1906.] Anniversary Address by Lord Rayleigh. 89 



imbued with certain images clearly formed, and yet appear hazardous or 

 even irrelevant to another exercised in a different order of ideas ? Merely 

 as an example, there are theorems known as " existence-theorems " having 

 physical interpretations, the object of which is to prove formally what to 

 many minds can be no clearer afterwards than it was before. The pure 

 mathematician will reply that even if this be so, the introduction of 

 electrical or thermal ideas into an analytical question is illogical, and from 

 his own point of view he is, of course, quite right. What is rather surprising 

 is that the analytical argument should so often take forms which seem to 

 have little relation to the intuition of the physicist. Possibly a better 

 approach to a reconciliation may come in the future. In the meantime we 

 must be content to allow the two methods to stand side- by side, and it will 

 be well if each party can admit that there is something of value to be 

 learned from the point of view of the other. 



In other branches, at any rate, the physicist has drawn immense 

 •advantage from the labours of the pure analyst. I may refer especially to 

 the general theory of the complex variable and to the special methods which 

 have been invented for applying it to particular problems. The rigorous 

 solution by Sommerfeld of a famous problem in diffraction, approximately 

 treated by Fresnel, is a case in point. We have moved a long way from 

 the time when it was possible for the highest authority in theoretical optics 

 to protest that he saw no validity in Fresnel's interpretation of the 

 imaginary which presents itself in the expression for the amplitude of 

 reflected light when the angle of incidence exceeds the critical value. In 

 this connection it is interesting to remember that, in his correspondence 

 with Young, Laplace expressed the opinion that the theoretical treatment 

 of reflexion was beyond the powers of analysis. The obvious moral is that 

 we are not to despair of the eventual solution of difficulties that may be too 

 much for ourselves. 



As more impartially situated than some, I may, perhaps, venture to say 

 that in my opinion many who work entirely upon the experimental side 

 of science underrate their obligations to the theorist and the mathematician. 

 Without the critical and co-ordinating labours of the latter we should 

 probably be floundering in a bog of imperfectly formulated and often 

 contradictory opinions. Even as it is, some branches can hardly escape 

 reproaches of the kind suggested. I shall not be supposed, I hope, to under- 

 value the labours of the experimenter. The courage and perseverance 

 demanded by much work of this nature is beyond all praise. And success 

 often depends upon what seems like a natural instinct for the truth — one of 

 the rarest of gifts. 



