The problem on the terminal motion of a thin elliptic plate over a horizontal plane taking into account orthotropic friction forces has been considered. Differential equations of the movement of the plate have been derived. The system of equations has been numerically solved under various initial conditions. It has been shown that sliding and spinning cease simultaneously. It has been found that the limiting behavior of the plate is governed not only by the ratio of the moment of inertia to the coefficient of friction, but also by the orientation of the plate. A comparison of the behaviors of the elliptic and circular plates has been carried out. The results of the calculations can be used to describe the phenomena that occur in a rail–wheel contact in more detail.