510 ALGEBRA AND ANALYSIS 



These considerations sufficiently show the interest it may have 

 to be assured that all the integrals of a system of partial differential 

 equations continuous as well as all their derivatives up to a deter- 

 mined order in a certain field of real variables are analytic functions; 

 it is understood, we suppose, there are in the equations only analytic 

 elements. We have for linear equations precise theorems, all the 

 integrals being analytic, if the characteristics are imaginary, and 

 very general propositions have also been obtained in other cases. 



The conditions at the limits that one is led to assume are very 

 different according as it is question of an equation of which the 

 integrals are or are not analytic. A type of the first case is given 

 by the problem generalized by Dirichlet; conditions of continuity 

 there play an essential part, and, in general, the solution cannot 

 be prolonged from the two sides of the continuum which serves as 

 support to the data; it is no longer the same in the second case, 

 where the disposition of this support in relation to the characteris- 

 tics plays the principal role, and where the field of existence of the 

 solution presents itself under wholly different conditions. 



All these notions, difficult to make precise in ordinary language 

 and fundamental for mathematical physics, are not of less interest 

 for infinitesimal geometry. 



It will suffice to recall that all the surfaces of constant positive 

 curvature are analytic, while there exist surfaces of constant nega- 

 tive curvature not analytic. 



From antiquity has been felt the confused sentiment of a certain 

 economy in natural phenomena; one of the first precise examples 

 is furnished by Fermat's principle relative to the economy of time 

 in the transmission of light. 



Then we came to recognize that the general equations of mechanics 

 correspond to a problem of minimum, or more exactly of variation, 

 and thus we obtained the principle of virtual velocities, then Ham- 

 ilton's principle, and that of least action. A great number of problems 

 appeared then as corresponding to minima of certain definite in- 

 tegrals. 



This was a very important advance, because the existence of 

 a minimum could in many cases be regarded as evident, and con- 

 sequently the demonstration of the existence of the solution was 

 effected. 



This reasoning has rendered immense services; the greatest geo- 

 meters, Gauss in the problem of the distribution of an attracting 

 mass corresponding to a given potential, Riemann in his theory of 

 Abelian functions, have been satisfied with it. To-day our attention 

 has been called to the dangers of this sort of demonstration; it is 

 possible for the minima to be simply limits and not to be actually 

 attained by veritable functions possessing the necessary properties 



