Given on the reference ellipsoid the points Pj {(pi, Aj), Pj (02; Aj), 4' ^^ geodetic latitude, 

 A is longitude, Pj is west of Pj with west longitudes considered positive. 



With cf>^ = a/2) {cf>i + 4>^), A0^= (1/2) (<^2-0,), AA= A2-A,,AA^=(l/2) AA; 



Let: k = sin cos A0 , K = sin A0j^ cos (f)^^, 



H = cos ^A4>^ - sin >jj, = cos '0JJJ - sin 'A^^^j , 



L = sin 'A0^ + H sin 'AAjjj = sin'(d/2), 1 - L = cos'(d/2), cos d = 1 - 2L, t = sinM=4L(l-L), 



U = 2kV(l - U, V = 2KVL, X = U+V, Y = U-V, 



T = d/sin d = l+(t/6) +3(tV40) + 5(tVll2) +35(tVll52) + 63 (tV2816) + , 



E = 30 cos d, A = 4T (8 + TE/15), D = 4(6 + T'), B = - 2D, C = T - 'AiA + E), (110) 



S = a sin d [T - (f/4) (TX - 3Y) + (fV64) ! X(A + CX) + Y (B + EY) + DXY !] ; 

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



(%) ida, + 8ai) = - (f/2) H (T + I) sin {a, +ai), {'A) (8a, -8ai)= -(f/2) H (T- 1) sin (a^-a^), 

 <^i-2 ^ Oi ■*■ 8ai , a2_i = 02+ Sa, • 

 Note that the quantities H, T, L, k, K enter into both distance and azimuth formulas. 

 Figure (21) shows an arrangement of equations (110) for desk computing using an ordinary 

 ten bank electric desk calculator and Peters eight place tables of trigonometric functions. It 

 is arranged to show the contribution of both the first and second order terms in the flattening. 



Table 4 summarizes the results of computations over 17 lines of known lengths and 

 azimuths. The computations are given in Appendix 3. Part of these lines were used in the 

 computations of Appendix 2. The first 11 lines are from two ACIC publications [12], lines 12 

 through 17 are Coast and Geodetic Survey specially computed lines, [22]. 



Note that all distances are within one meter and azimuths are within one second which 

 was the objective since this is adequate for any operational requirement. Other advantages 

 are (l) no conversion to parametric latitudes, (2) no square root calculation, (3) for desk 

 computers the only tabular data required is a table of the natural trigonometric functions as 

 Peters eight place tables, (4) the formulas are adaptable to high speed computers, (5) about 

 the same accuracy is obtained over all lines in all azimuths and latitudes. 



EXPANSION TO SECOND ORDER TERMS IN f USING PARAMETRIC LATITUDE 



Forsyth [18], gave an expansion of the geodesic to first order in the elliptic modulus 

 c = (e^ cos^a)/(l - e^ sin^a) where a is the complement of the parametric latitude of the vertex 

 of the geodesic. (See pages 118-120 of his treatise). We will follow the Forsyth method and 



79 



