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.