Integrated spectra have been obtained for the elliptical galaxy M32 and for the ''metal-rich'' Galactic globular cluster 47 Tuc. The spectra cover the wavelength interval lambdalambda3800-4400 angstrom at a resolution of 2.5 angstrom FWHM and S/N ratio of approximately 100:1. Similar data have been acquired for a library of 191 individual stars, and, to support the 47 Tuc observations, integrated spectra of four additional metal-rich Galactic globular clusters have been obtained. These observations are used to compare in detail the integrated spectra of M32 (the most extensively studied elliptical galaxy) and 47 Tuc (the best-studied metal-rich Galactic globular cluster). Although M32 and 47 Tuc have similar optical broadband colors and overall spectral types, when viewed at 2.5 angstrom resolution spectra numerous subtle differences between their integrated are clearly visible. A system of 13 spectral indices, many of them originally defined in Rose [AJ, 89, 1238 (1984)], has been used to quantify these differences. Altogether twelve diagnostic diagrams are presented to illustrate the manner in which the integrated spectrum of M32 differs from that of 47 Tuc. These diagrams are used to place several strong constraints on the stellar populations in these two systems. Specifically, (1) The mean integrated light of the evolved population in M32 has a much later spectral type and is more metal rich than that of 47 Tuc. In particular, any contribution from a red horizontal branch (RHB) population (i.e., from RHB stars which are bluer than clump giants in old open clusters) in M32 must be negligible, whereas a strong RHB contribution is required to explain the integrated spectrum of 47 Tuc (which is consistent with the known RHB component in the color-magnitude diagram of 47 Tuc). (2) Given that M32 has a later type mean evolved population than 47 Tuc but has very similar spectral type and B-V color, then either the mean main-sequence light in M32 must have an earlier spectral type than in 47 Tuc or the main sequence must contribute more light relative to the evolved stars, or both. In the former case, the mean main sequence of M32 should be approximately 0.09 mag bluer in B-V than in 47 Tuc, whereas in the latter case giants should contribute only approximately 35% of the integrated light at 4000 angstrom in M32, as opposed to approximately 55% in 47 Tuc. Both of these alternatives are discussed, and it is concluded that the latter alternative is more consistent with the spectral index data and with UV studies of M32 and 47 Tuc. In either case, though, it is concluded that M32 contains a large intermediate-age stellar population. (3) Several spectral indices indicate that the integrated spectrum of M32 contains a significant contribution from stars that are stronger lined than the typical solar neighborhood K giant. That is, the location of M32 in several diagnostic diagrams is best explained by including a contribution from stars that have been labeled as ''super-metal-rich'' or ''deltaCN strong'' by other observers. Such a component is not evident in the integrated spectrum of 47 Tuc. (4) A small, approximately 5% contribution from hot stars is required to explain the integrated spectrum of M32, whereas no contribution from hot stars is evident in the spectrum of 47 Tuc. The hot star contribution in M32 can be readily explained by a small metal-poor population. (5) In general, the observed spectral indices of 47 Tuc can be readily reproduced by linear combinations of moderately metal-poor (i.e., [Fe/H] approximately -0.5) main-sequence, subgiant, giant branch, and RHB spectra in proportions that are entirely consistent with those predicted from the well established color-magnitude diagram of this cluster. Only the strong CN bands and the equivalent width of the Ca I lambda4226 line are difficult to reproduce. This success in synthesizing the spectral indices of 47 Tuc with only a few basic stellar components demonstrates that the contributions of several key populations to the integrated light of stellar systems at 4000 angstrom can be tightly constrained by the relatively straightforward application of the spectral index system.
