= Dipolar perturbations of a critically rotating cylinder
{title2=$u_R=a(\omega+kR),\quad W=iaR[(kR)^2-\omega^2]$}
The regular $m=1$ <velocity> family satisfies the displayed formulas. The surface <pressure> condition and gravitational matching give $W_s=-\Omega_s^2R_0\eta$, while the kinematic condition is $i(\omega+\Omega_s)\eta=u_R(R_0)$. If the material radial displacement $u_R/[i(\omega+kR)]$ is continued regularly to the boundary, then $\eta=-ia$ and $\omega^2=0$: translation of an isolated cylinder has no restoring force. Using only the Eulerian <velocity> conditions instead leaves $\omega^2(\omega+\Omega_s)=0$. The additional surface-corotation case has $\eta=0$, $\Phi'=0$, $u_R=a(kR-\Omega_s)$ and $W=iaR[(kR)^2-\Omega_s^2]$; it satisfies the <velocity> equations and zero surface <pressure>. Excluding that degenerate branch is an extra admissibility assumption, not a valid division by a nonzero frequency at every surface.
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