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diff --git a/39373-0.txt b/39373-0.txt new file mode 100644 index 0000000..8e40560 --- /dev/null +++ b/39373-0.txt @@ -0,0 +1,9284 @@ +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.) + + + + + + + + + +--------------------------------------------------------------------+ + | | + | | + | TRANSCRIBER'S NOTES: | + | | + | Mathematical expressions which are in italics in the original text | + | are displayed in non-italic characters to increase the readability | + | of the book. | + | | + | For super and subscripts unicode characters have been used except | + | a few cases where carat symbol and underline have been used. | + | | + | Mathematical notation in the body of text is duly conserved but | + | parantheses have been added to expressions to avoid any kind of | + | misenterpretation. | + | | + +--------------------------------------------------------------------+ + + + + ENGLISH MEN OF SCIENCE + + EDITED BY + J. REYNOLDS GREEN, Sc.D. + + LORD KELVIN + + + + ENGLISH MEN + OF SCIENCE + + EDITED BY + DR. J. REYNOLDS GREEN. + + _With Photogravure Frontispiece._ + _Small Cr. 8vo, 2s. 6d. net per vol._ + + SPENCER. By J. ARTHUR THOMPSON. + PRIESTLEY. By Dr. THORPE, C.B., F.R.S. + FLOWER. By Prof. R. LYDEKKER, F.R.S. + HUXLEY. By Prof. AINSWORTH DAVIS. + BENTHAM. By B. DAYDON JACKSON, F.L.S. + DALTON. By J. P. MILLINGTON, M.A. + + _J. M. DENT & CO._ + + + _All Rights Reserved_ + + + + [Illustration: Lord Kelvin] + + + + LORD KELVIN + + _AN ACCOUNT OF HIS SCIENTIFIC LIFE AND WORK_ + + + BY + + ANDREW GRAY + LL.D., F.R.S., V.-P.R.S.E. + + PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW + + + PUBLISHED IN LONDON BY + J. M. DENT & CO., AND IN NEW + YORK BY E. P. DUTTON & CO. + 1908 + + + + RICHARD CLAY & SONS, LIMITED, + BREAD STREET HILL, E.C., AND + BUNGAY, SUFFOLK. + + + + +PREFACE + + +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 _Glasgow +Herald_ 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. + +I am under obligations to the proprietors of the _Glasgow Herald_ 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. + + A. GRAY. + + _The University_, _Glasgow_, + _May_ 20, 1908. + + + + +CONTENTS + + + CHAP. PAGE + + I. PARENTAGE AND EARLY EDUCATION 1 + + II. CLASSES AT THE UNIVERSITY OF GLASGOW. FIRST SCIENTIFIC + PAPERS 13 + + III. UNIVERSITY OF CAMBRIDGE. SCIENTIFIC WORK AS UNDERGRADUATE 23 + + IV. THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM. + ELECTRIC IMAGES. ELECTRIC INVERSION 33 + + V. THE CHAIR OF NATURAL PHILOSOPHY AT GLASGOW. ESTABLISHMENT + OF THE FIRST PHYSICAL LABORATORY 61 + + VI. FRIENDSHIP WITH STOKES AND JOULE. EARLY WORK AT GLASGOW 79 + + VII. THE 'ACCOUNT OF CARNOT'S THEORY OF THE MOTIVE POWER OF + HEAT'--TRANSITION TO THE DYNAMICAL THEORY OF HEAT 99 + + VIII. THERMODYNAMICS AND ABSOLUTE THERMOMETRY 114 + + IX. HYDRODYNAMICS--DYNAMICAL THEOREM OF MINIMUM + ENERGY--VORTEX MOTION 153 + + X. THE ENERGY THEORY OF ELECTROLYSIS--ELECTRICAL + UNITS--ELECTRICAL OSCILLATIONS 176 + + XI. THOMSON AND TAIT'S 'NATURAL PHILOSOPHY'--GYROSTATIC + ACTION--'ELECTROSTATICS AND MAGNETISM' 194 + + XII. THE AGE OF THE EARTH 229 + + XIII. BRITISH ASSOCIATION COMMITTEE ON ELECTRICAL STANDARDS 244 + + XIV. THE BALTIMORE LECTURES 254 + + XV. SPEED OF TELEGRAPH SIGNALLING--LAYING OF SUBMARINE + CABLES--TELEGRAPH INSTRUMENTS--NAVIGATIONAL + INSTRUMENTS, COMPASS AND SOUNDING MACHINE 264 + + XVI. LORD KELVIN IN HIS CLASS-ROOM AND LABORATORY 279 + + XVII. PRACTICAL ACTIVITIES--HONOURS AND DISTINCTIONS--LAST + ILLNESS AND DEATH 299 + + CONCLUSION 305 + + INDEX 317 + + +CORRIGENDUM + + Page 105, line 9 from foot, for ∂e + O read ∂e + o + + + + +LIST OF ILLUSTRATIONS + + + _To face page_ + LORD KELVIN (photogravure) _Frontispiece_ + LORD KELVIN IN 1846 64 + VIEW OF OLD COLLEGE 70 + + + + +LORD KELVIN + + + + +CHAPTER I + +PARENTAGE AND EARLY EDUCATION + + +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. + +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 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."[1] 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. + +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 Principia had +tended to perpetuate in this country. Laplace was at the height of his +fame in France, and was writing the great analytical Principia, his +_Mécanique Céleste_, 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. + +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. + +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 papers. Even moderators are not made of +impenetrable stuff, though fenced with sevenfold Jacquier, and tough +bull-hide of Vince and Wood." + +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. + +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 to manage the +library and some other matters) to regulate and superintend teaching and +discipline. + +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:-- + +"Niebuhr, after examining a portion of the _Fasti Consulares_, arrived +at the conclusion that the _senatus populusque Romanus_ 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 gens 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." + +A former Professor of Natural Philosophy, Dr. Anderson,[2] 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"! + +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 +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. + +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 gens 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 +Glasgow, a connection only terminated by the funeral ceremony in +Westminster Abbey on December 23, 1907. + +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." + +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.[3] 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, 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. + +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 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 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. + +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.[4] "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 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." + +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. + + + + +CHAPTER II + +CLASSES AT THE UNIVERSITY OF GLASGOW. FIRST SCIENTIFIC PAPERS + + +In 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 _Dialogues of +the Gods_ "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 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. + +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: + +"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.... + +"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.... + +"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,[5] Dr. Nichol, added 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.... + +"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. That +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'! + +"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 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." + +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, _La Théorie Analytique de la +Chaleur_, 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 _Cambridge Mathematical Journal_ (vol. ii, May 1841). +The object of the paper is to show in what cases a function f(x), which +is to have certain arbitrary values between certain values of x, 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 x, between 0 and a, +say, and for the assigned values of the function, f(x) 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 x, which lies outside the +limits of that variable between which the values of the function are +assigned, the values of f(x) there are to be found from the expansion +adopted, by rules which are laid down in the paper. + +Fourier used sine-expansions or cosine-expansions as it suited him for +the function between the limits, 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 x 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. + +The next paper is of the same final date, but appeared in the +_Cambridge Mathematical Journal_ 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 F(x), which +expresses the temperature at distance x from the centre, can be +expanded in an infinite series of the form + + a₁ sin n₁x + a₂ sin n₂x + ... + +where a₁, a₂, ... are multipliers to be determined and n₁, n₂, ... +are the roots, infinite in number, of the transcendental equation +(tan nX)⧸nX = 1 － hX. + +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 equation lies between 0 and 1⧸2, the second between +1 and 3⧸2, the third between 2 and 5⧸2, 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 x the same +value as the series + + A₁ sin (1⧸2)x + A₂ sin (3⧸2)x + ... + +where A₁, A₂, ... are known in terms of a₁, a₂, ... 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. + +The next ten papers, with two exceptions, are all on the motion of heat, +and appeared in the _Cambridge Mathematical Journal_ 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. + +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 +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. + +Two other great works of his father's collection of mathematical books, +Laplace's _Mécanique Céleste_ and Lagrange's _Mécanique Analytique_, +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 _Natural Philosophy_. + +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 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. + +The _Cyclopædia of Physical Science_ 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 +_Differential and Integral Calculus_ 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. + +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 _Cambridge Mathematical Journal_ 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 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. + +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 Journal (vols. iii and v), and in the celebrated +memoir by Gauss "On General Theorems 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 +_Scientific Memoirs_ in 1842. These anticipations are again referred to +below. + + + + +CHAPTER III + +UNIVERSITY OF CAMBRIDGE. SCIENTIFIC WORK AS UNDERGRADUATE + + +Thomson 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 _Cambridge Mathematical Journal_, 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 +of the C.U.M.S., the Cambridge University Musical Society.[6] 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. + +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. + +At Cambridge Thomson made the acquaintance of George Gabriel Stokes, who +graduated as Senior 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. + +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 +_Cambridge Mathematical Journal_ between 1842 and 1845; but three +appeared in 1845 in Liouville's _Journal de Mathématiques_. 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. + +Another paper, published in the form of a letter, of date October 8, +1845, to M. Liouville, and published in the _Journal de Mathématiques_ +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. + +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. + +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 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! + +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 _Cambridge Mathematical Journal_; +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. + +The Tripos Examination was held in the early part of January. On the +25th of that month Thomson 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. +21). 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. + +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 _Cambridge Mathematical Journal_ 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 Journal. + +Soon after the Tripos and Smith's Prize Examinations 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, "_Vous avez un Mèmoire de Green! M. Liouville me l'a dit._" +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, "_Voila mon affaire!_" + +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. + +It is a striking testimony to Thomson's genius that, 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. + +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.[7] In 1850 he obtained the +republication of Green's 'Essay' in Crelle's Journal; in later years he +frequently expressed regret that it had not been published in England. + +In the commencement of 1845 Thomson told Liouville of the method of +_Electric Images_ 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 +_Philosophical Magazine_.[8] This had been preceded by a series of +papers in March, May, and November 1848, November 1849, and February +1850, in the _Cambridge and Dublin Mathematical Journal_, on various +parts of the mathematical theory of electricity in equilibrium,[9] in +which the theory of images is dealt with. The letters to Liouville +promptly appeared in the Journal, 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." + +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. + + + + +CHAPTER IV + +THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM. ELECTRIC IMAGES. +ELECTRIC INVERSION + + +In 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. + +The whole space in which a distribution of electricity produces any +action on electrified bodies is called the _electrical field_ 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 _field-intensity_ 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, X, Y, Z in +three chosen directions at right angles to one another. + +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 +V 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, _point-charges_ we shall call +them], their positions, and the position of the point at which the +field-intensity is to be found. If q₁, q₂, ... be the point-charges, and +be positive when the charges are positive and negative when the charges +are negative, and r₁, r₂, ... be their distances from the point P, +V is q₁⧸r₁ + q₂⧸r₂ + ... The field-intensity is the rate of diminution +of the value of V at P, taken along the specified direction. The three +gradients parallel to the three chosen coordinate directions are +X, Y, Z; but for their calculation it is necessary to insert the values +of r₁, r₂, ... in terms of the coordinates which specify the positions +of the point-charges, and the coordinates x, y, z which specify the +position of P. Once this is done, X, Y, Z are obtained by a simple +systematic process of calculation, namely, differentiation of the +function V with respect to x, y, z. + +This function V seems to have been first used by Laplace for +gravitational matter in the _Mécanique Céleste_; its importance for +electricity and magnetism was recognised by Green, who named it the +potential. 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 P from a point at an infinite distance from every part of the +distribution; or, in other words, what we now call the _potential +energy_ of a charge q situated at P is qV. The excess of the potential +at P, over the potential at any other point Q in the field, is the work +which must be spent in carrying a positive unit from Q to P against +electrical repulsion. Of course, if the force to be overcome from Q to P +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 Q is higher than +that of P. + +The difference of potential depends only on the points P and Q, and not +at all on the path pursued between them. Thus, if a unit of electricity +be carried from P to Q 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 _mutatis mutandis_ of gravitational action. + +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 sinks +(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 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. + +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. + +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. + +Now let heat be poured into the medium at constant rate by a single +point-source P (Fig. 1), and drawn off at a smaller rate by a single +point-sink P', 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 +S 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 a be the radius of +the sphere and f the distance of the source from the centre of the +sphere, the heat carried off by the sink is the fraction a⧸f of that +given out by the source. + +[Illustration: FIG. 1.] + +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 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. + +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. + +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. + +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. 