756 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
and for the case last considered 
, A 2 c 2 
a = __f/!±L_ ■ ' 2 ^ 2 +9 . i 
* 2 ? + 2 + K/ 2 --'^2g + 2 + i J c 0 /2 * 0 ’ 
H — i/ 2 (/ 2 + ^). _| — | # Y* 2 + fl) _ J_ ^ me f unc ti on w ith. £ for 0, 
2 ? + 2 + ix 0 / 2 ^2g + 2 + i Ko /* 2 x 0 ’ 
^0 = 
^9 • • • + 
2? + 2 + K/ 2 2q + 2 + ±x 0 h 2 x 0 ’ 
\r 
• • •+; 
0 2q + 2 + ±x 0 P • ' * r 2g + 2 + W i2 *o’ 
where * 0 is the root of the equation 2g + 2 ^ - / 2 . • • + 2g + 2 1 j _^ 2 + l=Q, 
?= l 1 -/-! • • -$)**•’ v=fl I^r (/ - • • w* . . *+*)-*. 
Annex YI. Examples of Theorem C. — Nos. 129 to 132. 
129. First example relating to the (s-fl) coordinal sphere x 2 . . . -\-z 2 -\-w 2 =f 2 . 
Assume 
TTI M TTIt M . . . . 
Y = y... + c»+^)*<- >’ v = ^” (a constailt )> 
these values each satisfy the potential equation. 
V' is not infinite for any point outside the surfaces, and for indefinitely large distances 
it is of the proper form. 
V" is not infinite for any point inside the surface; and at the surface Y'=Y". 
The conditions of the theorem are therefore satisfied ; and writing 
we have 
? rfS 
(a-a ?) . . . + {c-z)*+ (e-w) 2 
r(*«-*) (dW> dW"\ 
z— 4(r \ 
W'= 
M 
(a ? 2 ...+^ 2 + w; 2 )^-i’ 
w „ M , dW" A 
W"=^; hence -^-=0, 
&_ z d w d\ M 
M ~ V dx • * ' +/ ofe + / j ...** + le*)*"* 
(s-l)j(a? 2 ... + z 2 + w 2 ) M 
= (z 2 ... +z 2 + zv 2 ) is+i ’ 
where 
