The substitution of the values of x, z, x′, y′, z′ in this equation will give immediately the value of cot α; and if we put ζ, ζ’ for the corresponding azimuths on a sphere, or on the supposition e = 0, the following relations exist
| cot α − cot ζ = e² | cos φ Q |
| cos φ′ Δ |
| cot α′ − cot ζ′ = −e² | cos φ′ Q | |
| cos φ Δ′ |
Δ′ sin φ − Δ sin φ′ = Q sin ω.
If from B we let fall a perpendicular on the meridian plane of A, and from A let fall a perpendicular on the meridian plane of B, then the following equations become geometrically evident:
| k sin μ sin α = (a/Δ′) cos φ′ sin ω |
| k sin μ′ sin α′ = (a/Δ) cos φ sin ω. |
Now in any surface u = 0 we have
k² = (x′ − x)² + (y′ − y)² + (z′ − z)²
| −cos μ = [ (x′ − x) | du | + (y′ − y) | du | + (z′ − z) | du | ] / k ( | du² | + | du² | + | du² | ) | 1/2 |
| dx | dy | dz | dx² | dy² | dz² |
| cos μ′ = [ (x′ − x) | du | + (y′ − y) | du | + (z′ − z) | du | ] / k ( | du² | + | du² | + | du² | ) | 1/2 | . |
| dx′ | dy′ | dz′ | dx′² | dy′² | dz′² |