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Use the stellar position second-moment tensor $I_{ij}=\int \rho x_i{x}_j{d}^{3}x$. This is the $I$ entering the tensor virial theorem, rather than the engineering moment-of-inertia tensor $\mathrm{tr}(I)1-I$. For a density pattern rotating steadily about its third principal axis, choose $t=0$ when its in-plane axes are principal axes. Then $I(0)=\mathrm{diag}(I_1,I_2,I_3)$ andIn particular, $I_{11}=I_1{\cos }^2\Omega t+I_2{\sin }^2\Omega t$, $I_{22}=I_1{\sin }^2\Omega t+I_2{\cos }^2\Omega t$, and $I_{12}=(I_1-I_2)\sin \Omega t\cos \Omega t$, while $I_{33}=I_3$ and $I_{13}=I_{23}=0$. Two differentiations at $t=0$ prove the inertia-tensor acceleration of a steadily rotating stellar pattern:Both the off-diagonal acceleration and the in-plane trace acceleration vanish at this instant.

Write $\mathbf{u}=\overline{\mathbf{v}}$ for the local mean streaming. The stellar kinetic-energy tensor, stellar pressure-energy tensor, and stellar potential-energy tensor have the conventionsFor an isolated self-gravitating system, $\frac{1}{2}{\ddot{I}}_{ij}=2T_{ij}+{\Pi }_{ij}+W_{ij}$. Subtracting its second diagonal component from the first givessoThis quotient requires $I_{11}\ne I_{22}$. In an axisymmetric density distribution it becomes $0/0$: rotating the pattern about its symmetry axis does not change the density, and this equation cannot determine a pattern speed. Stellar streaming need not be rigid rotation with angular frequency $\Omega$.

Next add the first two diagonal virial equations and use the third. Their acceleration terms vanish, soFor $T_{33}=0$, ${\Pi }_{33}=-W_{33}$. With the planar virial trace ${\Pi }_{\perp }={\Pi }_{11}+{\Pi }_{22}$ and $W_{\perp }=W_{11}+W_{22}$, the prescribed mass-weighted quantities satisfy ${\Pi }_{\perp }=2M{\sigma }_0^2$, $2(T_{11}+T_{22})=M{V}_0^2$, and $1-\delta =2{\Pi }_{33}/{\Pi }_{\perp }$. Therefore $W_{33}=-(1-\delta )M{\sigma }_0^2$ and the planar equation becomesEliminating $M{\sigma }_0^2$ givesThe ${\nu }_0$ in the numerator of the PDF's displayed relation is a notation slip: the velocity subsequently defined is $V_0$. The relation follows from the traces, not from replacing a galaxy by a solid rotating body.

For the axisymmetric oblate spheroid, rotational symmetry gives ${\Pi }_{11}={\Pi }_{22}=M{\sigma }_0^2$, ${\Pi }_{33}=M{\sigma }_0^2(1-\delta )$ and zero integrated off-diagonal random stresses. A line-of-sight unit vector can be chosen as $\mathbf{n}=(\sin i,0,\cos i)$, with $i=0$ face-on. Contracting the random stress tensor with this vector gives the inclination projection of an axisymmetric stellar velocity tensor:This is a mass-weighted random velocity dispersion after subtracting streaming, with consistent aperture weights. Along the projected major axis, azimuthal streaming lies along the direction whose line-of-sight projection is $\sin i$; a rotation amplitude $V_0$ consequently projects asThere is a necessary observational convention here. The globally defined $V_0^2=2(T_{11}+T_{22})/M$ is a streaming mean square. The displayed amplitude relation holds if the rotation curve is constant, or if the measured rotation-amplitude statistic is weighted consistently to represent this intrinsic quantity. Axisymmetry alone does not equate an arbitrary observed peak rotation to the global $V_0$. If $V_r$ instead means the full-aperture line-of-sight streaming root-mean-square, azimuthal averaging yieldsand the corresponding root-mean-square is $V_0\sin i/\sqrt{2}$. Keeping amplitude and root-mean-square distinct prevents a spurious factor of two in the kinetic energy.

Finally, let the intrinsic equatorial and polar semiaxes be $a$ and $c$, and put $q_t=c/a=1-{ϵ}_t$. An intrinsic boundary has $(x^2+y^2)/a^2+z^2/c^2=1$. Its sky minor-axis coordinate is $Y=y\cos i-z\sin i$. Eliminating the line-of-sight coordinate, or maximizing $Y$ on this ellipse, gives the projected minor semiaxis $\sqrt{a^2{\cos }^{2}i+c^2{\sin }^{2}i}$; the major semiaxis remains $a$. Thus the projected axis ratio of an oblate spheroid isSubtracting from unity provesThe face-on limit is round with zero projected rotation amplitude; the edge-on limit reveals the intrinsic flattening and in-plane random velocity dispersion. These limits check both the geometry and the convention for inclination.