The disturbed motion of a liquid having a free surface in a vertical, rotating, circular cylinder and an equilibrium distribution of velocity corresponding to rigid body rotation is studied in linear approximation. It is shown that a vertical gradient of velocity necessarily generates vorticity, whence the disturbed motion cannot remain irrotational. (Earlier studies of this problem were based on shallow water theory, in which approximation the disturbed motion can remain irrotational.) The natural frequencies for the free surface oscillations are studied in some detail for a deep tank (say depth > diameter), and it is found that one effect of rotation is to split the pairs of frequencies (that would have been of equal magnitude and opposite sign) for the nonrotating liquid. In addition, new resonant frequencies are found that have no counterpart for a nonrotating liquid and for which the motion does not fall off exponentially with depth. The results are applied to forced, transverse oscillations of the tank, and explicit solutions are given for three special cases.