INVESTIGATION OF HIGHER ORDER TERMS 

 m ANDOYER-LAMBERT APPROXIMATION 



While either form of Andoyer-Lambert approximation is probably satisfactory in the "state 

 of the art" in hyperbolic navigational systems development, the question arises as to the higher 

 order terms in the flattening of the Andoyer-Lambert approximation and the possibility of a single 

 set of formulae which will give distance within one meter and azimuth within one second over all 

 geodetic lines on the spheroid. This would be a practical operational system particularly if it 

 maintained the several attributes of the Andoyer-Lambert first order approximation. 



HISTORICAL 

 Now Lambert, [13], never published his derivation but had equivalent formulae for a first 

 order approximation several years before the publication posthumously in 1932 of Andoyer's 

 sketch, [15], of the derivation of the form as given in equation (74). Andoyer's derivation 

 employs a differential oblique spherical triangle and it is not clear how one would proceed to 

 higher order terms in the flattening. It is believed that Andoyer's derivation is the only 

 recognized published one in existence. 



DERIVATION FROM THE GREAT ELLIPTIC ARC 

 Independent derivations of the Andoyer-Lambert approximations were sought in the hopes of 

 discovering a simple method of arriving at higher order terms in the flattening. It was noticed 

 that the computations using the Andoyer-Lambert approximations; the ratios (d — sin d)/(l + cos d), 

 (d + sin d)/(l - cos d) were being used in forming computational parameters, [16]. It was decided 

 to try the ratios 



(sin 6^ + sin ^2)7(1 + cos d),(sin 0, - sin 62)^/(1 - cos d) (76) 



with the hope of relating these to other parameters and identification of the Andoyer-Lambert 

 approximations in some other extant series expansion as the great elliptic arc approximation. 

 See equations (19) through (42). 

 From equations (42) we have 



sin 9i = sin do cos d^, sin 62 = sin 6^ cos dj. (77) 



From (77), by simple algebraic operations and trigonometric identities, we may express 

 (76) as 



(sin di + sin diY/il + cos d) = 2 sin^^o cos^ YiHi + dj) 



(sin ^1 - sin ^2)7(1 - cos d) = 2 sin'6lo sin '!/2(di + dj) , , (78) 



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