(1) tan^r 



SPHERICAL HYPERBOLA FORMULAS AND PLANE EQUIVALENTS, [7] 



SPHERICAL PLANE (60) 



tan^a (sin^c - sin^a) a (c - a ) 



r " 



sin^c cos^a - sin^a c cos a- a 



• 2 2 2 2 



Sin a cos c 2 ^ ^ 2 



(2) sin^x sin y + sin a x " — ^+a 



sin^c-sin^a c -a 



cos2c ± cos2a a - c 



(3) tan R = R = 



sin^c cosjfi + sin 2a c cosjS - a 



sin (c - a) sin (R + c + a) , , ^ (c-a) (R + c + a) 



(4) tan=(|8/2) tanM/8/2)= 



sin (c + a) sin (R - c + a) (c+a) (R - c + a) 



In (1) and (2) of equations (60), the origin of coordinates is the midpoint M,, of the segment 

 Q1Q2, see Figure 5. (3) and (4) are two polar forms with origin at a Focus Qj, see Figures (5) 

 and (6). 



REFERENCES 



[6] Chauvenet, Plane and Spherical Trigonometry, 1871, page 158. 



[7] Equations (32), (34), (42), (50) to spherical hyperbolas are essentially those given without 

 derivation in LORAN, Pierce, McKenzie, Woodward, McGraw Hill 1948, pages 173, 175. 



37 



