his ears to prevent +injury to the tympanum from the report, and advised +the spectators to do the same. Then, adjusting his +eye-glass, he bent down, placed the rifle in position, and +fired, and the solemn stillness with which the aiming +and adjustments had been witnessed was succeeded by +vociferous applause. The length of tape drawn out +under a light spring was read off by McFarlane, who +had already placed on the blackboard the formula for +calculation of the velocity, with the factor by which +the length of tape had to be multiplied to give the +velocity in feet per second. Then, with the intimation +that a question involving numerical calculation would +be set on the subject, in the ensuing Monday morning +examination paper, the lecture generally closed, or was +rounded off with some further observations on angular +(or, as Thomson always preferred to call it, moment +of) momentum.</p> + +<p>Long after in the course of a debate in the House +of Lords on a proposal to make the use of the metric +system of weights and measures compulsory, Lord +Kelvin told their lordships how he had weighed out the +powder to charge this rifle, and, mistaking the weights, +had loaded the rifle with an amount of powder which +would have been almost certain to burst the piece, but +had happily paused before firing it off.</p> + +<p>He often interrupted the course of a lecture with a<span class='pagenum'><a name="Page_290" id="Page_290">290</a></span> +denunciation of the British "no-system of weights and +measures"—"insane," "brain-wasting," "dangerous," +were among the mildest epithets he applied to it, and +he would deeply sympathise with the student whose +recollection of avoirdupois weight, troy weight, apothecaries' +weight, etc., was somewhat hazy. The danger +of the system consisted mainly in the fact that the +apothecaries' dram is 60 grains, while the avoirdupois +dram is 27<small><sup>1</sup>⁄<sub>3</sub></small> grains. Thus so many drams of powder +required to charge a rifle is a very much larger quantity +when reckoned in apothecaries' drams than when +reckoned in avoirdupois. As a rule he left the loading +of the rifle, like all the other lecture-room experiments, +to his assistants.</p> + +<p>Another experiment which caused a great sensation +was that known as the "dew-drop"! A funnel of +brass, composed of a tube about 30 inches long and an +inch wide, and a conical mouth about ten inches wide, +had a piece of stout sheet India-rubber stretched, as +tightly as it could be by hand, across its mouth, and +made water-tight by a serving of twine and cement +round the edge. A wire soldered round the outside +of the lip gave a good hold for this serving and made +all perfectly secure. On the plane surface of the sheet +geometrical figures were drawn in ink, so that their +distortion could be afterwards studied. The funnel +was then hung by a strong support in an inverted +position behind the table, and water poured gently +into it from a rubber supply pipe connected with the +water-main. As the water was allowed to accumulate—very +slowly at first—the sheet of rubber gradually +stretched and bulged out, at first to a flat lens-shape, +and gradually more and more, till an immense water<span class='pagenum'><a name="Page_291" id="Page_291">291</a></span>-drop had been formed, 15 or 18 inches in horizontal +diameter, and of still greater vertical dimensions. +The rubber film was now, at the place of greatest +tension, quite thin and transparent, and its giving way +was anticipated by the students with keen enjoyment. +A large tub had been placed below to receive the +water, but the deluge always extended over the whole +floor space behind the table, and was greeted with +rapturous applause.</p> + +<p>Before the drop burst, and while it was forming, +Thomson discoursed on surface tension, emphasising +the essential difference between the tension in the +rubber-film and the surface-film of a dewdrop, and +pointing out how the geometrical figures had changed +in shape. Then he would poke it with the pointer he +held in his hand, and, turning to the class, as the mass +quivered, remark, "The trembling of the dewdrop, +gentlemen!"</p> + +<p>Vibrations of elastic solids were illustrated in various +ways, frequently by means of a symmetrical shape of +calves'-foot jelly, at the top of which a coloured marble +had been imbedded as a molecule, the motions of which +could be followed. And then he would discourse on +the Poisson-Navier theory of isotropic solids, and the +impossibility of the fixed relation which that theory +imposed between the modulus of rigidity and the +modulus of compression; and refer with approval to +the series of examples of "perfectly uniform, homogeneous, +isotropic solids," which Stokes had shown +could be obtained by making jellies of different degrees +of stiffness. Another example, frequently adduced as +indicating the falsity of the theory, was the entirely +different behaviour of blocks of India-rubber and<span class='pagenum'><a name="Page_292" id="Page_292">292</a></span> +cork, under compression applied by a Bramah press. +The cork diminished in thickness without spreading +out laterally; the rubber, being very little compressible, +bulged out all round as its thickness was +diminished.</p> + +<p>The lectures on acoustics, which came late in the +course, were also exceedingly popular. Two French +horns, with all their crooks and accessories, were displayed, +and sometimes, to the great delight of the class, +Thomson would essay to show how the pitch of a note +could be modified by means of the keys, or by the +hand inserted in the bell. The determination by the +siren of the pitch of the notes of tuning-forks excited +by a 'cello bow, and the tuning of a major third by +sounding at the same time the perfect fifth of the lower +note, were often exhibited, and commented on with +acute remarks, of which it is a pity no statement was +ever published.<a name="FNanchor_25_25" id="FNanchor_25_25"></a><a href="#Footnote_25_25" class="fnanchor">25</a></p> + +<p>The closing lecture of the ordinary course was +usually on light, and the subject which was generally +the last to be taken up—for as the days lengthened in +spring, it was possible sometimes to obtain sunlight for +the experiments—was often relegated to the last day or +two of the session. So after an hour's lecture Thomson +would say, "As this is the last day of the session, I will +go on for a little longer, after those who have to leave +have gone to their classes." Then he would resume +after ten o'clock, and go on to eleven, when another +opportunity would be given for students to leave, and the +lecture would be again resumed. Messengers would +<span class='pagenum'><a name="Page_293" id="Page_293">293</a></span>be sent from his house, where he was wanted for +business of different sorts, to find out what had become +of him, and the answer brought would be, hour after +hour, "He is still lecturing." At last he would conclude +about one o'clock, and gently thank the small +and devoted band who had remained to the end, for +their kind and prolonged attention.</p> + +<p>In the course of his lectures Thomson continually +called on his assistants for data of all kinds. In the +busiest time of his life—the fifteen years from 1870 to +1885—he trusted to his assistants for the preparation of +his class illustrations, and it was sometimes a little +difficult to anticipate his wishes, for without careful +rehearsal it is almost impossible to make sure that in +an experimental lecture everything will go without a +hitch. The digressions, generally most interesting and +instructive, in which he frequently indulged, almost +always rendered it necessary to bring some experiment +before the class which had not been anticipated, and +all kinds of things were kept in readiness, lest they +should be wanted suddenly.</p> + +<p>It has often been asserted that Thomson appealed to +his assistant for information contained in the multiplication-table, +and could not perform the ordinary +operations of arithmetic. His active mind, working on +ahead of the statements he was making at the moment, +often could not be brought back to the consideration of +the value of 9 times 6, and the like; but it was quite +untrue that he was incapable of making calculations. +His memory was good, and though he never could be, +for example, sure whether the aqueous humour was +before or behind the crystalline in the eye, he was +generally able at once to tell when a misstatement had<span class='pagenum'><a name="Page_294" id="Page_294">294</a></span> +been made as to any numerical question regarding +the subject under discussion.