Now equation (40) factors into [tan R (sin 2c cos /3 + sin 2a) — (cos 2c + cos 2a)]. 



[tan R (sin 2c cos j3 — sin 2a) — (cos 2c — cos 2a)] = 0. (41) 



Whence 



cos 2c + cos 2a cos 2c — cos 2a 



tan R = , tan R = 



sin 2c cos /3 + sin 2a sin 2c cos j8 — sin 2a 



or 



cos 2c ± cos 2a , . 



tan R = — ■ , (42) 



sin 2c cos j3 ± sin 2a 



where either the (two plus signs) or (two minus) signs are taken together. 



Equation (42) is the polar equation to spherical hyperbolas referred to a focus as pole. 

 We now derive expressions for the spherical rectangular coordinates x, y as functions of the 

 polar coordinates R, jS. 



From right triangles WP "Q, WQQ„ Q.QQ' (Figure 6) find 



sin X = sin R cos (3, 



sin y = sin R sin ^. , . 



sin X = sin k cos y ; 



cos R = cos k cos y. (44) 



Equations (43) are similar to equations (14)and provide the conversions from polar to 

 rectangular coordinates, i.e. from (43) 



sin R = (sin'x + sinV) '^\ (45) 



tan P = sin y/sin x . 



Since moving the origin from M to Q, (see Figure 5) is only a translation along the x-axis, 

 there is no change in y, but x is changed. Hence from (44) and the relations (23) and (26) we 

 can write when the origin is at Qj, k = S - S,: 

 FORMULAS FOR COMPUTATION OF S,x,y, FROM AND A 



sin y = cos (?„ sin — sin 6^ cos 9 cos ( A „ — A ) 



sin do cos 6 sin ( Aq— A) cos sin ( Ao - A) 



tan S = = (46) 



sin 6 — cos do sin y cos f 



_ cos 9 sin ( Aq - A) 



sin 00 sin 9 + cos 9o cos 9 cos {\„~ A) 

 sin X = sin k cos y = sin (S — Sj) cos y 

 FORMULAS FOR COMPUTATION OF S, 9, A FROM x AND y 



Let C = cos y, D = sin y, E = sin x, A = C sin ^o, B = D cos 9o, then 



S = arc sin (E/C) + Sj 



33 



