An introduction to the mathematical theory of attraction ...

232 Electric Images.
Thus all the charges are expressed in terms of L and the
radii of the spheres, and we see that a potential L is produced
at each of the spherical surfaces by placing charges La, &c.,
at the four centres,
- Lab, &c., at the six points G12, &c.
v/a2 + b2
/Labe ), &c., at the four points K, &c.,
v (a2b2 + b2e2 + ca'2)y,and ____ - Labcd
v/(a2b2c2 b2c + b 22 + a2 + da2b2)
EXAMPLES.
1. An insulated conductor formed of the larger segments of three spheres
cutting orthogonally is charged to potential L; find the density o of the distribution at any point P on the surface of the sphere (A).
L t b3 c3 b3c3
Ans. 47rc =- 1 - - + -3 -}
a BBP3 CPt (b2 + c2) 3 G23P3
2. Find the density o at a point P on (A) when the conductor charged to
potential L is formed of the segments of four spheres cutting orthogonally.
Ans. 4ro = - 1 - - (2 b - -+
L 1 b3 C3 d3 b3c3 c3d3
a t Bp3 CP3 I)P3 (b + C2)i G23P3 (c2 + d2)2 G34P3
d 3b b3c3d3
+ 2 --- —----- - -- -
(d2 + b2)i G24P3 (b2c2 + c2I2 + d2b2)_ HP3
3. A conductor, formed of the segments of three spheres cutting orthogonally,
and having the centres of the spheres in its interior, is at potential zero under
the influence of an external electrified point P; show how to determine the
distribution of mass on the surface of the conductor, and the potential in external
space.
Letthe spheres be denoted by (A), (B), and (C). The plane of the intersection of (A) and (B) cuts (C) in a circle which meets the circle of intersection of (A)
and (B) in two points 0 and 02. Invert the system from one of these points 0,
then the inverses of the spheres are planes, (A'), (B'), (C') intersecting perpendicularly at the point O'2, and the inverse of P is a point P' in the region
bounded by the quadrants of the three planes. Let b, 1, be the perpendicular
distances of P' from the planes, then
R2 R2 R2
2=a' =2b' 2c
e = -, 'n =2 - e '