</p> + +<p>In the higher mathematical class, to which he +lectured on Wednesdays, at noon, Thomson was exceedingly +interesting. There he seemed to work at +the subject as he lectured; new points to be investigated +continually presented themselves, and the +students were encouraged to work them out in the +week-long intervals between his lectures. Always the +physical interpretation of results was aimed at, even +intermediate steps were discussed. Thus the meaning +of the mathematical processes was ever kept in view, +and the men who could follow were made to think while +they worked, and to regard the mathematical analysis +as merely an aid, not an end in itself. "A little expenditure +of chalk is a saving of brains;" "the art of +reading mathematical books is judicious skipping," were +remarks he sometimes made, and illustrate his view of +the relative importance of mathematical work when +he regarded it as the handmaid of the physical thinker. +Yet he valued mathematics for its own sake, and was +keenly alive to elegance of form and method, as +readers of such great mathematical discussions as the +"Appendix on Spherical Harmonics," in Thomson +and Tait, will observe. He spoke with unqualified +admiration of the work of Green and Stokes, of +Cauchy's great memoir on Waves, and of Hamilton's +papers on Dynamics. But no form of vector-analysis, +neither the Quaternions of Hamilton nor +the Vectors of Willard Gibbs and Heaviside, +appealed to him, and the example of his friend and co-worker, +Tait, had no effect in modifying his adverse +verdict regarding this department of mathematics,<span class='pagenum'><a name="Page_295" id="Page_295">295</a></span> +a verdict which in later years became only more +emphatic.</p> + +<p>One session he began the first lecture of the higher +class by writing <i>dx</i> ⁄ <i>dt</i> in the middle of the blackboard, +and demanding of each of the ten or a dozen students +present, some of them distinguished graduates, what it +meant! One student described it as the limiting value +of the ratio of the increment of the dependent variable +<i>x</i> to the increment of the independent variable <i>t</i>, when +the latter increment is made indefinitely small. He +retorted, "That's what Todhunter would say!" The +others gave various slightly different versions of the same +definition. At last he impatiently remarked, "Does +nobody know that <i>dx</i> ⁄ <i>dt</i> means velocity?" Here +the physical idea as a whole was before his mind; and +he did not reflect that if <i>t</i> denoted time and <i>x</i> distance +in any direction, the explanation given by the student +did describe velocity with fair accuracy.</p> + +<p>An embarrassing peculiarity of his mathematical +discussions was his tendency, when a difficulty of +symbolisation occurred, to completely change the +notation. Also he was not uniformly accurate in +analytical work; but he more than made up for this +by the faculty he had of devising a test of the accuracy +of the result and of divining the error which had crept +in, if the test was not satisfied.</p> + +<p>The subjects he treated were always such great +branches of mathematics as the theory of the tides—he +discussed the tidal phenomena of the English +Channel in one course—the general theory of vibrations, +Fourier analysis, the theory of waves in water, +etc., etc. A very good idea of the manner and matter<span class='pagenum'><a name="Page_296" id="Page_296">296</a></span> +of his mathematical prelections can be obtained from a +perusal of the <i>Baltimore Lectures</i>.</p> + +<p>In the physical laboratory he was both inspiring and +distracting. He continually thought of new things to +be tried, and interrupted the course of the work with +interpolated experiments which often robbed the preceding +sequence of operations of their final result. His +ideas were on the whole better worked out by a really +good corps of students when he was from home, and +could only communicate by letter his views on the work +set forth in the daily reports which were forwarded to +him.</p> + +<p>He insisted with emphasis that a student who found +that a quadrant electrometer would not work well should +take it to pieces to ascertain what was the matter. This +of course generally resulted in the return of the instrument +to White's shop to be put together again and +adjusted. But, as he said, there was a cause for every +trouble of that kind, and the great thing was to find +out at once what it was.</p> + +<p>Thomson's concentration on the work in hand, +and his power of simply taking possession of men, +even mere spectators, and converting them into assistants, +was often shown in the laboratory. Several men +who have since become eminent were among the +assistants enrolled from the laboratory students. Professor +W. E. Ayrton and, later, Professor John Perry, +were students at Glasgow for a time, and rendered the +most able and willing help in the researches which +were then proceeding. This power was, no doubt, +the secret of his success in gathering round him an +enthusiastic corps of laboratory workers in the early +years of his professorship, and it was shown also by<span class='pagenum'><a name="Page_297" id="Page_297">297</a></span> +the ease with which he annexed the Blackstone +examination-room and, later, various spaces in the new +University buildings. There, after a time, the Natural +Philosophy rooms were found by the senatus to include +not only the original class-room, laboratory, etc., but +also all the spare attics and corridors in the neighbourhood, +and even the University tower itself! One of +his colleagues, who venerated him highly, remarked +recently, "He had a great faculty for annexation!"</p> + +<p>The incident referred to occurred while he was +preparing the article on <i>Heat</i> for the ninth edition of +the <i>Encyclopædia Britannica</i>. It seemed at first a pity +that Thomson should undertake to write such +articles; but in the course of their preparation he +came upon so many points on which experimental +information was wanting, and instituted so many +researches to answer his questions, that the essays took +very much the character of original papers. In the +article on <i>Heat</i> (he also wrote <i>Elasticity</i>), will be found +a long account of "Steam Thermometry," that is, of +thermometers in which the indicating substance was to +be the saturated vapours of different substances, water, +sulphurous acid, etc., etc., for he did not limit the term +"steam" to water-vapour. For some time every one +in the laboratory was employed in making sulphurous +acid, by heating copper in sulphuric acid in the usual +way, and condensing the gas in tubes immersed in freezing +mixtures; and the atmosphere of the room was of +a sort which, however noxious to germs of different +kinds, it was a little difficult to breathe. One morning, +when all were thus occupied, an eminent chemist, who +had just come home from the south for a vacation, +called to pay his respects. After a word or two of<span class='pagenum'><a name="Page_298" id="Page_298">298</a></span> +inquiry as to how his young friend was prospering +in his new post, Thomson said, "We are all very busy +brewing liquid sulphurous acid, for use in sulphurous +acid steam thermometers; we want a large quantity +of the liquid; would you mind helping us?" So, +desiring an assistant to find a flask and materials, he +enrolled this new and excellent recruit on the spot; +and what was intended to be a mere call, was prolonged +into a long day of ungrudging work at an +elementary chemical exercise!</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_299" id="Page_299">299</a></span></p> + +<h3>CHAPTER XVII</h3> + +<h4>PRACTICAL ACTIVITIES—HONOURS AND DISTINCTIONS—LAST +ILLNESS AND DEATH</h4> + +<p><span class="smcap">It</span> remains to say something of Lord Kelvin's public +and practical activities. All over the world he came +ultimately to be recognised as the greatest living scientific +authority in almost all branches of physics. Every +existing learned society sought to make him a Fellow, +honorary degrees were showered on him from all +quarters. A list of some of the most important of +these distinctions is given in the Royal Society Year-Book +for 1907; it is doubtful if a complete list could +be compiled. He was awarded the Keith Medal and +the Victoria Jubilee Medal by the Royal Society of +Edinburgh, and received in succession the Copley and +Royal Medals of the Royal Society of London, of +which he was elected a Fellow in 1851, and was President +from 1890 to 1895. For several periods of years +he was President of the Royal Society of Edinburgh, to +which he communicated his papers on heat, dissipation +of energy, vortex motion, and many other memoirs.