A method is given which can be used for finding approximate solutions to problems in linear elasticity. It is an extension of the use of an extremum principle mentioned by Prager and Synge involving both the stresses and the strains. By using this principle Hooke's Law can be approximately imposed on statically admissible stress fields and kinematically admissible strain fields which contain adjustable parameters. The extension uses the same extremum principle to impose approximately the symmetry of the stress tensor as well. This allows the use of a vector stress function, applicable to 2D and 3D problems, which, while ensuring that the equations of force equilibrium are satisfied, does not impose the symmetry of the stress tensor. A mixed finite-element formulation is developed with the displacement vector and a vector stress function as nodal variables. A simple 2D finite-element program is described which implements this method and an example of its use is discussed. Advantages over some existing mixed formulations are: it allows permissible stress jumps across a surface; the global matrix is symmetric and positive-definite; only C0 continuity is needed between the elements for both types of variable.