Читать книгу Lord Kelvin: An account of his scientific life and work - Andrew Gray - Страница 11

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.