694 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
function (and that a continuous one in value) for all values of the coordinates (a ... c) 
and positive values of the coordinate e. 
If, moreover, we assume that the distribution which corresponds to the given potential 
V is a distribution of a finite mass §gdx . . . dz over a finite portion of the plane w=0, 
viz. over a portion or area such that the distance of a point within the area from a fixed 
point, or say from the origin {a ... c) = (0 ... 0), is always finite ; this being so, we 
have the further condition that the prepotential V must for indefinitely large values of 
all or any of the coordinates (a ... c, e) reduce itself to the form 
(j 'qdx . . . dz)+(a? . . . -\-c 2 +e 2 f +q . 
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 
(£. + *L + £- + -J— i\y_o 
[da 1 ’ ’ ■ +*'+**+ 2}+l de) V — U ' 
These conditions satisfied, to the given prepotential V, there corresponds on the plane 
w= 0, a distribution given by the foregoing formula, and which 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 2(7 + 1 d 
de 2 ^ e de 
w=o, 
or, what is the same thing, 
say, for shortness, □ W=0. 
Consider, in general, the integral 
dz de e 2q+ 
• • + 
/dwy , , 
(dwy] 
U) +l 
{ de ) } 
taken over a closed surface S lying altogether on the positive side of the plane c=0, 
the function W being in the first instance arbitrary. 
Writing the integral under the form 
dx ...dzde ( e 2q+ 
dW d W 
dx dx 
+ ^ +1 
dz dz ~ 
dW dW\ 
de de J' 
we reduce the several terms by an integration by parts as follows :■ 
