= Solution
Let $p(r)=\rho(r)\sigma_r^2(r)$ and use the <velocity-anisotropy parameter> $\beta=1-\sigma_\theta^2/\sigma_r^2$, with $\sigma_\theta^2=\sigma_\phi^2$. Here the tangential variance is per component; a definition using the sum of both tangential variances instead has an extra factor of two. The original PDF has this correct per-component ratio, which the TeX corrupts.
At a point along the <line of sight>, $r^2=R^2+z^2$. The radial direction makes an angle with the sightline satisfying $\sin^2\chi=R^2/r^2$. <Spherical symmetry> eliminates mixed moments, and projection gives
$$
\overline{v_{\rm los}^2}=\sigma_r^2\cos^2\chi+\sigma_\theta^2\sin^2\chi=\sigma_r^2\left(1-\beta\frac{R^2}{r^2}\right).
$$
Using both sides of the <galaxy> and $dz=r\,dr/\sqrt{r^2-R^2}$,
$$
I(R)\sigma_p^2(R)=2\int_R^\infty\left(1-\beta\frac{R^2}{r^2}\right)p(r)\frac{r\,dr}{\sqrt{r^2-R^2}}.
$$
For this formula use $I$ as the projected density of the same tracer represented by $\rho$. If $\rho$ is a <mass density> and $I$ an observed <surface brightness>, a constant <mass-to-light ratio> rescales $I$ by that factor; otherwise replace $\rho$ by the <luminosity density> throughout the tracer equation. The gravitating $M(r)$ remains the total <mass>, not necessarily the tracer's <mass>.
The <Spherical Jeans equation> is $p'+2\beta p/r=-\rho GM/r^2$. Eliminate $\beta p=-r[p'+\rho GM/r^2]/2$ from the projected integrand:
$$
2\left(1-\beta\frac{R^2}{r^2}\right)p=2p+\frac{R^2}{r}p'+\frac{R^2\rho GM}{r^3}.
$$
Rearranging proves
$$
\boxed{I\sigma_p^2-R^2\int_R^\infty\frac{\rho GM(r)}{r^2\sqrt{r^2-R^2}}\,dr=\int_R^\infty\left(2p+\frac{R^2}{r}p'\right)\frac{r\,dr}{\sqrt{r^2-R^2}}.}
$$
No constancy of $\beta$ was needed. Convergence of the projected moments and the steady spherical assumptions are required. This is the integral form of the <projected spherical Jeans residual>.
Although $\beta$ has disappeared explicitly, the unknown radial pressure $p$ still carries the orbital information. Thus the identity does not remove the <mass--anisotropy--density degeneracy>: projected brightness and one projected <second moment> do not uniquely determine both $M(r)$ and the radial/tangential partition. A central rise in <velocity dispersion> can reflect a compact central <mass>, radial orbital bias, or a changing <mass-to-light ratio>; flattening and departures from equilibrium add further ambiguities.
For example, within a point-mass-dominated region with tracer $\rho\propto r^{-k}$ and constant $\beta$, direct solution gives $\sigma_r^2=GM_\bullet/[(k+1-2\beta)r]$. For $k>1$ the <projected velocity dispersion of a Keplerian cusp> is
$$
\sigma_p^2(R)=\frac{GM_\bullet}{(k+1-2\beta)R}\frac{B(k/2,1/2)}{B((k-1)/2,1/2)}\left(1-\frac{\beta k}{k+1}\right).
$$
The <beta function> factors come from setting $r=R\sec\chi$ in the integrals along the <line of sight>. Different allowed $\beta$ generally change the <mass> needed to fit the same normalization; $k=2$ is a special cancellation, not the general case. Resolving the central <black-hole sphere of influence> and constraining orbital structure with higher <velocity> moments or additional kinematic directions strengthens a compact-mass inference. \b[Brightness and <velocity dispersion> profiles alone do not establish a unique central black-hole <mass>, still less an <event horizon>.]
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