The computation of the geodetic from the astronomical azimuths has been given above. From k we can now compute the length s of the vertical section, and from this the shortest length. The difference of length of the geodetic line and either of the plane curves is e^4 s^5 cos^4 [phi]0 sin^2 2[alpha]0/360 a^4. At least this is an approximate expression. Supposing s = 0.1a, this quantity would be less than one-hundredth of a millimetre. The line s is now to be calculated as a circular arc with a mean radius r along AB. If [phi]0 = 1/2([phi] + [phi]'), [alpha]0 = 1/2(180 deg. + [alpha] - [alpha]'), [Delta]0 = (1 - e^2 sin^2 [phi]0)^1/2, then 1/r = [Delta]0/a [1 + e^2/(1 - e^2) (cos^2 [phi]0 cos^2 [alpha]0)], and approximately sin (s/2r) = k/2r. These formulae give, in the case of k = 0.1a, values certain to eight logarithmic decimal places. An excellent series of formulae for the solution of the problem, to determine the azimuths, chord and distance along the surface from the geographical co-ordinates, was given in 1882 by Ch. M. Schols (_Archives Neerlandaises_, vol. xvii.). _Irregularities of the Earth's Surface._ In considering the effect of unequal distribution of matter in the earth's crust on the form of the surface, we may simplify the matter by disregarding the considerations of rotation and eccentricity. In the first place, supposing the earth a sphere covered with a film of water, let the density [rho] be a function of the distance from the centre so that surfaces of equal density are concentric spheres. Let now a disturbance of the arrangement of matter take place, so that the density is no longer to be expressed by [rho], a function of r only, but is expressed by [rho] + [rho]', where [rho]' is a function of three co-ordinates [theta], [phi], r. Then [rho]' is the density of what may be designated disturbing matter; it is positive in some places and negative in others, and the whole quantity of matter whose density is [rho]' is zero. The previously spherical surface of the sea of radius a now takes a new form. Let P be a point on the disturbed surface, P' the corresponding point vertically below it on the undisturbed surface, PP' = N. The knowledge of N over the whole surface gives us the form of the disturbed or actual surface of the sea; it is an equipotential surface, and if V be the potential at P of the disturbing matter [rho]', M the mass of the earth (the attraction-constant is assumed equal to unity) M M M ----- + V = C = -- - --- N + V. a + N a a^2