8t>4 



SCIENCE. 



[ISr. S. Vol. XX. No. 521. 



grange did not set himself the question of 

 convergence. 



In twenty or thirty years the exigencies 

 in the rigor of proofs had grown. One 

 Imew that the two preceding modes of dem- 

 onstration are susceptible of all the pre- 

 cision necessary. 



For the first, there was need of no new 

 principle; for the second it was necessary 

 that the theory should develop in a new 

 ■way. Up to this point, the functions and 

 the variables had remained real. The con- 

 sideration of complex variables comes to 

 extend the field of analysis. The func- 

 tions of a complex variable with iinique 

 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 evi- 

 dence in limiting oneself 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 

 vconvenience in calculation. 



The general theorems of the theory of 

 analytic functions permitted to reply with 

 precision to questions remaining up to that 

 time undecided, such as the degree of gen- 

 erality of the integrals of differential equa- 

 tions. It became possible to push even to 

 the end the demonstration sketched by La- 

 grange 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 mathematical 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 his- 

 toric 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 maintain the elements of the surface at 

 given temperatures; from the physical 

 point of view, it may be regarded as evi- 

 dent that the temperature, 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 emis- 

 sive power varying on the surface in ac- 

 cordance with a given law; in particular 

 the temperature may be given on one por- 

 tion, while there is radiation on another 

 portion. 



These questions, which are not yet re- 

 solved in their greatest generality, have 

 greatly contributed to the orientation of 

 the theory of partial differential equations. 

 They have called attention to types of de- 

 terminatidns of integrals, which would not 

 have presented themselves in remaining at 

 a point of view. purely abstract. 



Laplace's equation had been met already 

 in hydrodynamics and in the study of at- 

 traction inversely as the square of the dis- 

 tance. This latter theory has led to put- 

 ting in evidence the most essential elements 

 such as the potentials of simple strata and 

 of double strata. Analytic combinations 

 of the highest importance were there met, 

 which since have been tiotably generalized, 

 such as Green's formula. 



