902 



SCIENCE 



[N. S. Vol. XXXVIII. No. 991 



of published lectures attest Ms skill as a 

 teacher, the names of which I will not take 

 your time to rehearse, merely remarking 

 that in addition to the usual treatments of 

 electricity, optics, the conduction of heat, 

 thermodynamics, capillarity, elasticity and 

 hydrodji-namics there are several volumes 

 on the modern subjects of electrical oscilla- 

 tions and the interrelations of electricity 

 and optics. 



The work of the mathematical physicist 

 is of two sorts, according as the emphasis 

 is laid on the word physics or on the word 

 mathematical. In the latter case the in- 

 vestigator concentrates his attention upon 

 the attempt to demonstrate that certain 

 problems have solutions^ furnishing so- 

 called existence theorems. In the former 

 the attempt is made to find the solutions, 

 assuming that they exist, in a form suit- 

 able for numerical computation. Poincare 

 did both, and, although capable of the 

 highest flights into abstract mathematics, 

 was by no means insensible to the needs of 

 the practical man, meaning by that not 

 only the physicist, but even the telegraph 

 engineer. This is attested by the number 

 of articles that he wrote on the theory of 

 telegraphy, both with and without wires, 

 as well as by the courses of lectures that he 

 gave at the higher professional school of 

 posts and telegraphs. It is certainly a 

 very rare thing for a pure mathematician 

 of the highest ability to write an article on 

 the theory of the telephone receiver, yet 

 this was done by Poincare, while in a paper 

 on the propagation of current in the vari- 

 able period on a line furnished with a re- 

 ceiver' he attacked an almost untouched 

 field of very great mathematical impor- 

 tance in the theory of differential equa- 

 tions. 



What is particularly striking in all of 

 Poincare 's writings is not so much the 

 clearness of exposition or the elegance of 



arrangement, for his lectures possess many 

 of the faults of lectures published by stu- 

 dents and his short articles are often ex- 

 tremely difficult reading, but rather the re- 

 markable directness with which he pro- 

 ceeds to his results and the extraordinary 

 command of every resource of pure mathe- 

 matics, particularly of Cauchy's theory of 

 the functions of a complex variable. I 

 know of but one other author whose re- 

 sources in function-theory seem to be at 

 all comparable with those of Poincare, I 

 mean Professor Sommerfeld, who seems to 

 be able to communicate his powers in that 

 line to his students. The heart of mathe- 

 matical physics is, without doubt, composed 

 of partial differential equations, and in 

 this subject Poincare was, of course, a 

 master. It is in connection with the defi- 

 nite integrals appearing in their solutions 

 that there is great opportunity for the ap- 

 plication of function-theory. The great art 

 in mathematical physics is that of making 

 approximations and it is here that Poin- 

 care was particularly strong. It is fre- 

 quently not so difficult to obtain the solu- 

 tion of the differential equation as to inter- 

 pret its physical meaning. In this matter 

 Poincare reminds us of his great country- 

 man Cauehy. 



I shall not attempt to make an analysis 

 of the articles of Poincare, many of which 

 I have great difficulty in following and 

 many of which could be far better treated 

 by others here present. I shall merely 

 undertake to give a slight idea of the con- 

 tents of those which have particularly im- 

 pressed me. I presume his contributions 

 of most far-reaching importance from a 

 mathematical point of view are his articles 

 on the eqiiations of mathematical physics, 

 of which he wrote three. This is a subject 

 which has received an enormous amount of 

 attention during the last twenty-five years 

 and it may be undoubtedly said that in 



