The energy of a uniform electron gas can be specified completely in terms of its second-order density matrix. Mayer has therefore suggested that trial matrices satisfying all the usual conditions might be employed to determine variationally correlation energies and pair functions. Unfortunately, the particular choice made by Mayer did not satisfy all the Pauli conditions on the second-order matrix. However, matrices satisfactory from this point of view are presented here and the consequences of assuming such forms are investigated. Since one of the matrices is a direct generalization of the Hartree-Fock expression, a description of correlation effects is assured and it appears that this should be adequate for all densities.