An introduction to the mathematical theory of attraction ...

Spheres influencing each other. 247
and
_ 1 1 i Lav (1 - X2)
ai - Ap JBn j + (46)
an Av~ +.By-" l - X2 vs
If we divide the second of equations (44) by the first, and
put D = s, we get
rav + v-1 C2 - C2 - b2
-- - = - - = V + V-1;
n + 1 ab
and solving for n from this equation, we obtain n =- v2;
hence, we have
v2 c v3 C+D -c 1
C= (C+1D) D=
v2 _ ( + 1 Lab 1 - v2 1- v2 Lab 1 - v2'
and
i 1i Lab (1 - v2) (47)
Jn3 CGv+n _Dv,, - 1-v2'+2'
As the potential due to the charges i0, j, &c., is the same
on each of the surfaces (A) and (B) as that due to the actual
distributions on those surfaces, the total mass Ea on (A)
must be equal to the sum of the charges at the interior points
A, 1, I2, &c., and the total mass Eb on (B) to the sum of the
charges at JO, J1, &c. Hence
Ea = in = La (1 - X2) ^
Lab vn
Eb = n - (- +2
If we put
n ab 2 0 Vn
q1l = a (-12)o 1 X2- ) ql c -)212 V 2
(48)
we see that q12 is symmetrical with respect to a and b, and
we have
Ea = qlL. Eb = q12L. (49)
If the sphere (B) were at potential ill, and (A) at potential