and the last equation of (56) may be written 



sin do \/l + cos (d, - d^) 

 = arc tan 



V2-sin^eJl + cos (d.-d,)] . (58) 



It is known that Ho^ = PP '^ will be given by Ho^ = [(y - yj) r - (z - z,) q]^ + [(z - z,) p - 

 (x - x,)r]^ + [(x - Xj) q - (y - yjp]^, where x,y, z, are coordinates of P; Xj, y,, Zj are co- (59) 



ordinates of Oi and p, q, r are direction cosines of the chord c = QiQj, [ill. See Figure 14. 

 From (56) and (58) we can express the rectangular coordinates of P as 

 a cos 01+ cos 02^08 AA 



X = a cos d cos A 



y = a cos 6 sin A 



z = b sin 6 



yf2 Vl + COS (di + dj) 

 a cos 03 sin A A 



VT Vl + cos(d,+d2) (60) 



sin 1+ sin 0^ 



VT Vl + cos (d,+dj) 



If the coordinates from (l) are converted to parametric latitude they will be Qj (a cos 6^, 



0, b sin 0j); Qj ( a cos 0j cos A A, a cos 0^ sin A A, b sin 6^ whence the direction cosines of 



the chord c = QiQj are 



a 

 p = — (cos 62 cos A A — cos Sj) 

 c 



q =— cos $2 sin A A (61) 



c 



b 



r = — (sin 0, - sin 0.) 

 c 



From (60) and the coordinates of Q, (a cos di, 0, b sin d^) we have 



a 

 X - X, = (cos Oi + cos ^jCosAA) — a cos 6^ 



V2Ro 



y-y. = (a cos eij sin AA)/V'2Ro (62) 



_b_ 

 z - z, = (sin 0, + sin 0.) - b sin 0. 



W2R0 



Where R„ = V 1 + cos (d^ + dj = /2" cos H(d, + d,). 

 With the values from (61) and (62) the expression (59) is formed to give 

 2 / r^ _ p \2 



Ho'^ — cosXcos'eJb'(tan='0,+tan'6l2-2tan d^tand^ cosAA)+a'sin 'AAI (63) 



c K„ 



56 



