Читать книгу Lord Kelvin: An account of his scientific life and work - Andrew Gray - Страница 9

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.