August i8, 1904] 



NA TURE 



have suggested most wonderful and far-reaching specula- 

 tions, and, whatever be the future of these particular 

 theories, they are bound in any case deeply to influence 

 •our views on fundamental points of chemistry and physics. 

 No reference to this subject would, I think, be satis- 

 factory without a word of homage to the unsurpassed 

 patience and skill in the devising of new experimental 

 methods to meet new and subtle conditions which it has 

 evoked. It will be felt as a matter of legitimate pride by 

 many present that the University of Cambridge has been 

 so conspicuously associated with this work. It would there- 

 fore have been natural and appropriate that this Chait 

 ■should have been occupied, this year above others, by one 

 who could have given us a survey of the facts as they at 

 present stand, and of their bearing, so far as can be dis- 

 cerned, on other and older branches of physics. Whether 

 from the experimental or from the more theoretical and 

 philosophical standpoint, there would have been no diffi- 

 culty in finding exponents of unrivalled authority. But it 

 has been otherwise ordered, and you and I must make the 

 best of it. If the subject cannot be further dealt with for 

 the moment, we have the satisfaction of knowing that it 

 will in due course engage the attention of the Section, 

 and that we may look forward to interesting and stimulat- 

 ing discussions, in which we trust the many distinguished 

 foreign physicists who honour us by their presence will 

 take an active part. 



It is, I believe, not an unknown thing for your 

 President to look up the records of previous meetings in 

 •search of inspiration, and possibly of an example. I have 

 myself not had to look very far, for I found that when the 

 British Association last met in Cambridge, in the year 

 1862, this .Section was presided over by Stokes, and, more- 

 over, that the Address which he gave was probably the 

 shortest ever made on such an occasion, for it occupies only 

 half a page of the report, and took. I should say, some 

 three or four minutes to deliver. It would be to the 

 advantage of the business of the meeting, and to my own 

 great relief, if I had the courage to follow so attractive a 

 precedent ; but I fear that the tradition which has since 

 established itself is too strong for me to break without 

 presumption. I will turn, therefore, in the first instance, 

 to a theme which. I think, naturally presents itself — viz., 

 a consideration of the place occupied by Stokes in the 

 ■development of Mathematical Physics. It is not proposed 

 to attempt an examination or appreciation of his own 

 Individual achievements; this has lately been done bv more 

 than one hand, and in the most authoritative manner. But 

 it is part of the greatness of the man that his w^ork can be 

 reviewed from more than one standpoint. What I specially 

 ■wish to direct attention to on this occasion is the historical 

 or evolutionary relation in which he stands to predecessors 

 and followers in the above field. 



The early years of Stokes's life were the closing years 

 of a mighty generation of mathematicians and mathematical 

 physicists. When he came to manhood, Lagrange, Laplace, 

 Poisson, Fourier, Fresnel, Ampere, had but recently passed 

 dway. Cauchy alone of this race of giants was still alive 

 and productive. It is upon these men that we must look 

 as the immediate intellectual ancestors of Stokes, for, 

 although Gauss and F. Neumann were alive and flourishing, 

 the interaction of German and English science was at that 

 time not very great. It is noteworthy, however, that the 

 development of the modern German school of mathematical 

 physics, represented by Helmholtz and Kirchhoff, in linear 

 succession to Neuinann, ran in many respects closely parallel 

 to the work of Stokes and his followers. 



When the foundations of Analytical Dynamics had been 

 laid by Euler and d'Alembert, the first important applica- 

 tion was naturally to the problems of Gravitational 

 Astronomy ; this formed, of course, the chief work of 

 Laplace, Lagrange, and others. .\fterwards came the 

 theoretical study of Elasticity, Conduction of Heat, Statical 

 Electricity, and Magnetism. The investigations in Elasticity 

 were undertaken mainly in relation to Physical Optics, with 

 the hope of finding a material medium capable of convey- 

 ing transverse vibrations, and of accounting also for the 

 various phenomena of reflection, refraction, and double re- 

 fraction. It has often been pointed out, as characteristic of 

 the French school referred to, that their physical specula- 

 tions were largely influenced by ideas transferred from 



VO. 18 16, VOL. 70] 



