An introduction to the mathematical theory of attraction ...

112 The Potential.
If U = U' + e log -, where r is the distance of any point
from a point P inside the field, since r log r = O when r = O,
we obtain as the uniplanar equation corresponding to (10)
U 7J ds8 +j I UvVdxdy = V-d ds-27reVp +Jj VvU'dxdy
If(dUdV dUdV\
- jd d ~ + - -7 xdy, (14)
JJ\(x d dy dy (
where Vp is the value of V at the point P.
61. Constant Potential.-If a closed equipotential
surface have no mass inside it, the potential is constant for
the whole of the internal space.
For, since V is constant over the equipotential surface S,
and there is no mass inside it, by Art. 26 we have
{d V adS= 0,
dv
also V2 V= O at every point inside S; hence, by (9),
(dV\2 (d7V d V,(d o
J\dx+) +dy + dz =;
and since every term here is positive, we must have
dV dV dV
- = _ = - = 0
dx dy dz
throughout the whole space inside 8, whence V must have
the same value throughout this space as at the boundary.
A similar theorem holds good for an uniplanar distribution
of mass acting inversely as the distance.
Again, if the potential have a constant value C for any
finite portion A of unoccupied space, it must have the same
value for the whole of space which can be reached from A
without passing through mass.
For if in any portion B of space adjacent to A the potential be everywhere greater than C, we can describe a sphere,