Significant deviations of the velocity profiles from a Gaussian are present in all objects. The strongest deviations are asymmetries quantified by h_3 and are found primarily along the major axis. The amplitude of h_3 is typically |h_3| <= 0.15 and is correlated with V/sigma. Its sign always indicates an excess of stars at low rotation velocities, and results in the mean stellar streaming velocity being lower than the mean of the best-fitting Gaussian, the quantity conventionally used to characterize rotation. The difference is approximately 10 per cent for M31, M32 and NGC 3115, and approximately 20 per cent for NGC 4594. It is thus important that models take the observed velocity profile shapes into account.
We demonstrate that the observed deviations from Gaussian velocity
profiles are for the most part not the result of projection and seeing
convolution. We calculate the velocity profiles predicted by Tonry's
model for M32, taking seeing and projection into account, but assuming
the local (unprojected) velocity distributions to be Gaussian. The
predicted velocity profiles are nearly Gaussian, in conflict with our
data. Self-consistent dynamical models must thus be constructed to fit
the observed velocity profiles. Such models should constrain the
central structure quite strongly, and will hence provide improved
constraints on the mass and presence of a supermassive central object.