= Pressure equation for inertial oscillations of a barotropic star
{title2=$W=p^{\prime}/\rho$}
In the <Cowling approximation>, a mode with phase $e^{i\omega t+im\phi}$ has rotating-frame frequency $\sigma=\omega+m\Omega$. The horizontal momentum matrix has determinant $4\Omega^2-\sigma^2$. Eliminating <velocities> yields a pressure-potential equation with vertical coefficient $1-4\Omega^2/\sigma^2$ and density-gradient term $2m\Omega\rho_RW/(R\sigma)$. Frequencies zero and plus or minus $2\Omega$ need the original system rather than this inversion.
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