  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.
