Dividing respective members of (16) and (19) find 



tan S = cos 6 sin (A,, — A)/ cos f (21) 



where cos f is given by (18). 



From (17) and (18) we have sin da cos f = sin 6 - cos do sin y whence (21) may be 



written 



sin On cos 6 sin (Ao — A) , , 



tanS= "- (22) 



sin 6 — cos do sin y 



Referring now to Figures 1 and 5, it is seen that d = !VIQ'= S - /4(S, + S 2), where 

 S, and Sj are the distances from ( ^g , Ao) to Q, and Qj respectively. 



Hence given the spherical curvilinear coordinates 6, \ of a point Q{6,X), to find S, x 

 and y with Oo, Aq, Sj, Sj known, compute y and S from (17) and (21) or (22) and then x from (20), i.e. 



sin y = cos do sin 6 — sin Og cos 6 cos (A,, — A) 



sin 6„ cos 6 sin (A^— A) cos B sin (An — A) 



tan S = ■_ = (23) 



sin Q — cos 9o sin y cos f 



cos 6 sin ( Ao — A) 



sin Oo sin 6 + cos 6^ cos 6 cos (Ao - A) 

 sin X = sin d cos y = sinLS - /4(S, + S^)] (1 - sin^y) ^'^ 



COMPUTATION OF S, d, A FROM x AND y 



From equation (20) one has sin d = sin x / cos y or sin [ S - /^ (Sj + Sj )] = sin x / cos y 



whence 



S = arc sin (sin x / cos y) + ^(Sj + Sj). (24) 



From equations (13) page 27, 



sin (9 = A cos S + B (25) 



sin (Ao— ^)=C sin S / cos d 

 where A = C sin Oo, B = D cos do, C = cos y, D = sin y 



Hence to compute S, d, A from x and y, first compute S from (24) and then and A from 

 (25) i.e.: 



let C = cos y, D = sin y, E = sin x, A = C sin ^o, B = D cos ^o- 

 Then 



5 = arc sin (E/C) + /^ (S 1 + S 2) 



6 = arc sin (A cos S + B) (26) 

 A = Ao — arc sin (C sin S/cos 6) 



30 



