2. The system of first order differential equations of the plumbline and its numerical solutions

At a given point P in space, the gradient vector of the potential -- the direction of which coincides with the normal to the equipotential surface passing through the given point -- is tangential to the plumbline going through P.
The first order ordinary differential equations of the plumbline are : where g = < g_x, g_y , g_z > is the acceleration vector.This kind of system of differential equations has standard numerical solutions .
Euler's method and the Richardson extrapolation implemented in the Bulirsch-Stoer method  were used to accomplish the computational scheme above.
The application of the Frenet frame to describe locally the plumbline requires not only the first derivatives of the gravitational potential but also the second and third derivatives.