43 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 are willing to acquire, some elementary knowledge of +electricity. + +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." + +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 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. + +[Illustration: FIG. 2.] + +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 A and B (Fig. 2), and regard them for the moment as +mirrors. A first image of an object P between the two mirrors is +produced directly by each, but the image I₁ in A is virtually an object +in front of B, and the image J₁ in B an object in front of A, so that +a second image more remote from the mirror than the first is produced in +each case. These second images I₂ and J₂ 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. + +[Illustration: FIG. 3.] + +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 in Brewster's kaleidoscope, namely at the angular points +of a hexagon, the sides of which are alternately (as shown in Fig. 3) of +lengths twice the distance of the electrified point from A and from B. + +[Illustration: FIG. 4.] + +Now consider the spherical surface referred to at p. 37, which is kept +at uniform potential by a charge at the external point P, and a charge +q' at the inverse point P' within the sphere. If E (Fig. 4) be any point +whatever on the surface, and r, r' be its distances from P and P', it is +easy to prove by geometry that the two triangles CPE and CEP' are +similar, and therefore r' = ra⧸f. [Here a⧸f is used to mean a divided +by f. 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 q at another point, is equal +to the ratio of the charge q to the distance between the points. Thus +the potential at E due to the charge q at P is q⧸r, and that at E due to +a charge q' at P' is q'⧸r'. Thus if q' = －qa⧸f, q' at P' will produce +a potential at E = －qa⧸fr' = －q⧸r, by the value of r. Hence q at P +and -qa⧸f at P' coexisting will give potential q⧸r + －q⧸r or zero, +at E. Thus the charge －qa⧸f, at the internal point P' will in presence +of +q at P keep all points of the spherical surface at zero potential. +These two charges represent the source and sink in the thermal analogue +of p. 37 above. + +Now replace S 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 q at P +alone, a charge is induced upon it which, in presence of P, maintains +it at zero potential. The internal charge －qa⧸f, 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 q at P. 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 external 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. + +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 N, and s be the surface-density: then according to the +system of units usually adopted 4πs = N. + +Let now the rate of diminution of potential per unit of 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. + +Thus it is only necessary to find for －qa⧸f at P' and q at P, the +falling off gradient N of potential outside the spherical surface at +any point E, and to take N⧸4π, to obtain s the surface-density at E. +Calculation of this gradient for the sphere gives 4πs = －q(f² － a²)⧸ar³. +The surface-density is thus inversely as the cube of the distance PE. + +If the influencing point P 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 P will now +be outside, but given by the same relation. And a will now be greater +than f, and the density will be given by 4πs = －q(a² － f²)⧸ar³, +where, f and r have the same meanings with regard to E and P +as before. + +P' is in each case called the image of P in the sphere S, and the +charge －qa⧸f there supposed situated is the _electric image_ of the +charge q at P. It will be seen that an electric image is a charge, or +system of 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. + +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 D on the other side of the surface, this +surface distribution may be equivalent to several different arrangements +of D. Thus the point-charge at P' 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 P' (Fig. 4) would do as well, provided this +spherical surface were not large enough to extend beyond the surface S. + +In order to find the potential of the sphere (Fig. 4) when insulated +with a charge Q upon it, in presence of the influencing charge q at the +external point P, it is only necessary to imagine uniformly distributed +over the sphere, already electrified in the manner just explained, the +charge Q + aq⧸f. Then the whole charge will be Q, and the uniformity of +distribution will be disturbed, as required by the action of the +influencing point-charge. The potential will be Q⧸a + q⧸f. For a +given potential V of the sphere, the total charge is aV － aq⧸f, +that is the charge is aV over and above the induced charge. + +If instead of a single influencing point-charge at P 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 that obtained by +superimposing all the surface distributions found for the different +influencing points. + +The force of repulsion between the point-charge q and the sphere +(with total charge Q) can be found at once by calculating the sum +of the forces between q at P and the charges Q + aq⧸f at C +and －aq⧸f at 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 Q uninfluenced by an external charge, one +representing the energy on a small conductor (not a point) at P existing +alone, and a third representing the mutual energy of the electrification +on the sphere and the charge q at P 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 f by a small +amount df. This will be the product F.df of the force F required and the +displacement. + +Either method may be applied to find the forces of attraction and +repulsion for the systems of electrified spheres described below. + +The problem of two mutually influencing non-intersecting spheres, S₁, S₂ +(Fig. 5), insulated with given charges, q₁, q₂, 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 S₂, say, be kept unaltered, while that on S₁ is allowed +to change, the action of S₂ on S₁ is the same as if the charge q₂ were +at the centre C₂ of S₂. Thus if f be the distance between the centres +C₁, C₂, and a₁ be the radius of S₁, the distribution will be that +corresponding to q₁ + a₁q₂⧸f uniformly distributed on S₁ together with +the induced charge －a₁q₂⧸f, which corresponds to the image-charge at +the point I₁ (within S₁), the inverse of C₂ with respect to S₁. Now +let the charge on S₁ be fixed in the state just supposed while that +on S₂ is freed. The charge on S₂ will rearrange itself under the +influence of q₁ + a₁q₂⧸f ( = q') and －a₁q₂⧸f, considered as at C₁ +and I₁ respectively. The former of these will give a distribution +equivalent to q₂ + a₂q'⧸f uniformly distributed over S₂, and an +induced distribution of amount a₂q'⧸f at J₁, the inverse point of C₁ +with regard to S₂. The image-charge －a₁q₂⧸f at I₁ in S₁ will react +on S₂ and give an induced distribution －a₂(－a₁q₂⧸f)f', (I₁C₂ = f') +corresponding to an image-charge a₂a₁q₂⧸ff' at the inverse point J₂ +of P₁ with respect to C₂S₂. Thus the distribution on S₂ is equivalent +to q₂ + a₂q'⧸f － a₂a₁q₂⧸ff' distributed uniformly over it, together with +the two induced distributions just described. + +[Illustration: FIG. 5.] + +In the same way these two induced distributions on S₂ may now be +regarded as reacting on the distribution on S₁ as would point-charges +－a₂q₁⧸f and a₂a₁q₂⧸ff', situated at J₁ and J₂ respectively, and would +give two induced distributions on S₁ corresponding to their images +in S₁. + +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. + +If the final potentials, V₁, V₂, 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 a₁V₁, +a₂V₂ in the two cases. The uniform distribution on S₁ will raise the +potential of S₂ above V₂, and to bring the potential down to V₂ in +presence of this distribution we must place an induced distribution +over S₂, represented as regards the external field by the image-charge +－a₂a₁V₁⧸f (at the image of C₁ in S₂) where f is the distance +between the centres. The charge a₂V₂ on S₂ will similarly have an +action on S₁ to be compensated in the same way by an image-charge +－a₁a₂V₂⧸f at the image of C₂ in S₁. Now these two image-charges +will react on the spheres S₁ and S₂ 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 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. + +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. + +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. + +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 q at P between +the inner surface of the outer and the outer surface of the inner, as +shown in Fig. 7. There the influence of q at P, 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. + +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. 6 the distances OP, OP, OQ, OQ' fulfil +the relation OP.OP' = OQ.OQ' = a². Thus P' is (see p. 37) the inverse +of the point P with respect to a sphere of radius a and centre O +(indicated by the dotted line in Fig. 6), and similarly Q' is the +inverse of Q with respect to the same sphere and centre. O is called the +centre of inversion, and the sphere of radius a is called the sphere of +inversion. Thus the sphere of Figs. 1 and 4 is the sphere of inversion +for the points P and P', which are inverse points of one another. For +any system of points P, Q, ..., another system P', Q', ... 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 P', Q', ... be regarded as the direct system, P, Q, ... +will be the corresponding inverse system with regard to the same sphere +and centre. P' is the image of P, and P is the image of P', and so on, +with regard to the same sphere and centre of inversion. + +[Illustration: FIG. 6.] + +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 passing through the centre of inversion. Obviously the inverse +of a sphere concentric with the sphere of inversion is a concentric +sphere. + +The line P'Q' is of course not the inverse of the line PQ, which has +for its inverse the circle passing through the three points O, P', Q', +as indicated in Fig. 6. + +The following results are easily proved. + +A locus and its inverse cut any line OP at the same angle. + +To a system of point-charges q₁, q₂, ... at points P₁, P₂, ... on +one side of the surface of the sphere of inversion there is a system +of charges aq₁⧸f₁, aq₂⧸f₂, ... on the other side of the spherical +surface [OP₁ = f₁, OP₂ = f₂]. 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. + +If V, V' be the potentials produced by the whole direct system at Q, +and by the whole inverse system at Q', V'⧸V = r⧸a = a⧸r', where OQ = r, +OQ' = r'. + +Thus if V is constant over any surface S', V' is not a constant over the +inverse surface S', unless r is a constant, that is, unless the surface +S' 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. + +Further, if q be distributed over an element dS of a surface, the +inverse charge aq⧸f will be distributed over the corresponding element +dS' of the inverse surface. But dS'⧸dS = a⁴⧸f⁴ = f'⁴⧸a⁴ where f, f' +are the distances of O from dS and dS'. Thus if s be the density on dS +and s' the inverse density on dS' we have s'⧸s = a³⧸f'³ = f³⧸a³. + +When V is constant over the direct surface, while r has different +values for different directions of OQ, the different points of the +inverse surface may be brought to zero potential by placing at O a +charge －aV. For this will produce at Q' a potential －aV⧸r' which +with V' will give at Q' a potential zero. This shows that V' is the +potential of the induced distribution on S' due to a charge －aV at O, +or that －V' is the potential due to the induced charge on S' produced +by the charge aV at O. + +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. + +[Illustration: FIG. 7.] + +Now take the case of two concentric spheres insulated and influenced by +a point-charge q placed at a point P between them as shown in Fig. 7. We +have seen at p. 49 how the induced distribution, and the 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 S₁ and S₂, are respectively －q.OA.PB⧸(OP.AB), －q.OB.AP⧸(OP.AB). + +It may be noticed that the sum of the induced charges is －q, and that +as the radii of the spheres are both made indefinitely great, while +the distance AB is kept finite, the ratios OA⧸OP, OB⧸OP approximate +to unity, and the charges to －q.PB⧸AB, －q.AP⧸AB, that is, the +charges are inversely as the distances of P 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. + +Now let this induced distribution, on the two concentric spheres, be +inverted from P as centre of inversion. We obtain two non-intersecting +spheres, as in Fig. 5, 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 _Electricity and Magnetism_, 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. + +[Illustration: FIG. 8.] + +Again take the case of two parallel infinite planes under the influence +of a point-charge between them. This system inverted from P as centre +gives the equilibrium distribution on two charged insulated spheres in +contact (Fig. 8); 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°⧸n, where n is a whole number, and are +therefore bounded in one direction by the straight line which is their +intersection. The image points I₁, J₁, ..., of P placed in the angle +between the planes are situated as shown in Fig. 3, and are n － 1 in +number. This system inverted from P 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 +perpendicularly intersecting planes, for which Fig. 9 gives a diagram. + +[Illustration: FIG. 9.] + +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. + +As a starting point for this investigation the distribution on a thin +plane circular disk of radius a is required. This can be obtained by +considering the disk as a limiting case of an oblate ellipsoid of +revolution, charged to potential V, say. If Fig. 10 represent the disk +and P the point at which the density is sought, so that CP = r, and +CA = a, the density is V⧸{2π²√(a² － r²)}. + +The ratio q⧸V, of charge to potential, which is called the electrostatic +capacity of the conductor, is thus 2a⧸π, that is a⧸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. + +[Illustration: FIG. 10.] + +[Illustration: FIG. 11.] + +Now invert this disk distribution with any point Q as centre of +inversion, and with radius of inversion a. The geometrical inverse is +a segment of a spherical surface which passes through Q. The inverse +distribution is the induced distribution on a conducting shell +uninsulated and coincident with the segment, and under the influence of +a charge －aV situated at Q (Fig. 11). Call this conducting shell the +"bowl." If the surface-densities at corresponding points on the disk and +on the inverse, say points P and P', be s and s', then, as on page 51, +s' = sa³⧸QP'³. If we put in the value of s given above, that of s' 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. + +If v, v' be the potentials at any point R of space, due to the disk +and to its image respectively, －v' = av⧸QR. If then R be coincident +with a point P' on the spherical segment we have (since then v = V) +V' = aV⧸QP', which is the potential due to the induced distribution +caused by the charge －aV at Q as already stated. + +The fact that the value of s' 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 Q in the same plane but outside the +bounding circle. + +Now consider the two parts of the spherical surface, the bowl B, and the +remainder S of the spherical surface. Q with the charge －aV may be +regarded as situated on the latter part of the surface. Any other +influencing charges situated on S 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 S be uniformly electrified to +density s, and held so electrified, the inducing distribution will be +one given by integration over the whole of S, and the bowl B will be at +zero potential under the influence of this electrification of S, just as +if B were replaced by a shell of metal connected to the earth by a long +fine wire. The densities are equal at infinitely near points on the two +sides of B. + +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 f greater than that of the spherical surface, and let this +shell be rigidly electrified with surface density －s. 