Note in Fisnre 23 that t\\o values of longitude are given, A and As. A is the longitude 

 associated with the point where maximum separation of chord and arc occurs but corresponding 

 to the rectangular coordinate system as defined in say Figure 14. As is the true geodetic 

 longitude of the same point and is of course obtained by adding A to Aj since A^ is negative. 



While a continuous svstem based on either the great elliptic section as depicted by Figure 

 17, or the Forsyth- Andoyer-Lambert approximation. Figure 23, mil pro\"ide all the information 

 as indicated and accurate enough for hyperbolic electronic systems and any present operational 

 requirements, the Forsvth-.\ndover-Lambert is probably to be preferred because of computational 

 simplicitv and existence of programs already operating for high speed computers. Should the 

 need arise for accuracy of the order of 1 meter in distance and 1 second in azimuth over the 

 ellipsoid, the extension to second order terms in the flattening pro\'ided by equations (110) or 

 (137), will suffice. 



Manv formulae are available for geodetic lines and differential corrections are available for 

 transforming elements such as geodetic azimuths to normal section azimuths, etc. [24]. Lsually 

 these are complicated, involve tabular material for a particular spheroid of reference, require 

 extensive root computation, and acctiracv depends on line length. For instance, Bomford says 

 Rudoe's formulae for the reverse problem, are "Lnnecessarily complex for general use," [21], 

 page 108. Also thev give "Normal Section" distances — not geodetic. The difference between 

 the geodesic and the Normal Section distance is of 4th order in the eccentricity.- of the meridian 

 ellipse [24]. 



Finally this investigation has raised the question as to whether either .\ndoyer or Lambert 

 should share anv credit for the first order approximation formula in terms of parametric latitude 

 recoEnizable intact in Fors%"th's 1895 paper. While Forsyth had an erroneous second order term 

 to the same expansion in terms of geodetic latitude, his first order term was correct and he thus 

 had both so-called Andover-Lambert formulae. Gougenheim apparently had in 1950 the first 

 correct expansion in jHint in terms of geodetic latitude which included the second order terms in 

 the flattening. 



REFERENCES (Distance Investigation) 

 [8] Confonnal Projections, P. D. Thomas, C. & G. S. Special Publication No. 251, G.P.O. 1952, 



pages 63, 72. 

 [9] Helmert, F. R. Die mathematischen und physikalischen Theorieen der Hoheren Geodasie 1, 



chapters 5 and 7, Leipzig, 1880. 

 [10] Conic Sections, C. Smith, MacMillan 1930, page 164. 



96 



