An introduction to the mathematical theory of attraction ...

Examples. 233
By changing the sign of the coordinate of a point, we obtain the coordinates of the image point in the corresponding coordinate plane. Thus in the
present case, the whole system is found by taking all possible combinations of
algebraical signs prefixed to the coordinates e, vj, Ç. The sign of the charge to be
placed at any point is negative if it has an odd number of negative coordinates,
and positive if it has an even number. If the image of P' in the plane (A') be
denoted by P'a with a corresponding notation for the other points, we have the
system:Points P', P'a, Pbc, 'c, P'ab, Pac P'bc, P'abc;
Coordinates, ZoS(, -, - -, -7 -, - e - -O, - -, - C, — 5 - -, - e - -7 - C;
Charges, e', - e', - e', - e', e', e', e', - e'.
In the inverse spherical system, since O is a point common to the three
spheres, if we put e =,uOP, we have for the electrified points, which produce
in external space the actual potential:Points, P, Pa, Pb, Pc, Pab, Pac, Pbc, Pabc;
Charges, /uOP, -/zOPa, -j OPb, -,OPc, /OPab, t1OPac, pzOPbc, -JiOPabc,
where Pa is the image of P in the sphere (A), Pab the image of Pa in (B), or of
Pb in (A), &c.
Since the resultant force in the interior of the conductor is zero, the density
of the distribution at any point of its surface can be determined as in preceding
Examples.
4. A conductor, formed of the segments of four spheres cutting orthogonally
and having their centres 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 conductor, and the potential in external space.
Adopting the same notation as that of the last Example and calling the
fourth sphere (D), if we invert from 0, one of the points common to (A), (B),
and (C), we get three rectangular planes and a sphere (D') cutting them perpendicularly, and having, therefore, 0'2 as its centre. If P' be the inverse of P,
the successive images of P' in the planes form the same system P'a, &c., as that
considered in the last Example. These points lie on a sphere having 02 for
centre; and their images in the concentric sphere (-1') are equidistant from
0'2, and are obviously in reference to each other a complete system of images in
the planes (A'), (B'), and (C'). Hence, by 5~, Art. 119, in the inverse system of
four spheres, a potential zero is obtained at the surface of each by placing at the
points
P, Pa, Pb, Pc, Pab, Pac, Pbc, Pabc, Pd, Pad, Pbd, Pcd, Pabd, -Pacd, Pbcd, Pabcd,
charges
fUOP,- -.0Pa, - pJOPb, - UPOPc, tlOPab, uLOPac, fOPbîc, - -- OPac, - OPd,
vOPad, VOPbd, 'OPcd, - OPabd, - vOPcad, - YOPbcd, v OPabc,
where
d
jOP = e, and YOPd = - e.
DP