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 －2πfs, for the charge is －πf²s, and the +capacity of the shell is ½f. The potential of the bowl will now be +zero, and its electrification will just neutralise the potential +-2πfs, that is, will be exactly the free electrification required to +produce potential 2πfs. + +To find this electrification let the value of f be only infinitesimally +greater than the diameter of the spherical surface of which B is a part; +then the bowl is under the influence (1) of a uniform electrification of +density －s 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 S above. It is obvious that by (1) a +density s 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. + +Interesting results are obtained by diminishing S more and more until +the shell is a complete sphere with a circular hole in it. Tabulated +results for different relative dimensions of S 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 _Electricity and Magnetism_, loc. cit. + +It is not quite clear whether Thomson discovered geometrical 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 images." In Thomson and +Tail's _Natural Philosophy_, § 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" (Nature, 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 opinion most of these +discoveries were later than 1845, the date of Thomson's first paper.[10] + +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. + + + + + +CHAPTER V + +THE CHAIR OF NATURAL PHILOSOPHY AT GLASGOW. ESTABLISHMENT OF THE FIRST +PHYSICAL LABORATORY + + +The 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 gens, 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, 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. + +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. + +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 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. + +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. + +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 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, _De caloris +distributione per terræ corpus_, 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." + +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 _Litteræ Humaniores_. 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. + +[Illustration: PROFESSOR WILLIAM THOMSON, 1846] + +The subject chosen had reference no doubt to the 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: _Ego, Gulielmus Thomson, B.A., physicus professor in hac +Academia designatus, promitto sancteque polliceor me in munere mihi +demandato studiose fideliterque versaturum._" Professor Thomson was then +"solemnly admitted and received by all the Members present, and took his +seat as a Member of Faculty." + +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 +_Mathematical and Physical Papers_ (vol. i, 1882). + +"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, '_De +Motu Caloris per Terræ Corpus_'[11]: 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 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. XIV.] + +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. + +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 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,[12] the French +mathematician, has recently thought fit to publish to the scientific +world." + +Again, in April 1852, the Faculty agree to pay a sum of £137 6_s._ +1½_d._ 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. + +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 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."[13] + +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.! + +The _University Calendar_ 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. + +"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.[14]) + +"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. + +"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." + +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. + +[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.] + +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, 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.[15] 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. + +[Illustration: INNER COURT OF THE OLD COLLEGE + +Showing Natural Philosophy Rooms] + +[In the view of the inner court of the Old College given opposite, the +windows on the ground-floor to 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.] + +Here again we may quote from the Bangor address: + +"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 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.'" + +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 Arts. Here it has meant, since +the Chair of Natural Philosophy was founded in 1577, and held by one of +the Regents of the University, _Artes Liberales_ in the widest sense, +that is, the study of _Litteræ Humaniores_ (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. + +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 _Relations des Expériences_ 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. + +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 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. + +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 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. + +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 +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. + +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. + +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 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. + +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 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. 88 below.] + +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. + + + + +CHAPTER VI + +FRIENDSHIP WITH STOKES AND JOULE. EARLY WORK AT GLASGOW + + +During 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 _Cambridge +Mathematical Journal_. In 1846 "Cambridge and Dublin" was substituted +for "Cambridge" in the title of the Journal, 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. + +For the mathematical power and sure physical instinct of Stokes Thomson +had always the greatest admiration. When asked on one occasion who was +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." + +After the death of Stokes in February 1902, Lord Kelvin again referred, +in an enthusiastic tribute in Nature 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." + +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. 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. + +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. + +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 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. + +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. + +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 _Electrodynamic Qualities of Metals_--the fact that up to a +certain magnetising force the effect of pull, applied to a wire of soft +iron, is to increase the 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. + +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. + +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 _Natural +Philosophy_ to bring back the current of teaching and thought in +dynamical science to the ideas of the Principia, 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, within the category of dynamical action between +material systems. + +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. + +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 Nature +article quoted above, p. 80, and the _Baltimore Lectures_, p. 101). 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 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. + +A letter of Stokes to Sir John Lubbock, printed in the _Scientific +Correspondence of Sir George Gabriel Stokes_, 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 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.... + +"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." + +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. + +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 +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 _char-à-banc_, coming slowly up the hill behind him, +lest it should get broken. But there, comfortably and safely seated in +the _char-à-banc_, 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 +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." + +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 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. + +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 _Natural Philosophy_, which commences +with a long chapter devoted entirely to kinematics. + +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 +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 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. + +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. + +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 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. + +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 Outlines, 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. + +Home exercises were not prescribed as part of the 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. + +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, 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[16] 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. + +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 +conférence, 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 clearness and accuracy the knowledge he has +gained, but they should be once a week to be beneficial." + +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 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. + +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 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 famulus rather than the much more +measured respect paid by modern assistants to their chiefs. + +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. + +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. + + + + +CHAPTER VII + +THE "ACCOUNT OF CARNOT'S THEORY OF THE MOTIVE POWER OF HEAT"--TRANSITION +TO THE DYNAMICAL THEORY OF HEAT + + +The 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 _Journal de l'École Polytechnique_, 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 _vice +versa_. 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. 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 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. + +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. + +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 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. + +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. + +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. + +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 +p and the increase of volume be dv, the work done against the external +force is pdv. 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 is the specific heat Cp 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 Cv, the specific heat at constant volume. +Now Mayer assumed that the excess of the specific heat Cp above Cv was +the thermal equivalent of the work pdv done in the former case. Thus he +obtained the equation J(Cp － Cv) = pdv, where J denotes the dynamical +equivalent of heat and Cp, Cv are taken in thermal units. But if a be +the coefficient of expansion of the air under constant pressure (that is +1⧸273), and v₀ be the volume of the air at 0° C., we have dv = av₀, +so that J(Cp － Cv) = apv₀. Now if p be one atmosphere, say 1.014 × 10^6 +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 + + J(Cp － Cv) = (1.014 × 10^6)⧸(273 × .001293) + +from which, if Cp － Cv is known, the value of J can be found. + +In Mayer's time the difference of the specific heats of air was +imperfectly known, and so J 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 +Cp － Cv may be taken as .0686. Thus J works out to 42.2 × 10^6, +in ergs per calorie, which is not far from the true value. Mayer +obtained a result equivalent to 36.5 × 10^6 ergs per calorie. + +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 dW for pdv, the work done, e for the +internal energy before expansion, and dH for the heat given to the gas, +we have obviously the equation + + JdH = de + dW + +where de 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 +dW for that case is zero. Let ∂e and ∂H be the energy change and the +heat supplied, then in this case + + J∂H = ∂e + O + +Thus + + J(dH － ∂H) = de － ∂e + dW + +and the assumption is that de = ∂e, so that dW = J(dH － ∂H); that +is, dW = J(Cp － Cv), 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 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. + +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 _Réflexions sur la +Puissance Motrice du Feu, et sur les Machines Propres à Développer cette +Puissance_, 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 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.[17] + +The first law of thermodynamics based on Joule's proportionality of heat +produced to work expended, and the converse assumed and verified _a +posteriori_, 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. + +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 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. + +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 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. + +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. + +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 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" (_Phil. Mag._, 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." + +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 Proceedings in 1852. + +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." + +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 _Philosophical Magazine_ for December 1851, +to which we shall refer later. + +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 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. + +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. 99 above), who used it to illustrate an +account of Carnot's theory. + +There appeared in the issue of the Edinburgh _Philosophical +Transactions_ 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. + + + + +CHAPTER VIII + +THERMODYNAMICS AND ABSOLUTE THERMOMETRY + + +The 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 _Poggendorff's Annalen_ in March and April 1850, and in the +_Philosophical Magazine_ 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 (_Pogg. Ann._, Dec. 1854) he stated his fundamental principle +explicitly in the form: "Heat can never 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." + +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. + +In the _Transactions R.S.E._, 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. + +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 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. +116, the other enunciates the principle of Reversibility as a criterion +of "perfection" of a heat engine. We quote these propositions. + +"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." + +"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." + +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. + +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. 