DEVELOPMENT OF MATHEMATICAL ANALYSIS 509 



integral is subjected to conditions of continuity, or is to become in- 

 finite in a determined manner for certain elements; one may so be 

 led to regard as equivalent to an arbitrary function the condition 

 of continuity in a given space, and then we clearly see how badly 

 formulated is the question of giving the number of the arbitrary 

 functions. It is at times a delicate matter to demonstrate that con- 

 ditions determine in a unique manner a solution, when we do not 

 wish to be contented with probabilities; it is then necessary to make 

 precise the manner in which the function and certain of. its deriva- 

 tives conduct themselves. 



Thus in Fourier's problem relative to an indefinite medium cer- 

 tain hypotheses must be made about the function and its first 

 derivatives at infinity, if we wish to establish that the solution is 

 unique. 



Formulas analogous to Green's render great services, but the 

 demonstrations one deduces from them are not always entirely 

 rigorous, implicitly supposing fulfilled for the limits conditions 

 which are not, a priori at least, necessary. This is, after so many 

 others, a new example of the evolution of exigencies in the rigor of 

 proofs. 



We remark, moreover, that the new study, rendered necessary, 

 has often led to a better account of the nature of integrals. 



True rigor is fecund, thus distinguishing itself from another purely 

 formal and tedious, which spreads a shadow over the problems it 

 touches. 



The difficulties in the demonstration of the unity of a solution 

 may be very different according as it is question of equations of 

 which all the integrals are or are not analytic. This is an important 

 point, and shows that even when we wish to put them aside, it is 

 necessary sometimes to consider non-analytic functions. 



Thus we cannot affirm that Cauchy's problem determines in a 

 unique manner one solution, the data of the problem being general, 

 that is to say not being characteristic. 



This is surely the case, if we envisage only analytic integrals, 

 but with non-analytic integrals there may be contacts of order 

 infinite. And theory here does not outstrip applications; on the 

 contrary, as the following example shows: 



Does the celebrated theorem of Lagrange on the potentials of 

 velocity in a perfect fluid hold good in a viscid fluid? 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. 



