The collected mathematical papers of Arthur Cayley.

607] A MEMOIR ON PREPOTENTIALS. 369i
where the II-functions on the right hand are respectively
=2+f1 S+2-1 (12-)-q-t dx 2q+l f
K ( Jo (2 -J (IC _f 2)2q 2 _
=s+ [1 ~S2- (1 - x)-q-1 dx S+2q f
- J o (K)2 =(f2x)t s +_ (2t + f2) -K J2)- (t +f 2) ---2 dt,
-+1 X —2 (1 - l)s+q-1 d1 s-2q f _
= S 2 Jo ( - f x+ (KC- _f22) — _J t -2 (t +f - -2)f2- (t +f 2)-+q dt,
l-2 +1 S —+l (1 ( - x)i-4 dx- K- r _
J (C2 rf 2x)-q+i (K2 - q f) 2). - f2+f
We have then
Rinog-integral = - f (i - ) Fs ] ) J t-+q (t +f2 - 2)+q-1 (t +f2)-q — dt
_( _ _ _ _ _ _ r ( 2 2(1 -- ( t + f 2 - K 22-n ( t + f 2 ) - ts - - dt
(K= (ft r+F (Kls r- q F (~2( 2) tq - (t +f2 -,2)q-1 (t +f2) — dtq
Observe that in II. and III. the integrals, except as to the limits, are the same
as in the corresponding formulae for the interior point.
If in the t-integrals we put t+2-f2 in place of t, and ultimately suppose K
indefinitely large in comparison with f, they severally become
r(t + 2t (t +f ( t)-q-si, t + f)- d( = ) d S + K2 )-2s
jo (t(t + K2+1 f t (K d
Ifa+^-f (t + K2 f2)rtt (t + K2)-^s ( dt =r_/ t+l ( s+ 2)
_ /~ n- r fI
*' o ( (t+/K~-^( r(^S+./)
J (t+K K -f2)-28-2t2s-8 (t + 2K2)P2, d dt =-f2- r2(q - t) K(f 22 )
*'~ v Jo (< -(+ r(f2+ ) Q 2 A
and they all four give
If in the tRing-interalt K+2f2 in place of t, and ultimately suppose K
which agrees with the value
(K +fy+2 n ~ [ 2>+ q (Kj+)2) = -8+2q 11 S, s + q, 0),
when K isindefinitely large.
C. ix. 47