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. + +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 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. + +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. + +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 _vice versa_: the +substance will do work against external forces where in the direct +action it had work done upon it, and _vice versa_: 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 H taken in at the source into work than +would the reversible one working between the same temperatures. Thus if +h be the heat given to the refrigerator by A, and h' that given by B +when both work directly and take in H; h must be less than h'. Then +couple the engines together so that B works backwards while A works +directly. A will take in H and deliver h, and do work equivalent to +H － h. B will take h' from the refrigerator and deliver H to the source, +and have work equivalent to H － h' spent upon it. There will be no heat +on the whole given to or taken from the source; but heat h' － h will be +taken from the refrigerator, and work equivalent to this will be done. +Thus _by a cyclical process_, 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. + +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. + +[Illustration: FIG. 12.] + +In the Carnot cycle, the first operation is an isothermal expansion (AB +in Fig. 12), in which the substance increases in volume by dv, and takes +in from the source heat of amount Mdv. The second operation is an +adiabatic expansion, BC, in which the volume is further increased and +the temperature sinks by dt to the temperature of the refrigerator. The +third operation is an isothermal compression, CD, until the volume and +pressure are such that an adiabatic compression DA will just bring the +substance back to the original state. If ∂p⧸∂t 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 ∂p⧸∂t.dt; and +thus the area of the closed cycle in the diagram which measures the +external work done in the succession of changes is ∂p⧸∂t.dtdv. Now, +by the second law, the work done must be a certain fraction of the +work-equivalent of the heat, Mdv, taken in from the source. This +fraction is independent of the nature of the working substance, but +varies with the temperature, and is therefore a function of the +temperature. Its ratio to the difference of temperature dt 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 μ, we have +the equation + + ∂p⧸∂t = μM ... (A) + +which may be taken as expressing in mathematical language the second law +of thermodynamics. M is here so chosen that Mdv 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. + +The work done in the cycle becomes now μMdtdv, or if H denote Mdv, it +is μHdt. The fraction of the heat utilised is thus μdt. This is +called the efficiency of the engine for the cycle. + +From the first law Thomson obtained another fundamental equation. For +every substance there is a relation connecting the pressure p (or more +generally the stress of some type), the volume v (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 v and t as the quantities to be varied, and supposed them +to sustain arbitrary small changes dv and dt in consequence of the +passage of heat to the substance from without. The amount of heat taken +in is Mdv + Ndt, where Mdv and Ndt are heats required for the changes +taken separately. But the substance expanding through dv does external +work pdv. Thus the net amount of energy given to the substance from +without is Mdv + Ndt － pdv or (M － p)dv + Ndt; 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 +M － p when the temperature but not the volume varies, is equal to the +rate of variation of N 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 dv, dt of volume and +temperature, (2) a variation －dv of volume only, (3) a variation －dt of +temperature only. M － p and N 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 +(M － p)dv + Ndt 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. + +This result combined with the equation A derived from the second law, +gives an important expression for Carnot's function. + +We shall not pursue this discussion further: so much is given to make +clear how certain results as to 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. + +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 v₀ and that at any other temperature be +v₁ we define the numerical value t, this latter temperature, by the +equation v = v₀(1 + Et), where E 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 t has +the value (v⧸v₀ － 1)⧸E, and is the temperature (on the constant +pressure scale of the gas thermometer) corresponding to the volume v. +Equal differences of temperature are such as correspond to equal +increments of the volume at 0° C. + +Similarly, on the constant volume scale we obtain a definition of +temperature from the pressure p, by the equation t = (p⧸p₀ － 1)⧸E', +where p₀ is the pressure at 0° C., and E' is 1⧸100 of the change of +pressure produced by raising the temperature from 0° C. to 100° C. + +For air E is approximately 1⧸273, and thus t = 273(v － v₀)⧸v₀. +If we take the case of v = 0, we get t = －273. Now, although this +temperature may be inaccessible, we may take it as zero, and the +temperature denoted by t is, when reckoned from this zero, 273 + t. +This zero is called the absolute zero on the constant pressure air +thermometer. The value of E' is very nearly the same as that of E; 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, E +would not differ from E'. + +It was suggested to Thomson by Joule, in a letter dated December 9, +1848, that the value of μ might be given by the equation +μ = JE⧸(1 + Et). Here we take heat in dynamical units, and therefore +the factor J is not required. With these units Joule's suggestion is +that μ = E⧸(1 + Et), or with E = 1⧸273 μ = 1⧸(273 + t), that is, +μ = 1⧸T where T is the temperature reckoned in centigrade degrees from +the absolute zero of the constant pressure air thermometer. + +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. The heat H +absorbed by the air in expanding from volume V to another volume V' at +constant temperature is the integral of Mdv taken from the former volume +to the latter. But by the value of M given on p. 121, if W be the +integral of pdv, that is the work done by the air in the expansion, +∂W⧸∂t = μH. The equation fulfilled by the gas at constant pressure +(the defining equation for t), v = v₀(1 + Et), gives for the integral +of pdv, that is W, the equation W = pv₀(1 + Et)log(V'⧸V), so that +∂W⧸∂t = EW⧸(1 + Et). Thus μH = EW⧸(1 + Et). + +Hence it follows that if μ = E⧸(1 + Et), the value of H will be simply +W. 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. + +This result naturally suggests the formation of a new scale of +thermometry by the adoption of the defining relation T = 1⧸μ, where T +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 _Philosophical +Transactions_ 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 T. +This scale was the natural successor, in the dynamical theory, of one +which Thomson had suggested in 1848, and which 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 absolute scale, since it would be +independent of the physical properties of any substance. In the same +sense the scale defined by T = 1⧸μ is truly an absolute scale. + +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 H taken in from +the source in the isothermal expansion which forms the first operation +of the cycle (p. 120) is Mdv, and, as we have seen, the work done in the +cycle is ∂p⧸∂t.dtdv, or μHdt. If we adopt the expression 1⧸T for +μ, we may put dT for dt; and we obtain for the work done the +expression HdT⧸T. The work done is thus the fraction dT⧸T of the heat +taken in, and this is what is properly called the efficiency of the +engine for the cycle. + +If we suppose the difference of temperatures between source and +refrigerator to be finite, T － T', say, then since T is the temperature +of the source, we have for the efficiency (T － T')⧸T. If the heat taken +in be H, the heat rejected is HT'⧸T, so that the heat received by the +engine is to the heat rejected by it in the ratio of T' to T. 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, 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. + +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 v₁, v₂ 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 dm from the first state to +the second. Thus dv, the change of volume, is dm(v₂ － v₁). Also if L +be the latent heat of the substance in the second state, _e.g._ the +latent heat of water, Mdv = Ldm; so that M(v₂ － v₁) = L. If dp be the +step of pressure corresponding to the step dT of temperature, equation +(A) becomes + + dT⧸T = dp(v₂ － v₁)⧸L ... (B) + +In the case of coexistence of the liquid and solid phases, this gives us +the very remarkable result that a change of pressure dp 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, dT, +according as v₂ is greater or less than v₁ Thus a substance like +water, which expands in freezing, so that v₂ － v₁ 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. 113 above). +On the other hand, a substance like paraffin wax, which contracts in +solidifying, would have its melting point raised by increase of pressure +and lowered by a diminution of pressure. + +The same conclusions would be applicable when the phases are liquid and +vapour of the same substance, if there were any case in which v₂ － v₁ +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. + +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, dp⧸dT can be calculated, or obtained +graphically from a curve; and the volume v₂ per unit mass of the vapour +will be given in terms of dp⧸dT, the temperature T, and the volume v 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 v₂ for any +substance for which the relation between p and T is known. It is thus +that the density of saturated steam can best be found. + +We can obtain another important result for the case of the working +substance in two phases from equation (B). The relation is + + ∂L⧸∂T + c － h = L⧸T ... (C) + +where c and h are the specific heats of the substance in the two phases +respectively, and L is the latent heat of the second phase at absolute +temperature T. + +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 T +in the lower phase, and to be changed to the other phase at that +temperature. The heat taken in will be L and the expansion will be +v₂ － v₁. 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 dT. The heat +given out by the substance will be hdT, where h is the specific heat of +the substance in the second phase. Now at the new temperature T － dT let +the substance be wholly brought back to the second phase; the heat given +out will be L － ∂L⧸∂T.dT. Finally, let the substance, now again +all in the first phase, be brought to the original temperature: the heat +taken in will be cdt, where c is the specific heat in the first phase. +Thus the net excess of heat taken in over heat given out in the cycle is +(∂L⧸∂T + c － h)dT. This must, in the indicator diagram for the +changes specified, be the area of the cycle or (v₂ － v₁)∂p⧸∂T.dT. +But by equation (B) L⧸T(v₂ － v₁) = ∂p⧸∂T, and the area of the cycle is +(L⧸T)dT. Equating the two expressions thus found for the area we get +equation (C). + +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. + +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 L⧸T and +∂L⧸∂T to be calculated, and therefore that of h to be found. It +appears that h 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. + +It is convenient to notice here the article on Heat which Thomson wrote +for the ninth edition of the _Encyclopædia Britannica_. 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, 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. + +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. + +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. + +Now let the instrument contain in its bulb and stem, up to the mark 0, N +degree-spaces, and let v be the volume of a degree-space at that +temperature. The volume up to the mark 0 will be Nv, at that +temperature; and if the substance of the vessel be quite uniform in +quality and free from stress, N will be the same for all temperatures. +If v₀ be the volume of a degree-space at the temperature of melting ice +the volume of the mercury at that temperature will be Nv₀. If G be the +expansion of the glass when the volume of a degree-space is increased +from v₀ to v by the rise of temperature, then v = v₀(1 + G). The +volume of the mercury has been increased therefore to (N + n)v₀(1 + G) +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 n degree-spaces more than +before. But if E 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 Nv₀(1 + E). Hence N(1 + E) = (N + n)(1 + G). This gives +n = N(E － G)⧸(1 + G). + +If we take, as is usual, n as measuring the temperature, and substitute +for it the symbol t, we have, since N = 100(1 + G₁₀₀)⧸(E₁₀₀ － G₁₀₀), + + t = 100 {(1 + G₁₀₀)⧸(1 + G)} {(E － G)⧸(E₁₀₀ － G₁₀₀)} ... (D) + +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 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. + +In his article on Heat Thomson gave two definitions of the scale of +absolute temperature. One is that stated on p. 126 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. + +The other definition is better adapted for general use, as it applies to +any cycle whatever which is reversible. Let the working substance +expand under constant pressure by an amount dv (AB' in Fig. 12), and let +heat H be given to the substance at the same time. The external work +done is pdv. Thomson called pdv⧸H the work ratio. Now let the +temperature be raised by dT without giving heat to the substance or +taking heat from it, and let the corresponding pressure rise be dp; and +call dp⧸p the pressure ratio. The temperature ratio dT⧸T is equal to +the product of the work ratio and the pressure ratio, that is, + + dT⧸T = dvdp⧸H + +This is clearly true; for dvdp is the area of a cycle like AB'C'D, +represented in Fig. 12, for which an amount of heat H is taken in, +though not in this case strictly at one temperature. And clearly, since +in Fig. 12 the change from B' to B is adiabatic, H is the heat which +would have to be taken in for the isothermal change AB in the Carnot +cycle ABCD, which has the same area as AB'C'D. Thus the efficiency of +the cycle is dvdp⧸H, and this by the former definition is dT⧸T. + +Or we may regard the matter thus:--The amount of heat H which +corresponds to an infinitesimal expansion dv may be used in equation (A) +whether the expansion is isothermal or not, if we take T as the average +temperature of the expansion. Hence we have dp⧸dT = H⧸(dv.T), that is, +dT⧸T = dpdv⧸H. The theorem on p. 128 is obtained by what is virtually +this process. + + +COMPARISON OF ABSOLUTE SCALE WITH SCALE OF AIR THERMOMETER + +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 de － ∂e of p. 107 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. + +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 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. + +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. + +The theory of the matter is set forth in the original papers, and in a +very elegant manner in the article on Heat. The result of the analysis +shows that if ∂w 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 T be absolute temperature, and v volume of a gramme of +the gas at pressure p, and the difference of pressure on the two sides +of the plug be dp, the equation which holds is + + (1⧸T) (∂T⧸∂v) = 1⧸{v + (∂w⧸dp)} ... (E)[18] + +It was found by Joule and Thomson that ∂w was proportional to dp for +values of dp 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. + +If there is neither heating nor cooling ∂w = 0, and we obtain by +integration T = Cv, where C is a constant. + +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. + +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. + +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. + +Another idea, that of _thermodynamic motivity_, 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 be equalised +by perfect thermodynamic engines working between the final temperature +T₀, 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 T₀ is such that exactly the heat given +out at T₀ by certain engines, receiving heat from bodies of higher +temperature than T₀, is supplied to the engines which work between T₀ +and bodies at lower temperatures. The whole useful work obtained in this +way was called by Thomson the motivity 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 T, T' the temperature to which +they are reduced when their motivity is extracted is √(TT'). If the +temperatures are equalised by conduction the resulting temperature is +higher, being ½(T + T'). 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. + +A very suggestive paper on the subject was published by Lord Kelvin in +the _Trans. R.S.E._, vol. 28, 1877-8. + + +DISSIPATION OF ENERGY + +In connection with the theory of heat must be mentioned Thomson's great +generalisation, the theory of the dissipation of energy.