In this paper, the partial differential equations governing the motion of plates when transverse shear and rotary inertia are considered are solved for plates bounded by parabolic cylinders. Three cases have to be considered, depending on the shape of the plate. They are:Case I:a plate bounded by two parabolic arcs (v=v1,u=u1,u= −u1);Case II:a plate bounded by three parabolic arcs (v=v1,v=v2,u=u1,u= −u1);Case III:a plate bounded by four parabolic arcs (v=v1,v=v2,u=u1,u= −u2). After converting the partial differential equations and boundary conditions into parabolic coordinates, product solutions are assumed and solutions of the resulting ordinary differential equations appear as definite integrals. The eight different types of boundary conditions entering into the theory, for each case, are satisfied by taking linear combinations of products of these definite integrals. The elimination of the arbitrary constants in each boundary problem leads to the frequency equation for the normal modes of vibration. The frequency equations for Case II and Case III are given by determinants of the 12th order equaled to zero, whereas for Case I the determinant is of the 6th order. In the classical theory, the frequency equations far Case II and Case III are given by determinants of the 8th order equated to zero, whereas for Case I a determinant of the 4th order is required.