Linearized equations derived by Chandrasekhar (1952) describing convection in the presence of a magnetic field have been used to derive growth rates of convection cells (of various sixes and shapes) as a function of the magnetic number Q and the Rayleigh number R. From these growth rates, it is found that the onset of unstable modes (as contrasted with overstable modes) occurs along a curve in the R, Q plane which lies between the overstability-curve and the curve derived on the principle of exchange of stabilities. This curve is the locus of points where the frequency of oscillation of overstable `nodes becomes zero. These calculations are applied to the penumbra of sunspots, assuming a uniform magnetic field of 1000 gauss and physical parameters appropriate to the outer layers of the convection zone. It is found that convection rolls (long, narrow convection cells) will result if the magnetic field in the penumbra is horizontal or nearly horizontal, as the Evershed effect strongly suggests. These convection rolls are shown to be consistent with the known properties of the penumbral filaments. Some other plausible models of the penumbra are shown to be inadequate.