In the Cowan-Kirkwood treatment of an electron gas, one particle is singled out and the relative motion described. The density for the new problem is then related to the pair function for the original. Two effects omitted from this model, the considerable velocity of the chosen particle and its statistical interaction with the remainder, are included in the present treatment. The pair function is considerably altered in detail but a calculation at a typical metallic density indicates that the two effects compensate energetically. This explains the success of the simple Cowan-Kirkwood treatment in this region. There are reasons why the present formulation should have a wider range of applicability than the original method.