Inclination projection of an axisymmetric stellar velocity tensor (${\sigma }_{\mathrm{los}}^2={\sigma }_0^2(1-\delta {\cos }^{2}i)$)

For an axisymmetric random velocity dispersion tensor with ${\Pi }_{11}={\Pi }_{22}=M{\sigma }_0^2$ and ${\Pi }_{33}=M{\sigma }_0^2(1-\delta )$, contracting with the line-of-sight unit vector $(\sin i,0,\cos i)$ gives the displayed mass-weighted velocity dispersion. A major-axis rotation amplitude projects as $V_r=V\sin i$. By contrast, an azimuthally averaged line-of-sight streaming root-mean-square is $V\sin i/\sqrt{2}$ when $V^2$ is the full in-plane streaming mean square. Aperture weights and the definition of the measured rotation statistic must match the intrinsic statistic.