An introduction to the mathematical theory of attraction ...

114 The Potential.
surrounding the entire mass, C is zero, and the potential is
zero for the whole of space external to S.
These two theorems are an immediate consequence of
equation (9). The boundaries of the region ~ are in this
case the closed surface S, and a sphere S' at infinity. If
1R be the radius of this sphere, and M the total mass, we have
V- dS8 = -[R 1R2 do = O, since isinfinite.
J' dv JR
Hence, as V2 V = 0 throughout 5, we get
f{(11 ) 2(v)2 +(dV/2d =V ddS = O,
] x y ) dzj dv
when V is zero on S; whence, as in Art. 61, V has the same
value throughout ~ as on the surface 8, and is therefore zero.
HIence again at each point of any closed surface surrounding
dV
S we have = 0, and therefore, by Art. 26, the total mass
dv
is zero.
Again, if V be constant over S, and the total mass zero,
f dV dS= O,
J dv
and therefore, as before, V is constant throughout ~, and is
therefore zero since it is zero at infinity.
Another theorem, which is more general than those given
above, and which may be proved in a similar manner, may
be enunciated thus:-If the potential be zero at every point
of the boundaries of a region in which there is no mass, the
potential is zero throughout the region.
63. Zero Potential for lUniplanar Distribution.If at a point P the potential of a uniplanar distribution of
mass acting inversely as the distance be zero independently
of the absolute magnitude of the unit of length, the total
mass must be zero.