PROFESSOR CAYLEY ON PREPOTENTIALS. 
765 
j{Ap + Bi) 2 .. .+C? 2 + EW*— l;-' s ’ 
where the s+2 coefficients A, B . . . C, E, L are given by means of the identity 
-(fl+A)(*+B) . . . (3+C)(5+E)(3+L) 
=Disct.|(#XX, Y . . . Z, W, T) 2 + 0(X 2 +Y 2 . . . + Z 2 +W 2 -T 2 )} ; 
viz. equating the discriminant to zero, we have an equation in 0, the roots whereof are 
—A, — B . . . — C, — E, — L. 
The integral is 
j*T 
which is of the form 
(A-L)£ 2 + (B-L>, 2 . .. +(C— L)£ 2 + (E — L> 2 p s ’ 
dt 
' f. ^ , 
J {a^ + brf . . . +c ? 2 + ew 2 } 2 
where I provisionally assume that a,b ... c,e are all positive. 
145. To transform this, in place of the s-fl variables f, q . . . a connected by 
£ 2 +? 7 2 . . . +£ 2 4-‘*> 2 =l, we introduce the s+1 variables x,y . . ,z,w such that 
9 9 
t== KWc. 
where 
and consequently 
g 2 =a| 2 +5jj 2 . . . +c% 2 -\-ew 2 , 
X 2 +y 2 ... -\-Z 2 + W 2 = l. 
Hence writing d$ to denote the spherical element corresponding to the point 
(x,y...z, w), we have by a former formula 
^S= — M l \/7j .. .$\ / c,u s/e) d6 d ^ d , 
f +1 
( ab .. . ce ■)* 
dt 
or, what is the same thing, 
{a^ + br?... +c^ 2 + ecu 2 } 2<s+1) (ab...ce) 4 
dS. 
Hence integrating each side, and observing that J dS, taken over the whole spherical 
surface x 2 -\-y 2 . . . -{-z 2 -\-w 2 =l, is =2( r ^) s+1 -rT(^-s-f-^-), we have 
Ji 
Kny 
a^ + br )*. . .c ? 2 + m 2 p (s+ 1 ) “T(is + i) * (ab . . . cef 
5 I 2 
