The incipient buoyant thermal convection in a vertical cylindrical annulus when heated from below is examined. The ends are assumed to be free, and the sidewalls perfectly conducting. The temperature needed to initiate fluid motion is expressed nondimensionally in terms of the Rayleigh number. The analytical conflict that arises for annuli of infinite aspects ratios due to insufficient independent boundary conditions is resolved. Calculations for the critical Rayleigh numbers are presented for a variety of geometries, and the corresponding velocity and temperature perturbations are also shown. The number of cells increases as the aspect and radius ratio decrease with a strong bias towards the development of azimuthally varying cells. These changes in cellular behavior are expected based on physical justifications and comparisons with previous studies. © 1990 by ASME.