340 
A MEMOIR ON PREPOTENTIALS. 
[607 
itself out into coincidence with the plane e = 0, so that it comes to include the whole 
space above the plane e = 0: say the surface breaks up into the surface positive infinity 
and the infinite plane e = 0. 
r 
The integral I e 2q+1 W dS separates itself into two parts, the first relating to 
the surface positive infinity, and vanishing if W = 0 at infinity (that is, if all or 
any of the coordinates x, , z, e are infinite); the second, relating to the plane e = 0, 
[ [ dW\ 
is I W{e 2q+1 —j—)dx ...dz, W here denoting its value at the plane, that is, when e = 0, 
and the integral being extended over the whole plane. The theorem thus becomes 
/*’••• 
dz de. e 2q+1 
dW' 
dx 
+ ... + 
dW 
dz 
dWV 
de 
= - W e 2q+1 
dW' 
de . 
dx ... dz. 
Hence also, if W = 0 at all points of the plane e = 0, the right-hand side vanishes, 
and we have 
dx ... dz de. e 2q+1 
\(dW 
dx 
+ ... + 
dW 
dz 
+ 
dW 
de 
= 0. 
dW dW dW 
Consequently =0, .., = 0, = 0, for all points whatever of positive space ; and 
therefore also W = 0 for all points whatever of positive space. 
33. Take next U, W, each of them a function of (x, .., z, e), and consider the 
integral 
dx ...dzde. e 2q+1 
dU dW 
dU dW dU dW 
dx dx dz dz + de de 
taken over the space within a closed surface S; treating this in a similar manner, we 
find it to be 
= - [ e* q+1 W d ~dS~j dx ... dz de. e~ q+1 WUU, 
where the integration extends over the whole of the closed surface S; and by parity 
of reasoning it is also 
= -J e 2g+i U d J^dS-Jdx...dzde.e 2q+1 UO W, 
with the same limits of integration; that is, we have 
I e 2q+1 W ddd dS+J dx...dzde.e 2q+1 WO U=j e 2q+1 U d ^dS + J dx... dzde.e 2q+1 UO W, 
which, if U and W each satisfy the prepotential equation, becomes 
J e 2q+1 W d ~ dS = j e 2q+1 U ~ 
dS. 
And if we now take the closed surface S to be the surface positive infinity, together 
with the plane e = 0, then, provided only U and V vanish at infinity, for each integral