With the identity (cos dj - cos d^Y = [1 ~ cos (di + dj)] [1 - cos (d^ - d^)], 

 we can write (47) finally as 



C = a|il-cos (d, + d,)n2-kMl-cos (d,-d,)]l|'/^ . (48) 



Now (48) gives the chord length no matter which latitude is used, or d, since for ^: 

 dj = arc cos (Nj sin ^Si /No sin 0o)j dj = arc cos (Nj sin (^j/Nosin 0o)» 

 k^' = [e'(l - e=')/a')] N„'sin'<;6o; while for 0: 



d^ = arc cos (sin ^j/sin 6„), dj = arc cos (sin ^^/sin 6^), k^ = e^sin^^o . Also (41) and (48) 

 make it possible to prepare a computing form in terms of either cf) or 6 with corresponding 

 azimuth forms from equations (12), (13), (15), (16), (17), (18). 



THE ANGLE BETWEEN THE CHORD AND THE HORIZON AT A GIVEN POINT OF THE 

 ELLIPSOID 



Referring to Figure 13, it is seen that the angle /3 is determined by a perpendicular, u, 

 from Q2 upon the tangent at Qj and the chord c. That is sin B = u/c. 



Now the length of u is obtained by normalizing the equation of the tangent plane at Qi, 

 equation (4), and substituting the coordinates of the point Q2 from (1): 



u = — [a^ -NjNj cos 0, cos 02 cos AA- (1 - e^) I^ N2 sin , sin 02 ]• (49) 



We can express u in parametric latitude, 6, since (1 — e^) N,N2 sin 0isin 02 = 



a^ sin 01 sin $2, N^Nj cos 0i cos 02= a^ cos 6^ cos $2, Nj = (a/\/ 1 - e ) yj I - e^ cos^ d^ , 



1 - (sin 0, sin 6^ + cos 6^ cos 6^ cos A A) 



u = aVl-e' , (50) 



Vl-e'cos='0, 



Referring to equation (46) and the discussion there, 



cos (dj + d2) = sin ^^sin d^ + cos 0^ cos 6^ cos A A, 



sin di = sin d^ cos dj, k = e sin 0^ and (50) can be written in the form 



1 - cos (dj+dj) 



(1-e^ +k^cosM,)'/^ ' 



Where b = a \/l — e^ is the minor semiaxis of the reference ellipsoid. From (48) and (51) 

 we have then 



( (1 - e') [1 - cos (di + d2)] I 



sin « = — = 



)[2-k'jl-cos (d,-d2)n (1 -e' + k'cosM,)j 

 and thus sin j8 is expressed in the same quantities as the distance and chord lengths; see 

 equations (41) and (48). 



(51) 



(52) 



54 



