DEVELOPMENT OF MATHEMATICAL ANALYSIS 513 



is explained that, since forty years, the works on ordinary differ- 

 ential 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 pro- 

 blems profit by the progress accomplished. 



It would not be over-difficult to give some examples, and I will re- 

 call only those linear differential equations, where figure arbitrary 

 parameters whose singular values are roots of entire transcendent 

 functions ; which in particular makes the successive harmonics of 

 a vibrating membrane correspond to the poles of a meromorphic 

 function. 



It happens also that the theory may be an element of classifica- 

 tion 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 interesting discoveries 

 in this way, the case of the motion of a solid heavy body treated 

 by Madame de Kovalevski and where the Abelian functions were 

 utilized is a remarkable 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 geometry, which has proposed so 

 many celebrated problems; in many difficult questions, the happy 

 combination of calculus and synthetic reasonings has realized con- 

 siderable progress, as is shown by the theories of applicable surfaces 

 and systems triply orthogonal. 



It is another part of geometry which plays a grand role in certain 

 analytic researches, I mean the geometry of situation or analysis 

 situs. We know that Riemann made from this point of view a com- 

 plete study of the continuum of two dimensions, on which rests his 

 theory of algebraic functions and their integrals. 



When this number of dimensions augments, the questions of 

 analysis situs become necessarily complicated; the geometric intui- 

 tion ceases, and the study becomes purely analytic, the mind being 

 guided solely by analogies which may be misleading 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 analysis situs an aid so fruitful as does the 

 theory of functions of one variable, owes it, however, useful orient- 

 ations. 



There is also another order of questions where the geometry of 

 situation intervenes; in the study of curves traced on a surface and 



