On the Curvature and Torsion of the Gravity Field

Considering two points P1 and P2 of the Earth’s external gravity field, sufficiently close to each other, and the two (vertical) versors n1 and n2, directed upwards which are tangent to the lines of force at P1 and P2, then the axes of two such versors will not, in general, be coplanar. When the two points lie on the same equipotential surface ∑, such a defect of coplanarity appears, as is well known, as the presence of the geodesic torsion of ∑ in the direction P1P2.If we put P2—P1 = st, where t is a unit vector, and s the length of the segment P1P2 which we suppose to be small to a first order, we then have from the generalized formulae of Frenet