The equations of the Debye-Hückel theory, modified to include quantum statistics, are discussed. It is found that the non-linear equations used by Cowan and Kirkwood are not unique, and that the non-linear theory can be formulated in different ways to give different answers. The linearized equations of these alternative formulations are discussed, and the correct form of the linearized theory is established. From the linear theory, the Helmholtz free energy of a slightly degenerate plasma is derived and, from this result, useful formulae in the near-classical limit are obtained for the pressure and internal energy.