In a "second supplement", Fresnel eventually exploited two related facts: (i) the "wave surface" was also the ray-velocity surface, which could be obtained by sectioning the ellipsoid that he had initially mistaken for the surface of elasticity, and (ii) the "wave surface" intersected each plane of symmetry of the ellipsoid in two curves: a circle and an ellipse. Thus he found that the "wave surface" is described by the 4th-degree equation