An introduction to the mathematical theory of attraction ...

256 Electric Images.
when the radii of the spheres and the distances between their centres are such
that the successive images lie between A and B, and that the angle BCA is
obtuse.
If A and C denote the angles BAC and the supplement of BCOA, we have,
then
AXiC= C, BJ2G = -2C, AJ2C=2C, BJC= r- 3C, &c.,
BAC = A, aAIIC= r-C-A BIi - +, c = +, AI2C r- A - 2,&c.
If p denote the perpendicular CM on AB, we have
P P P
CA - sid^' I&0
sin A' sin AJi C sin BI &c.;
whence
ep( 1 1 epy
= a sin (A + C) &c o sin ( + nC)'
-ep e o 1
a = sin nC
These expressions are due to Mr. F. Purser.
6. From the results obtained in the last Example deduce expressions for the
total charges on two insulated spheres, one of which (A) is at potential L, and
the other (B) at potential zero.
When the spheres (A) and (B) do not intersect the point C, the perpendicular
p, and the angles A, B, and C become imaginary, and if i = V- 1, we may put
A = ia, C = iy, where a and y are real, then
a2 + c2- b2. (2(a2b2+ b2c c2a2 )a4-b4-4 b 4) i
cosha =cosA= isinha= sin = -
2ac 4,2c2
c2 a_ 2 b 2(2b2 b2c2 + 2a2) -a-b4-c4 b
coshy = cos = -2a —b i sinh y =sin 4C= b2- 4 a2b2 -
Hence, if
a4 + b4 + c4 -2(a2b2 + b2c2 + c2a2)
4c2
we have
k. c
sinh a = -, sinh =
a ab'
sin ( + nC) = sinh i (a + n) = i sinh (a + n^y), sinh nC = sinh iny = i sinh n;
ab iab
also, p s =- si=- nh 7 = ik.
c c
Hence, putting e = La, and substituting in the expressions for Ea and Eb
given in the last Example, we get
1 - 1
Ea = Lk 2, 'n Eb Lk:,
= sinh (a + n'y)' b= sinh 'y
This agrees with Ex. 3, where a has the same meaning as a in the present
Example with its sign changed, and == y.
The method adopted here is due to Mr. F. Purser.