To design and fabricate cone bits, one must know how to predict the distribution of the temperature field in the supports of the roller bits. Theoretical calculation of nonstationary temperature fields under a nonsymmetric thermal loading is a separate serious mathematical problem. The edge conditions, and also certain system parameters of this problem are not precisely defined, since the temperature of the surrounding medium during drilling at great depths, the coefficient of friction in the bearings, and a number of other parameters, which cannot be assigned with the accuracy required for its solution, are unknown. Since it is currently impossible to obtain the temperature distribution in the bearing during drilling at great depths by experimental means, we decided to develop a computational-experimental method of determining the temperatures on the slip surfaces. The first stage of this method co~ists in the building of a mathematical model. The second stage consists in its experimental confirmation, which can be done during drilling at shallow depths. The third stage requires the development of a procedure for measuring the temperatures at individual points of the bits at great depths. The fourth stage involves determination of the temperature field using experimentally obtained temperature values at individual points and their use as edge conditions in the mathematical model. The problem statement for building a mathematical model of the temperature field in the body of a roller bit and journal assumes the following form. A system of bodies (shank journal and roller bit), which are in contact via friction surfaces (Fig. 1), is given; the heat sources are situated on the contact boundaries of the bodies (heat liberation in the bearings occurs as a result of friction). The thermophysical properties of the bodies and the initial temperatures on their perimeters are given. It is necessary to find the temperature distribution during operation of the bit on the contact surfaces of the bodies (the contact in the bearings). The problem can be resolved in the plane statement. The equation of nonstationary heat conduction in plane rigid bodies is