PROFESSOR CAYLEY ON PREPOTENTIALS. 
695 
. dW 
term in ^ 
is = j dy.. 
.dzdeWe»+‘^-§dx.. 
dW 
dz 
is — j* dx . . 
dW C 
, . ..de We 2 * + 1 ^f— dx. , 
..dzde W^ | 
dW 
de 
is — \dx. . 
, . ..dzWe^~-^dx. , 
Write d$ to denote an element of surface at the point (x ... z, e); and taking 
a ... y, & to denote the inclinations of the interior normal at that point to the positive 
axes of coordinates, we have 
dy . . . dzde=—dS cos a, 
dx 
dx 
and the first terms are together 
de=—dS cosy, 
dz=—dS cosS ; 
dW 
dW 
• c °s 7+-^ C0S M 
W here denoting the value at the surface, and the integration being extended over the 
whole of the closed surface : this may also be written 
where * denotes an element of the internal normal. 
The second terms are together 
= -J* • • • • • • +1 t)+s(^‘- T)}=- j**.**W D W. 
We have consequently 
= (dx ...dzdee 2 « +1 W □ W. 
31. The second term vanishes if W satisfies the prepotential equation nW=0 ; and 
this being so, if also W=0 for all points of the closed surface S, then the first term also 
vanishes, and we therefore have 
where the integration extends over the whole space included within the closed surface ; 
whence, W being a real function, 
d W 
dx 
= 0 , 
^-0 ^-0 
U ’ de — u ’ 
