= Cubic axisymmetric inertial mode of a rigid sphere
{title2=$W\propto z(R^2+z^2/6-r_0^2/2),\quad\sigma^2=4\Omega^2/5$}
For an axisymmetric <normal mode> in a rigid rotating sphere, substitute $W=z(AR^2+Bz^2+C)$ into the pressure equation and its impermeability condition. With $\beta=1-4\Omega^2/\sigma^2$, the interior equation gives $2A+3\beta B=0$. The boundary polynomial gives $3\beta B=(2+\beta)A$ and $\beta C=-(2+\beta)Ar_0^2$. A nontrivial oscillatory solution has $\beta=-4$, hence the displayed frequency and $B=A/6$, $C=-Ar_0^2/2$. It satisfies both equations identically.
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