356 PROFESSOR A. R. FORSYTH OH 
The first term inside the bracket is the expansion in partial fractions of 
P(«,) 
Q(0 
and is therefore zero since s=p ; the second is 
r ~£ +1 f </>K+d 1 1 
r— 1 lQ'K+0 —«p+rj 
_ ) __ — i definition 
Q(0 
so that the equation with which we began leads to 
that is, to equation (14).] 
19. Now let 
bV_ 
ls a J-2 
bu s 
P '(a,) * 
k=r 
A=l- 
|* A 1 
A=p 
W=iS 
A = P 
rPk ^§,dx f 
a K \/ P(A J 
(13') 
so that V stands to the As and W to the w’s in exactly the same relation as U to 
the ids. 
Applying now the theorem in § 6 we have 
u+v+w=-epilog 
/ My + NA 
\M?/—N?/ 
My +S \My) + 
The n th term in this series gives 
= —Ob 
1 n /N\ 2 ""Y z \ 3 ““ 3 
2n-l L l\Mj \y). 
- Yu c ±{vY+ hi s her p° wers of ;} 
so that nothing is contributed except by the first term, and we have 
U+Y+W=-N 1 . 
(22). 
The 2 p quantities N x , No, ..., N p , M x , . . . , M p are determined by the equation 
N 1 x p ~ 1 s+No.t p_ ^+ . . . +N p zd-M 1 x p " 1 y+ . . . + M p y=—x p y 
