He solved that problem in a "supplement" signed on 15 January 1818 (mentioned above). In the same document, he accommodated Malus's law by proposing an underlying law: that if polarized light is incident on a birefringent crystal with its optic axis at an angle θ to the "plane of polarization", the ordinary and extraordinary vibrations (as functions of time) are scaled by the factors cos θ and sin θ, respectively. Although modern readers easily interpret these factors in terms of perpendicular components of a transverse oscillation, Fresnel did not (yet) explain them that way. Hence he still needed the phase-inversion rule. He applied all these principles to a case of chromatic polarization not covered by Biot's formulae, involving two successive laminae with axes separated by 45°, and obtained predictions that disagreed with Biot's experiments (except in special cases) but agreed with his own.