Generalized Legendre Expansions for Any Curve Whatever, Traced Out Upon Any Surface Whatever

The Legendre expansion, or also Delambre’s formula, are the names given in Geodesy to those series of the power of the arc s of a geodesic traced on the ellipsoid of rotation, which permit us to determine the geographic coordinates φ 1 and λ1 (latitude and longitude) of the extremity P 1 of the geodesic of finite length s, and of azimuth α1 at P 1, if we know the geographic coordinates φ 0 and λ0 of the other extremity P 0, and the azimuth of the geodesic also at P 0