Now let [phi], [phi]' be the latitudes of two stations A and B; [alpha], [alpha]^* their mutual azimuths counted from north by east continuously from 0 deg. to 360 deg.; [omega] their difference of longitude measured from west to east; and s the distance AB. First compute a latitude [phi]1 by means of the formula [phi]1 = [phi] + (s cos [alpha]) / [rho], where [rho] is the radius of curvature of the meridian at the latitude [phi]; this will require but four places of logarithms. Then, in the first two of the following, five places are sufficient-- s^2 s^2 [epsilon] = ------- sin [alpha] cos a, [eta] = ------- sin^2[alpha] tan[phi]1, 2[rho]n 2[rho]n s [phi]' - [phi] = ---- cos ([alpha] - 2/3[epsilon]) - [eta], rho0 s sin (alpha - 1/3[epsilon]) [omega] = ----------------------------, n cos ([phi]' + 1/3[eta]) [alpha]^* - [alpha] = [omega] sin ([phi]' + 2/3[eta]) - [epsilon] + 180 deg. Here n is the normal or radius of curvature perpendicular to the meridian; both n and [rho] correspond to latitude [phi]1, and [rho]0 to latitude 1/2([phi] + [phi]'). For calculations of latitude and longitude, tables of the logarithmic values of [rho] sin 1", n sin 1", and 2 n [rho] sin 1" are necessary. The following table contains these logarithms for every ten minutes of latitude from 52 deg. to 53 deg. computed with the elements a = 20926060 and a : b = 295 : 294 :-- +------+------------------+--------------+--------------------+ | | 1 | 1 | 1 | | Lat. | Log.------------.| Log.--------.| Log.--------------.| | | [rho] sin 1" | n sin 1" | 2[rho]n sin 1" | +------+------------------+--------------+--------------------+ |deg. '| | | | |52 0 | 7.9939434 | 7.9928231 | 0.37131 | | 10 | 9309 | 8190 | 29 | | 20 | 9185 | 8148 | 28 | | 30 | 9060 | 8107 | 26 | | 40 | 8936 | 8065 | 24 | | 50 | 8812 | 8024 | 23 | |53 0 | 8688 | 7982 | 22 | +------+------------------+--------------+--------------------+ The logarithm in the last column is that required also for the calculation of spherical excesses, the spherical excess of a triangle being expressed by a b sin (C/2[rho]n) sin 1".