INVERSE METHOD OF DEFINITE INTEGRALS. 363 



and by comparing the corresponding terms 



- ^ + 1 ^_ (« + l)(w + 2) _ {n + l)(w + 2)(w + 3) . 



therefore, 

 rr-7--L'' + ^ T , (w +!)(« + 2) . (w + 1)(w + 2)(m + 3) 



Ly„ — J „ -I J . ^ „+i + r — . -I „+2 i r — - — . X „+3, Cue. 



31. To express the function Un which is reciprocal to (h. I. t)" in 

 a finite form, and also the function which Un generates. 



l.2.3...nU„ = 1.2.3...nT„+2.3. 4...(w + l) T„+,+3 . 4 . 5...(w + 2) . T,.+,+&cc. 



=^ { r„ + r,A + 7;a^+ ... T^h'+T^^.h"^^ + &,c.], 



h being put equal to unity after the differentiation. 

 But by Section v, we have 



h 



J— ^ = T,+ T,h + T^h' + &c. ad inf. ; 



"■■(A) 



therefore, U„ = - — — — -jt when A = 1. 



1.2. 3...ndh" 



Now by Taylor's Theorem, this quantity is the coefficient of k" in 



the expansion of - — j~r^ the latter is therefore the function which 

 U„ generates. 



32. Properties of Un- 



.1. jiUn (h. 1. ty = f,T„ (h. 1. ^)" = 1 . 2 . 3...W, by Sect. v. 



II. Changing the sign of k in the quantity which U„ generates, 

 we get 



Vol. V. Paet III. sB 



