An attempt is made to study in a simplified model of a binary substitutional alloy the influence of the differences in atomic radii of the constituent atoms and in the potential energies of the various possible pairs of atoms. The theory is developed for linear, simple cubic, and body centred cubic lattices, and it is found that the order-disorder transition in the three-dimensional lattices can be either first or second order depending on the differences between the various potential energies. The results are applied to the case of -brass. The agreement of theory with experiment is as good as can be expected from a theory involving necessarily a large number of simplifying assumptions. The effect of lattice vibrations on our results is considered briefly.