An introduction to the mathematical theory of attraction ...

238 Electric Images.
is plain that a + c = CD, p = ep, and that o may be expressed by the equation
4. Find the distribution of mass on a spherical bowl (B) at potential zero
under the influence of a charge e situated at any point O outside the bowl.
Invert the system from 0, selecting the radius of inversion R so that the
sphere (A), of which the bowl is part, may be inverted into itself. The bowl
(B) is inverted then into another portion of (A) bounded by the intersection of
(A) with the sphere which is the inverse of the base of (B), that is, (B) is
inverted into another bowl (B') belonging to the same sphere.
If (B') be at constant potential L' due to a distribution on (B'), by the last
Example the density a' of the distribution at any point P' of the inner surface
of (B') is given by the equation
Draw a plane through D'P' meeting the edge of (B') in the points i', E';
then the points D', H', P', K' being concyclic, so also are the inverse points
1D, if, P, K; and we have, as in Ex. 2,
eD' D'H'. D'K' D D _ D. DK
çp, Pif'.1K P r p P=i. PK'
where ÇD and.p are perpendiculars from D and P on the base of (B).
Again, by similar triangles, as in 2~, Art. 119,,I'H' _= OD D'K' = 0 'K,
OP' 02'
P'H' = -_ PH, P'K' = - lPK,
OH O0K
and therefore,.D';'. D'K' O)D'2 DE. DK OP2 DU. 9DK.P 'H'. PK'T Op'2 PH.. PK O D02 PU..PK
If o be the density, corresponding to -', of the distribution on (B), by (22),
Art. 119, we have =,( R
and (B) is at potential zero under the influence of a charge e at 0, provided
that RL' = - e.
Hence, substituting in o', we get
=-e R o: 2 (gD -tam-l ~P e D'J
2r2f OP3 OD *4\CP O19 ' p