Long-slit spectra of eleven bright elliptical galaxies are used to investigate the radial behavior of various atomic and molecular absorption features in these objects. The majority of features weaken with radius, with the Mg2 and Na5895 indices showing the strongest gradients. However, the strengths of H-beta and H-alpha absorption typically remain constant. The rate of change of the Mg2 index with log(r), DELTA-Mg2/DELTA-log(r), shows a galaxy-to-galaxy dispersion which is larger than expected from experimental uncertainties, indicating that the galaxies in this study have not been selected from a sample in which all systems have identical radial properties. This suggests that there are system-to-system differences in the star formation histories of elliptical galaxies. Moreover, the rates with which most atomic and molecular features change with respect to Mg2 in the outer envelopes of these systems appears to be in better agreement with trends defined by globular clusters than galaxy nuclei. Gorgas et al. [MNRAS, 245, 217 (1990)] reached similar conclusions in their study of early type galaxies. Values of DELTA-Mg2/DELTA-log(r) derived in this study are combined with those published in the literature to search for correlations with various galaxy properties. DELTA-Mg2/A log(r) appears to be loosely correlated with the central velocity dispersion, sigma-0 and the dynamical anisotropy parameter, (upsilon/sigma)*; although both relations are significant at roughly the 95% confidence level, a larger body of observations is required to establish unambiguously the reality of these relations. A significant relation does not exist between DELTA-Mg2/DELTA-log(r) and the integrated galaxy brightness M(B).