being the same in those special cases). Later commentators put the equation in the more compact and memorable form Earlier in the "second supplement", Fresnel modeled the medium as an array of point-masses and found that the force-displacement relation was described by a symmetric matrix, confirming the existence of three mutually perpendicular axes on which the displacement produced a parallel force. Later in the document, he noted that in a biaxial crystal, unlike a uniaxial crystal, the directions in which there is only one wave-normal velocity are not the same as those in which there is only one ray velocity. Nowadays we refer to the former directions as the optic axes or binormal axes, and the latter as the ray axes or biradial axes . Fresnel's "second supplement" was signed on 31 March 1822 and submitted the next day – less than a year after the publication of his pure-transverse-wave hypothesis, and just less than a year after the demonstration of his prototype eight-panel lighthouse lens . Second memoir (1822–26) Having presented the pieces of his theory in roughly the order of discovery, Fresnel needed to rearrange the material so as to emphasize the mechanical foundations; and he still needed a rigorous treatment of Biot's dihedral law. He attended to these matters in his "second memoir" on double refraction, published in the Recueils of the Académie des Sciences for 1824; this was not actually printed until late 1827, a few months after his death. In this work, having established the three perpendicular axes on which a displacement produces a parallel reaction, and thence constructed the surface of elasticity, he showed that Biot's dihedral law is exact provided that the binormals are taken as the optic axes, and the wave-normal direction as the direction of propagation. As early as 1822, Fresnel discussed his perpendicular axes with Cauchy. Acknowledging Fresnel's influence, Cauchy went on to develop the first rigorous theory of elasticity of non-isotropic solids (1827), hence the first rigorous theory of transverse waves therein (1830) — which he promptly tried to apply to optics. The ensuing difficulties drove a long competitive effort to find an accurate mechanical model of the aether. Fresnel's own model was not dynamically rigorous; for example, it deduced the reaction to a shear strain by considering the displacement of one particle while all others were fixed, and it assumed that the stiffness determined the wave velocity as