approximation formulas, one in terms of parametric latitude, the other in terms of geodetic 

 latitude. Since this Office uses the Andoyer-Lambert form in terms of parametric latitude, in 

 which geographic latitudes must first be converted to parametric, an investigation was made to 

 see if use of the parametric form to first order in the flattening was justified or necessary in 

 terms of operational requirements. This was done in connection with a review of an extensive 

 study by USAF (ACIC) of geodetic lines up to 6000 miles in length where the Andoyer-Lambert 

 approximation was recommended for such tasks as LORAN computing, since the errors in the 

 very near geodetic distances obtained are fairly constant on lines 50 to 6000 miles in length 

 and in all azimuths. The comparisons are given in tables 1 — 3, pages 65 to 57. 



Since some of the excursions in the first order form were of the order of 10 meters, the 

 problem of obtaining the expansion of the geodesic to second order terms in the flattening was 

 examined. By introducing two parameters X and Y, in terms of the latitude of the vertex of the 

 great elliptic arc, it was found that the great elliptic arc approximation produced the so-called 

 Andoyer-Lambert first order approximations. (See pages 68 — 69.) Similarly they could be 

 produced by modification of the differential equation to the geodesic (See pages 69 to 74). 



In review of an 1895 paper by the British Mathematician, A. R. Forsyth, by identifying his 

 fundamental approximation parameter as the vertex of the great elliptic arc, it was found that 

 he actually had both so-called Andoyer-Lambert first order expansions in the flattening, but 

 it had apparently not been recognized. Furthermore, he had an expansion to second order terms 

 in the flattening and in terms of geodetic latitude but it had two errors in the second order term. 

 After these had been detected and corrected, computations based on the resulting equations 

 give distances within a meter on all lines computed from 50 to 6000 miles. See pages 75 to 81. 



Forsyth did not have the expansion to the geodesic in terms of parametric latitude to second 

 order terms in the flattening, so his results were extended to second order terms. See pages 

 79 to 90. Then transformation equations were developed to convert one form to the other as far 

 as second order terms in the flattening, pages 90 to 92, and finally the difference formulas for 

 the principal parameters, pages 92 to 93. As a result of this study, distance and azimuth 

 formulas are available in terms of easily computed parameters, in terms of either parametric 

 or geodetic latitude which will give distances over all lines within a meter and azimuths within 

 a second which is adequate for any operational requirement. A more detailed summary of the 

 investigations of this section with a bibliography of references is given on pages 93 to 97. 



