1883.] certain Definite Integrals. 471 



suppose that f(z) can be expanded in terms of z by a converging 

 series in other cases. The series TJ 1 + U 2 z + . . . wonld, I believe, be 

 convergent in many cases, where 6' is large when compared with 

 0, 0', . . . ; bnt the subject requires further consideration. 



If we put x=0, z=~. 



a 



Hence we immediately deduce from the equation 



x=V +V 1 z+V 2 ^+,.. .... (235) 



the theorem U 6+ U^+^ +r J^ * + ... =0. 



Since \ \ dxx n ~ l = , 



JoJ n(n + Y) 



JoJ UT ^ n(n + l)(n + 2)' 



JoI^ (1 ~^ >K " 1= n(^ + lKi'2 3 )(^ + 3)' 

 and, therefore, as before, 



J 1 ^dxx n -^{l— x) 



= ^P_ + ^uJL? + A 2 • 1 ; 2 -Jj + ... . (236) 



n(n + l) n(*+l)C«+2) n(n+l)(n+%)(n+3) K 



with precisely similar formulas for j j*j*cZ^ re " 1 0(l— ;c) and other like 



integrals. 



We also have 



Jo </l—a? 



= (^ + ^1- + hdd +,&c.U 1 ^- . (237), 

 I ° T 2w + 2 (2»+2)(2rc+4) ' J WE-? v 



It is easily seen that these investigations extend to a great multi- 



2 i 2 



