Two versions of the principle of minimum potential energy for the frictionless adhesive contact between a smooth-surfaced elastic and a smooth-surfaced rigid body are set up. The non-variational first-order equalities equivalent to the variational principles are given, as well as a condition for stability. They lead to the conclusion that in the case of isotropy the intensity of the inverse square root singularity of the surface traction at the edge of contact is a constant composed of the material constants of the bodies. This property appears to be a powerful tool in solving special problems, notably in the case of the Hertz problem where it is shown that, except in the axisymmetric case, the contact area is not elliptic. Although the theory has been set up for the contact of an elastic body with a rigid body, the theory is easily extended to cover the contact of two elastic bodies as well.