520 LAPLACE. 



although he assumes the truth of this on his page 134). It may 

 be shewn by adopting the usual mode of proving Stirling's Theo- 

 rem. For by using Euler's theorem for summation, given in 

 Art. 334, it will appear that 



where t (^) = 9^ " ^l73 + ' 



26^ 3.4/ ' 5.Qs' ' 



the coefficients being the well-known numhers of Bernoidli. 



Thus 'i/r(5)+i/r(-5)=0; 



therefore e^^'^ x e^^"^ = e' ^1, 



that is </) (5) <^ (— 5) = 1. 



964. Laplace, after investigating a formula sometimes de- 

 duces another from it by passing from real to imaginary quantities. 

 This method cannot be considered demonstrative ; and indeed 

 Laplace himself admits that it may be employed to discover new 

 formulae, but that the results thus obtained should be confirmed 

 by direct demonstration. See his pages 87 and 471; also Art. 920. 



Thus as a specimen of his results we may quote one which he 

 gives on his page 134. 



Let Q = cos CT 



(^^ + ^r 





then 



/<00 



('00 • 



J 



A memoir by Cauchy on Definite Integrals is published in the 

 Journal de VEcole Folytechnique, 28^ Cahier ; this memoir was 

 presented to the Academy of Sciences, Jan. 2nd, 1815, but not 

 printed until 1841. The memoir discusses very fully the results 

 given by Laplace in the Chapter we are now considering. Cauchy 

 says, page 148, 



...je suis parvenu a qiielques resultats nouveaux, ainsi qii'a la 

 demonstration directe de plusieurs formules, que M. Laplace a deduites 