[19] Most people +have some 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 universe 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. + +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. + +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 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. + +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. + +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. + +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. + +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. + + +THERMOELASTICITY AND THERMOELECTRICITY + +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 may substitute dS +for dp and Ads for dv; where A is such a factor that AdSds is the work +done in the displacement ds by the stress dS. As an example consider a +wire subjected to simple longitudinal stress S. Longitudinal extension +is produced, but this is not the only change; there is at the same time +lateral contraction. However, s within certain limits is proportional to +S. + +Let heat dH in dynamical measure be given to the wire while the stress S +is maintained constant, and let the extension increase from s to s + ds. +The stress S will do work ASds _on the wire_, and the work ratio will be +-ASds⧸dH. Now let the stress be increased to S + dS while the extension +is kept constant, and the absolute temperature raised from T to T + dT. +The stress ratio (as we may call it) is dS⧸S and the temperature ratio +dT⧸T. Thus we obtain (p. 134 above) + + －(dS⧸dT) = (1⧸TA) (dH⧸ds) + +In his Heat article Thomson used the alteration e of strain under +constant stress (that is ds⧸l, where l is the length of the wire) +corresponding to an amount of heat sufficient to raise the temperature +under constant stress by 1°. Hence if K be the specific heat under +constant stress, and le be put for ds in the sense just stated, we have + + dT = －(TedS⧸Kρ) ... (F) + +where ρ is the density, since dH = KρlA. + +The ratio of dH to the increase ds of the extension is positive or +negative, that is, the substance absorbs or evolves heat, when strained +under the condition of constant stress, according as dS⧸dT 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, ds is negative or positive, in other words the wire shortens or +lengthens, according as dS⧸dT is positive or negative, that is, +according as the stress for a given strain is increased or diminished by +increase of temperature. + +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. + +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 ds of s is taken as before under +constant stress, or at uniform temperature, or whether the change dS of +S is effected adiabatically or at constant strain. + +In all these cases the same equation + + dT = －T (edS⧸Kρ) ... (G) + +applies, with the change of meaning of dT involved. + +This equation differs from that of Thomson as given in various places +(_e.g._ in the _Encyclopædia Britannica_ article on Elasticity which he +also wrote) in the negative sign on the right-hand side, but the +difference is only apparent. According to his specification a pressure +would be a positive stress, and an expansion 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 dS 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). + +This equation was applied by Thomson (article Elasticity) 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. + +The augmentation of the strain produced by raising the temperature 1° +is e, and therefore edT, that is, －Te²dS⧸Kρ, is the increase of strain +due to the sudden rise of temperature dT. This added to the isothermal +strain produced by dS will give the whole adiabatic strain. Thus +if M be the static or isothermal modulus, the adiabatic strain +is dS⧸M － Te²dS⧸Kρ. If M' denote the kinetic or adiabatic modulus +its value is dS divided by the whole adiabatic strain, that is, +M' = M⧸(1 － MTe²⧸Kρ) and the ratio M'⧸M = 1⧸(1 － MTe²⧸Kρ). + +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 K of the substance, under the +condition of constant stress, to the specific heat N under the condition +of constant strain of the corresponding type. This, indeed, is +self-evident if two changes of stress, one isothermal the other +adiabatic, _which produce the same steps of displacement ds_, be +considered, and it be remembered that the step ∂T of temperature which +accompanies the adiabatic change may be regarded as made up of a step +－dT of temperature, accompanying a displacement ds effected at constant +stress, and then two successive steps dT and ∂T effected, at constant +strain, along with the steps of stress dS. The ratio M'⧸M is easily seen +to have the value (∂T + dT)⧸dt, and since －KdT + N(∂T + dT) = 0, by +the adiabatic condition, the theorem is proved. + +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. + +Thomson showed in the Elasticity article how, by the value of M'⧸M, +derived as above from thermodynamic theory, the value of K⧸N 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). + +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 _Philosophical Magazine_ for January 1878. This was in +the main a reprint of an article entitled, "On the Thermoelastic and +Thermomagnetic Properties of Matter, Part I," which appeared in April +1855 in the first number of the _Quarterly Journal of Mathematics_. 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 _Cyclopædia of Physical Science_, edited +and in great part written by Professor J. P. Nichol, and published in +1860. For the same Cyclopædia Thomson also wrote an article entitled, +"Thermo-electric, Division I.--Pyro-Electricity, or Thermo-Electricity +of Non-conducting Crystals," and the enlarged _Phil. Mag._ article also +contained the application of thermodynamics to this kind of +thermoelectric action. + +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. + +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 _Mathematical and Physical Papers_. This series begins with the +Bakerian Lecture (published in the _Transactions of the Royal Society_ +for 1856) which includes an account of the remarkable 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. + +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. + +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 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. + +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. + +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. + +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. + +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, 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. + +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." + +Part VI (_Trans. R.S._, 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. + +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. + +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 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. + + + + +CHAPTER IX + +HYDRODYNAMICS--DYNAMICAL THEOREM OF MINIMUM ENERGY--VORTEX MOTION + + +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 molecules of a +fluid; even when we deal with infinitesimal elements, in the analysis of +fluid motion, we are 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 vortices are formed, and hence the name vortex motion has been +given to all those cases in which the condition of irrotationality is +not satisfied. + +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 _Transactions of the Royal Society of Edinburgh_. 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. + +In Chapter VI reference has been made to the "Notes on Hydrodynamics" +published by Thomson in the _Cambridge and Dublin Mathematical Journal_ +for 1848 and 1849. These Notes were not intended to be entirely +original, but were composed for the use of students, like Airy's Tracts +of fifteen years before. + +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. + +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. + +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. + +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 A, B, within the space, to be +joined by two lines ACB, ADB both lying within the space. These two +lines will form a circuit ACBDA. 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 A +to B round the ring one way, and the other from A to B the other way. +The circuit could not then be contracted towards a point without passing +out of the ring. + +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. + +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 line in the body ever changes its direction. +The body has translation, but no spin. + +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. + +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. + +This theorem was afterwards generalised by Thomson (_Proc. R.S.E._, +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 +by Lagrange in the _Mécanique Analytique_ as a generalisation of a +theorem given by Euler for a rigid body set into rotation by an impulse. + +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. + +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. + +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 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 1⧸8mv², where m denotes +the mass of the rod and v the velocity of the end. If the motion taken +were one of rotation about a point of the rod at distance x from the end +struck, the kinetic energy would be m(4l² － 6lx + 3x²)v²⧸6x², where +2l is the length of the rod, and this has its least value 1⧸8mv² for +x = 4l⧸3. For example, x = 2l gives 1⧸6mv², which is greater than the +value just found. + +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 produced is one-half the product of the impulse +into the velocity of the point struck, that is ½Iv, and it has just +been seen that this is the product of (1⧸6)mv² by the factor +(4l² － 6lx + 3x²)⧸x². This factor is 3I⧸mv, and is a minimum when +x = 4l⧸3. Thus for a given I, v will have its maximum value when the +factor referred to is least, and ½Iv will then be a maximum. + +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. + +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⧸v. Thus when v is given, I is increased by any constraint +compelling the rod to rotate about a particular axis, and so ½Iv, or +the kinetic energy, is increased. On the other hand, when I is given the +same constraint diminishes v, and so ½Iv is diminished. + +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 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. + +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. + +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. 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. + +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. + +Thomson's great paper on vortex-motion was read before the Royal Society +of Edinburgh in 1867, and was recast and augmented in the following +year. It will be possible to give here only a sketch of its scope and +main results. + +The fluid is supposed contained in a closed fixed vessel which is either +simply or multiply continuous (see p. 156), 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. + +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. + +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 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 x, y, z, about the axes of x, y, z are shown to be +½(∂w⧸∂y － ∂v⧸∂z), etc. + +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. + +Thomson then proves that if the angular momentum 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. + +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 P, Q in the fluid. Let q be the velocity of the +fluid at any element of length ds of such a path, and θ be the +angle between the direction of ds (taken positive in the sense from P +to Q) and the direction of q: q cos θ.ds is the flow along ds. If u, +v, w be the components of q at ds, parallel to the axes, and dx, dy, dz +be the projections of ds on the axes, udx + vdy + wdz is the same thing +as q cos θ.ds. The sum of the values of either of these expressions +for all the elements of the path between P and Q is the flow along the +path. The statement that u, v, w are the space-rates of variation of a +function φ (of x, y, z) parallel to the axes, or that q cos θ is +the space-rate of variation of φ along ds, merely means that this +sum is the same for whatever path may be drawn from P to Q. This, +however, is only the case when the paths are so taken that in each case +the value of φ 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. + +Since the flow from P to Q is the same for any two paths which fulfil +this condition, the flow from P to Q by any one path and from Q to P 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. + +The general theorem which he established may now be stated. Consider +any path joining PQ, and moving with the fluid, so that the line +contains always the same fluid particles. Let u̇, v̇, ẇ be the +time-rates of change of u, v, w at an element ds of the path, at +any instant, and du, dv, dw the excesses of the values of u, v, w +at the terminal extremity of ds above the values at the other +extremity; then the time-rate of variation of udx + vdy + wdz +is u̇dx + v̇dy + ẇdz + udu + vdv + wdw or u̇dx + v̇dy + ẇdz + qdq, +where q has the meaning specified above. Thus if S be the flow for +the whole path PQ, and Ṡ its time-rate of variation, S' denote the +sum of u̇dx + v̇dy + ẇdz along the path from P to Q, and q₁, q₀ the +resultant fluid velocities at Q and P, we get Ṡ = S' + ½(q₁² － q₀²). +This is Thomson's theorem. If the curve be closed, that is, if P and Q +be coincident, q₁ = q₀ and Ṡ = S'. But in certain circumstances S' is +zero, and so therefore is also Ṡ. Thus in the circumstances referred to, +as the closed path moves with the fluid Ṡ is continually zero, and it +follows that if Ṡ is zero at any instant it remains zero ever after. But +Ṡ is only zero if u, v, w 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. + +The circumstances in which S' 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 S' 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 S' which depends on pressure and +density vanishes for the circuit. + +It is to be clearly understood that the motion of a fluid may be +irrotational although the value of S does not vanish for every 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 S shall be zero; but +what is necessary for irrotational motion within any space is that S +should vanish for all paths which are capable of being shrunk down to +zero without passing out of that space. S need not vanish for a path +which cannot be so shrunk down, but it must, if the 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 cyclic, in the former acyclic. + +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 dS, a line perpendicular to the surface +towards the side chosen as the positive side, and calculate the angular +velocity ω, say, of the fluid about that normal from the components of +angular velocity determined in the manner explained at p. 164. This +Thomson called the rotation of the element. Now take the product ωdS 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 dS may change, but the circulation +round its edge by Thomson's theorem remains unaltered. Thus ω alters in +the inverse ratio of dS, and the line drawn at right angles to the +surface at dS, if kept of length proportional to ω, will lengthen or +shorten as dS contracts or expands. + +Now sum the values of ωdS for the finite surface enclosed by the +bounding curve. It follows from the fact that ωdS is equal to half the +circulation round the edge of dS, that this sum, which is usually +denoted by ΣωdS, 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 +ΣωdS 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. + +The equality of 2ΣωdS 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 _Natural Philosophy_, § 190. In the second edition of +the _Natural Philosophy_ 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. + +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. + +Take now any small plane area dS moving with the fluid, and draw axial +lines through every point of its boundary. These will form an axial tube +enclosing dS. If θ be the angle between the direction of resultant +rotation and a perpendicular to dS, the cross-section of the tube at +right angles to the normal, and to the axial lines which bound it, is +dS.cosθ. Let these axial lines be continued in both directions from the +element dS. 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. + +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 ΣωdS, +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 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. + +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. + +If any surface be drawn cutting a vortex-tube, it is clear from the +definition of the tube that the value of ΣωdS for every such +surface must be the same. This Thomson calls the "rotation of the tube." + +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. + +Examples of vortex-tubes are indicated on p. 154; 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. + +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 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. 13.) 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. + +[Illustration: FIG. 13.] + +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. + +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 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. + +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. + +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 P, and doubling +back so as to return to P 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 P is 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. + +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. + +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. + +Two vortex-rings projected one after the other also 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. 