In the present case, if we put
| 1 − | xx′ | − | zz′ | = U, |
| a² | b² |
then
| k² | = 2U − e² ( | z′ − z | ) | ² |
| a² | b |
cos μ = (a/k) ΔU; cos μ′ = (a/k) Δ′U.
Let u be such an angle that
| (1 − e²)½ sin φ = Δ sin u |
| cos φ = Δ cos u, |
then on expressing x, x′, z, z′ in terms of u and u′,
U = 1 − cos u cos u′ cos ω − sin u sin u′;
also, if v be the third side of a spherical triangle, of which two sides are ½π − u and ½π − u′ and the included angle ω, using a subsidiary angle ψ such that