In a recent paper, the solubility of hydrogen in palladium was discussed. The statistical theory employed was essentially the same as that given by Fowler† for critical adsorption, the only modification necessary being to take into account the dissociation of the hydrogen molecules on absorption. In the present paper Peierls' theory‡ is applied to the same problem in order to enable a comparison to be made between the theories. According to both theories we must postulate that the absorbed (adsorbed) particles go into a definite number of potential energy holes and that the heat of absorption increases as the number of holes filled increases. Fowler uses the Bragg and Williams type of approximation to determine the energy of the particles in the holes; while Peierls supposes that there is an interaction energy V for each pair of neighbouring adsorbed particles and that all those arrangements of the particles in the holes having the same number of nearest neighbours have the same energy. The calculation of the dependence of the energy on the arrangements is approximated to by a method due to Bethe.