J (R) = I (R) σ_{los}^{2} (R)

and

p_{*} (r) = ν (r) σ_{r}^{2} (r)

. The two preceding projections have the same Abel integral equation kernel, giving

ν (r) = - \frac{1}{π L} \int_{r}^{\infty} \frac{I ^{'} ( R )}{R ^{2} - r ^{2}} d R, p_{*} (r) = - \frac{1}{π L} \int_{r}^{\infty} \frac{J ^{'} ( R )}{R ^{2} - r ^{2}} d R .

(1)

Their ratio recovers

σ_{r}^{2} = p_{*} / ν

, and the isotropic Jeans equation gives

M (\leq r) = - \frac{r ^{2}}{G ν ( r )} \frac{d p _{*}}{d r} .

(2)

Equivalently put

A (r) = \int_{r}^{\infty} I^{'} (R) / R^{2} - r^{2} d R

and

B (r) = \int_{r}^{\infty} J^{'} (R) / R^{2} - r^{2} d R

. Then

M = - r^{2} B^{'} (r) / (G A (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.