A method for assessing the presence of a non-double-couple component in seismic sources is presented. The equation for the principal values of the moment tensor is a cubic equation with coefficients which are polynomial combinations of the tensor components. These coefficients are invariants of the tensor that relate to its symmetry and do not depend on the coordinate system used to describe the source. Each coefficient has a physical interpretation: The constant coefficient is determined by how well one or two double-couples describe the source, and the quadratic coefficient is the volume change associated with the source. If volume change is assumed to be absent, the constant term represents a measure of the double-couple nature of the source. A seismic waveform inversion method is presented which exactly explores the range of values of these properties. The method of extremal models is used to determine minimum and maximum values of the invariants subject to the constraint that the data must be satisfied within prescribed errors. For the invariant associated with the source volume change, this constraint results in a linear programming problem and a global solution may be found. For the other invariants a nonlinear programming problem results which may be solved by a reduced gradient algorithm. Two events were examined: A deep earthquake from the Bonin Islands and the Harzer nuclear explosion. A double-couple mechanism is essentially compatible with the Bonin Islands waveforms. Models with a positive volume change were required to fit the Harzer data. Some variation in the mechanism is possible due to gaps in the station distribution and possibly due to significant scattering from lateral heterogeneities. However, in spite of a range in the isotropic component, the event could not be fit with a double-couple mechanism.