13, 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. + +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 _Natural Philosophy_ in +the latest edition of Lamb's Hydrodynamics, and that many of these are +concerned with general theorems and results of great value. + + + + +CHAPTER X + +THE ENERGY THEORY OF ELECTROLYSIS--ELECTRICAL UNITS--ELECTRICAL +OSCILLATIONS + + +ELECTROLYSIS AND ELECTRICAL UNITS + +In December 1851 Thomson communicated an important paper to the +_Philosophical Magazine_ on "The Mechanical Theory of Electrolysis," and +"Applications of Mechanical Effect to the Measurement of Electromotive +Forces, and of Galvanic Resistances, in Absolute Units." + +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 W was supposed to be spent in a +given time, during which a quantity of heat H was evolved in the +circuit, and a certain amount of work M spent in the chemical apparatus +in effecting chemical change. If H be taken in dynamical units, W = H + +M. + +The work done in driving the disk, if the intensity of the field is I, +the current produced c, the radius of the disc r, and the angular +velocity of turning w, is ½Ir²cw. + +Thomson assumed that the work done in the electrochemical apparatus was +equal to the heat of chemical 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 M represented this resultant heat of combination. + +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. + +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 θ, say, for the whole chemical +change in the cell corresponding to the action on unit mass of that +element. Thus if E denote the whole quantity of that element operated on +the heat of combination in the vessel was θE. If E be taken for unit of +time, and ε denote the quantity set free by the passage of unit quantity +of electricity, then E = εc, since a current conveys c units of +electricity in one second. The number ε is a definite quantity of the +element, and is called its electrochemical equivalent. Again, from +Joule's experiments, H = Rc², if R denote the resistance of the current, +and so + + ½Ir²cw = Rc² + θεc + +and + + c = (½Ir²w － θε)⧸R + +The quantity ½Ir²w is the electromotive force due to the disk. + +Thus c was positive or negative according as ½Ir²w was greater or less +than θε, and was zero when ½Ir²w = θε. Thus the electromotive force +of the disk was opposed by a back electromotive force θε due to the +chemical action in the voltameter or battery, to which the wires from +the disk were connected. + +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. + +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 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. + +In the next paper, which appeared in the same number (December 1851) of +the _Philosophical Magazine_, 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. XIII.) + +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 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, c, +say, in strength, flowing in the circuit, renders a dynamical force cIl +necessary to move the slider of length l across the lines of force of +the field of intensity I, and if the speed of the slider required for +the current c be v, the rate at which work is done in moving the slider +is cIlv. 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 cIlv = Rc², or Ilv = Rc, so that Ilv is the +electromotive force. Any electromotive force otherwise produced, which +gave rise to the same current, must obviously be equal to Ilv, so that +the unit of electromotive force can thus be properly defined. + +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 is one C.G.S. unit, l one centimetre, and v 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 c be unity as well as +Ilv. + +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, 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. + + +ELECTRICAL OSCILLATIONS + +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. + +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 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. + +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 N 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, 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 Q². We suppose it to be ½Q²⧸C, where C is a constant depending on +the condenser and called its capacity. + +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. + +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 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. + +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 c denote the current, that is, the rate, +-dQ⧸dt, at which the charge of the condenser is being changed, and L a +quantity called self-inductance, depending 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 ½Lc². +This is analogous to the kinetic energy ½mv² of a body (say a pendulum +bob) of mass m and velocity v, so that L represents a quantity for the +conducting arrangement analogous to inertia, and c is the analogue of +the velocity of the body. The whole energy at any instant is thus + + ½Q²⧸C + ½Lc², or ½Q²⧸C + ½L(dQ⧸dt)². + +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 R 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) relatively to one +another and so cause frictional generation of heat. + +The time rate of generation of heat is thus Rc², or R(dQ⧸dt)², when the +units in which R and c 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 + + ½d/dt{(Q²⧸C) + L(dQ⧸dt)²} = －R(dQ⧸dt)² + +holds, or leaving out the common factor dQ⧸dt, the equation + + L(d²Q⧸dt²) + R(dQ⧸dt) + Q⧸C = 0 + +This last equation was established by Thomson, and is precisely that +which would be obtained for a pendulum bob of mass L, pulled back +towards the position of equilibrium with a force Q⧸C, where Q is the +displacement from the middle position, and having its motion damped out +by resisting force of amount R per unit of the velocity. + +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 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. + +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. + +The differential equation is applicable to the spring if Q is again +taken as displacement from the equilibrium position, L as the inertia of +the vibrating body, 1⧸C as the pull exerted by the spring per unit of +its extension (that is, the stiffness of the spring), and R has the same +meaning as before. + +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 +equation, which in that respect is imperfect. There is no difficulty, +only the addition of a little complication, in supplying the omission. + +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. + +It has been indicated that the differential equation does not represent +oscillatory motion if the value of R is too great. The exact condition +depends on the roots of the quadratic equation Lx² + Rx + 1⧸C = 0, +obtained by writing 1 for Q, and x for d⧸dt, and then treating x as a +quantity. These roots are －R⧸2L ± √(R²⧸4L² － 1⧸CL), and are +therefore real or imaginary according as 4L⧸C is less or greater than +R². If the roots are real, that is, if R² be greater than 4L⧸C, 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 R is sufficiently +great, or L⧸C sufficiently small (a weak spring and a small mass, or +both), to enable the condition to be fulfilled. + +If, however, the roots of the quadratic are imaginary, that is, if 4L⧸C +be greater than R² (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 Q disappears by oscillations about zero, of continually +diminishing amplitude. A complete discussion gives for the period of +oscillation 4πL⧸√(4L⧸C － R²), or if R be comparatively small, 2π√(LC). +The charge Q falls off by the fraction e^{－RT⧸2L} (where e is the +number 2.71828...) in each period T, and so gradually disappears. + +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 L, C, and R 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. + +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⧸C, of the capacity of the condenser. + +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 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. + +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. + +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 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. + +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." + +This anticipation was verified by experiments made by Feddersen, and +published in 1859 (_Pogg. Ann._, 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 (_Pogg. Ann._, 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.] + +The particular case of non-oscillatory discharge obtained by supposing C +and Q both infinitely great and to have a finite ratio V (which will be +the potential, p. 34, 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 c be put as before for the current －dQ⧸dt, and the +constant value V for Q⧸C, the equation is + + L(dc⧸dt) + Rc = V + +which gives, since c = 0 when t = 0, + + c = (V⧸R)[1 － e^{－(R⧸L)t}]. + +Thus, when an infinite time has elapsed the current has become V⧸R, the +steady value. + +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. + + + + +CHAPTER XI + +THOMSON AND TAIT'S 'NATURAL PHILOSOPHY'--GYROSTATIC +ACTION--'ELECTROSTATICS AND MAGNETISM' + + +THE 'NATURAL PHILOSOPHY' + +Professor Tait 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 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. + +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. + +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 +_Elements of Quaternions_, 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 as non-scientific and fruitless, which no doubt they were +from the physical point of view. + +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. + +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. + +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 _Treatise +on Natural Philosophy_, the second was entitled _Elements of Natural +Philosophy_, and consisted in the main of part of the non-mathematical +or large type portions of the Treatise. 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 Elements, to a student whose +mathematical reading was wide enough to tackle the Treatise, 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. + +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 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. + +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 _Theory of Sound_, Lamb's Hydrodynamics, and Routh's +_Dynamics of a System of Rigid Bodies_. 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. + +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 to that due to Newton's Principia. Three +other works, perhaps, have had the same degree and kind of influence on +mathematical thought--Laplace's _Mécanique Céleste_, Lagrange's +_Mécanique Analytique_, and Fourier's _Théorie Analytique de la +Chaleur_. + +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. + +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. + +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-dash" 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 short original § 343, he greatly +extended that in the first instance, and proceeded from section to +section until additions numbered from § 343a to § 343p, 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. + +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. + +There can be no doubt that greater clearness is obtained by the process +afterwards insisted on by Tait, 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 Elements by the +method here indicated. + +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 speed of the point (it does influence the velocity), 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.[20] + +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. + +In the next chapter, which is entitled "Dynamical Laws and Principles," +the authors at the outset declare their intention of following the +Principia 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 +_Natural Philosophy_ in Nature 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?" + +This innate power of resisting is merely the _materiæ vis insita_ of +Newton's "Definitio III," given in the Principia, 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. + +The relativity of motion is not emphasised so greatly in the _Natural +Philosophy_ 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. + +The treatment of force in the first and second laws of motion is frankly +causal. Force is there the cause 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. + +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 Scholium 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 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. + +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 _Mécanique +Céleste_, the _Mécanique Analytique_, and the memoirs of Jacobi and +Hamilton--all of which were an outcome of the Principia, and from which, +with the Principia, the authors of the _Natural Philosophy_ drew their +inspiration. + +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 _Natural +Philosophy_. 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 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." + +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 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. 214 below.) + +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." + +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 a discussion of technicalities beyond the reading of +ordinary students of dynamics. + +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. + +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 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. + +[Illustration: FIG. 14.] + +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 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 y. It was also proportional to the speed of turning of +the disk. + +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 y. 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 a distance y, 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 y, the +ordinate of the curve, and for any displacement dx along the datum line, +the registering cylinder is turned through an angle proportional to ydx. +Thus any finite angle turned through is proportional to the integral of +ydx 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 +y sin nx.dx, where y is the ordinate of the curve, and n a number +inversely proportional to the period of the tidal constituent +represented by the term. + +All that was necessary, in order to give the integral of a term +y sin nx.dx, was to make the disk oscillate about its axis as the +paper-cylinder was turned through an angle proportional to x. Thus one +disk, globe, and cylinder was arranged exactly as has been described for +the integral of ydx, 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 y from the centre of the disk. + +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. + +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. + +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. + +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, +and, finally, to the integration of any differential equation of any +order. + +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. + +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 _Natural Philosophy_. + + +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 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 _Natural Philosophy_ in Professor Karl +Pearson's continuation of Todhunter's _History of Elasticity_. 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. + +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. + + +GYROSTATIC ACTION + +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 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. + +[Illustration: FIG. 15.] + +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. + +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 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. + +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. + +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. + +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. + +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.