TRANSACTIONS OF SECTION A. 4.25 
things. Even the student whose tastes lie mainly in the direction of pure mathe- 
watics would profit, I think, by a wider scientific traiing. A long list of 
instances might be given to show that the most fruitful ideas in pure mathematics 
have been suggested by the study of physical problems. In the words of Fourier, 
who did so much to fulfil his own saying, ‘ L’étude approfondie de la nature est la 
source la plus féconde des découvertes mathématiques. Non-seulement cette 
étude, en offrant aux recherches un but déterminé, a l’avantage d’exclure les 
questions vyagues et les calculs sans issue; elle est encore un moyen assuré de 
former l’analyse elle-méme, et d’en découvrir les éléments qu'il nous importe le plus 
de connaitre, et que cette science doit toujours conserver : ces éléments fondamentaux 
sont ceux qui se reproduisent dans tous les effets naturels.’ 
Another characteristic of the past century of applied mathematics is that 
it was, on the whole, the age of linear equations. The analytical armoury 
fashioned by Lagrange, Poisson, Fourier, and others, though subject, of course, 
to continual improvement and development, has served the turn of a long line of 
successors. The predominance of linear equations, in most of the physical sub- 
jects referred to, rests on the fact that the changes are treated as infinitely 
small, The theory of small oscillations, in particular, runs as a thread 
through a great part of the literature of the period in question. It has suggested 
many important analytical results, and still gives the best and simplest in- 
tuitive foundation for a whole class of theorems which are otherwise hard to 
comprehend in their various relations, such as Fourier’s theorem, Laplace’s 
expansion, Bessel’s functions, and the like. Moreover, the interest of the subject, 
whether mathematical or physical, is not yet exhausted ; many important problems 
in Optics and Acoustics, for example, still await solution. The general theory has 
in comparatively recent times received an unexpected extension (to the case of 
‘latent motions’) at the hands of Lord Kelvin; and Lord Rayleigh, by his con- 
tinual additions to it, shows that, in his view, it is still incomplete. 
When the restriction to infinitely small motions is abandoned, the problems 
become of course much more arduous. The whole theory, for instance, of tho 
normal modes of vibration which is so important in Acoustics, and even in Music, 
disappears. The researches hitherto made in this direction have, moreover, 
encountered difficulties of a less patent character. It is conceivable that the 
modern analytical methods which have been developed in Astronomy may have 
an application to these questions. It would appear that there is an opening here 
for the mathematician; at all events, the numerical or graphical solution of any 
one of the numerous problems that could be suggested would be of the highest 
interest. One problem of the kind is already classical—the theory of steep 
water-waves discussed by Stokes; but even here the point of view has perhaps 
been rather artificially restricted. The question proposed by him, the determina-~ 
tion of the possible form of waves of permanent type, like the problem of periodic 
orbits in Astronomy, is very interesting mathematically, and forms a natural 
starting-point for investigation ; but it does not exhaust what is most important for 
us to know in the matter. Observation may suggest the existence of such waves as 
a fact ; but no reason has been given, so far as I know, why free water-waves should 
tend to assume a form consistent with permanence, or be influenced in their 
progress by considerations of geometrical simplicity. 
I have tried to indicate the kind of continuity of subject-matter, method, and 
spirit which runs through the work of the whole school of mathematical physicists 
of which Stokes may be taken as the represeutative. It is no less interesting, 
I think, to examine the points of contrast with more recent tendencies. These 
relate not so much to subject-matter and method as to the general mental 
attitude towards the problems of Nature. Mathematical and physical science 
haye become markedly introspective. The investigators of the classical school, as 
it may perhaps be styled, were animated by a simple and vigorous faith; they 
sought as a matter of course for a mechanical explanation of phenomena, and had 
no misgivings as to the trustiness of the analytical weapons which they wielded. 
But now the physicist and the mathematician alike are in trouble about their 
souls, We have discussions on the principles of mechanics, on the founda- 
