= Solution
At fixed radius, use spherical coordinates in velocity space with polar angle $\alpha$ measured from the radial direction:
$$
v_r=v\cos\alpha,
\qquad
v_t=v\sin\alpha,
\qquad
L=rv\sin\alpha.
$$
The <galactic distribution function> and volume element give angular weight
$$
L^{p-2}d^3v
\propto \sin^{p-1}\alpha\,d\alpha\,d\varphi.
$$
All dependence on $v$, $r$, and $g(E)$ cancels from ratios of second moments. Symmetry in the tangential plane gives
$$
\frac{\sigma_\theta^2}{\sigma_r^2}
=\frac{\sigma_\phi^2}{\sigma_r^2}
=\frac{\frac12\int_0^\pi\sin^{p+1}\alpha\,d\alpha}
{\int_0^\pi\cos^2\alpha\sin^{p-1}\alpha\,d\alpha}.
$$
Writing the angular integrals as <beta function> integrals and using the <Gamma function recurrence>,
$$
\frac12
\frac{B((p+2)/2,1/2)}{B(p/2,3/2)}
=\frac p2.
$$
Therefore, for every admissible energy factor $g$,
$$
\boxed{\sigma_\theta^2=\sigma_\phi^2
=\frac p2\sigma_r^2}.
$$
Equivalently, the <velocity-anisotropy parameter> is the constant $\beta=1-p/2$.
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