As far as is known, no explicit solution exists in the literature for the displacement equations in u, v and w, for a uniform cylinder of arbitrary cross section subjected to a lateral pressure loading. However, the advent of Ref. 1 now makes available an admirable treatise devoted entirely to the analysis of thin elastic shells. The equations developed in this reference apply only to linear problems, i.e. the displacements are assumed to be small in comparison with the thickness of the shell, but they are general enough to include all shells of arbitrary curvature. Unfortunately the generality of these equations inhibits their immediate use to cylindrical shell problems, and it is the purpose of this note to present the essential features of the theory for non-circular cylinders.