By introducing complex variable techniques in the theory of shallow shells, the problem is reduced to the solution of one fourth-order differential equation, and the displacement components and stress resultants and couples are expressed in terms of four complex potentials. Two of these complex potentials are those of thin plate theory, thus admitting of deducing the solution of a shallow shell problem from the limiting case of the circular flat plate. These results are applied to a case of non-rotational symmetry, a special case of which has practical significance in the design of tanks and boilers.