The effect of a density stratification on the steady, mechanically driven motion of a viscous fluid in a rotating cylinder with axis aligned with the rotation and gravity vectors and with parallel top and bottom surfaces that slope with respect to the plane perpendicular to the rotation vector is studied by a linear theory. Primary attention is given to a study of the alteration of the characteristics of the flow of a homogeneous fluid by the addition of a weak stratification. It is found, for example, that in the range $E^{\frac{3}{2}} < \sigma S < E$, where E = ν/ΩL2 and σS = ναgΔT0/κΩ2L, and with a homogeneous boundary condition on the perturbation temperature, the interior velocity is parallel to the direction perpendicular to the plane determined by the vector normal to the top surface and the rotation vector. The circulation closes in an inviscid, but heat-conducting, boundary layer of thickness E¾(σS)−½ on the side wall. Thus, with stratification, the steady flow in this configuration differs markedly from the corresponding flow in a cylinder where the top and bottom surfaces lie in planes perpendicular to the rotation vector. The difference is caused by the fact that in the container with sloping surfaces the basic stratification interacts with the geostrophic flow whereas, in the other case, the interaction is with the much smaller Ekman layer suction velocities.