where d = dj - dj. 



From (78) by adding and subtracting respective members, we write 



(sin d, + sin d^Y (sin 0, — sin diY 

 X = — + = 2 [sin' e^] 



1 + cos d 



1 - cos d 



(79) 



(sin d, + sin d.Y (sin 6, - sin d^Y 



Y = — = 2[sin'0o cos (d, + dj)], 



1 + cos d 1 - cos d 



where d = d2 - d,. 



The Andoyer-Lambert forms can now be written in terms of X and Y of (79) as 



S = a[d - (f/4) (Xd - Y sin d)], 



S = a[d - (f/4) (Xd - 3Y sin d)], 



where in the second equation, the geodetic latitude, <f), is used in forming the X and Y of 



(79). 



If in the expansion of the great elliptic arc, equation (41), we place d, = to ~d, , and then 



d = dj - di , k = e sin dg , we obtain as far as sixth order terms in e: 



1 - '4 e' sin ^6a [d - sin d cos (d^ + d^)] 



- (l/128)e'' sin''6lo[ 6d - 8 sin d cos (d, + dj) + sin2d cos 2(d, + d^)] 



(80) 



(l/1536)e' sin'(9o 



30d - 45 sin d cos (dj + d^) +9 sin 2d cos 2(di + dj) 

 - sin 3d cos 3(di + dj) ; 



(81) 



Using relations (79), equation (81) can be written: 



S = 



d - (eV8) (Xd - Y sin d) 

 - (eV512) [(6d - sin 2d) X' - 8(sin d) XY + 2(sin 2d) Y^ 



(eVl2,288) 



3(10d - 3 sin 2d) X' - 3(15 sin d - sin 3d) X'Y 

 L + 18(sin 2d) XY' - 4(sin 3d) Y' 



(82) 



Note in (82) that if all terms above the first power in f are ignored (e' = 2f) equation (82) reduces 

 directly to the Andoyer-Lambert form as given by the first of (80). Now it is known that the 

 difference in lengths of the great elliptic arc and the geodesic is of 4th order in e, [17], but the 

 6th order term will be useful for comparison later in the investigation. 



DERIVATION FROM MODIFIED DIFFERENTIAL EQUATIONS 

 The corresponding differential triangles, auxiliary sphere, spheroid, where geodetic latitude 

 has been converted to parametric are, as abstracted from Figure (20): 



69 



