516 LAPLACE. 



Then we sliall find that 



n //DiT A72N 1 /{d'^Yd'Y fd'Y\\ 



And the limits of t and t' will be — (X) and + co ; thus finally 

 we have approximately 



I \y dx dx — 



/{d'Y d'Y 

 y \dd' da!' 



2irY^ 



( d'Y 



jda da I 

 See Art. 907. 



961. The second Chapter of the second part of Livre l. is 

 entitled De ^integration par approximation, des Equations lineaires 

 aux differences finies et infiniment petites : this Chapter occupies 

 pages 110 — 125. 



This Chapter exemplifies the process of solving linear differential 

 equations by the aid of definite integrals. Laplace seems to be 

 the first who drew attention to this subject : it is now fully dis- 

 cussed in works on differential equations. See Boole's Differential 

 Equations. 



962. The third Chapter of the second part of Livre L is 

 entitled App)lication des m^thodes prdc^dentes, a V ap>proximation 

 de diverses fonctions de tres-grands nonibi^es: this Chapter oc- 

 cupies pages 126 — 177. 



The first example is the following. Suppose we have to in- 

 tegrate the equation in Finite Differences, 



Assume y^— ix'ipdx, where ^ is a function of x at present 



undetermined, and the limits of the integration are also unde- 

 termined. 



Let By stand for x" ; then -~^ = sx^~'^. Hence the proposed 



equation becomes 



