The spherical length, d, is determined from formulae as given with Figure 16, 

 (d = d, +d,); 



cot A = (cos di tan 62 - sin di cos AA)/sin AA 

 cot B = (cos 62 tan 6^ — sin 62 cos AA)/sin AA 

 sin d = cos ^ sin AA/sin B = cos^jsin AA/sin A; 

 these will compensate for any unfavorable triangle geometry. 

 The Andoyer-Lambert Formulae are taken in the form [13] 

 Sdj. = - (f/8) (VQVsin^yad + URVcos' Vid) 



(1) s = a(d + Sd ), Q = sin (9, - sin 6^ , R = sin 6^ + sin 62. 



V = d + sin d, U = d — sin d. 



With corresponding geodetic azimuths computed from 

 T = (f/2) d'Vsin d, SA"= T cos % sin 2B, 



(2) SB " = T cos '6I1 sin 2A; ga^^g = 180° - A + SA; gag^ = 180° + B - 5B 

 The Normal Section Azimuths may be written 



(3) cotjjO^g = - (cot A)/T, + (e'Q cos 0,)/(sin AA)T, cos 6^ 



cot ag A = (cot B/Tj + (e^Q cos ^^^/(sin AA)T2 cos d^ 

 T. = (1 - e' cos '9, Y" T, = (1 - e' cos % Y'^ 



The chord length becomes 



(4) c = a(4sinM/2-e^0')'^' 



The angle of dip of the chord may be written 



(5) ^ = arc sin [2b (sin ='d/2)/cTJ 



b = semiminor axis of ellipsoid, c = chord length, Tj = (1 - e^ cos^^i)''^. 

 The maximum separation of chord and arc becomes 



(6) H = (aVc) (1 - cos '/2d) [4 sin M/2 (cosM/2 - M) - e'0']'/Vcos V2A 



a = the semimajor axis of ellipsoid, c = chord length, M = e^ sin d^ sin 62, 

 Q = sin 62 ~ sin 6^ , e = eccentricity of the spheroid. 

 The geographic coordinates of the point where maximum separation of chord and arc occurs 



(7) tan A = (cos 62 sin AA)/(cos di + cos 62 cos AA) 



tan = R/(0.996609925) V 4 cos' 'Ai - R' 

 where R = sin 6^ + sin 02- 



Figure 23, shows the above formulae arranged in a computing form and the computations done 

 over the line MOSCOW TO CAPE OF GOOD HOPE. See line No. 12, TABLE 1, and Figure 17. 



94 



