514 LAPLACE. 



/, 



dx-=^ ^ 



^ ^ 



see D. Bierens de Haan, page 268. 



where a is supposed positive ; these seem due to Laplace ; see 

 D. Bierens de Haan, page 282, TMorie...desProh., pages 99—134. 

 We may remark that these two results, together with 



/■ 



J 



sin ax dx it ,^ „•. 



1 -i- X X Z 



are referred by D. F. Gregory, in his Examples ofthe.,.Diffe7'ential 

 and Integral Calcidus, to Laplace's memoir of 1782 ; but they are 

 not explicitly given there : with respect to the last result see 

 D. Bierens de Haan, page 293. 



959. Since the integral le~^' dt occurs in the expressions of 



Art. 957, Laplace is led to make some observations on modes of 

 approximating to the value of this integral. He gives the follow- 

 ing series which present no difficulty : 



/, 



t' 1 t' 1 t' 

 e-'V< = T-3+^g-^^ + ...; 



j;>.,.„,-..(..K.jiS!.,5^*...)> 



In the memoir of 1782 the second of these three expressions 

 does not occur. 



Laplace also gives a development of I e'^"^ dt into the form of 



J T 



a continued fraction, which he takes from his MScanique Celeste, 

 Livy^e x. See also De Morgan's Differential and Integral Calculus, 

 page 591, for this and some similar developments. 