</p> + +<p>He was President of the British Association at the +Edinburgh meeting in 1871, when he delivered a +presidential address, noteworthy in many respects, but +chiefly remarkable in the popular mind on account of +his suggestion that life was conveyed to the earth by a +seed, a germ enclosed in a crevice of a meteorite. This +was understood at the time by many people as an attempt<span class='pagenum'><a name="Page_300" id="Page_300">300</a></span> +to explain the origin of life itself, instead of what it +was intended to be, an explanation of the beginning of +the existence of living things on a planet which was +originally, on the completion of its formation by the +condensation of nebular matter, red hot even at its +surface. On several occasions he was president of +Section A, and he was constant in attendance at the +Association meetings, and an eager listener and participator +in the discussions and debates. His scientific +curiosity was never at rest, and he dearly liked to meet +and converse with scientific workers.</p> + +<p>Lady Thomson, who had been long an invalid, died +in 1870, and in 1874 Sir William Thomson was married +to Miss Frances Anna Blandy (daughter of Mr. Charles +R. Blandy of Madeira) who survives him as Lady +Kelvin. To her tender solicitude he owed much of +his constant and long-continued activity in all kinds of +work. She accompanied him on all public occasions, +and he relied greatly on her helpfulness and ever +watchful care.</p> + +<p>In 1892 Sir William Thomson, while President of +the Royal Society, was raised to the Peerage, with the +title of Baron Kelvin of Netherhall, Largs; and more +lately he was created a member of the Order of Merit +and a G.C.V.O. His foreign distinctions were very +numerous. He was a Knight of the Order <i>Pour le +Mèrite</i> of Prussia, a Foreign Associate of the Institute +of France, and a Grand Officer of the Legion of +Honour. But no public honour or mark of royal +favour could raise him in the estimation of all who +know anything of science or of the labours of the scientific +men to whom we owe the necessities and luxuries +of our present civilisation.<span class='pagenum'><a name="Page_301" id="Page_301">301</a></span></p> + +<p>In 1896 the City and University of Glasgow +celebrated the jubilee of his Professorship of Natural +Philosophy. The rejoicings on that occasion will +never be forgotten by those whose privilege it was +to take part in them. Delegates came from every +country in the world, and kings and princes, universities +and learned societies, colleges and scholastic +institutions of every kind, vied with each other in doing +honour to the veteran who had fought for truth and +light for so many years, and won so many victories. +A memorial volume of the proceedings was published, +including a review of Lord Kelvin's work by the +late Professor FitzGerald, and a full report appeared in +<i>Nature</i> and other journals at the time, so that it is +unnecessary to give particulars here. And indeed it is +impossible by any verbal description to convey an idea +of the enthusiasm with which the scientific world +acclaimed its leader, and of the dignity and state of the +ceremonies.</p> + +<p>In 1899, at the age of seventy-five, Lord Kelvin +resigned the Chair of Natural Philosophy, and retired, +not to rest, but to investigate more vigorously than ever +the properties of matter. One remarkable fruit of his +leisure we have in his great book, the <i>Baltimore +Lectures</i>, in which theories of light are discussed +with a power which excites the reverence of all +engaged in the new researches and which recent +discoveries have called into existence. And it is not +too much to say that the means of discussing and +extending these discoveries are in great measure due +to Lord Kelvin.</p> + +<p>During the year 1907 Lord Kelvin performed +many University duties and seemed to be in unusually<span class='pagenum'><a name="Page_302" id="Page_302">302</a></span> +good health. He presided as Chancellor at the +installation of Mr. Asquith as Lord Rector on January +11, and in the same capacity attended a few days later +the funeral of Principal Story, the Vice-Chancellor, +who died on January 13. On April 23 he presided at +the long and arduous ceremonies of honorary graduation, +and the public opening of the new Natural +Philosophy Institute and the new Medical Buildings, +by the Prince of Wales. As Chancellor he conferred +the degree of Doctor of Laws on the Prince and +Princess, and took the chair at the luncheon which +followed the proceedings, when he proposed in a short +and graceful speech the health of the Princess.</p> + +<p>He was able to take part also in various political and +social meetings, and to give attention to the work in +progress at the factories of his firm in Cambridge +Street. Lady Kelvin and he left Netherhall, Largs, +for Aix les Bains, at the end of July, but visited the +British Association at Leicester in passing. There he +heard the presidential address of his old friend, Sir +David Gill, to whom he moved a vote of thanks in his +usual vivacious manner.</p> + +<p>Lord Kelvin had been accustomed for a good many +years to spend a month or six weeks in summer or +early autumn at the famous French watering-place, +from which he seemed always to receive much benefit. +For a long time he had suffered from an intermittent +and painful form of facial neuralgia, which, except +during its attacks, which came and passed suddenly, +did not incapacitate him from work. With the exception +of a rather serious illness in 1906, this was the +only ailment from which he had suffered for many years, +and his general health was otherwise uniformly good.<span class='pagenum'><a name="Page_303" id="Page_303">303</a></span></p> + +<p>Lord and Lady Kelvin returned to Netherhall on +September 14, with the intention of going in a day or +two to Belfast, to open the new scientific buildings of +Queen's College. But, unfortunately, on the day of +their arrival Lady Kelvin became very seriously ill, and +the visit to Ireland had to be abandoned. His address +was, however, read by his nephew, James Thomson, +son of his elder brother, and was a tribute to the city +of his birth, and the memory of his father.</p> + +<p>The illness of Lady Kelvin caused much anxiety +for many weeks, and this, and perhaps some incautious +exposure, led to the impairment of Lord Kelvin's +health. A chill caught on November 23 caused him +to be confined to bed; and though he managed for +a week or two still to do some work on a paper +with which he had been occupied for a considerable +time, he became worse, and gradually sank, until his +death at a quarter-past ten o'clock on the evening of +December 18.</p> + +<p>The keen sorrow which was universally felt for +Lord Kelvin's death was manifested by all classes of +the community. In Glasgow every one mourned as +for the greatest of the land, and the testimony to the +affection in which he was held, and the reverence for +his character and scientific achievements, was extraordinary. +And this feeling was universal; from all +parts of the world poured in telegrams of respectful +sympathy with Lady Kelvin and with the University +of Glasgow in their bereavement.</p> + +<p>The view was immediately and strongly expressed, +both privately and by the press, that the most illustrious +natural philosopher since Newton should rest beside +the great founder of physical science in Westminster<span class='pagenum'><a name="Page_304" id="Page_304">304</a></span> +Abbey, and a requisition was immediately prepared +and forwarded by the Royal Society of London to +the Dean of Westminster. The wish of the whole +scientific world was at once acceded to, and on December +23, at noon, the interment took place, with a state +and yet a simplicity which will never be forgotten by +those who were present.</p> + +<p>Nearly all the scientific notabilities of the country +were present, and the coffin, preceded by the choristers +and the clergy, while the hymn, "Brief life is here our +portion," was sung, was followed round the cloistered +aisles from St. Faith's chapel to the choir, by the +relatives, representatives of His Majesty the King and +the Prince of Wales, by the Royal Society, by delegates +from the Institute of France, representatives of the +Universities of Cambridge, Oxford, Glasgow, and other +universities, of the Royal Society of Edinburgh (of +which Lord Kelvin was president when he died), and +of most of the learned societies of the kingdom. +Then, after a short service, the body was followed to +the grave in the cloisters by the same company of +mourners, and to the solemn words of the Burial Service +was laid close by where rests all that was mortal +of Isaac Newton. There he sleeps well who toiled +during a long life for the cause of natural knowledge, +and served nobly, as a hero of peace, his country and +the world.