A MEMOIR ON PREPOTENTIALS. 
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607] 
Write dS to denote an element of surface at the point (x,..,z, e). Then taking 
a, .., 7, 8 to denote the inclinations of the interior normal at that point to the positive 
axes of coordinates, we have 
dy ... dz de = — dS cos a, 
dx de = — dS cos 7, 
dx dz = — dS cos 8 ; 
and the first terms are together 
'dW 
—j— cos a + ... + 
dx 
J e^ +1 
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 
J e^ +1 W 
where « denotes an element of the internal normal. 
The second terms are together 
dx.. dzde Wd W. 
We have consequently 
dW r 
e *q+i w dS— dx ... dz de e^ +1 WU W. 
31. The second term vanishes if W satisfies the prepotential equation □ W — 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 
j dx ... dz de. e 2<?+1 
where the integration extends over the whole space included within the closed surface; 
whence, W being a real function, 
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 Q!F=0, we may imagine the closed 
surface to become larger and larger, and ultimately infinite, at the same time flattening 
43—2