An introduction to the mathematical theory of attraction ...

124 The Potential.
If v be selected in such a manner as to be zero at each of
the boundaries, and so as to make lI v v2u d5 positive, and
if a be positive and so small that its square is negligible
compared with its first power, we have Q+av < Qu. Hence,
if,t' = tu + av, the function u' satisfies the same boundary conditions as u, but Qu, < QU. From this it follows that if a function u satisfy the boundary conditions, we can always find
another function u' satisfying the same conditions, and such
that Qu, < Q, unless V2a = O throughout the field of integration. Now Q, is essentially positive and cannot therefore be
diminished without limit. Hence a function < exists which
satisfies the boundary conditions, and is such that Q&, cannot
be made less, but if this be so we must have v29 = O throughout
the field 5.
Again there is only one such function. For if there were
two, p and 0', and if we put < - p' = X, we should have X = 0
on the boundary, and V2X = throughout the field, and therefore have X zero. Hence < = <'.
A similar theorem holds good for a plane.
It is to be observed that the validity of the proof given
above for the first part of Thomson's theorem is not admitted
by Weierstrass and other eminent mathematicians.
71. Surface Distribution.-It is always possible to
distribute mass over a closed surface S so as to produce the
same potential in external space as a given distribution of
mass M[ which is inside the surface, or the same potential
in internal space as a given distribution M which is outside.
To prove this, let ~i be the region outside S, and ~2 the
region inside, v, the normal to S drawn into @i, and v, the
normal drawn into 2; then, if V be the potential due to M,
by Art. 70, there is a function < such that < = V at S, and is
of the order at a point P at infinity, where R is the distance of P from the origin, and that v2< = O throughout 5i,
and a function 4 such that = V at S, and V24 = 0 throughout ~2.