868 



SCIENCE. 



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



sides in the very ealcixlations of the investi- 

 gation of variations useful indications may 

 appear, relative to the conditions at the 

 limits ; a beautiful example of it was given 

 by Kirchoff in the delicate investigation 

 of the conditions at the limits of the equi- 

 librium of flexure of plates. 



VI. 



I have been led to expand particularly 

 on partial differential equations. 



Examples chosen in rational mechanics 

 and in celestial mechanics would readily 

 show the role which ordinary differential 

 equations play in the progress of these sci- 

 ences whose history, as we have seen, has 

 been so narrowly bound to that of analysis. 



When the hope of integrating with sim- 

 ple functions was lost, one strove to find 

 developments permitting to follow a phe- 

 nomenon as long as possible, or at least to 

 obtain information of its qualitative bear- 

 ing. 



For practise, the methods of approxima- 

 tion form an extremely important part of 

 mathematics, and it is thus that the highest 

 parts of theoretic arithmetic find them- 

 selves connected with the applied sciences. 

 As to series, the demonstrations themselves 

 of the existence of integrals furnish them 

 from the very first; thus Cauchy's first 

 method gives developments convergent as 

 long as the integrals and the differential 

 coefficients remain continuous. 



When any circumstance permits our 

 foreseeing that such is always the case, we 

 obtain developments always convergent. 

 In the problem of n bodies, we can in this 

 way obtain developments valid so long as 

 there are no shocks. 



If the bodies, instead of attracting, repel 

 each other, this contingency need not be 

 feared and we would obtain developments 

 valid indefinitely; unhappily, as Fresnel 

 said one day to Laplace, nature is not con- 

 cerned about analvtic difficulties and the 



celestial bodies attract instead of repelling 

 each other. 



One would even be tempted at times to 

 go further than the great physicist and say 

 that nature has sown difficulties in the 

 paths of the analysts. 



Thus to take another example, Ave can 

 generally decide, given a system of dif- 

 ferential equations of the first order, 

 whether the general solution is stable or 

 not about a point, and to find developments 

 in series valid for stable solutions it is 

 only necessary that certain inequalities be 

 verified. 



But if we apply these results to the equa- 

 tions of dynamics to discuss stability, we 

 find ourselves exactly in the particular case 

 which is unfavorable. Nay, in general here 

 it is not possible to decide on the stability ; 

 in the case of a function of forces having' 

 a maximum, reasoning classic, but indirect, 

 establishes the stability which can not be 

 deduced from any development valid for 

 every value of the time. 



Do not lament these difficulties ; they will 

 be the source of future progress. 



Such are also the difficulties which still 

 present to us, in spite of so many works, 

 the equations of celestial mechanics; the 

 astronomers have almost drawn from them, 

 since Newton, by means of series prac- 

 tically convergent and approximations 

 happily conducted all that is necessary for 

 the foretelling of the motions of the heav- 

 enly bodies. 



The analysts would ask more, but they no 

 longer hope to attain the integration by 

 means of simple functions or developments 

 always convergent. 



What admirable recent researches have 

 best taught them is the immense difficulty 

 of the problem; a new way has, however, 

 been opened by the study of particular 

 solutions, such as the periodic solutions 

 and the asymptotic solutions which have 

 already been utilized. It is not perhaps 



