ROEESSOR CAYLEY ON PREPOTENTIALS. 
737 
99. We have W the same function of (x . . . z, e) that V is of [a ... c, e ) ; viz. writing 
for the positive root of 
f + l'" +F+A+X = 1 ’ 
the value of W is 
= f # _?_1 (^+/ 2 - • • t + h^dt. 
z^ 
Considering the formula which involves e 2? W, — first, if -p . . .+^>1, then when e is 
= 0 the value of X is not =0 ; the integral W is therefore finite (not indefinitely large), 
and we have e 2? W=0, consequently g=0. 
x i z i 
But if to . . . 1, then when e is indefinitely small, X is also indefinitely small; 
/p2 ^2 
viz. we then have - = 1 — . . . — the value of W is 
w =(/. . . A)-'f t-’-'dt, =(/. . 
and hence 
r(|g+g ) i 
e-(ri)<r 9 -} 
r(jg+g) 
~(nyT(q+i) 
(/.../»)-'( 10 ... 
100. Again, using the formula which involves (e 2q+i ; we have here = — 0 
or substituting for © and j e their values and multiplying by e 2q+1 , we find 
dV 
e 2q+i y L =2e 2q+2 Q-'J l -'Q, 
de ’ 
:2^ +2 S ? 2 [ ( y 2 + ( jp • ■ • + (0+/*- • • 0 + #) K 
and therefore 
e * q+X de — 2e * ,+i ' A V 2 [(/ 2 + A) 2 * • • +(A 2 + A) 2 +A 2 ] 1 ( A +/ 2 - • • *• 
Hence, writing e=0, first for an exterior point or p . .. X is not =0, and 
the expression vanishes in virtue of the factor e 2q+2 ; whence also g = 0; next for an 
QC^ Z ^ . 6^ 1 / X ‘ ^ 
interior point or j 2 - • • +^<1, X is =0, hence also — 2 =- ^1— j 2 . . . —jpj is infinite ; 
x 2 
and neglecting in comparison with it the terms &c., the value is 
2(0 (/-/0- 
=2(10, 
