A numerical scheme, suitable for digital computer use, is presented for the solution of the elasto-hydrodynamic problem. The variation of viscosity with pressure, compressibility, and, of course, surface deformations are included in this analysis. The method is applicable over a wide range of loads—from the lightly loaded situation where there is simple hydrodynamic lubrication up to the heavily loaded cases where the pressure distribution is closely related to the Hertzian pressure distribution for the contact of unlubricated surfaces. A series of solutions are presented which show the importance of the various effects and a study is made showing the transition region between the hydrodynamic and Hertzian pressure distributions. ¹ Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Carnegie Institute of Technology. Contributed by the ASLE Technical Committee on Gears and Gear Lubricants and presented at the Annual Meeting of the American Society of Lubrication Engineers held in St. Louis, Missouri. May 1962.