Fod ( + 1)2 2f(f + 1) d dVV' ( F,'


t J&,b,FT.. Y* &n(0l, y),
 (5-38race)
 (5-138)


where dQ' sin O'dO'do', and R = 2 since the scalar curvature R = 2/r2 for a
two-sphere of radius r and our two-sphere has the unit radius [11, 19].
 Now, substituting Eqs. (5-132) and (5-133) into Eqs. (5-128) and (5-129) along
with Eqs. (5-137) and (5-138), respectively, we have


 1
.1)(f + 1)(f + 2)


SdQ'~ F/,b,


1 N
2 ci/b'i Trace)
2


x ( csc2 0a -cot 0 ) [ Y*C (0,/ /') Y (0, ) (5-139)


 (S l1)(f + 1)( + 2) d a' i a)
^^Trw f F -"


x 2 csc 0 (0^ cot O) Y y* m (i,0)Yem(0,) ,
 Tn I


(5-140)


where we used the equality 2( 2 + 1)2 2( + 1) = (- 1)( + 1)( +
Eq. (5-63) we may derive

 24f (0 csc2 o2 cot 00e) P(cos y)


 (0 -csc2 0,2 cot 00o) P y*(c (o ', s')Y (0, O) ,
 2+ 1
 -- csc 0 (a0a9 cot 09) P (cos 7)
 27x


2 csc 0 (e90 cot 80o ) [Y Y*'m(0 U)Yem(0, )
 Tn


2). From





 (5-141)




 (5-142)


where cos 7 = cos 0 cos 0' + sin 0 sin 0' cos(q '). In the coincidence limit xa x',
we have cos7 -+ sin0' cos((' C*) cos where 0* -= o JrA/f (r2 + J2),
with the definition of the rotated angles (6, 4) given in Eq. (4-151). Using the
chain rule along with Eq. (4-151), we may rewrite Eqs. (5-141) and (5-142) in the


S ,