  We study pairs of co-axial vortex rings starting from the action for a
classical bosonic string in a three-form background. We complete earlier work
on the phase diagram of classical orbits by explicitly considering the case
where the circulations of the two vortex rings are equal and opposite. We then
go on to study perturbations, focusing on cases where the relevant
four-dimensional transfer matrix splits into two-dimensional blocks. When the
circulations of the rings have the same sign, instabilities are mostly limited
to wavelengths smaller than a dynamically generated length scale at which
single-ring instabilities occur. When the circulations have the opposite sign,
larger wavelength instabilities can occur.
