696 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
for all points within the closed surface ; consequently, since W vanishes at the surface, 
W=0 for all points within the closed surface. 
32. Considering W as satisfying the equation □ W=0, we may imagine the closed 
surface to become larger and larger, and ultimately infinite, at the same time flattening 
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. 
C dW 
The integral le 2?+1 W ~^g~dS separates itself into two parts, the first relating to the 
surface positive infinity, and which vanishes 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 is 
jw (^ 2g+l ~[pJ 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 
= -Jw ^ +1 ^Pj dx . . . dz. 
Hence also if W = 0 at all points of the plane e=0, the right-hand side vanishes, 
and we have 
J*’- 
dz de e 2q+l 
dw y 
dx ) ' 
■ + 
+ 
Consequently a ^=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 
I' 
7 7 7 , , dV dW 
rfW rfU dW^ 
dz dz 
de de /’ 
taken over the space within a closed surface S ; treating this in a similar manner, we 
find it to be 
= -Je 22+1 W ^ cZS-Jcte ...dz 
de e 2q+l WqU, 
where the integration extends over the whole of the closed surface S ; and by parity of 
= -jV +1 U^S-j^. . . dzdee 2q+l UnW, 
with the same limits of integration ; that is, we have 
|V' +1 W^dS+jjdx...dzde. e 2q+ \W □ U=jV s+1 U ^ dS +jjdx ...dzde. e 2?+, U □ W, 
