A new Reynolds equation, based on the Eyring theory of non-Newtonian flow, is derived for flow in one dimension. It is shown that this new equation reduces to the traditional Reynolds equation as the Eyring model approaches the Newtonian model in the limit. Numerical solutions are presented for a selected oil at two different temperatures. The central film thickness decreases with increasing dimensionless viscosity parameter and slide/roll ratios. A transition zone is noted through which the ratio of minimum to central film thickness passes as the pressure distribution goes from near Hertzian to a distribution that appreciably deviates from Hertzian.