From pages 111, 117 of Forsyth find: 

 tan ^I'sin Ug = tan i/j, cos ^i = tan Oq tan Ij, cos v^ cos an = sin Ij, 

 tan (^2 sin ao= tan 1/2, cos <j!)2= tan Oq tan I2, cos 1^2 cos Oo = sin Ij, 

 whence 



cos li sin 0,' sin 1, 



= 1 , (108) 



sin i^j cosi/iCosoq 



cos Ij sin (p2 sin Ij 



sin 1^2 cos 1^2 cos Co 



The values from (108) placed in (107) prove the second of (106) and thus Gougenheim's paper 

 provides an independent check of the corrections given here to Forsyth's second order term. 

 Gougenheim also gave formulae for azimuths, convergence of the meridians, and difference in 

 longitude between the spheroidal and spherical (elliptical) vertices of geodesies in terms of the 

 same variables. The importance of Gougenheim's work has not been recognized. He has had a 

 correct expansion including the second order term in the flattening, in print since 1950. 



THE FORSYTH-ANDOYER-LAMBERT TYPE APPROXIMATION IN GEODETIC LATITUDE WITH 

 SECOND ORDER TERMS 



With the corrections to (102), i.e. with the numerical coefficient of *1 as 15/64 and the term 

 *2 omitted, equation (102) may be written, with relations (103) and (95), as 



S = a[d - (f/4) (Xd - 3Y sin d ) + (fVl28) (AX + BY + CX' + DXY + EY^)], (109) 



where a, f are the semimajor axis and flattening of the reference ellipsoid; d is the spherical 

 distance between the points Pj {<f)^, X^,), Pj {(^21 ^2) on the ellipsoid given by some spherical 

 formula as cos d = sin (p^ sin (f)2 + cos <^i cos c/Sj cos AX; is geodetic latitude, A is longitude, 

 AA = Aj - A, ; A = 64d + 16d' cot d, D = 48 sin d + 8d' esc d, B = - 2D, E = 30 sin 2d, 



(sin (/>! + sin ^j)^ (sin <f>i - sin (^12)^ 



C = -(30d +8d' cotd +E/2), X 



1 + cos d 1 - cos d 



(sin 01 + sin ^2^^ (sin 0i - sin ^j)^ 



Y = - ; d = d2 - d,, where di and d2 are spherical distances 



1 + cos d 1 - cos d 



from the vertex of the great elliptic arc to the points Pi (cjSi , A,), Pj (02 » ^2)- 



Now by factoring sin d out of every term of (109) and using the azimuth formulae as given by 



Lambert, [13], we can, by means of trigonometric identities, arrange equations (109) in a form 



more convenient for computing as follows: 



78 