[21] + +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 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. + +[Illustration: FIG. 16.] + +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 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. + +[Illustration: FIG. 17.] + +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. + +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 +arrow-heads 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. + +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. + +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. + +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 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. + +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 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. + +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.[22] + + +ELECTROSTATICS AND MAGNETISM + +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 _Natural Philosophy_, and just before the appearance of the Elements, +Messrs. Macmillan & Co. published for him a collection of memoirs +entitled _Reprint of Papers on Electrostatics and Magnetism_. 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 measuring 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. + +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 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. + +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. + +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 by +an attached attracted-disk electrometer, the principle of which we shall +merely indicate. + +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. + +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. + +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 _Electrostatics and Magnetism_. These results have been +much referred to in later researches. + +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. + +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." + +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 led to the machines with revolving +carriers, to which reference has been made above. + +The mathematical theory of magnetism which Thomson gave in 1849, in the +_Phil. Trans. R.S._, 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 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. + + + + +CHAPTER XII + +THE AGE OF THE EARTH + + +From 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 _Natural Philosophy_, and +in vol. ii of his _Popular Lectures and Addresses_, which is devoted to +geology and general physics. + +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. 65 above) in his inaugural dissertation "_De Caloris distributione in +Terræ Corpus_"; 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 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. + +The address began with a protest against the old uniformitarian view of +geological changes as expressed by Playfair in his _Illustrations of the +Huttonian Theory_. 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. + +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 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. + +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 +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. + +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 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. + +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 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. + +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. + +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 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 _Transactions, R.S.E._, 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. + +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 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. + +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. + +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 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. + +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 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. + +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 1⧸10 to 1⧸50 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. + +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 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." + +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 Nature (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 (_American Journal of +Science_, January 1893) on experiments on the physical properties of +rocks at high temperatures. + +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 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. + +In Nature 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. + +After the publication of the paper on Geological Time a reply to it was +made by Professor Huxley, in an address to the Geological Society of +London, delivered on February 19, 1869. He adopted the rôle 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 _Glasgow Herald_ 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 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." + +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 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." + + + + +CHAPTER XIII + +BRITISH ASSOCIATION COMMITTEE ON ELECTRICAL STANDARDS + + +When 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. + +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 to do for electricity and +magnetism what had been done by Regnault and others for heat. + +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 _Intensitas vis +magneticæ terrestris ad mensuram absolutam revocata_, 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. + +Methods of measuring electric quantities in absolute units were +described by W. Weber, in Parts II and III of his _Elecktrodynamische +Maassbestimmungen_, 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. 74-76) 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 +be sought which would have a temperature effect of small amount, and +which were stable and durable in all their properties. + +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. + +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. + +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 m be +the mass of a ball, g gravity in cm. sec. units, d the distance in +cms. of the centres of the balls apart, and l the length in cms. of +a thread, the charge q, say, on each ball is easily found to be +√[mgd³⧸√{4^(l² － d²)}]. Thus the charge is got in absolute +centimetre-gramme-second units in terms of the mass m obtained by +ordinary weighing, and l and d obtained by easy and exact measurements. + +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 easily observed), +the charge on the ball, and the value of m. The electric field-intensity +is obtained by dividing the value of the force by q. The field intensity +multiplied by D, 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 D gives the charge on the +conductor. + +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. + +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 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. + +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. + +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 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. + +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 AH, where A denotes the effective area of the +coils, and H 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 _relatively to the coil_ 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. + +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 H. But this was the field cut by the coil as +it turned, and therefore (except for a small term depending on the +turning of the coil in the field of the needle) the value of H did not +appear in the result, and did not require to be known. + +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[23]; 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. + +A unit called the B.A. unit, which was intended to contain 10^9 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. + +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 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. + +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 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. + +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. + +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--10^9 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. + +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. 251). + + + + +CHAPTER XIV + +THE BALTIMORE LECTURES + + +The 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 _The Baltimore Lectures_ 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. + +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 3 × 10^10 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. + +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 _non-magnetic; nothing magnetic_." 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." + +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 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 rationale +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. + +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 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. + +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 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. + +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. + +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. + +Dynamical illustrations, which old Glasgow students will recognise, +appear continually in the lectures. They will remember, almost with +affection, the system of three particles (7 lb. or 14 lb. weights!) +joined 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. + +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 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. + +[Illustration: FIG. 18.] + +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 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. + +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 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 _Popular Lectures and Addresses_, 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." + +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. + +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 _suo more_ he endeavoured to +frame dynamical explanations of electronic (or, as he 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. + + + + +CHAPTER XV + +SPEED OF TELEGRAPH SIGNALLING--LAYING OF SUBMARINE CABLES--TELEGRAPH +INSTRUMENTS--NAVIGATIONAL INSTRUMENTS, COMPASS AND SOUNDING MACHINE + + +THEORY OF SIGNALLING + +When 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 +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 n, the changes of potential are propagated with velocity +2√(πn⧸cr), where c and r are the capacity and resistance of the cable, +each taken per unit length. In this case, for a long cable, there 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. + +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. + + +LAYING OF FIRST AMERICAN CABLES + +The first cable of 1858 was laid by the U.S. frigate Niagara and H.M.S. +Agamemnon, 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 practical cases of +importance--for example, in the lifting of a cable for repairs. + +A first Atlantic cable had been partly laid in 1857 by the Niagara, 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 +Agamemnon towards Valentia, the Niagara towards Newfoundland. Professor +Thomson was engineer in charge of the electrical testing on board of the +Agamemnon. 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. + +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. + +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. + +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. + +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. + + +MIRROR GALVANOMETER AND SIPHON RECORDER + +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 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. + +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 in the U.S. frigate Niagara) is in the collection of +historical apparatus in the Natural Philosophy Department of the +University of Glasgow. + +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. + +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. + +The arrangement of magnet and suspended coil in this instrument has +become widely known as that of the "d'Arsonval galvanometer." This +application was 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. + +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. + + +MARINERS' COMPASS AND SOUNDING MACHINE + +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. + +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. + +The weight of the card is extremely small--only 170½ grains; that is +less than ⅖ 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 the card also causes the error due +to friction on the point of support to be very small. + +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. + +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. + +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. + +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. + +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 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. + +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 _Admiralty Manual of Deviations of the Compass_, +led to Lord Kelvin's inventions. + +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. + +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 +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. + +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.[24] + +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 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). + +For many years Sir William Thomson was a keen yachtsman, and his +schooner yacht, the _Lalla Rookh_, 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. + +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 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. + +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. + + + + +CHAPTER XVI + +LORD KELVIN IN HIS CLASS-ROOM AND LABORATORY + + +It 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 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. + +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 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. + +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 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 _Divina Commedia_ of Dante. + +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. + +McFarlane used to tell of an incident which illustrated the ingenuity +with which it was sometimes attempted to evade the ordeal of the _viva +voce_ 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 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. + +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. + +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. + +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. + +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 +Logik, 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 philosopher 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. + +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 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. + +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. + +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 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. + +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, [:θ] + μθ = 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. + +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 _Natural Philosophy_--"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 Barlow to make +some calculation, and of course forget to return it. Next morning some +number would be wanted from Barlow in a hurry, and the book would be +missing. Then Thomson would order that Barlow should be chained to the +lecture-table, and enjoin his assistant to see that that was done +without an hour's delay! + +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 hundred pieces of chalk on this ledge +to-morrow; remember, a hundred 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. + +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 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. + +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. + +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, 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 Ｖ 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. + +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. + +He often interrupted the course of a lecture with a 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 +27⅓ 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. + +Another experiment which caused a great sensation was that known as the +"dewdrop"! 