An introduction to the mathematical theory of attraction ...

260 Systems of Conductors.
CHAPTER VII.
SYSTEMS OF CONDUCTORS.
127. Distribution on Charged Conductors.-If a
system of charged insulated conductors be in a state of electric
equilibrium, the potential is constant on the surface of each
conductor, and there is only one possible distribution of
electricity which produces a potential having given values at
these surfaces, since (Art. 64) there is only one possible
potential in external space, and its differential coefficient
determines the density of the distribution at any point on a
conductor. Again, if the total charge on each conductor be
given, there is only one possible distribution of electricity
consistent with equilibrium. This proposition is a generalization of that given in Art. 75, and is proved in a similar
manner.
In fact, if Si, S2, &c. be the surfaces of the conductors,
e,, e2, &c., the given charges, V and V' the potentials due to
two supposed distributions of these charges consistent with
equilibrium, we have V= Ci, V'= C'0, at 8,; V= C2, V = C',
at S2, &c.; and v being the normal to a surface drawn into
external space,
d- ' dS = -47re, = d S a, &c.;
Jdv J v
whence
} (V ( -- ) (- ')dS, = 0, &c.
Again, throughout the whole of space 5 outside the
conductors V2 V = V2 V = 0; hence, if Y = Y - V', we have
an there + ( 9) dS, + &c. c+ s ho2ut = ~ h
and therefore, by (9), Art, 58, ( is constant throughout the