If it be necessary to determine the geographical latitude and longitude as well as the azimuths to a greater degree of accuracy than is given by the above formulae, we make use of the following formula: given the latitude φ of A, and the azimuth α and the distance s of B, to determine the latitude φ′ and longitude ω of B, and the back azimuth α′. Here it is understood that α′ is symmetrical to α, so that α* + α′ = 360°.
Let
θ = sΔ / a, where Δ = (1 − e² sin² φ)1/2
and
| ξ = | e² θ² | cos² φ sin 2α, ξ′ = | e² θ³ | cos² φ cos² α; |
| 4 (1 − e²) | 6 (1 − e²) |
ξ, ξ′ are always very minute quantities even for the longest distances; then, putting κ = 90° − φ,
| tan | α′ + ξ − ω | = | sin ½(κ − θ − ξ′) | cot | α |
| 2 | sin ½(κ + θ + ξ′) | 2 |
| tan | α′ + ξ − ω | = | cos ½(κ − θ − ξ′) | cot | α |
| 2 | cos ½(κ + θ + ξ′) | 2 |
| φ′ − φ = | s sin ½(α′ + ξ − α) | ( 1 + | θ² | cos² | α′ − α | ); |
| ρ0 sin ½(α′ + ξ + α) | 12 | 2 |
here ρ0 is the radius of curvature of the meridian for the mean latitude ½(φ + φ′). These formulae are approximate only, but they are sufficiently precise even for very long distances.