Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2005.In studies of triaxial elliptical galaxies, one of the least observationally studied phenomena is figure rotation. Figure rotation has important consequences for the orbital structure and could explain the survival of steep nuclear cusps. For this project, we thus wish to investigate the possibility of measuring the figure rotation of an elliptical galaxy for which the geometry is approximately known using the Tremaine-Weinberg (TW) method. Originally meant for measuring the pattern speed of barred disk galaxies, we test the validity of the method using NGC 5266, a minor-axis dust-lane elliptical. In the process, the galaxy's line-of-sight velocity distribution (LOSVD) is measured along several slit positions. Measurements of the velocity v, velocity dispersion a, skewness /J3, and kurtosis h^ are derived using the Fourier Correlation Quotient method and a Gauss-Hermite series. This work represents the most detailed stellar kinematic measurements of NGC 5266 to date and confirm that it is one of the fastest rotating elliptical galaxies known today (Varnas et al. 1987). We find a maximum velocity of about 167 km s_1 at both a PA of 274° and 304°. This is compared to a maximum of 212 ± 7kms~1 at a PA of 287° found elsewhere (Varnas et al. 1987). The TW method yields significantly different values for the pattern speed. These vary between -19 and 22kms Wcsec"1. The discrepancy between the results casts doubt on the ability to straightforwardly apply the TW method to elliptical systems, but the study provides some insight into how the method may be more successfully implemented in the future