</p> + +<hr /> + +<p><span class='pagenum'><a name="Page_305" id="Page_305">305</a></span></p> + +<h3>CONCLUSION</h3> + +<p><span class="smcap">The</span> imperfect sketch of Lord Kelvin's scientific +life and work which this book contains can only give +a faint notion of the great achievements of the long +life that has now ended. Beyond the researches +which he carried out and the discoveries he made, +there is the inspiration which his work and example +gave to others. Inspired himself by Lagrange, Laplace, +Ampère, and Fourier, and led to experimental research +by the necessity for answers to the questions +which his mathematical expression of the discoveries +of the twenty-five years which preceded the establishment +of his laboratory had suggested—the theories of +electricity and magnetism, of heat, of elasticity, his +discoveries in general dynamics and in fluid motion, +the publication of "Thomson and Tait," all made him +the inspirer of others; and there was no one, however +eminent, who was not proud to acknowledge his +obligations to his genius. Clerk Maxwell, before he +wrote the most original treatise on electricity that has +ever appeared, gave himself to the study of Faraday's +Experimental Researches and to the papers of Thomson. +And if some, like FitzGerald and others, have +regretted that the electromagnetic theory of light to +which Maxwell was led by Faraday, and, indeed, by +Thomson himself, did not meet with a more sympathetic +reception at his hands, they have not been<span class='pagenum'><a name="Page_306" id="Page_306">306</a></span> +unmindful of the source from which much of their +illumination has come.</p> + +<p>He has founded a school of thought in mathematical +physics, of men in whose minds the symbol is always +the servant of the ideas, whose motto is interpretation +by dynamical processes and models as far as that is +possible, who shirk no mathematical difficulties when +they have to be encountered, but are never led away +from the straight road to the goal which they seek +to reach—the systematic and clear formulation of the +course of physical action.</p> + +<p>And in Lord Kelvin's mind there was blended with +a clear physical instinct which put aside all that was +extraneous and unessential to the main issue an extraordinary +power of concentration on the problem in +hand, and a determination that was never daunted by +failure, which consented to postponement but never to +relinquishment, and which led often after long intervals +of time to success in the end. He believed that light +would come at last on the most baffling of problems, +if only it were looked at from every point of view and +its conditions were completely formulated; but he +could put what was for the time impossible aside, and +devote himself to the immediately possible and realisable. +And as often happens with every thinker, his +mind, released from the task, returned to it of itself, +and what before appeared shrouded in impenetrable mist +stood out suddenly sharp and distinct like a mountain-top +before a climber who has at last risen above the clouds.</p> + +<p>With the great mathematical power and sure instinct +which led him to success in physical research was +combined a keen perception of the importance of practical +applications. Sometimes the practical question<span class='pagenum'><a name="Page_307" id="Page_307">307</a></span> +suggested the theoretical and experimental research, as +when the needs of submarine telegraphy led to the +discussion of the speed of signalling and the evolution of +the reflecting galvanometer and the siphon recorder. On +the other hand, the mathematical theory of electricity +and magnetism had led to quantitative measurement and +absolute units at an earlier time, when the need for these +was beginning to be felt clearly by scientific workers and +dimly by those far-sighted practical men who dreamed—for +a dream it was thought at the time—of linking the +Old World with the New by a submarine cable. But +the quantitative study of electricity in the laboratory +threw light on economic conditions, and the mass +of data already obtained, mainly as a mere matter of +experimental investigation of the properties of matter, +became at once a valuable asset of the race of submarine +cable engineers which suddenly sprang into existence.</p> + +<p>And so it has been with the more recent applications +of electricity. The induction of currents discovered +by Faraday could not become of practical importance +until its laws had been quantitatively discussed, a much +longer process than that of discovery; and we have +seen how the British Association Committee, led +by Thomson and Maxwell, brought the ideas and +quantities of this new branch of science into numerical +relation with the units of already existing practical +enterprise. The electrical measuring instruments—first +the electrometers, and more recently the electric +current balances and other beautiful instruments for +the dynamo-room and the workshop—which Lord +Kelvin invented have brought the precision of the +laboratory into the everyday duties of the secondary +battery attendant and the wireman.<span class='pagenum'><a name="Page_308" id="Page_308">308</a></span></p> + +<p>And as to methods of measurement, those who +remember the haziness of even telegraph engineers +as to the measurement of the efficiency of electrical +currents and electromotive forces in the circuits of +lamps and dynamos, in the early days of electric +lighting, know how much the world is indebted to +Thomson.<a name="FNanchor_26_26" id="FNanchor_26_26"></a><a href="#Footnote_26_26" class="fnanchor">26</a> He it was who showed at first how +cables were to be tested, as well as how they were +to be worked; it was his task, again, to show how +instruments were to be calibrated for practical +measurement of current and energy supplied by the +early contractors to consumers. He had in the quiet +of his laboratory long before elaborated methods of +comparing resistances, and given the Wheatstone +balance its secondary conductors for the comparison +of low resistances; he now showed how the same +principles could be applied to measure the efficiencies +of dynamos and to make up the account of charge +and discharge for a secondary battery.</p> + +<p>And if the siphon-recorder and the mariners' compass +and the sounding machine proved pecuniarily +profitable, the reward was that of the inventor, who +has an indefeasible right to the fruit of his brain and +his hand. But Lord Kelvin's activity was not confined +merely to those practical things which have, to use the +ordinary phrase, "money in them"; he gave his time +and energies freely to the perfecting of the harmonic +analysis of the tides, undertook again, for a Committee +of the British Association, the investigation of the tides +<span class='pagenum'><a name="Page_309" id="Page_309">309</a></span>for different parts of the world, superintended the +analysis of tidal records, and invented tide-predicting +machines and improved tide-gauges.</p> + +<p>Lord Kelvin's work in the theory of heat and in the +science of energy generally would have given him a +title to immortality even if it had stood alone; and +there can be no doubt, even in the mind of the most +determined practical contemner of the Carnot cycle, +of the enormous importance of these achievements. +Here he was a pioneer, and yet his papers, theoretical +and yet practical, written one after another in pencil +and despatched, rough as they were, to be printed by +the Royal Society of Edinburgh, form, as they are +collected in volume i of his <i>Mathematical and Physical +Papers</i>, in some respects the best treatise on thermodynamics +at the present time! There are treatises +written from a more general standpoint, which deal +with complex problems of chemical and physical change +of means of thermodynamic potentials, and processes +which are not to be found set forth in this volume of +papers; but even these are to a great extent an outcome +of his "Thermoelastic, Thermomagnetic and +Thermoelectric Properties of Matter."</p> + +<p>In hydrodynamics also Lord Kelvin never lost sight +of practical applications, even while pursuing the most +intensely theoretical researches into the action of vortices +or the propagation of waves. In his later years he +worked out the theory of ship-waves with a power +which has made more than one skilful and successful +cultivator of this branch of science say that he was +no mere mathematician, but a man who, like the +prophets of old, could divine what is hid from the eyes +of ordinary mortals. Of the ultimate importance of<span class='pagenum'><a name="Page_310" id="Page_310">310</a></span> +these for practical questions of the construction of +ships, and the economy of fuel in their propulsion, +there can be little doubt. Unhappily, the applications +will have now to be made by others.