754 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
125. We have to find the value of § corresponding to the foregoing value of V ; viz. 
W being the value of V, on writing therein (x, y,. . .z) in place of (a, b , . . . c), then 
(theorem A) 
r(j»+g) / t ,w\ 
?- 2(ri)T(j+l)^ it); 
Take X the same function of (x, y,. . .z,e) that 6 is of (a, b , . . .c, e), viz. A the positive 
root of 
. i y I i 
/ 2 + a “^ 2 + a* • • t A 2 + a a - 
and (|, 7>i, . „ . r) corresponding to (a, (3, . . . 7, s), viz. 
S— ,/.7'o t /3-r T‘--£— j h i , T> T — V^ 1 ' 
■ 2 T yi ^.2 
" t / 2 + A~/ + A * ’ * _ A 2 + A’ 
V^+a’ V/+a'.‘ v^ 2 +Y 
so that W is the same function of (|, r„...X) that Y is of (a, (3, . . . 0): say this is 
t-i-'dt 
--CAxf/j^ t 2 V/ 2 + ^./ + ^.“ 
then we have for § the value 
A 2 + *’ 
r(^+ g ) 
e-2(r*)T(j+i) 
/ i 2 ? 2 \ - 1 / 1 
1 ^dW n dW 
/i 2 + A / •\/ 2 + A < = ^•••+/i 2 + A^’ J 
where e is to be put =0. 
126. Suppose e is =0, then if ^+^...+^>1, X is not =0, hut is the positive root 
of 
- + 
/ 2 + A / + A" A 2 + A 
P ' f 
V 
2 ~2 / /v>2 ,,2 
rv • • • + =1> r 5 —\f 1 — r 'i , 7 TTTTT" * • 
/ 2 + a # 2 +a A 2 +a 
a ? 2 ,?/ 2 
is = 0 , and we have 
f= 0 , viz. g is =0 for all points outside the ellipsoid + ^ • •+T 2 = 1 » 
J 9 ' 1 
But if then on writing e=0, we have A=0, r*=- 
J 9 K ; 
0 B(^ + g) . . q+1 e- q+2 A / 1 £ dW _i_ 1 dW , ■ 1 9 * 9 dW \ 
^~2^T(q + l)- A A« +1 V\/ 2? € V ^ ^ d? " A =0 
- r( i*+ g ) ^ 
2^r(g+i)- A • \/ 2 5 d£ ^ </? ^A =( ,’ 
= -CA 0 ^ 0 .+ 
2A-?- 1 
A 0 2 /^...A 
4>o i 
Aofg...h’ W +l ’ 
where term in ( ) is 
