[epsilon] [sigma] /m^2 - b^2 [b] - k \ [B] - B = --------- + -------k ( ---------k + -------- ), 3 3 \ 20 4k / [epsilon] [sigma] /m^2 - c^2 [c] - k \ [C] - C = --------- + -------k ( ---------k + -------- ), 3 3 \ 20 4k / in which [epsilon] = [sigma] k {1 + (m^2k / 8)}, 3m^2 = a^2 + b^2 + c^2, 3k = [a] + [b] + [c]. For the ellipsoid of rotation the measure of curvature is equal to 1 / [rho]n, [rho] and n being the radii of curvature of the meridian and perpendicular. It is rarely that the terms of the fourth order are required. As a rule spheroidal triangles are calculated as spherical (after Legendre), i.e. like plane triangles with a decrease of each angle of about [epsilon] / 3; [epsilon] must, however, be calculated for each triangle separately with its mean measure of curvature k. The geodetic line being the shortest that can be drawn on any surface between two given points, we may be conducted to its most important characteristics by the following considerations: let p, q be adjacent points on a curved surface; through s the middle point of the chord pq imagine a plane drawn perpendicular to pq, and let S be any point in the intersection of this plane with the surface; then pS + Sq is evidently least when sS is a minimum, which is when sS is a normal to the surface; hence it follows that of all plane curves on the surface joining p, q, when those points are indefinitely near to one another, that is the shortest which is made by the normal plane. That is to say, the osculating plane at any point of a geodetic line contains the normal to the surface at that point. Imagine now three points in space, A, B, C, such that AB = BC = c; let the direction cosines of AB be l, m, n, those of BC l', m', n', then x, y, z being the co-ordinates of B, those of A and C will be respectively-- x - cl : y - cm : z - cn x + cl': y + cm': z + cn'. Hence the co-ordinates of the middle point M of AC are x + 1/2c(l' - l), y + 1/2c(m' - m), z + 1/2c(n' - n), and the direction cosines of BM are therefore proportional to l' - l : m' - m : n' - n. If the angle made by BC with AB be indefinitely small, the direction cosines of BM are as [delta]l : [delta]m : [delta]n. Now if AB, BC be two contiguous elements of a geodetic, then BM must be a normal to the surface, and since [delta]l, [delta]m, [delta]n are in this case represented by [delta](dx/ds), [delta](dy/ds), [delta](dz/ds), and if the equation of the surface be u = 0, we have d^2x / du d^2y / du d^2z / du ---- / -- = ---- / -- = ---- / --, ds^2 / dx ds^2 / dy ds^2 / dz which, however, are equivalent to only one equation. In the case of the spheroid this equation becomes d^2x d^2y y ---- - x ---- = 0, ds^2 ds^2