  We revisit the somewhat classical problem of the linear stability of a
rigidly rotating liquid column in this communication. Although literature
pertaining to this problem dates back to 1959, the relation between inviscid
and viscous stability criteria has not yet been clarified. While the viscous
criterion for stability, given by $We = n^2+k^2-1$, is both necessary and
sufficient, this relation has only been shown to be sufficient in the inviscid
case. Here, $We = \rho \Omega^2 a^3/\gamma$ is the Weber number and measures
the relative magnitudes of the centrifugal and surface tension forces, with
$\Omega$ being the angular velocity of the rigidly rotating column, $a$ the
column radius, $\rho$ the density of the fluid, and $\gamma$ the surface
tension coefficient; $k$ and $n$ denote the axial and azimuthal wavenumbers of
the imposed perturbation. We show that the subtle difference between the
inviscid and viscous criteria arises from the surprisingly complicated picture
of inviscid stability in the $We-k$ plane. For all $n >1$, the viscously
unstable region, corresponding to $We > n^2+k^2-1$, contains an infinite
hierarchy of inviscidly stable islands ending in cusps, with a dominant leading
island. Only the dominant island, now infinite in extent along the $We$ axis,
persists for $n= 1$. This picture may be understood, based on the underlying
eigenspectrum, as arising from the cascade of coalescences between a retrograde
mode, that is the continuation of the cograde surface-tension-driven mode
across the zero Doppler frequency point, and successive retrograde Coriolis
modes constituting an infinite hierarchy.
