866 



SCIENCE. 



[N. S. Vol. XX. No. 521. 



into which enter the most profound notions 

 of modern analysis. 



V. 



I will enter into certain analytic details 

 especially interesting for mathematical 

 physics. 



The question of the generality of the 

 solution of a partial differentia] equation 

 has presented some apparent paradoxes. 

 For the same equation, the nimiber of arbi- 

 trary 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 indefinite medium, considers as evident 

 that a solution will be determined if its 

 value for < = 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 question, 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 num- 

 ber of functions of any number of inde- 

 pendent variables presents, from the arith- 

 metical point of view, no greater generality 

 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 integral is subjected to condi- 

 tions of continuity, or is to become infinite 

 in a determined manner for certain ele- 

 ments; 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 formu- 

 lated is the question of giving the number 

 of the arbitrary functions. It is at times 

 a delicate matter to demonstrate that condi- 



tions determine in a unique manner a solu- 

 tion, when we do not wish to be contented 

 with probabilities; it is then necessary to 

 make precise the manner in which the func- 

 tion and certain of its derivatives conduct 

 themselves. 



Thus in Fourier's problem relative to an 

 indefinite medium certain hypotheses must 

 be made about the function and its first 

 derivatives at infinity, if we wish to estab- 

 lish 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 differ- 

 ent 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 can not 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 character- 

 istic. 



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 



