TRANSFORMATION FROM SECOND ORDER FORM IN GEODETIC LATITUDE 

 TO SECOND ORDER IN PARAMETRIC 



In terms of geodetic latitude, the equations corresponding to (132) are: 



- = d'-(f/4) (X'd'-3Y'sind') 

 a 



+ (fVl28) (AX'+ BY'+ CX" + DX'Y'+ EY") 



A = 64d ' + 16d " cot d ', B = - 96 sin d ' - 16d " esc d ', (138) 



C =- 30d'- 15 sin 2d'- 8d " cot d', 



D = 48 sin d'+ 8d" CSC d', E = 30 sin 2d' 

 (See Equation (109), page 78. 

 Equation (132) is written here in the form: 



l=d- (f/4) (Xd - Y sin d) (139) 



a 



+ (fVl28) (AoX + BoY + C„X= + D„XY + EoY^) 

 Ao = led"" cot d, Bo = - 16d^ CSC d, Cj, = 2d + sin 2d - Sd'' cot d. 

 Do = 8d^ CSC d, Eo = - 2 sin 2d 

 Now we should be able to find transformation equations of the form: 



d'=d'(d, X, Y), X'=X'(X, Y, d), Y'= Y'(Y, X, d) (140) 



which when substituted in (138) should produce equations (139). 

 The definitions of X', Y ' and X, Y are: 



X'= 2 sin'<;6o, X = 2sin='0o (141) 



Y'= 2 sin ^00 cos (di+ d/), Y = 2 sin ^do cos (di + dj) 

 where 00 > ^o ^e respectively geodetic, parametric latitude of the vertex of the great 

 elliptic arc. From the equation tan = (1 - f) tan 0, or equivalent, we find: 



00 = 6lo + f sin ^ cos doil + i cos %). (142) 



From the values indicated by Forsyth on page 120, of his treatise, to first order in f, 

 extending the results to second order in f we find: 



d ' = d - (f/2) Y sin d + (fVl6) [4Y (X - 3) sin d + (2Y' - X') sin 2d] (143) 



and to first order in f, 



cos (dj'+ d 2) = cos (d, + d2) + f cosd sirf ^o ~ ^ "^o^ ^ ^'^^ ^0 cos^ (d, + dj) . (144) 



From (142), to first order in f, find 



2 sin '00 = 2 sin 'd^ (1 + 2f cos %). (145) 



90 



