From lllh, lllo, lllr we write a second formula for tan $ : 



tan $ = tan j/'csc a^ - cA (esc V'+ cot^tto) tan i/' esc a,, 



+ c tan V CSC a^ 



V sec v' CSC v' - B cot Cq + (9/64) +(1/32) sin^j/' 



+ —(2 - csc^i^') - (1/16) sec^Op 

 4 



+ A^ (cscVcsc^Co + cot''ao + /^ cot^ap) 



From lllg, llle, 111k, lllp, lllq, lilt we can write: 

 cos 6 - cos a„ cos v' + c -0 



(118) 



(119) 



+ c cos an cos 1/ 



— cos 2 I/'- V tan ly'- (5/64) sin V- (3/32) sinV 

 4 



- B tan Oo - AMI + ¥2 cot^^oo + ¥2 cot^i/') / 



Now in (119), the coefficient of c was zero as it should be and the coefficient of c^ must 

 be zero since cos 6 = cos Oo cos v' . Placing the coefficient of c^ in (119) equal to zero find: 



- B cot ao = A^(l + % cot^ ao + Vi cot^i/') cot^Oo cos 2 i/' cot^Oo 



4 



(120) 



+ V tan i/'cot^Qo + (5/64) sinVcot^Oo + (3/32) sin''i/'cot^ao 



With the value of - B cot Oq from (120) placed in the second order term of (118) and with 

 some manipulation through the identities Ills, we can write (118) as: 

 tan $ = tan v ' esc Oq — c A cot v ' esc Oo sec ' 



+ c' esc oo sec'(^'/A' cot i^'d + (3/2) cot'a„) + V \ (121) 



+ — (sin 2 v' - cot v') + (1/16) sin 1v' 

 4 



- (3/256) sin 4i/'- (1/32) sin 1v' tan 'ao 



From (117) and (121), since tan 0'= tan i/'csc a^ from Ills, the coefficients of the terms 

 in c and c^ must be respeceively equal. Equating the second order terms in (117) and (121) and 

 solving for V we find: 



V = »P sin ao - '/2A' cot v' cot ^a^ (122) 



^ [ 2i/'tan ^ao(l + csc^ao)-sin 2i/'+ cot v' {\ - 2 tan ^Cq)] 



4 



' sin 1v' 3 sin ^v' tan a,, sin 1v 



+ — tan^a (1 - 6 tan^a ) 



16 16 



256 



32 



From llli, 111b, 111m, lllp, lllq, the value of U in terms of A from lilt, and V fror 

 (122) we may write: 



84 



