= Solution
For a uniform sphere, $M=4\pi\rho R^3/3$ and its <gravitational potential energy> is $\Omega=-3GM^2/(5R)$. The ideal-gas <kinetic energy> is $E_{\rm kin}=3Mk_BT/(2\bar m)$. Equating these magnitudes using the <virial theorem> gives $GM/R=5k_BT/\bar m$. Eliminating $R$ yields the <uniform-sphere pressure-work collapse criterion>
$$
M_{\rm crit}=\left(\frac{5k_BT}{G\bar m}\right)^{3/2}
\left(\frac3{4\pi\rho}\right)^{1/2}.
$$
In particular, for fixed composition,
$$
\boxed{M_J\propto T^{3/2}\rho^{-1/2}.}
$$
The same <Jeans mass> scaling follows from $c_s^2k^2\sim4\pi G\rho$ in the <Jeans instability> dispersion relation and a characteristic size $k^{-1}$. Pressure communicates support in a sound-crossing time; sufficiently strong self-gravity wins when the gravitational time is shorter.
The numerical coefficient depends on the equilibrium, boundary and wavelength-to-cloud conventions. A sphere with both exactly uniform density and uniform temperature is not an exact hydrostatic self-gravitating ideal-gas equilibrium without additional stresses. The uniform-sphere argument establishes the characteristic mass scaling; it does not prove an exact stability boundary for that nonexistent equilibrium.
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