612 Mr. Savidge on the Integration of a Class of 



(9), (17), which hold good for this solution. The result may 

 be expressed generally as follows : 



g =o L n(s) n(a+?2; j 



* I |2n [grc + l "*" | 2n + 2 ••"/* * ■ <^U 

 where 



m = 1 — • . ax, and 



Jo 1 ~ {e 



o / , , o\ 2w + 4 # (a— ?i)(a — w — 1) 

 2m 2 =(n+a + 2)m l - 2^2 • |2 ' 



a •/ , , on 2?i + 6 (a-w)(a-?i — l)(a—w — 2) . 

 3??i 3 =(n + a + 3)m 2 ~^-— g. v ^ ^-^ &c. 



When (a — n) is a positive integer, there are a finite 

 number of these coefficients, the rest vanishing ; they may be 

 expressed as follows : 



s—a—n 1 



s— a— » o 1 



V s=2 |l(2?Z+5)(2?Z + 5+l) 



■-a—n 



m 1 =(n + a+l)'(n+a+2) 2 , 9 , 9 - ^ i )( ? ^i , , 9V 

 v /v y s =3 ]2(2w + s)(2n + s + l)(2n + s + 2) 



and so on. 



(As before, it is to be understood that when the upper limit sum- 

 mation is less than the lower there no terms under the sign.) 

 If n + a is —1 or less, the above constants are infinite ; 

 in this case, and generally when a is less than — J-, the 

 following solution, which may be obtained as before, or 

 immediately from the above by (19), is suitable : 



s=o L U(s)IL(n — a — 1) J 



\2n \ 2n+l \ 2n + 2 '" V } 



where the constants jo , Pi are obtained from m , m 1 



by substituting — (a + 1) for a in the expressions determining 

 the latter coefficients. 



