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In this paper we form relations for the determination of the elements of theE\"otv\"os matrix of the Earth's normal gravity field. In addition a relationbetween the Gauss curvature of the normal equipotential surface and the Gausscurvature of the actual equipotential surface both passing through the point Pis presented. For this purpose we use a global Cartesian system (X, Y, Z) anduse the variables X, and Y to form a local parameterization a normalequipotential surface to describe its fundamental forms and the plumblinecurvature. The first and second order partial derivatives of the normalpotential can be determined from suitable matrix transformations between theglobal Cartesian coordinates and the ellipsoidal coordinates. Due to thesymmetry of the field the directions of the local system (x, y, z) areprincipal directions hence the first two diagonal elements of the E\"otv\"osmatrix with the measure of the normal gravity vector are sufficient to describethe Gauss curvature of the normal equipotential surface and this aspect givesus the opportunity to insert into the elements of the E\"otv\"os matrix theGauss curvature.

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