1876.] Evaluation of a class of Definite Integrals. 9- 



(althougli a^""^ is not a function of x — oT^), because «^"~^— a;^""^... 

 — a;"^""^^ is a function of ic — x'^, and all the terms except the first 

 cancel one another by the integration. 



Similar remarks of course apply to Cauchy's formula (1). 



§ 5. The formula (2) can be deduced from Boole's general 

 theorem 



/> (-)/(- - i) '^ -fj'f (") ® [-^ (^)] ^^3V« - 



{Phil. Trans. 1857, p. 782) in exactly the same way as Boole him- 

 self obtained Cauchy's formula (1)\ 



In Boole's formulae the quantity- subject to the functional sign 

 is a; — ax~^ {a positive), and it is worth noticing that (1) and (2) 

 can readily be so transformed as to assume the more general form. 



Take, for example, (2), replace ^ {x) by ^ [ax), and transform 

 the left-hand integral by taking x = x' : b ; thus 



= n\ x<f) (ax) dx + ^^ -^ ' I o^^ {ax) (Za; + &c. 



= -J x<^{x)dx^-- '■^. 'A x^(j){x)dx + &c. 



cf/ J Q a , o I J Q 



Taking b = a, and replacing a^ by a, we have 

 |"'^-V(«,- 2) da: = m-f . + '■" + '^l"" - ^\ '-'P, + &c.; 

 while (1) treated in the same way gives 



fx'y(x-^dx = a"A, + ^^'l^''a'^-'A, + &c., ■' 

 which is Boole's formula (loc. cit. p. 783). 



§ 6. As an example of (2) I give the evaluation of the integral 



f 



'I n 



x'" ^ cosech [x \dx 



\ xj 



(where cosech m denotes the hyperbolic cosecant of u). 



^ T may here note an erratum in Caiichy's third corollary (loc. cit. in § 1, 2'- 56), 

 viz. in (17) and the left-hand side of (18) cos tx should be cos t (,r - x"'). 



