DERIVATIONS FROM EXPANSIONS OF FORSYTH 

 In reviewing the literature on geodetic computation one finds that A. R. Forsyth, [18], as 

 early as 1895 had given some series expansions for geodetic arc length in terms of the flattening 

 and certain spherical and elliptic parameters. On page 120 of his treatise one finds the expression 

 S,2/a = i^j'- i/i- !4 cd/j'- v{) + (1/8) c (sin 21^2'- sin 2i/,') . (101) 



Now the correspondences between the parameters as used by Forsyth in deriving (101) and 

 those used above in this investigation are to first order in f: 

 V2 = dj, Vi = dj, Vj — i^J= dj — di = d, c = 2f sin^ 9^, 



sin 21/2'- sin 2i/{ = sin 2d2 - sin 2di = 2 sin (dj - d^) cos (dj + dj) = 2 sin d cos (d^ + dj) 

 so that equation (101) becomes equivalently 



S = a[d - (f/2) id[sin %] - sin d [sin^^^ cos (d^ + dM, 

 which in turn by means of relations (79) can be written S = a[d - (f/4) (Xd - Y sin d)] , and 

 identified as the first Andoyer-Lambert form of equations (80). 

 On page 116 of Forsyth's treatise one finds the expression 



Sij/a = 1^2 ~ 1^1 + ^1(3/4) cos^ ati(sin 2i/2 - sin 2i/j) - (¥2) (fj" i^i^ cos^a,, S 

 + f ^ {Vt) (j^2~ i^i'^ cos ^Oo sin ^aosin 0i' sin cjj^/sm 2(^o 



+ (1/2 -1^1) [(1/16) cos" Co + cos^oo sin^Qo] (102) 



+ (3/8) sin'oo cos^Oo (sin 2(^2' ~ sin 2^1*) 

 - (3/4) cos^Qo sin^Oo (sin 2v2 - sin Iv^ 

 + (23/64) cos^a,, (sin ^v^ - sin 4t^j) 



Now the equivalent relationships between Forsyth's parameters as used in (102) and the ones 



used in this investigation are: 



1^1 = d,, 1^2 = <l2> I'a- 1^1 = d2 - di = d, f = f, 1, = <;6i, Ij = 4>i, 



2cf)o = 4>2~ 4>i - 4'2~ 01 = '^ ~ ^1 = ^^' '^os (f>i = cot <^o tan 0i= cos ^ cos d, sec 0i 



sin (f>i = sin d^ sec (fe^, cos (f)2 = cot <^(, tan 02 = cos 00 cos dj sec 02 (103) 



sin 02' = sin d2 sec 02, cos v 1= cos d j = sin 0i/sin 00 , 



cos 1^2 = cos d2 = sin 02/sin 00, 00 = "^ — 00, the relationship sin a osin (1^2 - h) 



= cos Ij cos Ij sin 20o given on pages 106, 121 of Forsyth, [18], 



becomes cos 0^^ sin d = cos 0^ cos 0^ sin A A in the notation of this investigation. 



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