A definition of the ideally polarisable electrode which allows a general statistical mechanical treatment is given, comparisons with the usual thermodynamic approach being made. It is shown that, with this definition, the system behaves as a pure capacitance to an imposed alternating potential of sufficiently low frequency. The equilibrium states of the system are described by the grand canonical distribution. Thermodynamic functions, given in terms of the grand canonical partition function, can be computed in molecular terms. In particular, general expressions for the surface tension and the pressure are obtained, and compared to those proposed by various authors. The separation of the pressure into electrical and non-electrical parts is analyzed. Expressions are given for the electrochemical potentials of charged and of dipolar species. With these microscopic definitions, the Gibbs adsorption isotherms and the Lippmann equation are derived. For several simple models, including that of Gouy and Chapman for an ionic solution and a metal electrode, these relations are explicity considered. The inadmissability, in a coherent model, of reducing the solvent to a medium of fixed dielectric constant, is emphasized.