184 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
a constant value = A at all points of S, then at every point O of 
external infinite space T’ the potential will be, 
1. If A=0, likewise = 0. 
2. If A is not = 0, it will be less than A, and will have the 
same sign as A. 
Demonstration—I. We must demonstrate that the potential 
can have no value at O beyond the limits 0 and A. Let us 
assume such a value B for the potential at O, and let us denote 
by C an arbitrary quantity between B and 0, and also between 
Band A. Straight lines being drawn in every direction from 
O there will be on each of them a point O', at which the potential 
will be = C, so that the whole line O O! belongs to the space T’. | 
This follows directly from the continuous variation of the poten- 
tial, which, if the straight line be sufficiently prolonged, must 
either pass from B to A, or must decrease infinitely, according 
as the straight line meets or does not meet the surface S (com- 
pare the remark at the close of Art. 21). The locus of all the 
points O’ forms then a closed surface; and as the potential is 
constant in it and=C, so, according to the theorem in the 
preceding article, it must have the same value at all points of 
the space enclosed by the surface, and yet it has at O the value 
B, which is different from C. Thus the assumption necessarily 
leads to a contradiction. 
Thus, for the case of A = 0, our proposition is completely 
proved; for the second case of A not being = 0, so far is evident 
that the potential can at no point of T’ be greater than A, or 
have a contrary sign. 
II. To make our demonstration complete for the second case, 
let us describe round QO, as a centre, a spherical surface, having 
a radius R less than the least distance of the point O from § ; 
let the spherical surface so described be resolved into elements 
ds, and let the potential in each element be denoted by V; and 
the potential at O be again called B. According to the theorem 
in Art. 20, the integral extended over the whole spherical sur- 
face, will then be 
fy ds=47 R?B, and consequently fw — B)ds=0. 
But this equality can only subsist, either if V is constant = B~ 
at all points of the spherical surface, or if at different points ‘of 
the spherical surface, V differs from B in opposite directions. 
In the first supposition, according to Art. 25, the potential 
