e = arc sin (A cos S + B) (47) 



A = Ao - arc sin (C sin S/cos 6) 

 AN ALTERNATIVE EQUATION TO THE SPHERICAL HYPERBOLA WITH ORIGIN AT A FOCUS 

 If S = /^(ao+ bo + C(,) in the spherical triangle 



bo = 2c 



Figure 7. 



then tan^ /^A 



sin (s - bo) sin (s - c „) 



,[6]. 



sin S sin (s - ao) 

 Referring to figure 6, ao = CTj , bo = 2c, Co = R: and from (27) we have the conditions 



ai - R = 2a, CTj + R = 2(R +a). 

 Hence 



s = '/4(ffi +R) + c = R+a + c, 



s - ao = !4(R -ffi) + c = c - a, 



s-bo = R+a-c, S-Co = c+a 



A = 77 - ^, tan HA = tan (77/2 - /S /2) = cot j8/2 



(48) 



(49) 



With the values from (49) placed in (48) find 



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



tan'j8/2 = - -_ , 



sin(c + a) sin (R — c + a) 



which is the desired alternative form, [7] . 



CORRESPONDING PLANE HYPERBOLA EQUIVALENTS 



For the plane case and analogous reference system. Figure 5 becomes 



(50) 



34 



