LAPLACE. 513 



then J = J = ^, + 2i?,^ + 353^+...; 



[ydx = YJe-^' {B^ + 2B,t + SBf + ...) dt. 



Such is the method of Laplace. It will be practically advan- 

 tageous in the cases where B^, B^, B^, ... form a rapidly converging 

 series; and it is to such cases that we shall have to apply it, when 

 we give some examples of it from Laplace's next Chapter. In 

 these examples there will be no difficulty in calculating the terms 

 B^, B^, B^, ..., so far as we shall require them. An investigation of 

 the general values of these coefficients as far as B^ inclusive will be 

 found in De Morgan's Differential and hitegral Calculus, page 602. 



If we suppose that the limits of x are such as to make the cor- 

 responding values of y zero, the limits of t will be — co and + oo . 



Now if r be odd I e-^'fdt vanishes, and if r be even it is equal to 



^ —00 



(^_1)(^_3)... 3.1 

 Thus we have 



i^ 



3 5 3 



ydx= Ys/ir\B^-]r-^B^-^-i^ B^-^.,.\. 



Besides the transformation y = Ye~^^ Laplace also takes cases 

 in which the exponent of e instead of being — f has other values. 

 Thus on his page 88 the exponent is — t, and on his page 93 

 it is — f'' ; in the first of these cases Y is not supposed to be a 

 maximum value of y. 



958. Some definite integrals are given on pages 95 — 101, in 

 connexion with which it may be useful to supply a few references. 



The formula marked {T) on page 95 occurs in Laplace's memoir 

 of 1782, page 17. 



COS rx e** ^ dx = -^r- e ia" . 



Jo 2« 



this was given by Laplace in the Memoires...de llnstitut for 

 1810, page 290 ; see also Tables dlntegrales Definies, 1858, by 



D. Bierens de Haan, page 376. 



S3 



