An introduction to the mathematical theory of attraction ...

246 Electric Images.
In like manner we obtain
n = =.Cvn + Dv-. (42)
Jn
A and B are determined from the values of i, and i, and
C and D from those of jo and ji. Since io = La, we have
b. Lab - a La2b
So= - C - = ~1 = b Yo = C2 5 b2
jo-co c b2o= b,
c —
c
-b. b(c2 - b2) La'b - La2b2
i a2 cc _ a2 _ b2( c - a2- b2 - c(c2 - a - b2)
C- b2
G --
C
Hence we obtain
i c2 - b2
+ B= aO= - v A+ Bv-l = a=l (43)
C+D)= - (30 C =-D-ov- -(c~2 - a2 - b2) (44)
+1=[0=Lab' Cv. —D- 1 La2b
Dividing the second of equations (43) by the first, and
A.
putting =, we have
5v + v-1 c2 - h2 a2
Ç + 1 -ab ab
Solving for E, we obtain
v(a + vb) v(a + vb)2 v(a + vb)2
b + va abv2 + (a2 + b2) v + ab vc2
hence
A a + vb
- À2, where À= -. (45)
Taking v as that root of the equation for x which is less
than unity, we have a + vb < c, and X < 1, then
A2 A +B
A= (a +B), B= -
À2 - I i 1 À29