It is frequently necessary to obtain the co-ordinates of one point with reference to another point; that is, let a perpendicular arc be drawn from B to the meridian of A meeting it in P, then, [alpha] being the azimuth of B at A, the co-ordinates of B with reference to A are AP = s cos ([alpha] - 2/3[epsilon]), BP = s sin ([alpha] - 1/3[epsilon]), where [epsilon] is the spherical excess of APB, viz. s^2 sin [alpha] cos [alpha] multiplied by the quantity whose logarithm is in the fourth column of the above table. If it be necessary to determine the geographical latitude and longitude as well as the azimuths to a greater degree of accuracy than is given by the above formulae, we make use of the following formula: given the latitude [phi] of A, and the azimuth [alpha] and the distance s of B, to determine the latitude [phi]' and longitude [omega] of B, and the back azimuth [alpha]'. Here it is understood that [alpha]' is symmetrical to [alpha], so that [alpha]^* + [alpha]' = 360 deg. Let [theta] = s [Delta] / a, where [Delta] = (1 - e^2 sin^2 [phi])^1/2 and e^2 [theta]^2 [xi] = ------------- cos^2 [phi] sin 2[alpha], (4 (1 - e^2) e^2 [theta]^3 [xi]' = ------------- cos^2 [phi] cos^2 [alpha]; (6 (1 - e^2) [xi], [xi]' are always very minute quantities even for the longest distances; then, putting [kappa] = 90 deg. - [phi], [alpha]' + [xi] - [omega] sin 1/2([kappa] - [theta] - [xi]') [alpha] tan------------------------- = ---------------------------------- cot ------- 2 sin 1/2([kappa] + [theta] + [xi]') 2 [alpha]' + [xi] + [omega] cos 1/2([kappa] - [theta] - [xi]') [alpha] tan------------------------- = ---------------------------------- cot ------- 2 cos 1/2([kappa] + [theta] + [xi]') 2 s sin 1/2([alpha]' + [xi] - [alpha]) / [theta]^2 [alpha]' - [alpha]\ [phi]' - [phi] = ----------------------------------------- ( 1 + ---------cos^2 ------------------ ); [rho]0 sin 1/2([alpha]' + [xi] + [alpha]) \ 12 2 / here [rho]0 is the radius of curvature of the meridian for the mean latitude 1/2([phi] + [phi]'). These formulae are approximate only, but they are sufficiently precise even for very long distances. For lines of any length the formulae of F.W. Bessel (_Astr. Nach._, 1823, iv. 241) are suitable. If the two points A and B be defined by their geographical co-ordinates, we can accurately calculate the corresponding astronomical azimuths, i.e. those of the vertical section, and then proceed, in the case of not too great distances, to determine the length and the azimuth of the shortest lines. For _any_ distances recourse must again be made to Bessel's formula.[4]