x1 = rλ1, x2 = rλ2 ... xn = rλn, where Σλ² = 1.
Hence
ds² = (Radr / a² − r²)² + R²r²dΔ² / (a² − r²).
where
dΔ² = Σdλ².
Also calling ρ the geodesic distance from the origin, we have
| cos h (ρ/R) = | a | , sin h (ρ/R) = | r | . |
| √(a² − r²) | √(a² − r²) |
Hence
ds² = dρ² + (R sin h (ρ/R))² dΔ².
Putting