Dkcember 23, 1904.] 



SCIENCE. 



867 



on the potentials of velocity in a perfect 

 fluid hold good in a viscid fluid 1 Examples 

 have been given where the coordinates of 

 different points of a viscous fluid starting 

 from rest are not expressible as analytic 

 functions of the time starting from the 

 initial instant of the motion, and where the 

 nul rotations as well as all their derivatives 

 with respect to the time at this instant are, 

 however, not identically nul; Lagrange's 

 theorem, therefore, does not hold true. 



These considerations sufficiently show the 

 interest it may have to be assured that all 

 the integrals of a system of partial differ- 

 ential equations continuous as well as all 

 their derivatives up to a determined 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 prob- 

 lem generalized by Dirichlet; conditions of 

 continuity there play an essential part, and, 

 in general, the solution can not be pro- 

 longed 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 characteristics plays the 

 principal role, and where the field of ex- 

 istence of the sohition presents itself under 

 wholly different conditions. 



All these notions, difiicult to make pre- 

 cise in ordinary language and fundamental 

 for mathematical physics, are not of less 

 interest for infinitesimal geometry. 



It will suffice to recall that all the sur- 

 faces of constant positive curvatiire are 



analytic, while there exist surfaces of con- 

 stant negative curvature not analytic. 



From antiquity has been felt the con- 

 fused sentiment of a certain economy in 

 natural phenomena; one of the first precise 

 examples is furnished by Fermat's prin- 

 ciple relative to the economy of time in the 

 transmission of light. 



Then we came to recognize that the gen- 

 eral equations of mechanics correspond to 

 a problem of minimum, or more exactly of 

 variation, and thus we obtained the prin- 

 ciple of virtual velocities, then Hamilton's 

 principle, and that of least action. A great 

 number of problems appeared then as corre- 

 sponding to minima of certain definite 

 integrals. 



This was a very important advance, be- 

 cause the existence of a minimum could in 

 many cases be regarded as evident, and con- 

 sequently the demonstration of the ex- 

 istence of the solution was effected. 



This reasoning has rendered immense ser- 

 vices ; the greatest geometers. Gauss in the 

 problem of the distribution of an attracting 

 mass corresponding to' a given potential, 

 Rieniann 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 of con- 

 tinuity. We are, therefore, no longer con- 

 tent with the probabilities offered by the 

 reasoning long classic. 



Whether we proceed indirectly or 

 whether we seek to give a rigorous proof 

 of the existence of a function corresponding 

 to the minimum, the route is long and 

 arduous. 



Further, not the less will it be always 

 useful to connect a question of mechanics 

 or of mathematical physics with a problem 

 of minimum ; in this first of all is a source 

 of fecund analytic transformations, and be- 



