178 GAUSS ON GENERAL PROPOSITIONS REGARDING FORCES 
21. 
THEOREM.—tThe potential V of masses, which are all situ- 
ated externally in reference to a coherent space, cannot have a 
constant value in one part of this space, and a different value 
in another part of the same space. 
Demonstration.—Let us assume, in every point of the space A, 
the potential constant to be = a, and in every point of another 
space B adjoining A, and not containing any mass, to be greater 
algebraically than a. Construct a sphere partly in B, and the 
remainder with the centre in A, which construction will always 
be possible. Now R being the radius of this sphere, and ds 
an indeterminate element of its surface, then, according to the 
theorem in the preceding article, / Vds = 42 R*a, and 
i (V —a) ds=0, which is impossible, as for the part of the 
surface which is in A, V — a= 0, and for the remaining part is 
by hypothesis not = 0, but is positive. 
In like manner the impossibility of V being less than a at all 
points of a space adjoining A will be clearly seen. But it is 
manifest that one at least of these two cases must exist if our 
theorem were false. 
This theorem includes the two following propositions :-— 
I. If the space containing the masses include a space devoid 
of mass, and the potential in a part of this space have a con- 
stant value, this value will be the same for all points of the 
whole included space. 
II. If the potential of the mass included in a finite space 
have a constant value in any part of external space, this value 
will be the same for all infinite space. 
At the same time it is easy to perceive that in the second 
case the constant value of the potential can be no other than 0. 
For if we designate by M the aggregate of all the masses if 
they have all one sign, or if they have different signs, the aggre- — 
gate, either of all the positive, or of all the negative masses, — 
taken exclusively according as the first or the last-named prepon- — 
derate, then the potential, at a point of which the distance from : 
the nearest element = 7, is always taken absolutely less than 
x 
ro) which fraction may manifestly be less in external space 
than any assignable quantity. 
