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.

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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

CORRIGENDUM

[Page 105], line 9 from foot, for ∂e + O read ∂e + o

LIST OF ILLUSTRATIONS

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.

Fig. 1.

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.

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.

Fig. 2.

Fig. 3.

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.

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.

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' at the inverse point J₂ of P₁ distributed uniformly over it, together with the two induced distributions just described.

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.

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.

Fig. 7.

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.

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.

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.

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.

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.

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 6s. 1½d. 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.

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.