Formulae for the determination of the elements of the E\"otvos matrix of the Earth's normal gravity field and a relation between normal and actual Gaussian curvature

In this paper we form relations for the determination of the elements of the E\"otv\"os matrix of the Earth's normal gravity field. In addition a relation between the Gauss curvature of the normal equipotential surface and the Gauss curvature of the actual equipotential surface both passing through the point P is presented. For this purpose we use a global Cartesian system (X, Y, Z) and use the variables X, and Y to form a local parameterization a normal equipotential surface to describe its fundamental forms and the plumbline curvature. The first and second order partial derivatives of the normal potential can be determined from suitable matrix transformations between the global Cartesian coordinates and the ellipsoidal coordinates. Due to the symmetry of the field the directions of the local system (x, y, z) are principal directions hence the first two diagonal elements of the E\"otv\"os matrix with the measure of the normal gravity vector are sufficient to describe the Gauss curvature of the normal equipotential surface and this aspect gives us the opportunity to insert into the elements of the E\"otv\"os matrix the Gauss curvature.