Lamb's circular vortex-pair was shown by the author  to be the unique maximiser, up to axial translations, of a certain integral functional of the stream function subject to a kinetic energy constraint. The present paper proves that the axial translations of Lamb's circular vortex-pair are the unique maximisers of kinetic energy subject to constraints on the vorticity and impulse. This formulation is more natural, in that it involves only conserved quantities. The result is deduced from  by non-convex duality. The analogous theory for Hill's spherical vortex is also discussed briefly.