= Solution
For the <isotropic Jeans-Abel mass inversion>, let $J(R)=I(R)\sigma_{\rm los}^2(R)$ and $p_*(r)=\nu(r)\sigma_r^2(r)$. The two preceding projections have the same <Abel integral equation> kernel, giving
$$
\boxed{\nu(r)=-\frac1{\pi L}\int_r^\infty\frac{I'(R)}{\sqrt{R^2-r^2}}\,dR},\qquad
\boxed{p_*(r)=-\frac1{\pi L}\int_r^\infty\frac{J'(R)}{\sqrt{R^2-r^2}}\,dR}.
$$
Their ratio recovers $\sigma_r^2=p_*/\nu$, and the isotropic Jeans equation gives
$$
\boxed{M(\le r)=-\frac{r^2}{G\nu(r)}\frac{dp_*}{dr}}.
$$
Equivalently put $A(r)=\int_r^\infty I'(R)/\sqrt{R^2-r^2}\,dR$ and $B(r)=\int_r^\infty J'(R)/\sqrt{R^2-r^2}\,dR$. Then $M=-r^2B'(r)/(GA(r))$: the common stellar <luminosity> cancels from the inferred <mass>. Differentiating these singular-kernel integrals is understood through suitable regularity or a nonsingular variable change, not by naively keeping an infinite lower-end boundary term.
The inversion assumes adequate radial coverage, convergence and a smooth spherical equilibrium. In practice derivatives amplify noise and the unobserved outer tail needs treatment or a model. Unknown anisotropy reintroduces a <mass>–anisotropy degeneracy; the uniqueness just derived is conditional on the explicitly isotropic velocity distribution.
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