axes separated by 45°, and obtained predictions that disagreed with Biot's experiments (except in special cases) but agreed with his own. Fresnel applied the same principles to the standard case of chromatic polarization, in which one birefringent lamina was sliced parallel to its axis and placed between a polarizer and an analyzer. If the analyzer took the form of a thick calcite crystal with its axis in the plane of polarization, Fresnel predicted that the intensities of the ordinary and extraordinary images of the lamina were respectively proportional to where is the angle from the initial plane of polarization to the optic axis of the lamina, is the angle from the initial plane of polarization to the plane of polarization of the final ordinary image, and is the phase lag of the extraordinary wave relative to the ordinary wave due to the difference in propagation times through the lamina. The terms in are the frequency-dependent terms and explain why the lamina must be thin in order to produce discernible colors: if the lamina is too thick, will pass through too many cycles as the frequency varies through the visible range, and the eye (which divides the visible spectrum into only three bands) will not be able to resolve the cycles. From these equations it is easily verified that for all so that the colors are complementary. Without the phase-inversion rule, there would be a plus sign in front of the last term in the second equation, so that the -dependent term would be the same in both equations, implying (incorrectly) that the colors were of the same hue. These equations were included in an undated note that Fresnel gave to Biot, to which Biot added a few lines of his own. If we substitute and then Fresnel's formulae can be rewritten as which are none other than Biot's empirical formulae of 1812, except that Biot interpreted and as the "unaffected" and "affected" selections of the rays incident on the lamina. If Biot's substitutions were accurate, they would imply that his experimental results were more fully explained by Fresnel's theory than by his own. Arago delayed reporting on Fresnel's works on chromatic polarization until June 1821, when he used them in a broad attack on Biot's theory. In his written response, Biot protested that Arago's attack went beyond the proper scope of a report on the nominated works of Fresnel. But