730 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
87. The third of these possesses a remarkable property: write mf instead of f, and 
at the same time change t into m 2 t, the integral becomes 
f( i + q ) f s+ 1 £ t~ g -*\m 2 (t-\-f 2 )—z 2 \~ q+i (t+f 2 )~ is+q - 1 dt—term. in g ; 
and hence writing or m= 1+jj and therefore m 2 == 1 + 2 |f, the value is 
- 2+1 
m-q)V(ls+g) f S+ J 0 *"*"*{ t+r-* 2 +jr( t+ f^\ in e. 
Hence the term in (f is 
=S/into expression /* £ 
where the factor which multiplies If is, as it should be, the ring-integral ; it in fact 
agrees with one of the expressions previously obtained for this integral. 
88. Similarly for an exterior point x>f', starting in like manner from, Disk-integral 
2«— 2 fs 
x ^rr/i n i „ i, 
= (X __ g ^ (x +/)'+*-* 1 ~ n(ia, is, ls+q, w) + n (-l 5 , Is, ls+g-1, u) 
and reducing in like manner, the term in \ } may be expressed in the four forms 
' ( *+f ) S+2q ~ 2 into 
r (is + ?)T(i-?) k s+2 *~ 2 
[-( 1 - J 2)n(i«+ff, i-q, 1 n(ls+g-l, f-g, -i+g,{J)], 
9I-. r+i+ ( x +/)- +2 ^- 2 . 
" m + Q)r( x 2 s-q) 
[— C 1 — 5) n (*+& * s -*' S)+i~ n(-i+g,is-g+i,is+g-i,Q] ; 
gi-# -.- r jsTj : (x+fY-w.-fy^ { 
[-n (i-q, h+q, is-q, £) + (l-g) i^- 1 n(*-g, is+q, \s-q, Q], 
2J + 1 
91- Mr into 
rc^“g)r(*+g) U / l J 
[-n(ls-?,i+ ? ,i-2,£W(l-f) n(| S -2+l, -1+2, 1-2,5)]- 
