  We study the stability of density stratified flow between co-rotating
vertical cylinders with rotation rates $\Omega_o < \Omega_i$ and radius ratio
$r_i/r_o=0.877$, where subscripts $o$ and $i$ refer to the outer and inner
cylinders. Just as in stellar and planetary accretion disks, the flow has
rotation, anticyclonic shear, and a stabilizing density gradient parallel to
the rotation axis. The primary instability of the laminar state leads not to
axisymmetric Taylor vortex flow but to the non-axisymmetric {\it
stratorotational instability} (SRI), so named by Shalybkov and R\"udiger
(2005). The present work extends the range of Reynolds numbers and buoyancy
frequencies ($N=\sqrt{(-g/\rho)(\partial \rho/\partial z)}$) examined in the
previous experiments by Boubnov and Hopfinger (1997) and Le Bars and Le Gal
(2007). Our observations reveal that the axial wavelength of the SRI
instability increases nearly linearly with Froude number, $Fr= \Omega_i/N$. For
small outer cylinder Reynolds number, the SRI occurs for inner inner Reynolds
number larger than for the axisymmetric Taylor vortex flow (i.e., the SRI is
more stable). For somewhat larger outer Reynolds numbers the SRI occurs for
smaller inner Reynolds numbers than Taylor vortex flow and even below the
Rayleigh stability line for an inviscid fluid. Shalybkov and R\"udiger (2005)
proposed that the laminar state of a stably stratified rotating shear flow
should be stable for $\Omega_o/ \Omega_i > r_i/r_o$, but we find that this
stability criterion is violated for $N$ sufficiently large; however, the
destabilizing effect of the density stratification diminishes as the Reynolds
number increases. At large Reynolds number the primary instability leads not to
the SRI but to a previously unreported nonperiodic state that mixes the fluid.
