This thesis presents a systematic method for determining the forces on a simplified roller bit. The kinematics of an actual roller bit drilling in rock is complex; nevertheless, it is desirable to obtain an accurate analytical model. As a preliminary study, the roller bit is replaced in this thesis by a simple, two-dimensional bit and the rock is replaced by a Coulomb plastic material. Results from previous work, plus additional assumptions, were incorporated into a computer program to determine the forces required to move the bit in a specified manner. Twelve problems were worked to test the program. By studying similar simplified drilling problems, accompanied by supporting laboratory work, one could eventually obtain an accurate analytical model for drilling of an actual roller bit. This could lead to the compilation of a catalog of results which, in turn, would allow selection of the optimum drill bit for any given situation.