= Solution
Consider an element displaced upward by $\xi$ in <hydrostatic equilibrium>. Assume it quickly reaches the ambient <pressure>, but moves adiabatically and retains its composition. Put $\nabla=d\log T/d\log P$, $\nabla_\mu=d\log\mu/d\log P$, and let $\nabla_{\rm ad}$ be the <adiabatic temperature gradient>. The <equation of state> has differential
$$
d\log\rho=\alpha_P\,d\log P-\delta\,d\log T+\phi\,d\log\mu,
$$
where $\delta=-(\partial\log\rho/\partial\log T)_{P,\mu}$ and $\phi=(\partial\log\rho/\partial\log\mu)_{P,T}$.
At the new <pressure>, the parcel-to-environment density difference is
$$
\frac{\rho_{\rm parcel}-\rho_{\rm env}}\rho
=[\delta(\nabla-\nabla_{\rm ad})-\phi\nabla_\mu]\,\Delta\log P.
$$
With positive pressure scale height $H_P=-(d\log P/dr)^{-1}$, $\Delta\log P=-\xi/H_P$. The <buoyancy> acceleration is therefore $\ddot\xi=-N^2\xi$, with <stellar buoyancy frequency>
$$
N^2=\frac g{H_P}[\delta(\nabla_{\rm ad}-\nabla)+\phi\nabla_\mu].
$$
A negative $N^2$ amplifies the displacement. For a radiative background, $\nabla=\nabla_{\rm rad}$, so the <Ledoux criterion> for instability is
$$
\boxed{\nabla_{\rm rad}>\nabla_{\rm ad}+\frac\phi\delta\nabla_\mu.}
$$
For uniform composition, $\nabla_\mu=0$, giving the <Schwarzschild criterion>
$$
\boxed{\nabla_{\rm rad}>\nabla_{\rm ad}.}
$$
Equality is marginal stability. A <mean molecular weight> increasing inward has $\nabla_\mu>0$ and supplies a stabilizing composition term when $\phi,\delta>0$.
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