</p> + +<p>It is interesting to note that the investigation of waves +in canals with which Lord Kelvin recently enriched +the <i>Proceedings of the Royal Society of Edinburgh</i> have +been carried out by a strikingly ingenious adaptation of +the Fourier solution of the differential equation of the +diffusion of heat along a bar, or of electricity along a +slowly worked cable. Thus, beginning with Fourier +mathematics in his earliest researches, he has in some +of his last work applied the special exponential form +of Fourier solution of the diffusion equation to a +case, that of wave propagation, essentially different +in physical nature, and distinct in mathematical +signification, from that for which it was originally +given.</p> + +<p>Lord Kelvin's written work consists of the <i>Electrostatics +and Magnetism</i>, three volumes of <i>Collected +Mathematical and Physical Papers</i>, three of <i>Popular +Lectures and Addresses</i>, the <i>Baltimore Lectures</i>, a very +considerable number of papers as yet uncollected, +and the <i>Natural Philosophy</i>. But this, great as it was, +represented only a relatively small part of his activities. +He advised public companies on special engineering and +electrical questions, served on Royal Commissions, +acted as consulting engineer to cable companies and +other corporations, was employed as arbiter in disputes +when scientific questions were involved, advocated distinctive +signalling for lighthouses and devised apparatus +for this purpose, and he was, above all, a great +inventor. His patents are many and important. One<span class='pagenum'><a name="Page_311" id="Page_311">311</a></span> +of them was for a water-tap warranted not to drip, +another, for electrical generating machines, meters, +etc., was perhaps the patent of largest extent ever +granted.</p> + +<p>To Lord Kelvin's class teaching reference has +been made in an earlier chapter. He was certainly +inspiring to the best students. At meetings of the +British Association his luminous remarks in discussion +helped and encouraged younger workers, and his +enthusiasm was infectious. But with the ordinary +student who cannot receive or retain his mental nutriment +except by a carefully studied mode of presentation, +he was not so successful. He saw too much +while he spoke; new ideas or novel modes of viewing +old ones presented themselves unexpectedly, associations +crowded upon his mind, and he was apt to be +discursive, to the perplexity of all except those whose +minds were endued also with something of the same +kind of physical instinct or perception. Then he was +so busy with many things that he did not find time to +ponder over and arrange the matter of his elementary +lectures, from the point of view of the presentment +most suitable to the capacity of his hearers. To the +suggestion which has lately been made, that he should +not have been obliged to lecture to elementary +students, he would have been the first to object. As +a matter of fact, in his later years he lectured to the +ordinary class only twice a week, and to the higher +class once. The remainder of the lectures were given +by his nephew, Dr. J. T. Bottomley, who for nearly +thirty years acted as his deputy as regards a great part +of the routine work of the chair.</p> + +<p>It is hardly worth while to refute the statement<span class='pagenum'><a name="Page_312" id="Page_312">312</a></span> +often made that Lord Kelvin could not perform the +operations of simple arithmetic. The truth is, that in +the class-room he was too eager in the anticipation of +the results of a calculation, or too busy with thoughts +of what lay beyond, to be troubled with the multiplication +table, and so he often appealed to his assistants +for elementary information which at the moment his +rapidly working mind could not be made to supply for +itself.</p> + +<p>To sum up, Lord Kelvin's scientific activity had +lasted for nearly seventy years. He was born four +years after Oersted made his famous discovery of the +action of an electric current on a magnet, and two +years before Ampère, founding on this experiment, +brought forth the first great memoir on electromagnetism. +Thus his life had seen the growth of +modern electrical science from its real infancy to its +now vigorous youth. The discoveries of Faraday in +electrical induction were given to the world when +Lord Kelvin was a boy, and one of the great tasks +which he accomplished was to weave these discoveries +together in a uniform web of mathematical theory. +This theory suggested, as we have seen, new problems +to be solved by experiment, which he attacked with +the aid of his students in the small and meagrely +equipped laboratory established sixty years ago in the +Old College in the High Street. It was his lot to live +to see his presentations of theory lead to new developments +in his own hands and the hands of other men +of genius—Helmholtz and Clerk Maxwell, for example—and +to survive until these developments had led to +practical applications throughout our industries, and in +all the affairs of present-day life and work. His true<span class='pagenum'><a name="Page_313" id="Page_313">313</a></span> +monument will be his work and its results, and to only +a few men in the world's history has such a massive +and majestic memorial been reared.</p> + +<p>He was a tireless worker. In every day of his life +he was occupied with many things, but he was never +cumbered. The problems of nature were ever in his +mind, but he could put them aside in the press of +affairs, and take them up again immediately to push +them forward another stage towards solution. His +"green book" was at hand on his table or in his +pocket; and whenever a moment's leisure occurred he +had pencil in hand, and was deep in triple integrals +and applications of Green's Theorem, that unfailing +resource of physical mathematicians.</p> + +<div class="poem"> +<div class="stanza"> +<span class="i0">Saepe stilum vertas quae digna legi sint</span> +<span class="i0">Scripturus,</span> +</div> +</div> + +<p>the motto which Horace recommends, was his, and he +would playfully quote it, pointing to the eraser-pad in +the top of his gold pencil-case. He erased, corrected, +amended, and rewrote with unceasing diligence, to the +dismay of his shorthand-writing secretary.</p> + +<p>The theories and facts of electricity and magnetism, +the production and propagation of waves in water or +in the luminiferous ether, the structure and density of +the ether itself, the relations of heat and work, the +motions of the heavenly bodies, the constitution of +crystals, the theory of music, the practical problems +of navigation, of telegraphing under the sea, and of +the electric lighting of cities—all these and more came +before his mind in turn, and sometimes most of them +in the course of a single day. He could turn from +one thing to another, and find mental rest in diversity +of mental occupation.<span class='pagenum'><a name="Page_314" id="Page_314">314</a></span></p> + +<p>He would lecture from nine to ten o'clock in the +morning to his ordinary class, though generally this +was by no means the first scientific work of the day. +At ten o'clock he passed through his laboratory and +spoke to his laboratory students or to any one who +might be waiting to consult him, answered some urgent +letter, or gave directions to his secretary; then he +walked or drove to White's workshop to immerse himself +in the details of instrument construction until he +was again due at the university for luncheon, or to +lecture to his higher mathematical class on some such +subject as the theory of the tides or the Fourier +analysis.</p> + +<p>As scientific adviser to submarine telegraph companies +and other public bodies, and more recently as +President of the Royal Society of London, he made +frequent journeys to London. These were arranged +so as to involve the minimum expenditure of time. +He travelled by night when alone, and could do so +with comfort, for he possessed the gift of being able to +sleep well in almost any circumstances. Thus he +would go to London one night, spend a busy day in +all kinds of business—scientific, practical, or political—and +return the next night to Glasgow, fresh and eager +for work on his arrival. Here may be noticed his +power of detaching himself from his environment, and +of putting aside things which might well have been +anxieties, and of becoming again absorbed in the problem +which circumstances had made him temporarily +abandon.