508 ALGEBRA AND ANALYSIS 



all that telegraphy by cables owes to the profound discussion of a 

 Fourier's equation carried over into electricity. 



The equations long ago written of hydrodynamics, the equations 

 of the theory of electricity, those of Maxwell and of Hertz in electro- 

 magnetism, have offered problems analogous to those recalled above, 

 but under conditions still more varied. Many unsurmounted diffi- 

 culties are there met with; but how many beautiful results we owe 

 to the study of particular cases, whose number one would wish to 

 see increase. To be noted also as interesting at once to analysis and 

 physics are the profound differences which the propagation may 

 present according to the phenomena studied. With equations such 

 as those of sound, we have propagation by waves; with the equa- 

 tion of heat, each variation is felt instantly at every distance, but 

 very little at a very great distance, and we cannot then speak of 

 velocity of propagation. 



In other cases of which Kirchoff 's equation relative to the propa- 

 gation of electricity with induction and capacity offers the simplest 

 type, there is a wave front with a velocity determined but with a 

 remainder behind which does not vanish. 



These diverse circumstances reveal very different properties of 

 integrals; their study has been delved into only in a few particular 

 cases, and it raises questions into which enter the most profound 

 notions of modern analysis. 



I will enter into certain analytic details especially interesting for 

 mathematical physics. 



The question of the generality of the solution of a partial differential 

 equation has presented some apparent paradoxes. For the same 

 equation, the number of arbitrary functions figuring in the general 

 integral was not always the same, following the form of the integral 

 envisaged. Thus Fourier, studying the equation of heat in an indefin- 

 ite medium, considers as evident that a solution will be determined 

 if its value for ^=0 is given, that is to say one arbitrary function of 

 the three coordinates x, y, z ; from the point of view of Cauchy , we 

 may consider, on the contrary, that in the general solution there are 

 two arbitrary functions of the three variables. In reality, the ques- 

 tion, as it has long been stated, has not a precise signification. 



In the first place, when it is a question only of analytic functions, 

 any finite number of functions of any number of independent vari- 

 ables presents, from the arithmetical point of view, no greater gen- 

 erality than a single function of a single variable, since in the one 

 case and in the other the ensemble of coefficients of the development 

 forms an enumerable series. But there is something more. In reality, 

 beyond the conditions which are translated by given functions, an 



