The Schwarzschild criterion for convection is generalized to include departures from adiabatic motion. It is demonstrated that a thermally unstable atmosphere is also convectively unstable, regardless of the atmospheric temperature graqient. If the latter is sufficiently subadiabatic (e.g., if the temperature increases rapidly with height), convection sets in as exponentially growing oscillations. In the presence of a magnetic field, a thermally unstable atmosphere is monotonically unstable, although overstability is also possible if the temperature gradient is subadiabatic. The effects of conduction, viscosity, opacity, and rotation are evaluated. In this paper the assumption is made that the radiative cooling (or the source function) depends only o the local values of density and temperature.