i 
350 PROFESSOR A. R. FORSYTH ON 
according as s< or =p — 1, the summation being extended over the p roots of 
-%(x) = 0; and therefore 
• r Z p a/- 1 
Thus the left-hand side of (A) 
r=p 
— V 
= 1 
all the terms disappearing except the first and so verifying (A) and proving that 
x 2 , x. 2 , . . . , x p are the roots of (11). 
15. Taking now our set of integrals u we have 
_ } Pfy A ) bd\ 
{j ~ fy=i fyx — Oi)y/ H(®*) blip 
0 _fy| p Pfo) 
A=i (z K — a^y/R(x .0 buy, 
\ = p 
l = is 
A 
PfeO 
2 =i fyx—« ( x)v / li(« A ) 
( 12 ). 
A=p 
A=1 
PQA) 
(*a— a p )v/B(ajA) bu IM 
/ Cll ^ J, CtJ, ^ 
Multiply these respectively by ... and add; then, in virtue of 
equation (11), 
_w!o_ p (fy) a,- a 
P'(« M ) \=i fyihfy) ^ 
so that if we write 
we have 
Q _ fyfy pA. P (x)dx 
~ K- I J \/ B(*) 
(13) 
Ipul^ £>U 
P'(«A 
(14). 
U is obviously a symmetric function of the x’s, and is therefore a function of the ms. 
