DEVELOPMENT OF MATHEMATICAL ANALYSIS 507 



Laplace's equation had been met already in hydrodynamics and 

 in the study of attraction inversely as the square of the distance. 

 This latter theory has led to putting 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 notably generalized, such as Green's 

 formula. 



The fundamental problems of static electricity belong to the 

 same order of ideas, and that was surely a beautiful triumph for 

 theory, the discovery of the celebrated theorem on electric phe- 

 nomena in the interior of hollow conductors, which later Faraday 

 rediscovered experimentally, without having known of Green's 

 memoir. 



All this magnificent ensemble has remained the type of the theories 

 already old of mathematical physics, which seem to us almost to 

 have attained perfection, and which exercise still so happy an in- 

 fluence on the progress of pure analysis in suggesting to it the most 

 beautiful problems. The theory of functions offers us another mem- 

 orable affiliation. 



There the analytic transformations which come into play are not 

 distinct from those we have met in the permanent movement of 

 heat. Certain fundamental problems of the theory of functions of 

 a complex variable lost then their abstract enunciation to take a 

 physical form, such as that of the distribution of temperature on 

 a closed surface of any connection and not radiating, in calorific 

 equilibrium with two sources of heat which necessarily correspond 

 to flows equal and of contrary signs. Transposing, we face a ques- 

 tion relative to Abelian integrals of the third species in the theory of 

 algebraic curves. 



The examples which precede, where we have envisaged only the 

 equations of heat and of attraction, show that the influence of 

 physical theories has been exercised not only on the general nature 

 of the problems to be solved, but even in the details of the analytic 

 transformations. Thus is currently designated in recent memoirs on 

 partial differential equations under the name of Green's formula, 

 a formula inspired by the primitive formula of the English physicist. 

 The theory of dynamic electricity and that of magnetism, with 

 Ampere and Gauss, have been the origin of important progress; the 

 study of curvilinear integrals and that of the integrals of surfaces 

 have taken thence all their developments, and formulas, such as 

 that of Stokes which might also be called Ampere's formula, have 

 appeared for the first time in memoirs on physics. The equations 

 of the propagation of electricity, to which are attached the names of 

 Ohm and Kirchoff, while presenting a great analogy with those of 

 heat, offer often conditions at the limits a little different; we know 



