An introduction to the mathematical theory of attraction ...

244 Electric Images.
Again, as the potential at (B) is zero, the total charge Eb
on its surface is equal and opposite in algebraical sign to the
sum of the interior masses; hence
Eb ( )=-(e+Ea) -
The total charges E, and Eb may be expressed by the
equations
-e ab-f -e f-a
Ea =b - ---- -/-
1 -i b f ' 1- f
hence, if A and B denote the points in which OC meets the
spheres (A) and (B), we have
Ea a OB = - e
b ~ -b OA'
If we put b - a = c, and suppose c to remain finite while
a and b become infinite, the spheres become parallel planes,
and we find for the total charges E1 and E, on parallel planes
at potential zero under the influence of a charge e situated at
a point O between the planes, the equations
E = ep ep (37)
c c
where p, and p2 are the distances of O from the planes, and
c is the distance between them.
126. Spheres influencing each other.-If a sphere
(B), at potential zero, be in the presence of an insulated
sphere (A) at potential L, the total mass on each sphere can
be expressed by an infinite series deducible by the method of
images. This mode of obtaining the series was first employed
by Thomson. The special form of investigation here adopted
is due to Kirchhoff.
Let the centres of the spheres be denoted by A and B,
their radii by a and b, and the distance AB by c.
A charge La placed at A produces a potential L at the
surface of (A); but in order to have the potential zero at (B),
a charge must be placed at J0, the image of A in (B). To
render the addition to the potential zero at (A) another
charge must be supposed at I1 the image of Jo in (A), and so