of continuity. We are, therefore, no longer content with the prob- 

 abilities offered by the reasoning long classic. 



Whether we proceed indirectly or whether we seek to give a rigor- 

 ous proof of the existence of a function corresponding to the mini- 

 mum, the route is long and arduous. 



Further, not the less will it be always useful to connect a ques- 

 tion of mechanics or of mathematical physics with a problem of 

 minimum; in this first of all is a source of fecund analytic trans- 

 formations, and besides in the very calculations of the investigation 

 of variations useful indications may appear, relative to the condi- 

 tions at the limits; a beautiful example of it was given by Kirchoff 

 in the delicate investigation of the conditions at the limits of the 

 equilibrium 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 sciences whose history, as we have seen, 

 has been so narrowly bound to that of analysis. 



When the hope of integrating with simple functions was lost, one 

 strove to find developments permitting to follow a phenomenon as long 

 as possible, or at least to obtain information of its qualitative bearing. 



For practice, the methods of approximation form an extremely 

 important part of mathematics, and it is thus that the highest parts 

 of theoretic arithmetic find themselves connected with the. applied 

 sciences. As to series, the demonstrations themselves of the exist- 

 ence 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 pro- 

 blem 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 contin- 

 gency need not be feared and we should obtain developments valid 

 indefinitely; unhappily, as Fresnel said one day to Laplace, nature 

 is not concerned about analytic 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, we can generally decide, given a 

 system of differential equations of the first order, whether the gen- 



