result as being contained in a note appended to his "second memoir" on double refraction. No such note appeared in print, and the relevant manuscripts found after his death showed only that, around 1824, he was comparing refractive indices (measured by Fraunhofer) with a theoretical formula, the meaning of which was not fully explained. In the 1830s, Fresnel's suggestion was taken up by Cauchy, Baden Powell, and Philip Kelland, and it was found to be tolerably consistent with the variation of refractive indices with wavelength over the visible spectrum for a variety of transparent media . These investigations were enough to show that the wave theory was at least compatible with dispersion; if the model of dispersion was to be accurate over a wider range of frequencies, it needed to be modified so as to take account of resonances within the medium . Conical refraction The analytical complexity of Fresnel's derivation of the ray-velocity surface was an implicit challenge to find a shorter path to the result. This was answered by MacCullagh in 1830, and by William Rowan Hamilton in 1832. Hamilton went further, establishing two properties of the surface that Fresnel, in the short time given to him, had overlooked: (i) at each of the four points where the inner and outer sheets of the surface make contact, the surface has a tangent cone (tangential to both sheets), hence a cone of normals, indicating that a cone of wave-normal directions corresponds to a single ray-velocity vector; and (ii) around each of these points, the outer sheet has a circle of contact with a tangent plane, indicating that a cone of ray directions corresponds to a single wave-normal velocity vector. As Hamilton noted, these properties respectively imply that (i) a narrow beam propagating inside the crystal in the direction of the single ray velocity will, on exiting the crystal through a flat surface, break into a hollow cone (external conical refraction), and (ii) a narrow beam striking a flat surface of the crystal in the appropriate direction (corresponding to that of the single internal wave-normal velocity) will, on entering the crystal, break into a hollow cone (internal conical refraction). Thus a new pair of phenomena, qualitatively different from anything previously observed or suspected, had been predicted by mathematics as consequences of Fresnel's theory. The prompt experimental confirmation of those predictions by Humphrey Lloyd brought Hamilton a prize that had never