S = a d - (eV2) / ' (2 sin ^o - 3 sin'^S) D5d 

 d, 



- (3eV8) f^^ sin'9!.(4 - 5 sin'<;6) D5d 



- (5eVl6) f^^ sinV(6 - 7 sin^cf} DSd 



From (77), with replaced by tp, we have sin^c/) = — (1 + cos 2d), and with the aid of 



trigonometric identities we can find expressions for sin''0 and sin'e/), i.e. 

 sin^^o 



(91) 



sin (^ ■■ 



(1 + cos 2d), 



sin <^(, 



(3 + 4 cos 2d + cos 4d), 



(92) 



sin' 00 



sin*<i = (10 + 15 cos 2d + 6 cos 4d + cos 6d). 



32 



The values of sin ^(f>, sin''0, sin *(;6 from (92) placed in (91) give 



d - (e74) sinVo / Ml - 3 cos 2d) DSd 



d, n 



- (3eV64) sin ^<f>a j 



(5eV512) sinVo/ = 

 di 



(16 - 15 sin'0„) + (16 - 20 sin^o) cos 2d1 DSd 

 — 5 sin ^(pg cos 4d 



172 - 70 sin'^o) + (96 - 105 sin'^So) cos 2d' 



+ (24 - 42 sin^^o) cos 4d 

 _ - 7 sin ^00 cos 6d 



(93) 



DSd 



Integration of (93) with respect to d leads to: 



(94) 



S=a 



d-(e74) id [sin'^ol - 3 sin d [sin'cjSo cos (d^ + d^)]! 



- (3eVl28) 



- (5eVl536) 



32d [sin'<;6o] - 30d [sin'^o] ' + 32 sin d [sin''<;6o cos (d, + d^)] 



- 40 sin d [sin^^ol [sin^c^pcos (d, + dj)] 



- 10 sin 2d [sin^0o cos (di + di)]^ + 5 sin 2d [sin^^o^ 



'216d [sin'<;io]' - 210d [sin'<^or+ 288 sin d[sin'(^o] [sin'(?So cos (dj+dj)] 

 -315 sin d[sin^<;6ol' [sin^c^o cos (dj + d2)] + 72 sin 2d[sin'0o cos(d,+ d2)]' 

 - 126 sin 2d [sin'^J [sin'^So cos (d, + dj)]'- 36 sin 2d [sin' cf)oY 

 + 63 sin 2d [sin'<;6or- 28 sin 3d [sin '<^o cos (dj + d^)]' 

 _+ 21 sin 3d [sin'^o]'' [sin^<?i>o cos (d, + d^)] . 



73 



