A spherical rectangular coordinate system with a great circle base line as an axis. 



Figure 5, page 29, shows, pictorially, tMs coordinate system developed on the great circle 

 base line referenced to the vertex of the great circle base line. The conversion equations are 

 developed in equations(14)to(26), pages 28 to 30. 



Derivation of the equations of spherical hyperbolas and their plane equivalents. 



Having established a spherical rectangular coordinate system we are in a position to derive 

 the equations of spherical hyperbolas referenced to the system. This is done in both spherical 

 rectangular coordinates and spherical polar form, equations (27) to (50), pages 31 to 34. See 

 also figures 5, 6, and 7, pages 29, 32,34. 



The plane hyperbola equivalents are developed in equations (51) to (59), pages 35 and 36 

 and a comparison table of the spherical and plane equivalents is given as equation (60), page 

 37. See also Figures (8) and (9), pages 35 and 36. 



An example of computations using the plane hyperbola approximation is given as Appendix 1, 

 pages 99 to 110. 



Distance computations and conversions; Azimuths; Associated geometrical quantities. 



The classical ' inverse' problem of geodesy was considered here since it is inherent in the 

 electronic navigational systems problem. In the ' inverse" problem, the latitudes and longitudes 

 of each of two points are given from which the distance between the points and the azimuths at 

 the two given points are to be determined. 



The geodesic on the reference ellipsoid, other than meridians and circular equator, is a 

 space curve, and its vertex (the latitude where it is orthogonal to a meridian) is not easily ex- 

 pressible in terms of the geographical coordinates (latitude and longitude) of two points on it. 

 The actual length involves the evaluation of an elliptic integral, whose modulus depends on the 

 latitude of the vertex of the geodesic. Iterative solutions have been devised as Helmert's, 

 based on the earlier work of Bessel. 



Approximations based on plane curves which are near the geodesic in length as the normal 

 sections and the great elliptic arc have been devised. An investigation of these was made, 

 including some extensions for instance in the series development for the great elliptic arc 

 approximation. See pages 48 to 51 and Figure 15, page 50. Also their use and expression in 

 terms of common computational parameters with some associated geometrical quantities useful 

 in operational applications as the angle of depression of the chord below the horizon, the 

 maximum separation between the chord and the surface, and the geographic coordinates of the 

 point on the sxu'face where maximum separation occurs. 



An investigation of the expansion of the geodesic length in powers of the flattening was 

 made which to first order in the flattening are the well-known, so-called Andoyer-Lambert 



