The present paper develops a nonlinear theory for the deformation of an elastic surface by assuming the existence of a strain energy function and postulating a principle of virtual work which governs its mechanical behavior. By considering the strain energy function to depend on the first- and second-order deformation gradients, the field equations and the general constitutive relations are obtained. In addition to the conventional couple stresses, there are shown to exist energetically undetermined double stresses without moment.