728 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
that is 
=(1— a? a_1 (l— x.l — vxy~ l dx— 2 J vx.X°-~ l {\ - x.l—vxf hJx, 
=2^ # a-1 (l — a+ -1 (l — vxfdx. 
«-l 
(i-v)U{u, 3, 1 -/ 3 , v)+-j-n{*-l, 3 + 1 , - 3 , «)= 2 n(a, 3, - 3 , «). 
We have therefore 
(i-p)n (is+q, is, |- 2 , -i+ 2> |) 
=2Tl++ 2 , i- 2 , — i+ 2 , ; 
and from the same equation written in the form 
n(«— 1, 3+1, —3, v)+~j(l—v)n(u, 3, 1-3, v)= 2 ^ r - l n(a, 3, —3, v), 
we obtain 
n^s -g, i+fi', i— S 'j j2^+T^r+(l— 2+ 1, — t+ 2> f— 2>p) 
= 2( r g l H g ? )n (l g -g+ 1 ^ -i+^ i-?» /*)• 
84. Hence the terms in [ ] are 
_ 2 =-> r*sri (/+*)* +s *- 2 i_„ _ ij _„ *!\ 
~ r(i«+g)r(i- ff ) ' / s+2 *- 2 ’ n \ 2 + ^ 2 ^ 2+?, /7’ 
-r(is-q+mi+g) 7 ^ A V q+ ’ 2+ ^’ 2 q, fV‘ 
respectively, and the corresponding values of the disk-integral are 
Y{§-q)T(\s+q)f l 2!? -n(2S+£, 2 <b .»+&/*) 
-r+ri ff-^y- 29 tWi , 1 , 1 * a \ e'- 2? r+rf 
r(i« ,g+i)r(T+ff) W ) • n V s ~ 2+1 ’ “ 2+ ^’ 2_?? ’/y ~i-< z^FH)’ 
which we may again verify by writing therein «=0, viz. the Tl-functions thus become 
r(j*+g).r(j-y) d r(+- g +i)r(-i+ g ) 
r(++i) r (*.+*) 
and consequently the integral is 
— _1 r ^ r 2 / g l-2 ? \ 
s 1 - 2 ? r$«r£ 
■wr(*.+tf 
< 2 \ g i-2 ? m«r-i 
