PROFESSOR CAYLEY ON PREPOTENTIALS. 
739 
and, as before, 6 is the positive root of the equation 
/ >2 /»2 0*2 
T — 1 __Z _ — 0 
j*+r mm h*+d v — u - 
^s-\-q is positive in order that the integral may be finite ; also m is positive. 
104. In order to show that V satisfies the prepotential equation □ V = 0, I shall, in 
the first place, consider the more general expression, 
V=r dt I m T, 
Jo+ri 
where tj is a constant positive quantity which will be ultimately put =0. The functions 
previously called J and 0 will be written J 0 and 0 O , and J, 0 will now denote 
J, =1-, 
+ >)+/ 2 "' 
c z e * 
1 + ij + h~ + 
0, =(Q+v)-*-'(0+, + f . . . ; 
whence also, subtracting from J the evanescent function J 0 , we have 
say this is 
=’(i 
+f '.«+*+/ 
. . . + 
6 + n? .6 j-y + Ii 2 
and we have thence, by former equations and in the present notation, 
a dd 
d + ri+pda 
_ | 6 — — P 
2 • • • T A _I_ _j_ J.2 r/„~r A ■ J- T • i , 
S + >i + A 2 dc'd + v\ de J 0 ‘ 
V^=f, 
v n 
□ 0= 
J 0 '©0 * 
In virtue of the equation which determines 0, we have 
cl\ 
^=J 
-J"0 
*+/ 
da 
and thence 
MDCCCLXXV. 
4a 2 
, \ dd i 
(“8+1+/V 0 * | 
rffl I 
da J 
— J m 0 
rfa 2 ’ 
with like expressions for . . . 
5 F 
