338 
A MEMOIR ON PREPOTENTIALS. 
[607 
The assumptions upon which this last condition is obtained are perhaps unnecessary ; 
instead of the condition in the foregoing form we, in fact, use only the condition that 
the prepotential vanishes for a point at infinity, that is, when all or any one or more 
of the coordinates (a, .., c, e) are or is infinite. 
Again, as we have seen, the prepotential V must satisfy the prepotential equation 
/ d 2 d? d 2 2q + 1 d \ rr n 
(& + - + * i+ S» + — T“ de) V -°- 
These conditions satisfied, to the given prepotential V there corresponds, on the 
plane w = 0, a distribution given by the foregoing formula ; it will be a distribution 
over a finite portion of the plane, as already mentioned. 
30. The proof depends upon properties of the prepotential equation 
d 2 
\dx 2 + 
+ A + — + 2 g —- — ) W = 0 
+ dz 2+ de 2 + e de) 
or, what is the same thing, 
dW 
A 
dx 
g2 3 +l 
dx 
+ • • • + 
dz 
g2(?+l 
d 
dz J de 
+ -y e 2q+1 
dW 
de 
= 0, 
say, for shortness, □ W = 0. 
Consider, in general, the integral 
dx...dzde. e 2q+1 
dW 
dx 
+ ... + 
dWY fdWy 
+ 
dz 
de 
taken over a closed surface S lying altogether on the positive side of the plane e = 0, 
the function W being in the first instance arbitrary. 
dW dW | c 2 ? +i dW dW 
dz dz de de 
Writing the integral under the form 
f j j j ( dW dW , 
dx... dzde (e 2q+1 j— . -=—I-... + e q+ 
J \ dx dx 
we reduce the several terms by an integration by parts as follows:— 
The term in 
dW 
dx 
is =J dy.. 
dW 
,.dzdeWe 2q+ '~~ 
dx 
f dx. 
..dzdeWÎ- 
dx 
Af) 
dW 
dz 
is = J dx .. 
dW 
de We 2q+1 - 
dz 
j dx.. 
.. dzdeW 
dz 
(-*© 
dW 
de 
is — J dx.. 
.... dz 
f dx.. 
. dz de 
de 
HI)