From (79), with 6 replaced by (p, we have 



(sin 01 + sin 0j)^ (sin <;6i - sin (fi^^ 



1 + cos d 1 - cos d 



(sin 00 + sin <ji^^ (sin 0i — sin (f)^' 



1 + cos d 



1 — cos d 



= 2[sin'0j, 



= 2[sin^0o cos (di + d^)] , 



(95) 



Substituting from (95) in (94) we obtain finally 



S = a d - (eV8) (Xd - 3Y sin d) 



- (3eV512) f64(Xd + Y sin d) + (5 sin 2d - 30d) X' (96) 



-40 (sin d)XY -10(sin2d)Y■ 

 - (5eVl2,288) [T432d - 72 sin 2d) X' + 576 (sin d) XY - 144 (sin 2d) Y"^ 

 + (63 sin 2d - 210 d) X' + (21 sin 3d - 315 sin d) X'Y 

 _- 126 (sin 2d) XY^' - 28(sin 3d) Y' 



If, in (96), we place e^ = 2f, ignoring all terms above first order in f, one obtains the second 

 of equations (80), or the Andoyer-Lambert approximation in terms of geodetic latitude, 0. 



Now the Andoyer-Lambert forms can be obtained from other modifications of differential 

 equations. For instance if the differential for arc length along the geodesic is taken in the form, 

 [8] page 64, 



ds = (N' cos='0/No cos 0o) dA, N = a/(l - e' sin^)'/^; (97) 



if the differential of arc length from (84), after converting to geodetic latitude is written 



ds = [(1 - e'sin>)-'/^ - (1 - e^ sin^^o)'"/^] DSd; (98) 



and if (97) and (98) are combined with the relationship dX = = (sin a^/cos <f) DSd = (cos 0a/cos^0) DSd 

 from the differential right triangles above with 6 replaced by 0, one can write 



(ds/a) = DSd + 



a - e' sin^ )"■' (1 - e^' sin^o)'/'- 1 

 + (1 - e') '/' id - e' sinV)~'^'- (1 - e ^sin '0„)-'/^ 



DSd 



(99) 



Expanding the expressions in (99) to first order terms in f, e^ = 2f, equation (99) can be written 

 in the integral form 



S = a [d - f / ' (2 sin'0o - 3 sin^) DSd]. (100) 



di 



Comparison of equations (100) and (91) (with e^ = 2f) shows that (100) will again give the 

 second of equations (80) or the Andoyer-Lambert Approximation in terms of geodetic latitude. 



74 



