MATHEMATICAL PHYSICS. 513 



ceived an unexpected extension (to the case of 'latent motions') at 

 the hands of Lord Kelvin; and Lord Raylcigh, by his continual addi- 

 tions 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 the 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 various 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 determination of the possible forms 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 per- 

 manence, or be influenced in their progress by considerations of geomet- 

 rical 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 repre- 

 sentative. 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 have be- 

 come 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 ana- 

 lytical weapons which they wielded. But now the physicist and the 

 mathematician alike are in trouble about their souls. We have dis- 

 cussions on the principles of mechanics, on the foundations of geom- 

 etry, on the logic of the most rudimentary arithmetical processes, as 

 well as the more artificial operations of the calculus. These discus- 

 sions are legitimate and inevitable, and have led to some results which 

 are now widely accepted. Although they were carried on to a great 

 extent independently, the questions involved will, I think, be found to 



VOL. lxv. — 33. 



