An introduction to the mathematical theory of attraction ...

128 The Potential.
surface so as to produce at each point a resultant force normal
to the surface, which is therefore an equipotential surface for
this distribution, and the electric mass constitutes what is
termed a couche de niveau.
If the total amount M1 of mass be given, there is only one
possible distribution consistent zoith electric equilibrium.
For, if there be two possible distributions, let their
potentials in external space be U and V; then at the surface
S of the conductor we have U = Ci, V= C2; and if v be
the normal to S drawn outwards,
4r-= - U dS=_ dVdS.
dJ v J dv
Hence
dU dV
(U - V) -- dS = 0;
J \ dv dv
also if X = U -, by equation (9), Art. 58, X is constant
throughout the whole of space outside 8, and therefore
~ is zero at every point of S; whence the density of the disdv
tribution producing U is everywhere the same as that of the
distribution producing V.
It can be shown, as in Art. 70, that there is only one
possible distribution of mass over a closed surface S producing
a given potential at every point of this surface.
Hence, if an insulated conductor be charged to given potential, the quantity and distribution of electricity on its surface is
determined.
EXAMPLES.
1. The potential V is c (x, y, z) throughout the region inside a closed
surface S, and is zero throughout external space; find the corresponding distribution of mass.
In order that it should be possible to fulfil the required conditions P (x, y, z)
must be zero at the surface S; then inside 8, the density p is given by the
equation 42rp + v2p) = 0, and the surface density a on S by the equation
47ro + - = O, where v is the normal to S drawn inwards. Since S is an equidv
potential surface corresponding to F, we have
d dq ) (dy2 ) ( dq )2
di, +x Tïj d