The scalar potentials Aev and Aod must satisfy the Poisson's equations
V6VaAev Vhs -cot OhfS + aohs + csc2 000h, (5-85)
a tb t o t)
V VaAod -EbVh -=- cse 0 (Oohs Oh8) (5-86)
according to Eqs. (5-53) and (5-54) along with Eq. (5-80). However, one can also
decompose Aev and Aod into spherical harmonic components,
Aev ,ZA^k(0, ), (5-87)
Aod A7 Y(Od(0, (5-88)
Then, we may solve the Poisson's equations (5-85) and (5-86) in terms of spherical
harmonics via Eqs. (5-87) and (5-88), and obtain the expressions for A'T and AXm
Atn + 1) dQ' (cot O'hs, + /o/ + csc2 O'ahs, Y*'m( / ),(5-89)
A 1 1)- d'1 csc' (8oh, t,,hsy,)Y* ',(O'), (5-90)
where dQ' = sin O'dO'd.'.
Now, substituting Eqs. (5-87) and (5-88) into Eqs. (5-83) and (5-84) along
with Eqs. (5-89) and (5-90), respectively, we have
hev 1) 1 dQ' (cot O'ht, + Oohe, + csc2 0'/h)
hS od __ JdQ CsC O (0oh 0,ht,)
6 xsY 0) (5-92))
xsin 0 yy-t.. ^Y^n(,) (5-92)
m0T