Where Ro = [1 + cos (d. + d^)]'''' = /2 cos 'Ad, + dj). 



Using the relationships (42), (48), (57) equation (63) can be solved for Hq in any of the 

 following several forms: 



bo (V2-\/l + cos (d, +dj) , ^ 



H„ = - , (64) 



V2-kMl -cos(d, -dj)! 



abo /(V2 \ 

 = — ^ — -lsin(d, +d,), 



c \ Ro / 



2abo 

 = sin ^(d, + dj) [1 - cos '/^(d, + dj)] , 



Where Ro = V 1 + cos (d^ + dj) = V2 cos VM, + d^) 



bo = \/l - k^ ^ a\/l - eo^ = minor semiaxis of the great elliptic arc — see Figure 15. Thus 

 Ho is also expressed in quantities common with other elements of the great elliptic arc — see 

 equations (41), (48), and (52). 



A COMPUTING FORM FOR GREAT ELLIPTIC ARC LENGTH AND ASSOCIATED ELEMENTS 



Since the computations to be discussed with the great elliptic arc approximation and the 

 Andoyer-Lambert approximation both involve corrections to spherical elements, the basic spherical 

 approximation is reviewed in Figure 16, and basic spherical formulae listed. 

 Now from (42) write 



sin'6»o = K/(K + 1), 



K = (A tan d, + B tan 6^)/ sin^' A A (65) 



A = tan 01 - tan ^j cos A A , B = tan ^2 - tan 6 , cos A A . (66) 



Azimuth equations (17) become 



cot aAB= D, (R, - B), cot aBA = Dj (A - Rj) 



D, - cos e,/T, sin A A, Dj = cos d^/T^ sin A A (67) 



R, - C/cos 02, Rj " ~ C/cos 6i 



C = e^ (sin $2 ~ sin 0,) 



Tj = (1 - e^ cos 'd,) '/' , T, = (1 - e^ cos %) 'I' 

 Equation (41) becomes 



s = a (H + U, + Uj + U3) (68) 



where U. = -N^ (H - Q,), U^ = - N^ (6H - 8Q, + Q^), 



U3 = - N3 (30H - 45Q, + 9O2 - Q3) 



k^ = e^ sin ^ 00 " ^o^ (eccentricity of the great elliptic arc). 



57 



