An introduction to the mathematical theory of attraction ...

224 Electric Images.
5~. If two points A, and A2 belonging to the system E
are images of each other in a surface 8, the inverse points
A'1 and A'2 are images of each other in the surface S' which
is the inverse of S.
This follows from the consideration that the joint potential due to ei at Ai and e2 at A2 is zero at 8, and therefore,
by 1~, (a) so also is the potential at S' due to e', at A', and
e'2 at A'2. Hence, A', and A'2 are images of each other with
respect to S'.
As a particular case of the above, we have the result that,
if A be the centre of the sphere 8, the point A' is the image
of the origin in the surface S' which is the inverse of S.
This is obvious if we remember that the image of A in S is
a point at infinity whose inverse is the origin.
6~. If t and t' be the tangents from O to inverse spheres
whose radii are a and a', and the distances of whose centres
from O are a and a', we have
a a a t R2 R2 t2 a2 _ a2
ta a __ R (3
a' a' t' t2 a'-a'2 R- R2
7~. If p be the perpendicular distance of O from a plane
whose inverse is a sphere having a for radius, a is given by
the equation
2pa = R2. (24)
EXAMPLES.
1. Find the distribution of mass on a sphere whose centre is A, and which
is at potential zero under the influence of a charge e at a point 0.
Invert the sphere from 0, and we obtain a sphere at constant potential L,
where RL = - e. The density a' at any point P' of this sphere is given by the
equation
L e R2 e
L= 4. =- _r, then, by (22), r=47r,% 47ra'R' OP3 47ra"
a OA2 - 2 e 0A2 -a2
but, by (23), =, whence o 4 = - OP
a 22 47ra 0_P3
2. Show that a solid sphere S whose density varies inversely as the fifth
power of the distance r from a point O is centrobaric, and find its baric centre.