An introduction to the mathematical theory of attraction ...

84 The Potential.
whence dv + d' + 47rc = 0. (12)
Equation (12) is called the characteristic equation at the
charged surface. When the surface is closed, the form of V
on one side is in general different from its form on the other.
Of this we have had an example in the case of a thin
spherical shell, Art. 42.
47. Differential Equations for Uniplanar Distribution.-In the case of a uniplanar distribution, V is a
function of x and y;
d2 d2
then v2 becomes + d
(1d2 2>
and ffv2 Vdx dy, taken through the space bounded by a closed
curve s, is equal to - f Nds taken round the curve; this again,
by Art. 36, equals 27rl; whence, taking for s the boundary
of an element where the uniplanar density is r, we have
V2 V+ 2r = 0. (13)
Again, by Art. 36, we see that at a curve on which there is a
mass distribution of density v, we have
d:V dV
dv+ dv 2r. (14)
48. Transformation of Coordinates.-It has been
shown, in Art. 45, that
v2 Vd = d, (15)
J~ J~=S d~;Tdn
where the first integral is taken through the volume bounded
by the closed surface 8, over which the second integral is
taken, and n is the normal to the element dS drawn outward.
From equation (15), we can readily deduce the form of V2V
for any coordinates.
In the case of polar coordinates, we have
d( = r2 sin 0 dr dO do;