An introduction to the mathematical theory of attraction ...

Examples. 249
Substituting in (48) and (50) for 1 - X2, 1 - v', 1 -,i2, and
the infinite series, in accordance with the equations obtained
above, we get
oo V 2n+l. - 2k = l -— ^. (52)
q22 19+ 2k$ v0 1 )
r,, = +2/e s
Other methods of finding the values of q1n, &c., will be
found in the Examples.
EXAMiPLES.
1. Obtain directly, in terms of a, b, and c, the first three terms in qil, qlo,
and q22.
Since qi2L = io + il + i2 + &c., and q12L = jo +jl + i2 + &c., we have to
find io, jo, &c. We have then
b Lab
'o = La, BIo = BA = c, jo io = -
b2 C2 - b2 a La2b
BJo=, AJo o= -Bo, i = j = _b
ig2 a1;2C C(C2,LC2, b
i = a2 a2c c2 -a -- b2)
JO C2 - b2>' CBIc- c-b2
b La2b2 b2 b2(2 - b2)
J~1 =- t1 - =(_ a2 _ b2)' B = — C(2 =- 2 _- b2)
A(J2=- - b2 )2 - a22 a a. b2
(c2-a2 c - b2) 2 =J (c2-b2 + ac)(c2- b2- ac)
a a2(c2 - a2 - b2)
AI2 =~ =
A2J1 (e2 - b2)3_ -a22'
Bi2 = e- 4A1 = eC{c +4 a4 + b4 2ac2 - 2622 + a22}
(z2 - b2)2_ - a2c2
b La3b3
J -2 BI2 c(c2 - a2- - b2 + ab)(c2 - a - ba - ab)
Hence we obtain
a2b a3b2
ql = a +2 - b + (c2 - b2 + ac)(c2 -b2 -ac)'
ab a2b2 a3b3
ql ( -- a2 - 2) c(c2 - a2 _ b2 ab)(c - a2 - b2 - ab) '