The work reported in this thesis is based on the vierbein field formulation of gravitational theory, used in conjunction with the method of the compensating field. It is shown that the most general linear equations of second order for a tensor field, which are invariant under orientations of the local inertial frame and under gauge transformations of the vierbein field components are identical with Einstein's field equation written down in the weak field approximation. An attempt is made to take into account any possibly existing gravitating effect of gravitation by applying the method of the compensating field to the weak field Lagrangian, resulting in a set of nonlinear field equations. The invariance properties of the modified field equations are examined, and some special solutions are exhibited.