342 
A MEMOIR ON PREPOTENTIALS. 
[607 
where J denotes the surface of the s-coordinal unit sphere a 2 + ... + 7 2 = 1, and the 
r-integral is to be taken from r = 0 to r = oo ; the values of the two factors thus are 
[¿a 2 ( r i)* f r*- 1 dr r (g + 1) 
J r (is) ’ J (1 + r 2 f s+q+1 r (is + q + 1) ■ 
Hence the expression in question is 
and we have 
“ 2 (£* + ?) V o 
2JJÏI 
T is 
i_ y £ r ( q + 1 ) 
Ï 1 (2 s 4* q + 1) 
-2(r^T(y+l) 
F (is + q) 
dx... dz 
{(a — x) 1 + ... + (c — zf + e 2 )^ s+q 
-2(r^r( g + l) 
F (is + q) 
or, what is the same thing, 
i 
tcP 
Co 
+ 
dx ... dz 
0 
2(ri) s r(? + l) 
[ e de) 
{(a — xf + ... + (c — zf + e 2 \& +q 
35. Take now V a function of (a, .., c, e) satisfying the prepotential equation in 
regard to these variables, always finite, and vanishing at infinity; and let W be the 
same function of (x, .., z, e), W therefore satisfying the prepotential equation in regard 
to the last-mentioned variables. Consider the function 
r (£s + q) 
0 
2(Tiyr(q + l) 
V de ) 
{(a — x) 2 + ...+(c — z) 2 + 
g2p+g 
where the integral is taken over the infinite plane e = 0; then this function (V — the 
integral) satisfies the prepotential equation (for each term separately satisfies it), is 
always finite, and it vanishes at infinity. It also, as has just been seen, vanishes for 
any point whatever of the plane e = 0. Consequently it vanishes for all points whatever 
of positive space. Or, what is the same thing, if we write 
V 
=/ 
p dx ... dz 
{(a — x) 2 + ... + (c — z) 2 + e 2 }* s+ 9 
•(A), 
where p is a function of (x, .., z), and the integral is taken over the whole infinite 
plane, then if F is a function of (a, .., c, e) satisfying the above conditions, there 
exists a corresponding value of p ; viz. taking W the same function of (x,.., z, e) 
which V is of (a, .., c, e), the value of p is 
P = 
P ($s + q) 
2(riYV(q+l) 
e 2q+1 
dW 
de 
(A), 
where e is to be put =0 in the function e 2q+1 
dW 
de 
This is the prepotential-plane 
theorem; viz. taking for the prepotential in regard to a given point (a, .., c, e) a 
function of (a, .., c, e) satisfying the prescribed conditions, but otherwise arbitrary, 
there exists on the plane e = 0 a distribution p given by the last-mentioned formula.