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.