In order to take advantage of this identity, we may rewrite Eqs. (5-145)

as


S ev




hSod
 o a, sm


and (5-146)


 S2+ 1 2 d 1 H2}2
 2-+ jtJ2II dX (t X2)2
 2(f- 1)(f + t)( + 2) o 27 f, )
 F(O, 4) cos(24) d2
x PX X2 (X), (5-150)
 1 X2 dX2
 _2f+ 1 2+ J 1 dX (t X2)2
 2(- X)(f + 1)( + 2) o 2 x- )
 G(e, )) sin(2 _) d2
x P (X ) (5-151)
 1 X2 dX2


Now, we separate the variables in F(O, 4) cos(24)/(1 X2) and

G(H, 4) sin(24)/(1 X2) and expand them in d2P(X)/dX2,


F(e, 4) cos(24)
 1 X2
G(O, 4) sin(24)
 1 X2


 d2
 dX2
 d2
Sn ) dX2 P(X).
n


Then, substituting Eqs. (5-152) and (5-153) into Eqs. (5-150) and (5-151), respec-

tively, we obtain via Eq. (5-149),


hSev

hSod
O0 Xa,__a
 t "o


 S 2W d () (
 2bx(u)

#ri^)


(5-154)


(5-155)


5 (ga)).


 However, Trace race in Eq. (5-127) must be treated as a scalar according

to Eq. (5-130). As we did previously, we first rotate the coordinates (0, () ,- ( D, )
in the expression of hTrace so that

 horace ,) (h0+csc2 Ohs) (0, 4). (5-156)
 ST2 .... (e6_( o~ c


(5-152)

(5-153)