512 ALGEBRA AND ANALYSIS 



eral 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 equations of dynamics to dis- 

 cuss 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 cannot 

 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 astro- 

 nomers have almost drawn from them, since Newton, by means of 

 series practically convergent and approximations happily con- 

 ducted, all that is necessary for the foretelling of the motions of the 

 heavenly bodies. 



The analysts would ask more, but they no longer hope to attain 

 the integration by means of simple functions or developments al- 

 ways 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 so much because of the needs of practice 

 as in order not to avow itself vanquished, that analysis will never 

 resign itself to abandon, without a decisive victory, a subject where 

 it has met so many brilliant triumphs; and again, what more beau- 

 tiful field could the theories new-born or rejuvenated of the modern 

 doctrine of functions find, to essay their forces, than this classic 

 problem of n bodies? 



It is a joy for the analyst to encounter in applications equations 

 that he can integrate with known functions, with transcendents 

 already classed. 



Such encounters are unhapily rare; the problem of the pendulum, 

 the classic cases of the motion of a solid body around a fixed point, 

 are examples where the elliptic functions have permitted us to effect 

 the integration. 



It would also be extremely interesting to encounter a question 

 of mechanics which might be the origin of the discovery of a new 

 transcendent possessing some remarkable property; I should be 

 embarrassed to give an example of it unless in carrying back to the 

 pendulum the de"but of the theory of elliptic functions. 



The interpenetration between theory and applications is here 

 much less than in the questions of mathematical physics. Thus 



