506 ALGEBRA AND ANALYSIS 



analysis. The functions of a complex variable with unique derivative 

 are necessarily developable in Taylor's series; we come back thus 

 to the mode of development of which the author of the theory of 

 analytic functions had understood the interest, but of which the 

 importance could not be put fully in evidence in limiting one's self 

 to real variables. They also owe the grand role that they have not 

 ceased to play to the facility with which we can manage them, and 

 to their convenience in calculation. 



The general theorems of the theory of analytic functions permitted 

 to reply with precision to questions remaining up to that time un- 

 decided, such as the degree of generality of the integrals of differential 

 equations. It became possible to push even to the end the demon- 

 stration sketched by Lagrange for an ordinary differential equa- 

 tion. For a partial differential equation or a system of such equations, 

 precise theorems were established. It is not that on this latter point 

 the results obtained, however important they may be, resolve 

 completely the diverse questions that may be set; because in mathe- 

 matical physics, and often in geometry, the conditions at the limits 

 are susceptible of forms so varied that the problem called Cauchy's 

 appears often under very severe form. I will shortly return to this 

 capital point. 



IV 



Without restricting ourselves to the historic order, we will follow 

 the development of mathematical physics during the last century, 

 in so far as it interests analysis. 



The problems of calorific equilibrium lead to the equation already 

 encountered by Laplace in the study of attraction. Few equations 

 have been the object of so many works as this celebrated equation. 

 The conditions at the limits may be of divers forms. The simplest 

 case is that of the calorific equilibrium of a body of which we main- 

 tain the elements of the surface at given temperatures; from the 

 physical point of view, it may be regarded as evident that the tem- 

 perature, continuous within the interior since no source of heat is 

 there, is determined when it is given at the surface. 



A more general case is that where, the state remaining permanent, 

 there might be radiation toward the outside with an emissive power 

 varying on the surface in accordance with a given law; in particular 

 the temperature may be given on one portion, while there is radiation 

 on another portion. 



These questions, which are not yet resolved in their greatest gen- 

 erality, have greatly contributed to the orientation of the theory of 

 partial differential equations. They have called attention to types of 

 determinations of integrals, which would not have presented them- 

 selves in remaining at a point of view purely abstract. 



