N. = k74, N, = kVl28 = 1/8 N.\ N3 = kVl536 = (1/3) N.N^, 



Q, = sin H cos P, Qj = sin 2H cos 2P, Q3 = sin 3H cos 3P, H = d^ + dj, P = d, - dj. 

 di and dj are computed from 



cos 2di = 2(1 - cos^0,)/sin^9o " 1 

 cos 2d2 = 2(1 - cos^02)/sin^0o - 1 

 since cos^0, and cos^^j are already needed for T, and Tj, (67) above, and the use of sin^^^ 

 eliminates the computation of the square root of K/(K + 1). A check is provided by 

 sin (di + dj) = sin 6^ sin 62 + cos d^ cos d^ cos A A. 



From (48) the equation of the chord may be written 



c = a(VW) '/^ V = (1 - cos H), W = 2 - k'R, R = (1 - cos P). 

 From (51) and (52) in terms of the symbols used above find 



b / V 



u = bV/Ti sin B = bV/cT, = — , / — . 



From (64) in terms of the above symbols find Ho 



2ab„ 



(69) 



(sin YzE) (1 - cos '/2H), 



(70) 



(71) 

 (72) 



hg = a\J 1 - k , k^ = e^ sin do- 

 Figure 17, shows equations (65) through (72) arranged for computing and a computation 

 performed on the line Moscow to Cape of Good Hope. On the form find the geodetic distance, 

 the normal section azimuths, the chord distance, the angle between the chord and the horizon 

 at P,, and the maximum separation of the chord and surface. The following table lists these 

 values and gives a comparison with the distances computed by the rigorous Helmert method and 

 the Andoyer-Lambert Approximation. Note that the geographic coordinates of the point 

 P(<^,A) where the maximum chord separation from the surface occurs may be computed from (56), 

 (58), and already computed quantities in Figure (17). 



MOSCOW TO CAPE OF GOOD HOPE 



DISTANCE AZIMUTHS 



Meters n.m. Method Forward Back Type 



10,102,069.91 5454.6814 Great Elliptic 15°46' 56';744 190°39' 27':350 Great Elliptic Section 



15° 49' 57 '1607 190° 41' 29" 799 Normal Section 



15° 48' 17'1674 190° 39' 32';208 Geodetic 



15° 48* 17':518 190° 39' 32':il0 Geodetic 



10,102,069.06 5454.6809 Helmert 



10,102,065.28 5454.6789 Andoyer- 

 Lambert 



CHORD DISTANCE 

 (MAXIMUM CHORD SEPARATION) 

 CHORD DEPRESSION ANGLE 



meters 

 9,068,419.05 

 1,906,854.55 

 45° 32' 37':462. 



n.m. 

 4896.5546 

 1029.6191 



59 



