Emission line profiles of the red giant alpha Tau, obtained by Carpenter et al. (1991) using the echelle and medium dispersion gratings of the Goddard High Resolution Spectrograph, are re-examined. Ratios of monochromatic flux densities of lines of the C II] 2s(2)2p2P-degrees --> 2s2p(2)4P multiplet near 2325 angstrom, well-known diagnostics of electron densities in the chromospheres of cool evolved stars, change systematically with relative wavelength across the line profiles. With the justifiable assumption that these lines are optically thin, this implies that the electron density varies systematically across the line profiles. This is the first time that traditional electron density diagnostic line ratios have been successfully applied to monochromatic line flux densities in cool stars. The C II] data are examined together with lines of Co II and Si II] also observed by Carpenter et al. (1991) to infer (1) possible causes of the apparent net downflow in C II] lines, and (2) other basic properties of the gas flows in the chromosphere of alpha Tau. These conclusions are based upon emission measure analysis as well as electron density determinations. This work clearly demonstrates the power of very high signal-to-noise, high-resolution spectra in the UV. The monochromatic density diagnostic technique holds promise as a powerful tool for studying flows in a variety of astrophysical objects, including the Sun. The paper concludes with a compilation of ions from the boron and aluminum sequences for which this technique can be expected to produce valuable results using astronomical data from the GHRS and solar data from instruments on SOHO.