An introduction to the mathematical theory of attraction ...

Variation of Potential in Space unoccupied by Mass. 119
and therefore so also is d- dS taken over the surface; but
j dr
dS = - 41rM, where M1 is the mass inside the sphere;
dr
hence M is positive; and as the sphere may be diminished
without limit, there must be positive mass at P. If V be a
minimum, it can be shown in a similar manner that there is
negative mass at P.
It follows from what has been proved above that V
cannot be a maximum or a minimum in unoccupied space. The
same theorem can be proved in a similar manner for a uniplanar distribution of mass acting inversely as the distance.
66. Variation of the Potential in Space unoccupied by Mass.-The potential of masses which are
outside a closed surface S has at all points inside this surface
a value which lies between the extreme values on the surface.
For, let A be the greatest and B the least value of the
potential on the surface S; then if anywhere inside the surface V be greater than A, or less than B, there must be a
point where it is a maximum or a minimum, which is impossible.
It follows, as a corollary, that if V be constant over S, it
is constant throughout its interior.
Similar results hold good for uniplanar mass.
If the potential V of masses inside a closed surface S has
at all points of S the same algebraical sign, it has the same
sign at all points outside S, and its greatest magnitude irrespective of sign in external space is less than its greatest
magnitude on S.
For, if V were positive at every point of 8, and negative
at any point outside 8, since V is zero at infinity, there must
be a point outside S at which V is a minimum, but this is
impossible. In like manner it can be shown that if V be
negative at every point of S it cannot be positive at any
point in external space. Again, if V have at any point
outside S a value of the same sign, but greater in magnitude
than any of its values on 8, it must be a maximum or a
minimum at some point in external space which is impossible.
Lastly, if A be the greatest value of V irrespective of sign