Dkcember 23. 1904.] 



SCIENCE. 



869 



so much because of the needs of practice as 

 in order not to avow itself vanquished, that 

 analysis will never resign itself to abandon, 

 without a decisive victory, a subject where 

 it has met so many brilliant triumphs; 

 and again, what more beautiful field could 

 the theories new-born or rejuvenated of the 

 modern doctrine of functions find, to essay 

 their forces, than this classic problem of 

 n bodies? 



It is a joy for the analyst to encounter 

 in applications equations ' that he can 

 integrate with known functions, with tran- 

 scendents already classed. 



Such encounters are unhappily rare ; the 

 problem of the pendulum, the classic cases 

 of the motion of a solid body around a 

 fixed point are examples where the elliptic 

 functions have permitted us to effect the 

 integration. 



' It would also be extremely interesting to 

 encounter a question of mechanics which 

 might be the origin of the discovery of a 

 new transcendent possessing some remark- 

 able property; I would be embarrassed to 

 give an example of it unless in carrying 

 back to the pendukim the debut of the 

 theory of elliptic functions. 



The interpenetration between theory and 

 applications is here much less than in the 

 questions of mathematical physics. Thus 

 is explained, that, since forty years, the 

 worlss on ordinary differential equations 

 attached to analytic functions have had in 

 great part a theoretic character altogether 

 abstract. 



The pure theory has notably taken the 

 advance ; we have had occasion to say that 

 it was well it should be so, but evidently 

 there is here a question of measure, and we 

 may hope to see the old problems profit 

 by the progress accomplished. 



It would not be over difficult to give some 

 examples, and I will recall only those linear 

 differential equations, where figure arbi- 

 trary parameters whose singular values are 



roots of entire transcendent functions; 

 which in particular makes the successive 

 harmonics of a vibrating membrane corre- 

 spond to the poles of a meromorphie func- 

 tion . 



It happens also that the theory may be 

 an element of classification in leading to 

 seek conditions for which the solution falls 

 under a determined type, as for example 

 that the integral may be uniform. There 

 have been and there yet will be many inter- 

 esting discoveries in this way, the case of 

 the motion of a solid heavy body treated 

 by Mme. de Kovalevski and where the 

 Abelian functions were utilized is a remark- 

 able example. 



VII. 

 In studying the reciprocal relations of 

 analysis with mechanics and mathematical 

 physics, we have on our way more than 

 once encountered the infinitesimal geom- 

 etry, which has proposed so many cele- 

 brated problems; in many difficult ques- 

 tions, the happy combination of calculus 

 and synthetic reasonings has realized con- 

 siderable progress, as show the theories of 

 applicable surfaces and systems triply or- 

 thogonal. 



It is another part of geometry which 

 plays a grand role in certain analytic re- 

 searches, I mean the geometry of situation 

 or analysis situs. We know that Riemann 

 made from this point of view a complete 

 study of the continuum of two dimensions, 

 on which rests his theory of algebraic func- 

 tions and their integrals. 



When this number of dimensions aug- 

 ments, the questions of analysis situs be- 

 come necessarily complicated; the geomet- 

 ric intuition ceases, and the study becomes 

 purely analytic, the mind being guided 

 solely by analogies which may be mislead- 

 ing and need to be discussed very closely. 

 The theory of algebraic functions of two 

 variables, which transports us into a space 

 of four dimensions, without ■ getting from 



