and satisfy [28]


 d ) d 26, ( + 1)!
dX2 (X) P2(X) +( )!
 dX dX 2f + 1 (f 1)!


(5-101)


In order to take advantage of this identity, we may rewrite Eqs. (5-97) and (5-98)

as


 2 +C t j 2' A ) f dX (1



 2 2L + t) j2 A ) J 1dX (1
 2(f ) 2
E 2(+ 1) ) 27x _1
g^T~A L 1


S2)F(, ) sin cos d
 1 X2 dX '
 (5-102)
 SG(e, ,) sin 0 sin d
X2) Pe(X).
 1 X2 dX (
 (5-103)


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

G(, )) sin e sin 4/(1 X2) and expand them in dP,(X)/dX,


F(O, 4) sin O cos 4
 1 X2
G(H, 4) sin O sin 4


 d
n

n


Then, substituting Eqs. (5-104) and (5-105) into Eqs. (5-102) and (5-103), respec-

tively, we obtain via Eq. (5-101),


^ 27

 *27(^ )


(5-106)


(5-107)


5 (ga)).


From these we find our f-mode singular fields hSev and hs d,
 rt6' = -(6


(5-108)

(5-109)


 AhSod

and by combination

 hse h ieve iSod


(5-110)


 dX(1
-1


hSod
t6 ,,a


(5-104)

(5-105)


hSev

hSod
t6 y a.-


(9(4))> ,


({(W )) + (+ ( ) ,