</p> + +<p>Genius has been said to be the power of taking +infinite pains: it is that indeed, but it is also far more. +Genius means ideas, intuition, a faculty of seizing by<span class='pagenum'><a name="Page_315" id="Page_315">315</a></span> +thought the hidden relations of things, and withal the +power of proceeding step by step to their clear and +full expression, whether in the language of mathematical +analysis or in the diction of daily life. Such +was the genius of Lord Kelvin; it was lofty and it +was practical. He understood—for he had felt—the +fascination of knowledge apart from its application to +mechanical devices; he did not disdain to devote his +great powers to the service of mankind. His objects +of daily contemplation were the play of forces, the +actions of bodies in all their varied manifestations, or, +as he preferred to sum up the realm of physics, the +observation and discussion of properties of matter. But +his eyes were ever open to the bearing of all that he +saw or discovered on the improvement of industrial +appliances, to the possibility of using it to increase the +comfort and safety of men, and so to augment the sum +total of human happiness.</p> + +<p>His statement, which has been so often quoted, that +after fifty-five years of constant study he knew little +more of electricity and magnetism than he did at the +beginning of his career, is not to be taken as a confession +of failure. It was, like Newton's famous declaration, +an indication of his sense of the vastness of the +ocean of truth and the manifoldness of the treasures +which still lie within its "deep unfathomed caves." +Like Newton, he had merely wandered along the shore +of that great ocean, and here and there sounded its +accessible depths, while its infinite expanse lay unexplored. +And also like Newton—indeed like all +great men—he stood with deep reverence before the +great problems of the soul and destiny of man. He +believed that Nature, which he had sought all his life<span class='pagenum'><a name="Page_316" id="Page_316">316</a></span> +to know and understand, showed everywhere the +handiwork of an infinite and beneficent intelligence, +and he had faith that in the end all that appeared +dark and perplexing would stand forth in fulness +of light.</p> + +<hr /> +<h3>FOOTNOTES</h3> +<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">1</span></a> Lord Kelvin's address on his installation as Chancellor of the +University of Glasgow, November 29, 1904.</p></div> + +<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">2</span></a> Successor of Dr. Dick, the Professor of Natural Philosophy +who induced the Faculty to grant a workshop to James Watt when +the Corporation of Hammermen prevented him from starting +business in Glasgow, and for whom Watt was repairing the +Newcomen engine when he invented the separate condenser.</p></div> + +<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">3</span></a> A model steam-engine which he made in his youth was carefully +preserved by his brother in the Natural Philosophy Department. It +was homely but accurate in construction: the beam was of wood, and +the piston was an old thick copper penny!</p></div> + +<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">4</span></a> Proceedings on the occasion of the Presentation to the University +of Glasgow of the Portrait of Emeritus Professor G. G. Ramsay. +November 6, 1907.</p></div> + +<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">5</span></a> Apparently for a short time in 1841, when Dr. Meikleham was +laid aside by illness.</p></div> + +<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">6</span></a> The C.U.M.S. began as a Peterhouse society in 1843, and after a +first concert, which was followed by a supper, and that by "certain +operations on the chapel roof," the Master would only give permission to +hold a second concert in the Red Lion at Cambridge, there being no +room in College, on condition that the society called itself the University +Musical Society. The new society was formed in May 1844; +the first president was G. E. Smith, of Peterhouse, the second was +Blow, also of Peterhouse, a violin player and 'cellist, and the third was +Thomson. [See <i>Cambridge Chronicle</i>, July 10, 1903, and <i>The Cambridge +Review</i>, Feb. 20, 1908.]</p></div> + +<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">7</span></a> It is rather strange that the ninth edition of the <i>Encyclopædia +Britannica</i> contains no biography of Green. Born in the year 1793 at +Nottingham, the son of a baker, he assisted his father, who latterly +acquired a miller's business at the neighbouring village of Sneinton. In +1829 his father died, and he disposed of the business in order that he +might have leisure to give to mathematics, in which, though entirely +self-taught, he had begun to make original researches. His famous +'Essay' was published by subscription in 1828, and attracted but little +attention. In 1833, at forty years of age, Green entered at Gonville and +Caius College, and obtained the fourth place in the mathematical tripos +of 1837, the year of Griffin, Sylvester, and Gregory. His university +career, whatever else it may have done, apparently did not tend to make +his earlier work much better known to the general scientific public, and +he died in 1841 without the scientific recognition which was his due. +That came later when, as stated below, Thomson discovered him to the +French mathematicians and republished his 'Essay.'</p></div> + +<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">8</span></a> January 1869, <i>Reprint</i>, etc., Article XV.</p></div> + +<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">9</span></a> <i>Reprint</i>, Article V.</p></div> + +<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">10</span></a> The geometrical idea was, however, given and applied at least as +early as 1836 by Bellavitis, for a paper entitled "Teoria delle figure +inversa" appears in the <i>Annali delle Scienze del Regno Lombardo-Veneto</i> +for that year. It was also described as an independent discovery +by Mr. John Wm. Stubbs, in a paper in the <i>Philosophical Magazine</i> +for November 1843. In a note on the history of the transformation +in Taylor's <i>Geometry of Conics</i> the date (without reference) of +Bellavitis is given, and it is stated that the method of inversion +was given afresh by Messrs. Ingram and Stubbs (Dublin, <i>Phil. Soc. +Trans.</i> I). The note also mentions that inversion was "applied by +Dr. Hirst to attractions," but contains no reference to Thomson's +papers!</p></div> + +<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">11</span></a> "<i>De Caloris distributione per Terræ Corpus</i>" in the Faculty minute, +as stated above.</p></div> + +<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">12</span></a> <i>Sic.</i> Without doubt a mistake of the scribe for "Liouville."</p></div> + +<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">13</span></a> <i>North Wales Chronicle</i>, Report, Feb. 7, 1885.</p></div> + +<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">14</span></a> Published: <i>Treatise on Natural Philosophy</i>, vol. i in 1867; <i>Elements +of Natural Philosophy</i> in 1873.</p></div> + +<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">15</span></a> The exact date at which this was done cannot be determined from +the Minutes of the Faculty, as they contain no reference to the appropriation +of space for the purpose. In his <i>Oration on James Watt</i>, delivered +at the Ninth Jubilee of the University of Glasgow, in 1901, Lord Kelvin +referred to the Glasgow Physical Laboratory as having grown up between +1846 and 1856; and elsewhere he has referred to it as having +been "incipient" in 1851.</p></div> + +<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">16</span></a> There are now in Glasgow in the winter session alone about 360 +elementary students and 80 advanced students, and about 250 taking +practical laboratory work.</p></div> + +<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">17</span></a> Before his death (in 1832) Carnot had obtained a clear perception +of the true state of the case, and of the complete doctrine of the +conservatism of energy. [See extracts from Carnot's unpublished +writings appended, with a biography, to the reprinted Memoir, by his +younger brother, Hippolyte Carnot.]