Astronomy ; as, for instance, in the conception of a solid 

 body as made up of discrete particles acting on one 

 another at a distance with forces in the lines joining them, 

 which formed the basis of most of their work on Elasticity 

 and Optics. The difliculty of carrying out these ideas in 

 a logical manner was enormous, and the strict course of 

 mathematical deduction had to be replaced by more or less 

 precarious assumptions. The detailed study of the geoinetry 

 of a continuous deformable medium which was instituted 

 by Cauchy was a first step towards liberating the theory 

 from arbitrary and unnecessary hypothesis ; but it was 

 reserved for Green, the immediate predecessor of Stokes 

 among English mathematicians, to carry out this process 

 completely and independently, with the help of Lagrange's 

 general dynamical methods, w-hich here found their first 

 application to questions of physics outside the ordinary 

 Dynamics of rigid bodies and fluids. The modern school 

 of English physicists, since the time of Green and Stokes, 

 have consistently endeavoured to rnake out, in any given 

 class of phenomena, how much can be recognised as a 

 manifestation of general dynamical principles, independent 

 of the particular mechanism which may be at work. One 

 of the most striking examples of this was the identifica- 

 tion bv Maxwell of the laws of Electromagnetism with the 

 dynamical equations of Lagrange. It would, however, be 

 going too far to claim this tendency as the e.xclusive 

 characteristic of English physicists; for example, the elastic 

 investigations of Green and Stokes have their parallel in 

 the independent though later work of Kirchhoff ; and the 

 beautiful theory of dynamical systems with latent motion 

 which we owe to Lord Kelvin stands in a very similar 

 relation to the work of Helmholtz and Hertz. 



But perhaps the most important and characteristic 

 feature in the mathematical work of the later school is 

 its increasing relation to and association with experiment. 

 In the davs when the chief applications of Mathematics 

 were to the problems of Gravitational Astronomy, the 

 mathematician inight well take his materials at second 

 hand; and in soine respects the division of labour was, and 

 still mav be, of advantage. The same thing holds, in a 

 measure, of the problems of ordinary Dynamics, where 

 some practical knowledge of the subject-matter is within 

 the reach of evervone. But when we pass to the more 

 recondite phenomena of Physical Optics, Acoustics, and 

 Electricity, it hardly needs the demonstrations which have 

 involuntarilv been given to show that the theoretical treat- 

 ment must tend to degenerate into the pursuit of mere 

 academic subtleties unless it is constantly vivified by direct 

 contact with reality. Stokes, at all events, with little 

 guidance or encouragement from his immediate environ- 

 ment, made himself from the first practically acquainted 

 with the subjects he treated. Generations of Cambridge 

 students recall the enthusiasm which characterised his 

 experimental demonstrations in Optics. These appealed to 

 us all ; but some of us, I am afraid, under the influence 

 of the academic ideas of the time, thought it a little 

 unnecessary to show practically that the height of the 

 lecture-room could be measured by the barometer, or to 

 verify the calculated period of oscillation of water in a 

 tank' by actually timing the waves with the help of the 

 image of a candle-flame reflected at the surface. 



The practical character of the mathematical work of 

 Stokes and his followers is shown especially in the constant 

 effort to reduce the solution of a physical problem to a 

 quantitative forrn. A conspicuous instance is furnished by 

 the labour and skill which he devoted, from this point of 

 view, to the theory of the Bessel's Function, which pre- 

 sents itself so frequently in important questions of Optics, 

 Electricitv, and Acoustics, but is so refractory to ordinary 

 methods of treatment. It is now generally accepted that 

 an analytical solution of a physical question, however 

 elegant it may be made to appear by means of a judicious 

 notation, is not complete so long as the results are given 

 merely in terms of functions defined by infinite series or 

 definite integrals, and cannot be exhibited in a numerical 

 or graphical form. This view did not originate, of course, 

 wMth Stokes; it is clearlv indicated, for instance, in the 

 works of Fourier and Poinsot, but no previous writer had, 

 I think, acted upon it so consistently and thoroughly. 



We have had so many striking examples of the fruitful- 

 ness of the combination of great mathematical and experi- 



