in Definite Integrals. 55 



and 



r°° cos ax 



Jo 



1 + cT 2 



IH+o- -)-r~- • • w 



Similarly, by taking #„== v - , we obtain the correct value of 



f 



00 ^ sin am? _ 



The peculiarity of (2) and (3) consists in the appearance on 

 the right-hand side of terms with fractional arguments. In such 

 an equation as (4), where one side is a function of « 2 , while the 

 other involves uneven powers of a } it seems as though it would 

 be impossible to evaluate the integral by any direct procedure ; 

 for a priori it would appear that no method of expansion and 

 integration term by term could transform a function of a 2 

 into one of a, and thus, as it were, extract the square root 

 of a constant involved. The way in which the symbolic pro- 

 cess introduces ^E, and so actually does effect this conver- 

 sion, is interesting : when I first applied the identity (1) to 

 the integral in (4), I scarcely expected to obtain any result 

 capable of interpretation. 



Whenever (2) and (3) admit of interpretation, it is highly pro- 

 bable that the result so given will be the true one ; e. g., taking 



"* = i>TTr wefind 



J i+^-2tr(i) r(|) + r(2) r(i) + "*/ 



_tt r 2a%_ _ 2af a 2 "1 



~2\ \A +fl 1.3Vtt + 1.2 ~'l 



the known value. But (2) and (3), as general formulae, are re- 

 markable ; and they would give results in very many cases where 

 it might not be easy to evaluate the integrals otherwise. ] 



Trinity College, Cambridge, 

 June 19, 1874. 



