An investigation has been made of the equilibrium of a thin incompressible elastic membrane lying between two rigid bodies, one fixed and the other mobile. The present paper concerns the corresponding problem for a compressible membrane.Membranes are divided mathematically into four classes according as their compressibility is (i) finite, (ii) small, (iii) very small, (iv) so small that the membrane may be considered as incompressible. The movable boundary is assigned a rigid body displacement. By the introduction of approximation based on the thinness of the membrane a determination of the resulting stress, strain and displacement is made for each of the above four classes of membranes.