2.3 Computation of the Frenet frame of the plumbline

        Restricting the investigations to the gravitational part of the gravity potential the actual orientation of the Frenet frame  depends on the direction of g*1 kg = gradV which gives the direction of the first base vector t in a position tangential to the curve, whereas gradV and the higher derivatives of the gravitational potential determine both the normal n and the binormal b base vectors of the frame:   In the Frenet frame <t, n, b> the local equation of plumbline is approximated by neglecting terms higher than order 3 :                                                                                                         X = Ds-Ds³k²/6
                                                                                                        Y = Ds²k/2                                        (4)
                                                                                                         Z = ktDs³/6   where both k, the curvature of the plumbline, and t, the torsion of the plumbline, are defined at the origin of the local Frenet frame by the expressions in Eqs. (5), where E is the Eötvös tensor,° denotes dyadic product and E_i is the i^th  row of the Eötvös tensor. The coordinates computed refer to the Frenet frame the origin of which is actually a running point along the plumbline. Therefore X,Y and Z have to be transformed to the x, y, and z coordinate system of the density model to obtain the coordinates of the extrapolated point of the plumbline being investigated.