Machine elements，such as gears and anti-friction bearings are subjected to rolling　and sliding contact loads which cause surface deterioration in the form of pitting or spalling，or dimensional change. The former occurs in hard materials and latter soft materials. Dimensional change occurs as plastic deformation which accumulates with increasing load cycles. One of the best-known results in mechanical engineering comes from the classical analysis by H. Hertz of the elastic contact. It has long been realized that surfaces are rough on a microscopic scale，and that this causes the real area of contact to be extremely small compared to the nominal area. It is usually assumed that the real area of contact between two nominally flat metal surfaces is determined by the plastic deformation of their highest aspeties. This leads at once to the result that the real area of contact is directly proportional to the load and independent of the apparent area. Archard pointed out that plastic deformation could not be the universal rule，and introduced a model which showed that，contrary to earlier ideas，the area of contact could be proportional to the load even with purely elastic contact. Most of the attention of the past work has been upon the determination of relationships between the real contact area and geometric parameters. However，little has been done to determine the effect of these parameters upon the state of stress within the bodies. Herin，a model is constructed that enables one to study the effect of asperity density in the rolling-sliding contact. The theoretical equations for the stress component in a semi-infinite elastic body by periodically spaced semi-ellipsoid pressures are derivated by using a Galerkin vector potential function.