In this paper we consider the effect of internal heat generation and a spatial variation of the gravity field on the onset of thermal convection in spherical shells. If the temperature gradient and gravity fields have the same spatial variation, then initially quiet fluids are subcritically stable. For these flows the effect of inertially non-linear disturbances is not destabilizing if the Rayleigh number is below the critical value set by linear theory plus ‘exchange of stabilities’. For subcritically-stable flows a principle of exchange of stabilities is not necessary; a stronger statement of stability for the same stability limit can be made. For the many cases calculated here in which subcritical instabilities can exist, the difference between the linear and energy limits is small and can be contracted only toward the energy limit by an improved linear theory.