</p></div> + +<div class="footnote"><p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">18</span></a> This equation for the porous plug experiment may be established +in the following manner, which forms a good example of +Thomson's second definition of absolute temperature. Take pressure +and volume of the gas on the supply side of the plug as <i>p</i> + <i>dp</i> +and <i>v</i>, and on the delivery side as <i>p</i> and <i>v</i> + <i>dv</i>, so that <i>dp</i> and <i>dv</i> are +positive. The net work done in forcing the gas through the plug += (<i>p</i> + <i>dp</i>) <i>v</i> − <i>p</i> (<i>v</i> + <i>dv</i>) = − <i>pdv</i> + <i>vdp</i>. Let a heating effect result +so that temperature is changed from <i>T</i> to <i>T</i> + <i>∂</i><i>T</i>. Let this be annulled +by abstraction of heat <i>C<sub>p</sub></i><i>∂</i><i>T</i> at constant pressure. (<i>C<sub>p</sub></i> = sp. heat press. +const.) [It is to be understood that <i>dv</i> is the total expansion existing, +<i>after</i> this abstraction of heat.] The energy <i>e</i> of the fluid has been +increased by <i>de</i> = − <i>pdv</i> + <i>vdp</i> − <i>C<sub>p</sub></i><i>∂</i><i>T</i>. +</p><p> +Now, since the original temperature has been restored, the same +expansion <i>dv</i> if imposed isothermally would involve the same energy +change <i>de</i>; but in that case heat <i>dH</i> (dynamical) would be absorbed, +and work <i>pdv</i> would be done by the gas. Hence <i>de</i> = <i>dH</i> − <i>pdv</i>. +This, with the former value of <i>de</i>, gives <i>dH</i> = <i>vdp</i> − <i>C<sub>p</sub></i><i>∂</i><i>T</i>. Thomson's +work-ratio is thus <i>pdv</i> ⁄ (<i>vdp</i> − <i>C<sub>p</sub></i><i>∂</i><i>T</i>). Now suppose <i>dp</i> imposed +without change of volume, and <i>dT</i> to be the resulting temperature +change. The temperature and pressure ratios are <i>dT</i> ⁄ <i>T</i>, <i>dp</i> ⁄ <i>p</i>. Thus +<i>dT</i> ⁄ <i>T</i> = <i>dp</i> <i>dv</i> ⁄ (<i>vdp</i> − <i>C<sub>p</sub></i><i>∂</i><i>T</i>), or +</p> +<div class="center"> +<img class="floatInsert35" src="images/ftn18a.png" alt="" title="" /> +</div> +<p> +which is Thomson's equation. The <i>minus</i> sign on the right arises from +a heating effect having been taken here as the normal case. +</p><p> +If the temperature <i>T</i> is restored by removing the heat at constant +volume, a similar process gives the equation +</p> +<div class="center"> +<img class="floatInsert45" src="images/ftn18b.png" alt="" title="" /> +</div> +<p> +where <i>dp</i> is the change of pressure <i>before</i> the restoration of the +temperature <i>T</i>, and <i>∂</i><i>T</i> ⁄ <i>∂</i><i>p</i> is the rate of variation of <i>T</i> with <i>p</i>, volume +constant.</p></div> + +<div class="footnote"><p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">19</span></a> "On a Universal Tendency in Nature to Dissipation of Energy," +<i>Proc. R.S.E.</i>, 1852, and <i>Phil. Mag.</i>, Oct., 1852.</p></div> + +<div class="footnote"><p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">20</span></a> To this may be added the extremely useful theorem for such +problems, that if any directed quantity <i>L</i>, say, characteristic of the motion +of a body, be associated with a line or axis <i>Ol</i>, which is changing in +direction, it causes a rate of production of the same quantity for a line +or axis instantaneously at right angles to <i>Ol</i>, towards which <i>Ol</i> is turning +with angular velocity <i>ω</i>, of amount <i>ωL</i>. If <i>M</i> be the amount of the +quantity already existing for this latter line or axis, the total rate of +growth of the quantity is there <i>M</i> + <i>ωL</i>. For example, a particle +moving with uniform speed <i>v</i> in a circle of radius <i>r</i>, has momentum +<i>mv</i> along the tangent. But the tangent is turning round as the particle +moves with angular speed <i>v</i> ⁄ <i>r</i>, towards the radius. The rate of growth +of momentum towards the centre is therefore</p> + +<p class="center"><i>mv</i> × <i>v</i> ⁄ <i>r</i> = <i>mv</i><sup>2</sup> ⁄ <i>r</i>.</p></div> + +<div class="footnote"><p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">21</span></a> See Gray's <i>Lehrbuch der Physik</i>, s. 278. Vieweg u. Sohn, 1904.</p></div> + +<div class="footnote"><p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">22</span></a> Gray, Royal Institution, Friday Evening Discourse, February 1898.</p></div> + +<div class="footnote"><p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">23</span></a> See the <i>Reports of the Committee on Electrical Standards</i>, edited by +Prof. Fleeming Jenkin, F.R.S., Maxwell's <i>Electricity and Magnetism</i>, and +Gray's <i>Theory and Practice of Absolute Measurements in Electricity and +Magnetism</i>, Vol. II, Part II.</p></div> + +<div class="footnote"><p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">24</span></a> The writer once, on a thick night, in a passenger steamer in the +Race of Alderney, when the engines were stopped and soundings were +being taken, saw the reel and cord go overboard, nearly taking one of +the men with it. A new hank of cord had to be got and bent on a new +reel; an operation that took a long time, during which the exact +locality of the ship was a matter of uncertainty. Comment is needless!</p></div> + +<div class="footnote"><p><a name="Footnote_25_25" id="Footnote_25_25"></a><a href="#FNanchor_25_25"><span class="label">25</span></a> The tuning of a major third, in this way, is described in the paper +entitled "Beats on Imperfect Harmonies," published in <i>Popular Lectures +and Addresses</i>, vol. ii.</p></div> + +<div class="footnote"><p><a name="Footnote_26_26" id="Footnote_26_26"></a><a href="#FNanchor_26_26"><span class="label">26</span></a> The writer well remembers meeting a man of some experience in +cable work who was on his way to measure the alternating currents in +a Jablochkoff candle installation by the aid of an Ayrton and Perry +galvanometer with steel needle!</p></div> + +<hr /> + +<p><span class='pagenum'><a name="Page_317" id="Page_317">317</a></span></p> + +<h3>INDEX</h3> + +<ul class="IX"> +<li> +Atlantic cables, <a href="#Page_267">267</a>, <a href="#Page_268">268</a> +</li><li> +Atmospheric electricity, <a href="#Page_226">226</a> +</li><li> +Atoms, size of, <a href="#Page_261">261</a> +</li><li> +Ayrton, W. E., <a href="#Page_296">296</a> +</li><li> +</li><li> +Baltimore lectures, <a href="#Page_254">254</a>-<a href="#Page_263">263</a> +</li><li> +Bertrand's theorem of maximum kinetic energy, <a href="#Page_158">158</a> +</li><li> +Bottomley, James Thomson, <a href="#Page_311">311</a> +</li><li> +Bottomley, William, <a href="#Page_7">7</a> +</li><li> +British Association, electrical standards, <a href="#Page_244">244</a>-<a href="#Page_253">253</a> +</li><li> +</li><li> +Cambridge University Musical Society, <a href="#Page_24">24</a> +</li><li> +<i>Cambridge and Dublin Mathematical Journal</i>, <a href="#Page_25">25</a>, <a href="#Page_31">31</a>, <a href="#Page_78">78</a> +</li><li> +Carnot, Sadi, <a href="#Page_77">77</a>, <a href="#Page_101">101</a> +</li><li> +Carnot's <i>Théorie Motrice du Feu</i>, <a href="#Page_87">87</a>, <a href="#Page_101">101</a>, <a href="#Page_108">108</a> <i>et seq.</i> +</li><li> +Cauchy, <a href="#Page_294">294</a> +</li><li> +Chasles, <a href="#Page_28">28</a>, <a href="#Page_43">43</a> +</li><li> +Clapeyron, <a href="#Page_101">101</a>, <a href="#Page_112">112</a> +</li><li> +Clausius, <a href="#Page_114">114</a> <i>et seq.</i> +</li><li> +College, the old, of Glasgow, <a href="#Page_10">10</a> +</li><li> +Compass, errors of, <a href="#Page_273">273</a> +</li><li> +</li><li> +"Dew-drop," artificial, <a href="#Page_290">290</a> +</li><li> +Dynamical theorems, Thomson's and Bertrand's, <a href="#Page_158">158</a> <i>et seq.</i> +</li><li> +</li><li> +Earth, the age of, <a href="#Page_196">196</a>, <a href="#Page_229">229</a>-<a href="#Page_243">243</a> +</li><li> +Earth, tidal retardation of, <a href="#Page_230">230</a> +</li><li> +Elasticity, Poisson-Navier theory of, <a href="#Page_291">291</a>; +<ul><li> +encyclopædia article on, <a href="#Page_297">297</a> +</li></ul></li><li> +Electrical oscillations, <a href="#Page_181">181</a> <i>et seq.</i> +</li><li> +Electricity, mathematical theory of, <a href="#Page_33">33</a> +</li><li> +Electrolysis, mechanical theory of, <a href="#Page_176">176</a> +</li><li> +Electrometers, <a href="#Page_223">223</a> <i>et seq.</i> +</li><li> +Electromotive forces, estimation of, by heats of combination, <a href="#Page_178">178</a> +</li><li> +Electromotive forces, measurement of, <a href="#Page_179">179</a> +</li><li> +<i>Electrostatics and Magnetism</i>, <a href="#Page_222">222</a> <i>et seq.</i> +</li><li> +Ellis, Robert Leslie, <a href="#Page_26">26</a> +</li><li> +Energy, dissipation of, <a href="#Page_139">139</a> +</li><li> +</li><li> +Faculty, the, of the University of Glasgow, <a href="#Page_4">4</a>, <a href="#Page_63">63</a>-<a href="#Page_67">67</a> +</li><li> +Faraday, <a href="#Page_61">61</a> +</li><li> +Faure, M., <a href="#Page_81">81</a> +</li><li> +FitzGerald, G. F., <a href="#Page_301">301</a>, <a href="#Page_305">305</a> +</li><li> +Fourier, <i>Théorie Analytique de la Chaleur</i>, <a href="#Page_16">16</a> <i>et seq.