  Although N-body studies of dark matter halos show that the density profiles,
rho(r), are not simple power-laws, the quantity rho/sigma^3, where sigma(r) is
the velocity dispersion, is in fact a featureless power-law over ~3 decades in
radius. In the first part of the paper we demonstrate, using the semi-analytic
Extended Secondary Infall Model (ESIM), that the nearly scale-free nature of
rho/sigma^3 is a robust feature of virialized halos in equilibrium. By
examining the processes in common between numerical N-body and semi-analytic
approaches, we argue that the scale-free nature of rho/sigma^3 cannot be the
result of hierarchical merging, rather it must be an outcome of violent
relaxation. The empirical results of the first part of the paper motivate the
analytical work of the second part of the paper, where we use rho/sigma^3
proportional to r^{-alpha} as an additional constraint in the isotropic Jeans
equation of hydrostatic equilibrium. Our analysis shows that the constrained
Jeans equation has different types of solutions, and in particular, it admits a
unique ``periodic'' solution with alpha=1.9444. We derive the analytic
expression for this density profile, which asymptotes to inner and outer
profiles of rho ~ r^{-0.78}, and rho ~ r^{-3.44}, respectively.