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-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. + +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!" + +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 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. + +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.[25] + +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 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. + +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. + +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 been made as to any numerical question regarding the +subject under discussion. + +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, a verdict which in later years became only +more emphatic. + +One session he began the first lecture of the higher class by writing +dx⧸dt 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 x to the increment of the +independent variable t, 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 dx⧸dt means velocity?" Here +the physical idea as a whole was before his mind; and he did not reflect +that if t denoted time and x distance in any direction, the explanation +given by the student did describe velocity with fair accuracy. + +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. + +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 of his mathematical prelections can be obtained +from a perusal of the _Baltimore Lectures_. + +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. + +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. + +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 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!" + +The incident referred to occurred while he was preparing the article on +Heat for the ninth edition of the _Encyclopædia Britannica_. 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 Heat (he also wrote +Elasticity), 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 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! + + + + +CHAPTER XVII + +PRACTICAL ACTIVITIES--HONOURS AND DISTINCTIONS--LAST ILLNESS AND DEATH + + +It 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. + +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 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. + +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. + +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 _Pour le Mèrite_ 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. + +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 Nature 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. + +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 _Baltimore Lectures_, 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. + +During the year 1907 Lord Kelvin performed many University duties and +seemed to be in unusually 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. + +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. + +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. + +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. + +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. + +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. + +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 +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. + +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. + + + + +CONCLUSION + + +The 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 unmindful of the source from +which much of their illumination has come. + +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. + +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. + +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 +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. + +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. + +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.[26] 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. + +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 for different parts +of the world, superintended the analysis of tidal records, and invented +tide-predicting machines and improved tide-gauges. + +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 +_Mathematical and Physical Papers_, 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." + +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 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. + +It is interesting to note that the investigation of waves in canals with +which Lord Kelvin recently enriched the _Proceedings of the Royal +Society of Edinburgh_ 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. + +Lord Kelvin's written work consists of the _Electrostatics and +Magnetism_, three volumes of _Collected Mathematical and Physical +Papers_, three of _Popular Lectures and Addresses_, the _Baltimore +Lectures_, a very considerable number of papers as yet uncollected, and +the _Natural Philosophy_. 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 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. + +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. + +It is hardly worth while to refute the statement 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. + +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 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. + +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. + + Saepe stilum vertas quae digna legi sint + Scripturus, + +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. + +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. + +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. + +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. + +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 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. + +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 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. + + + + +FOOTNOTES: + + + [1] Lord Kelvin's address on his installation as Chancellor of + the University of Glasgow, November 29, 1904. + + [2] 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. + + [3] 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! + + [4] Proceedings on the occasion of the Presentation to the + University of Glasgow of the Portrait of Emeritus Professor G. + G. Ramsay. November 6, 1907. + + [5] Apparently for a short time in 1841, when Dr. Meikleham was + laid aside by illness. + + [6] 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 _Cambridge Chronicle_, July 10, 1903, and _The + Cambridge Review_, Feb. 20, 1908.] + + [7] It is rather strange that the ninth edition of the + _Encyclopædia Britannica_ 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.' + + [8] January 1869, Reprint, etc., Article XV. + + [9] Reprint, Article V. + + [10] 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 _Annali delle + Scienze del Regno Lombardo-Veneto_ for that year. It was also + described as an independent discovery by Mr. John Wm. Stubbs, + in a paper in the _Philosophical Magazine_ for November 1843. + In a note on the history of the transformation in Taylor's + _Geometry of Conics_ 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, _Phil. Soc. + Trans._ I). The note also mentions that inversion was "applied + by Dr. Hirst to attractions," but contains no reference to + Thomson's papers! + + [11] "_De Caloris distributione per Terræ Corpus_" in the + Faculty minute, as stated above. + + [12] _Sic._ Without doubt a mistake of the scribe for + "Liouville." + + [13] _North Wales Chronicle_, Report, Feb. 7, 1885. + + [14] Published: _Treatise on Natural Philosophy_, vol. i in + 1867; _Elements of Natural Philosophy_ in 1873. + + [15] 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 _Oration + on James Watt_, 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. + + [16] 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. + + [17] 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.] + + [18] 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 p + dp and v, and on the delivery side as p and v + dv, so + that dp and dv are positive. The net work done in forcing the + gas through the plug = (p + dp)v － p(v + dv) = － pdv + vdp. + Let a heating effect result so that temperature is changed from + T to T + ∂T. Let this be annulled by abstraction of heat + Cp∂T at constant pressure. (Cp = sp. heat press. const.) + [It is to be understood that dv is the total expansion + existing, after this abstraction of heat.] The energy e of the + fluid has been increased by de = － pdv + vdp － Cp∂T. + + Now, since the original temperature has been restored, the + same expansion dv if imposed isothermally would involve the + same energy change de; but in that case heat dH (dynamical) + would be absorbed, and work pdv would be done by the gas. + Hence de = dH － pdv. This, with the former value of de, gives + dH = vdp － Cp∂T. Thomson's work-ratio is thus pdv⧸(vdp － Cp∂T). + Now suppose dp imposed without change of volume, and dT to be the + resulting temperature change. The temperature and pressure ratios + are dT⧸T, dp⧸p. Thus dT⧸T = dp dv⧸(vdp － Cp∂T), or + + (v⧸T)(dT⧸dv) = 1⧸[1 － (Cp⧸v)(∂T⧸dp)] + + which is Thomson's equation. The minus sign on the right arises + from a heating effect having been taken here as the normal + case. + + If the temperature T is restored by removing the heat at + constant volume, a similar process gives the equation + + (v⧸T)(dT⧸dv) = [1 + (∂T⧸∂p)(∂T⧸dp)]⧸[1 － (Cv⧸v)(∂T⧸dp)] + + where dp is the change of pressure before the restoration of + the temperature T, and ∂T⧸∂p is the rate of variation of T + with p, volume constant. + + [19] "On a Universal Tendency in Nature to Dissipation of + Energy," _Proc. R.S.E._, 1852, and _Phil. Mag._, Oct., 1852. + + [20] To this may be added the extremely useful theorem for such + problems, that if any directed quantity L, say, characteristic + of the motion of a body, be associated with a line or axis Ol, + 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 Ol, towards which Ol is turning with angular + velocity ω, of amount ωL. If M be the amount of the + quantity already existing for this latter line or axis, the + total rate of growth of the quantity is there M + ωL. For + example, a particle moving with uniform speed v in a circle of + radius r, has momentum mv along the tangent. But the tangent is + turning round as the particle moves with angular speed v⧸r, + towards the radius. The rate of growth of momentum towards the + centre is therefore + + mv × v⧸r = mv²⧸r. + + [21] See Gray's _Lehrbuch der Physik_, s. 278. Vieweg u. Sohn, + 1904. + + [22] Gray, Royal Institution, Friday Evening Discourse, + February 1898. + + [23] See the _Reports of the Committee on Electrical Standards_, + edited by Prof. Fleeming Jenkin, F.R.S., Maxwell's _Electricity + and Magnetism_, and Gray's _Theory and Practice of Absolute + Measurements in Electricity and Magnetism_, Vol. II, Part II. + + [24] 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! + + [25] The tuning of a major third, in this way, is described in + the paper entitled "Beats on Imperfect Harmonies," published in + _Popular Lectures and Addresses_, vol. ii. + + [26] 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! + + + + +INDEX + + + Atlantic cables, 267, 268 + + Atmospheric electricity, 226 + + Atoms, size of, 261 + + Ayrton, W. E., 296 + + + Baltimore lectures, 254-263 + + Bertrand's theorem of maximum kinetic energy, 158 + + Bottomley, James Thomson, 311 + + Bottomley, William, 7 + + British Association, electrical standards, 244-253 + + + Cambridge University Musical Society, 24 + + _Cambridge and Dublin Mathematical Journal_, 25, 31, 78 + + Carnot, Sadi, 77, 101 + + Carnot's _Théorie Motrice du Feu_, 87, 101, 108 _et seq._ + + Cauchy, 294 + + Chasles, 28, 43 + + Clapeyron, 101, 112 + + Clausius, 114 _et seq._ + + College, the old, of Glasgow, 10 + + Compass, errors of, 273 + + + "Dew-drop," artificial, 290 + + Dynamical theorems, Thomson's and Bertrand's, 158 _et seq._ + + + Earth, the age of, 196, 229-243 + + Earth, tidal retardation of, 230 + + Elasticity, Poisson-Navier theory of, 291; + encyclopædia article on, 297 + + Electrical oscillations, 181 _et seq._ + + Electricity, mathematical theory of, 33 + + Electrolysis, mechanical theory of, 176 + + Electrometers, 223 _et seq._ + + Electromotive forces, estimation of, by heats of combination, 178 + + Electromotive forces, measurement of, 179 + + _Electrostatics and Magnetism_, 222 _et seq._ + + Ellis, Robert Leslie, 26 + + Energy, dissipation of, 139 + + + Faculty, the, of the University of Glasgow, 4, 63-67 + + Faraday, 61 + + Faure, M., 81 + + FitzGerald, G. F., 301, 305 + + Fourier, _Théorie Analytique de la Chaleur_, 16 _et seq._ + + + Gauss, 28 + + Gauss and Weber, 245 + + Green, George, of Nottingham, 21, 30, 294 + + Gregory, J. W., 241 + + Goodwin, Harvey, 26 + + Gyrostats and gyrostatic action, 214, 284-286 + + + Hamilton, Sir William Rowan, 196, 294 + + Heat, encyclopædia article on, 297 + + Heaviside, Oliver, 294 + + Helmholtz, von, 113 + + Hertz, 191, 256 + + Hopkins, William, 23 + + Huxley, 77, 196, 242 + + Hydrodynamics, 153-175 + + + Images, electric, 31, 38-59 + + Inversion, electrical, 49 _et seq._ + + Inversion, geometrical, 59, 60 + + + Joule, James Prescott, 77, 86 _et seq._, 101 _et seq._ + + + Larmor, Joseph, 256 + + Lectures on Natural Philosophy at Glasgow, 279 _et seq._ + + Liouville, 31 + + Liouville's _Journal de Mathématiques_, 25, 26, 31 + + Loschmidt, 262 + + Lubbock, Sir John (Lord Avebury), 85 + + Luminiferous ether, motion of planets through, 256 + + + Magnetism, theory of, 227 + + Mariners' compass, 272 _et seq._ + + Maxwell, 117, 193, 305 + + Mayer, of Heilbronn, 105 + + McFarlane, Donald, 96, 287, 289 + + McKichan, Dugald, 193 + + _Mécanique Analytique_ of Lagrange, 199, 205 + + _Mécanique Céleste_ of Laplace, 199, 205 + + Meikleham, William, 61 + + Mirror galvanometer, 268 + + Motivity, thermodynamic, 138 + + + Natural Philosophy, Chair of, at Glasgow, 63 + + _Natural Philosophy_, Thomson and Tait's, 196 _et seq._ + + Navigational sounding machine, 272 + + Newton, 195, 202 + + Nichol, John, Professor of English Language and Literature, 5 + + Nichol, John Pringle, Professor of Astronomy, 5, 20, 61, 63 + + + Oersted, 61 + + Oscillations, electrical, 181 _et seq._ + + + Parkinson, Stephen, 27 + + Peltier, 148 + + Pendulum, ballistic, 288 + + Perry, John, 240, 296 + + Phosphorescence, dynamical theory of, 259 + + Physical laboratory, first, 70 + + Pickering, 217 + + Polarised light, rotation of plane of, 220 + + Principia, Newton's, 195, 202 + + + Ramsay, George Gilbert, Professor of Humanity, 11 + + Regnault, 29 + + Royal Society of Edinburgh, presidency of, 299 + + Royal Society of London, presidency of, 299 + + Rumford, Count, 103 + + + Seebeck, 148 + + Signalling, theory of telegraphic, 264 + + Siphon recorder, 268, 270 + + Smith, Archibald, 275 + + Spectrum analysis, dynamical theory of, 84 + + Stokes, Sir George Gabriel, 24, 79, 80, 81, 85, 291, 294 + + Stoney, Dr. Johnstone, 262 + + Sun's heat, age of, 232 + + + Tait, Peter Guthrie, 194 _et seq._ + + Temperature, absolute, 125 _et seq._; + comparison of, with scale of air thermometer, 135 + + Thermodynamics, 99-152 + + Thermoelasticity, 142 _et seq._ + + Thermoelectricity, 147 _et seq._ + + Thermometry, absolute, 114-152 + + Thomson, David, 61 + + Thomson, James, Professor of Mathematics, 1-4, 7 + + Thomson, James, Professor of Engineering, 113, 209; + integrating machine, 209, 303 + + Thomson and Tait's Natural Philosophy, 68, 196 _et seq._, 218 + + Thomson's theorem of minimum kinetic energy, 158 + + Thomson, Thomas, Professor of Chemistry, 6 + + + + Thomson, prevalence of name at Glasgow College, 5 + + Thomson, William, Lord Kelvin:-- + Parentage and early education, 1-12 + Career at Universities of Glasgow and Cambridge, 13-32 + Early researches, 16, 18, 31 + Election to Chair of Natural Philosophy at Glasgow, 64 + Scientific researches, passim; + Jubilee of, 301; + Chancellor of University of Glasgow, 302 + In class-room and laboratory, 279-298 + Practical activities, honours and distinctions, last illness and + death, 299-304; + funeral in Westminster Abbey, 304 + + Tidal Analyser, 211 + + Tide Predicter, 208 + + + Vortex-Motion, 161-175 + + + Waldstein sonata, 24 + + Weber, W., 193 + + Weights and measures, British, 289, 290 + + White, James, 276 + + Willard Gibbs, 294 + + + + RICHARD CLAY & SONS, LIMITED, + BREAD STREET HILL, E.C., AND + BUNGAY, SUFFOLK. + + + + + +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 39373-0.txt or 39373-0.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 the Department +of Mathematics at the University of Glasgow.) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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