PROFESSOR CAYLEY ON PREPOTENTIALS, 
and to this we may join, jj being arbitrary, 
[ M_2_ 
•» * _L r! o T I 
6 + + J 2 da 0 + v) + h 9 dc~^~ d -f-r] de J ^9 -\-J 9 . 6 + >] +F 
Again, defining V t 0, Ud as immediately appears, we have 
r7 „ A $\ 2 , (dty 1 , T , 4 
- \da) •••+(&)> -F • 4J ’ - J' 
and passing to the second differential coefficients, we have 
<m 2 8ffl 2 4a 2 J" 
^-J'(fl+/ 2 ) _ J /2 (S +/ 2 ) 3- J ,3 (0 +/ 2 ) 2 ’ 
+/: 
where 
J" =— 2 
(0 +/ 2 ) 3 + (0+A 2 ) 3 + 0 3 }’ 
with the like formulae for ... ^4, 44- Joining to these 2g + 1 ~ , we obtain 
rfe 2 de 9 ° e de J'0 
, rf 2 fl , <Z 2 0 , 2 g + l 
D ~ y«?a 2 ' • • + dc 2 +ie 2 + e de)’ 
2 f 1 1 l + ( 2 g + l) ) 
” J'{0+/ 2 --- “ i “fl + A 2 " t " 0 j 
8 4P' 
-F 2 (“i J, ')-F ( J ')’ 
where the last two terms destroy each other ; and observing that we have 
i/_L_ 
©— 2 \Q+f2- - • ^Q+h 9 ^ 0 )' 
the result is 
4©' 
J'©* 
97. First example. y?—d? . . . -fc 2 , and 0 is the positive root of ^p-Fg-=l. 
* V is assumed = j -\-f 2 )~ is dt, where g'+l is positive. 
Jd 
I do not work the example out ; it corresponds step by step with, and is hardly 
more simple than, the next example, which relates to the ellipsoid. The result is 
p = 0, if ,x 9 , . . -j-z 2 ''*/ 2 , 
r(fr + g) n s (- 
f -(ri) s r( ? +i)/ y 
l ...+z 9 \« . 
if w\..+zWf: 
? 1 
( x 9 ...+z 9 \ 
| dx . ..dz 
l f J 
J{(a 
—sc ) 9 . . . + (c— 
z) 9 + e 9 }^ 
hence the integral 
