k^2 = (x' - x)^2 + (y' - y)^2 + (z' - z)^2 _ _ | du du du | / / du^2 du^2 du^2 \ 1/2 -cos [mu] = |(x' - x) -- + (y' - y) -- + (z' - z) -- | / k ( ---- + ---- + ---- ) |_ dx dy dz_|/ \ dx^2 dy^2 dz^2 / _ _ | du du du | / / du^2 du^2 du^2 \ 1/2 -cos [mu]' = |(x' - x) --- + (y' - y) --- + (z' - z) --- | / k ( ----- + ----- + ----- ). |_ dx' dy' dz'_|/ \ dx'^2 dy'^2 dz'^2 / In the present case, if we put xx' zz' 1 - --- - --- = U, a^2 b^2 then k^2 /z' - z \ ^2 --- = 2U - e^2 ( ------ ) a^2 \ b / cos [mu] = (a/k) [Delta]U; cos [mu]' = (a/k) [Delta]'U. Let u be such an angle that (1 - e^2)^1/2 sin [phi] = [Delta] sin u cos [phi] = [Delta] cos u, then on expressing x, x', z, z' in terms of u and u', U = 1 - cos u cos u' cos [omega] - sin u sin u'; also, if v be the third side of a spherical triangle, of which two sides are 1/2[pi] - u and 1/2[pi] - u' and the included angle [omega], using a subsidiary angle [psi] such that sin [psi] sin 1/2v = e sin 1/2(u' - u) cos 1/2(u' + u), we obtain finally the following equations:-- k = 2a cos [psi] sin 1/2v cos [mu] = [Delta] sec [psi] sin 1/2v cos [mu]' = [Delta]' sec [psi] sin 1/2v sin [mu] sin [alpha] = (a/k) cos u' sin [omega] sin [mu]' sin [alpha]' = (a/k) cos u sin [omega]. These determine rigorously the distance, and the mutual zenith distances and azimuths, of any two points on a spheroid whose latitudes and difference of longitude are given. By a series of reductions from the equations containing [zeta], [zeta]' it may be shown that [alpha] + [alpha]' = [zeta] + [zeta]' + 1/4e^4[omega]([phi]' - [phi])^2 cos^4 [phi]0 sin [phi]0 + ..., where [phi]0 is the mean of [phi] and [phi]', and the higher powers of e are neglected. A short computation will show that the small quantity on the right-hand side of this equation cannot amount even to the thousandth part of a second for k < 0.1a, which is, practically speaking, zero; consequently the sum of the azimuths [alpha] + [alpha]' on the spheroid is equal to the sum of the spherical azimuths, whence follows this very important theorem (known as Dalby's theorem). If [phi], [phi]' be the latitudes of two points on the surface of a spheroid, [omega] their difference of longitude, [alpha], [alpha]' their reciprocal azimuths, tan 1/2[omega] = cot 1/2([alpha] + [alpha]') {cos 1/2([phi]' - [phi])/ sin 1/2([phi]' + [phi])}.