T = d/sin d = 1 + (t/6) + 3(tV40) + 5(t7ll2) + 35(tVll52) + 63(t72816) +,(1 radian - 206,264.8062 seconds 



E = 30 cos d , A - 4T (8 + TE/15), D = 4(6 + T^^), B - -2D, 



C = T - '/zCA + E), f/4 - 0.000847518825, fV64 - 0.179572039x10"' (Clarke 1866) 



S - a sin d [T - (f/4) (TX - 3Y) + (fV64) { X(A + CX) + Y(B + EY) + DXYi], 



sin (oj + Oi) = (K sin AA)A^, sin (oj- a^) = (k sin AA)/(1 - L), 



%(da2 + §c%) = -<f/2) H (T + 1) sin (a, + a,), 'AiSa^ - Sa.) = -(f/2) H (T - 1) sin (g, -a), 



''i-z = «! + 8fii, ctj^i = Cj + Scj. 



Additional check formulae 



(sin cf>. + sin ch.y (sin d>, — sin c6,)^ 



X = 1 -i- + Ll r!i = 2 sin ^ </.o = 2F/(F + sin^AA) 



1 + cos d 1 — cos d 



(sin <^i+ sin (jij)^ (sin <^, — sin ^2)^ 



Y = = 2 sin ^<f)a cos (di + d2) 



1 + cos d 1 - cos d 



F = tan ^ (fei + tan ^02 ~ 2 tan ^1 tan (f)^ cos AX 



cos (di + d^) = Y/X, 1 + cos d = 8kV(X + Y), 1 - cos d = 8KV(X - Y), 



/ k^ K'\ /k' kA 



cos d = 4 , 4 + = 1 . 



\X+Y X-Y/ yX+Y X-Y/ 



NOTE: If the second order term is ignored, the resulting equations are the equivalent of the so called 

 Andoyer-Lambert approximation in terms of geodetic latitude. 



The quantities H, T, L, k, K enter into both distance and azimuth formulas. Distances are given 

 within a meter and azimuths within a second over all lines in all latitudes and azimuths. Other advantages 

 are (1) no conversion to parametric latitudes, (2) no square root calculations, (3) for desk computers the 

 only tabular data required is a table of the natural trigonometric functions as Peter's eight place tables. 

 (4) the formulas are adaptable to high speed computers. See Table 4 page 81 and Appendix 3, lines 12 

 through 16, for desk computer sample computations based on these formulas as checked against 5 Coast 

 and Geodetic Survey specially computed lines. The mean difference for the 5 lines between true geodetic 

 lengths and computed values was 0.15 meter with a maximum difference of 0.24 meter. The mean difference 

 between true and computed azimuths was 0.59 second with a maximum difference of 0.93 second. 



GEODESIC IN TERMS OF GREAT ELLIPTIC ARC, IN PARAMETRIC LATITUDE WITH SECOND ORDER 

 TERMS IN THE FLATTENING 



Given on the reference ellipsoid the points Pi(^i, A,), Pi^^i^ '^2); P2 west of Pj , west longitudes 

 considered positive. (Geodetic latitudes are converted to parametric by the relation tan = (1 — f) tan cf) 

 or an equivalent formula). With d = Viid^ + dj, Mm = '/^(^2 - ^i)> AX = A^ - Ai, AA^, = AA/2; 



