Under certain conditions, the motion caused in an annulus of fluid by rotating it about its (vertical) axis of symmetry and at the same time subjecting it to a radial temperature gradient has been shown by Hide (1958) to be mostly concentrated in a narrow jet stream which meanders between the inner and outer cylindrical boundaries of the fluid in a regular wave pattern: this wave pattern has a small angular velocity relative to the cylindrical walls containing the fluid. A theoretical solution has been found by Davies (1959) which is valid in the main body of the fluid: this solution neglects viscosity (which is permissible except near the boundaries of the fluid), and is related to the absolute angular velocity of the wave pattern. The present paper introduces viscous boundary layers between the main body of the fluid and the cylindrical walls, in an attempt to find a relation between the angular velocity of the wave pattern and that of the walls. That this is only partially successful is due to the presence of the boundary layer at the rigid surface at the bottom of the fluid (which is rotating with the same angular velocity as the cylindrical walls): this layer is ignored in the present theory. In addition to this contribution towards a complete explanation of the steady motion, the theory describes qualitatively certain periodic oscillations (vacillation) which were observed by Hide in his experiments.