sin ψ sin ½v = e sin ½ (u′ − u) cos ½ (u′ + u),
we obtain finally the following equations:—
| k | = 2a cos ψ sin ½v |
| cos μ | = Δ sec ψ sin ½v |
| cos μ′ | = Δ′ sec ψ sin ½v |
| sin μ sin α | = (a/k) cos u′ sin ω |
| sin μ′ sin α′ | = (a/k) cos u sin ω. |
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 ζ, ζ′ it may be shown that
α + α′ = ζ + ζ′ + ¼e4ω (φ′ − φ)² cos4 φ0 sin φ0 + ...,
where φ0 is the mean of φ and φ′, 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 α + α′ on the spheroid is equal to the sum of the spherical azimuths, whence follows this very important theorem (known as Dalby′s theorem). If φ, φ′ be the latitudes of two points on the surface of a spheroid, ω their difference of longitude, α, α′ their reciprocal azimuths,
tan ½ω = cot ½ (α + α′) {cos ½ (φ′ − φ) / sin ½ (φ′ + φ)}.
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
e4s5 cos4 φ0 sin² 2α0/360 a4.