New low temperature measurements of the reflectivity of crystalline anthracene are reported. Attention is confined to the important b axis polarized absorption band in the vicinity of the 0-0 transition. In this way line broadening by "two particle" excitations are eliminated. It is shown how low temperature reflectivity data, even when limited to only a region of the spectrum, can provide interesting information concerning the exciton dynamics. For this crystal transition optical constants are obtained from the observed reflectance data by use of a Kramers-Kronig analysis and a General Dispersion analysis. Discussed in some detail are those methods of obtaining optical constants, the values of the various band parameters and the implicatio...