</i> +</li><li> +</li><li> +Gauss, <a href="#Page_28">28</a> +</li><li> +Gauss and Weber, <a href="#Page_245">245</a> +</li><li> +Green, George, of Nottingham, <a href="#Page_21">21</a>, <a href="#Page_30">30</a>, <a href="#Page_294">294</a> +</li><li> +Gregory, J. W., <a href="#Page_241">241</a> +</li><li> +Goodwin, Harvey, <a href="#Page_26">26</a> +</li><li> +Gyrostats and gyrostatic action, <a href="#Page_214">214</a>, <a href="#Page_284">284</a>-<a href="#Page_286">286</a> +</li><li> +</li><li> +Hamilton, Sir William Rowan, <a href="#Page_196">196</a>, <a href="#Page_294">294</a> +</li><li> +Heat, encyclopædia article on, <a href="#Page_297">297</a> +</li><li> +Heaviside, Oliver, <a href="#Page_294">294</a> +</li><li> +Helmholtz, von, <a href="#Page_113">113</a> +</li><li> +Hertz, <a href="#Page_191">191</a>, <a href="#Page_256">256</a> +</li><li> +Hopkins, William, <a href="#Page_23">23</a> +</li><li> +Huxley, <a href="#Page_77">77</a>, <a href="#Page_196">196</a>, <a href="#Page_242">242</a> +</li><li> +Hydrodynamics, <a href="#Page_153">153</a>-<a href="#Page_175">175</a> +</li><li> +</li><li> +Images, electric, <a href="#Page_31">31</a>, <a href="#Page_38">38</a>-<a href="#Page_59">59</a> +</li><li> +Inversion, electrical, <a href="#Page_49">49</a> <i>et seq.</i> +<span class='pagenum'><a name="Page_318" id="Page_318">318</a></span> +</li><li> +Inversion, geometrical, <a href="#Page_59">59</a>, <a href="#Page_60">60</a> +</li><li> +</li><li> +Joule, James Prescott, <a href="#Page_77">77</a>, <a href="#Page_86">86</a> <i>et seq.</i>, <a href="#Page_101">101</a> <i>et seq.</i> +</li><li> +</li><li> +Larmor, Joseph, <a href="#Page_256">256</a> +</li><li> +Lectures on Natural Philosophy at Glasgow, <a href="#Page_279">279</a> <i>et seq.</i> +</li><li> +Liouville, <a href="#Page_31">31</a> +</li><li> +Liouville's <i>Journal de Mathématiques</i>, <a href="#Page_25">25</a>, <a href="#Page_26">26</a>, <a href="#Page_31">31</a> +</li><li> +Loschmidt, <a href="#Page_262">262</a> +</li><li> +Lubbock, Sir John (Lord Avebury), <a href="#Page_85">85</a> +</li><li> +Luminiferous ether, motion of planets through, <a href="#Page_256">256</a> +</li><li> +</li><li> +Magnetism, theory of, <a href="#Page_227">227</a> +</li><li> +Mariners' compass, <a href="#Page_272">272</a> <i>et seq.</i> +</li><li> +Maxwell, <a href="#Page_117">117</a>, <a href="#Page_193">193</a>, <a href="#Page_305">305</a> +</li><li> +Mayer, of Heilbronn, <a href="#Page_105">105</a> +</li><li> +McFarlane, Donald, <a href="#Page_96">96</a>, <a href="#Page_287">287</a>, <a href="#Page_289">289</a> +</li><li> +McKichan, Dugald, <a href="#Page_193">193</a> +</li><li> +<i>Mécanique Analytique</i> of Lagrange, <a href="#Page_199">199</a>, <a href="#Page_205">205</a> +</li><li> +<i>Mécanique Céleste</i> of Laplace, <a href="#Page_199">199</a>, <a href="#Page_205">205</a> +</li><li> +Meikleham, William, <a href="#Page_61">61</a> +</li><li> +Mirror galvanometer, <a href="#Page_268">268</a> +</li><li> +Motivity, thermodynamic, <a href="#Page_138">138</a> +</li><li> +</li><li> +Natural Philosophy, Chair of, at Glasgow, <a href="#Page_63">63</a> +</li><li> +<i>Natural Philosophy</i>, Thomson and Tait's, <a href="#Page_196">196</a> <i>et seq.</i> +</li><li> +Navigational sounding machine, <a href="#Page_272">272</a> +</li><li> +Newton, <a href="#Page_195">195</a>, <a href="#Page_202">202</a> +</li><li> +Nichol, John, Professor of English Language and Literature, <a href="#Page_5">5</a> +</li><li> +Nichol, John Pringle, Professor of Astronomy, <a href="#Page_5">5</a>, <a href="#Page_20">20</a>, <a href="#Page_61">61</a>, <a href="#Page_63">63</a> +</li><li> +</li><li> +Oersted, <a href="#Page_61">61</a> +</li><li> +Oscillations, electrical, <a href="#Page_181">181</a> <i>et seq.</i> +</li><li> +</li><li> +Parkinson, Stephen, <a href="#Page_27">27</a> +</li><li> +Peltier, <a href="#Page_148">148</a> +</li><li> +Pendulum, ballistic, <a href="#Page_288">288</a> +</li><li> +Perry, John, <a href="#Page_240">240</a>, <a href="#Page_296">296</a> +</li><li> +Phosphorescence, dynamical theory of, <a href="#Page_259">259</a> +</li><li> +Physical laboratory, first, <a href="#Page_70">70</a> +</li><li> +Pickering, <a href="#Page_217">217</a> +</li><li> +Polarised light, rotation of plane of, <a href="#Page_220">220</a> +</li><li> +<i>Principia</i>, Newton's, <a href="#Page_195">195</a>, <a href="#Page_202">202</a> +</li><li> +</li><li> +Ramsay, George Gilbert, Professor of Humanity, <a href="#Page_11">11</a> +</li><li> +Regnault, <a href="#Page_29">29</a> +</li><li> +Royal Society of Edinburgh, presidency of, <a href="#Page_299">299</a> +</li><li> +Royal Society of London, presidency of, <a href="#Page_299">299</a> +</li><li> +Rumford, Count, <a href="#Page_103">103</a> +</li><li> +</li><li> +Seebeck, <a href="#Page_148">148</a> +</li><li> +Signalling, theory of telegraphic, <a href="#Page_264">264</a> +</li><li> +Siphon recorder, <a href="#Page_268">268</a>, <a href="#Page_270">270</a> +</li><li> +Smith, Archibald, <a href="#Page_275">275</a> +</li><li> +Spectrum analysis, dynamical theory of, <a href="#Page_84">84</a> +</li><li> +Stokes, Sir George Gabriel, <a href="#Page_24">24</a>, <a href="#Page_79">79</a>, <a href="#Page_80">80</a>, <a href="#Page_81">81</a>, <a href="#Page_85">85</a>, <a href="#Page_291">291</a>, <a href="#Page_294">294</a> +</li><li> +Stoney, Dr. Johnstone, <a href="#Page_262">262</a> +</li><li> +Sun's heat, age of, <a href="#Page_232">232</a> +</li><li> +</li><li> +Tait, Peter Guthrie, <a href="#Page_194">194</a> <i>et seq.</i> +</li><li> +Temperature, absolute, <a href="#Page_125">125</a> <i>et seq.</i>; +<ul><li> +comparison of, with scale of air thermometer, <a href="#Page_135">135</a> +</li></ul></li><li> +Thermodynamics, <a href="#Page_99">99</a>-<a href="#Page_152">152</a> +</li><li> +Thermoelasticity, <a href="#Page_142">142</a> <i>et seq.</i> +</li><li> +Thermoelectricity, <a href="#Page_147">147</a> <i>et seq.</i> +</li><li> +Thermometry, absolute, <a href="#Page_114">114</a>-<a href="#Page_152">152</a> +</li><li> +Thomson, David, <a href="#Page_61">61</a> +</li><li> +Thomson, James, Professor of Mathematics, <a href="#Page_1">1</a>-<a href="#Page_4">4</a>, <a href="#Page_7">7</a> +</li><li> +Thomson, James, Professor of Engineering, <a href="#Page_113">113</a>, <a href="#Page_209">209</a>; +<ul><li> +integrating machine, <a href="#Page_209">209</a>, <a href="#Page_303">303</a> +</li></ul></li><li> +Thomson and Tait's Natural Philosophy, <a href="#Page_68">68</a>, <a href="#Page_196">196</a> <i>et seq.</i>, <a href="#Page_218">218</a> +</li><li> +Thomson's theorem of minimum kinetic energy, <a href="#Page_158">158</a> +</li><li> +Thomson, Thomas, Professor of Chemistry, <a href="#Page_6">6</a> +<span class='pagenum'><a name="Page_319" id="Page_319">319</a></span> +</li><li> +<i>Thomson</i>, prevalence of name at Glasgow College, <a href="#Page_5">5</a> +</li><li> +Thomson, William, Lord Kelvin:— +<ul><li> + Parentage and early education, <a href="#Page_1">1</a>-<a href="#Page_12">12</a> + </li><li> + Career at Universities of Glasgow and Cambridge, <a href="#Page_13">13</a>-<a href="#Page_32">32</a> + </li><li> + Early researches, <a href="#Page_16">16</a>, <a href="#Page_18">18</a>, <a href="#Page_31">31</a> + </li><li> + Election to Chair of Natural Philosophy at Glasgow, <a href="#Page_64">64</a> + </li><li> + Scientific researches, <i>passim</i>; + <ul><li> + Jubilee of, <a href="#Page_301">301</a>; + </li><li> + Chancellor of University of Glasgow, <a href="#Page_302">302</a> + </li></ul></li><li> + In class-room and laboratory, <a href="#Page_279">279</a>-<a href="#Page_298">298</a> + </li><li> + Practical activities, honours and distinctions, last illness and death, <a href="#Page_299">299</a>-<a href="#Page_304">304</a>; + <ul><li> + funeral in Westminster Abbey, <a href="#Page_304">304</a> + </li></ul></li></ul></li><li> + +Tidal Analyser, <a href="#Page_211">211</a> +</li><li> +Tide Predicter, <a href="#Page_208">208</a> +</li><li> +</li><li> +Vortex-Motion, <a href="#Page_161">161</a>-<a href="#Page_175">175</a> +</li><li> +</li><li> +Waldstein sonata, <a href="#Page_24">24</a> +</li><li> +Weber, W., <a href="#Page_193">193</a> +</li><li> +Weights and measures, British, <a href="#Page_289">289</a>, <a href="#Page_290">290</a> +</li><li> +White, James, <a href="#Page_276">276</a> +</li><li> +Willard Gibbs, <a href="#Page_294">294</a> +</li></ul> + +<hr /> +<p class="center"> +<span class="smcap">Richard Clay & Sons, Limited</span>,<br /> +<small>BREAD STREET HILL, E.C., AND<br /> +BUNGAY, SUFFOLK.</small><br /> +</p> + + + + + + + + + +<pre> + + + + + +End of the Project Gutenberg EBook of Lord Kelvin, by Andrew Gray + +*** END OF THIS PROJECT GUTENBERG EBOOK LORD KELVIN *** + +***** This file should be named 39373-h.htm or 39373-h.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/9/3/7/39373/ + +Produced by Laura Wisewell, Turgut Dincer, Tamise Totterdell +and the Online Distributed Proofreading Team at +http://www.pgdp.net (The original copy of this book was +generously made available for scanning by 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