Since the study of energy transformations and the numerical integration of simplifiied equations are sometimes used as alternative approaches to the same physical problem, it is often desirable that the simplified equations conserve total energy under reversible adiabatic processes. Preferably, the equations should also conserve the sum of kinetic energy and available potential energy, and they should describe the tendency for static stability to increase as kinetic energy is released. It is found that if the equation of balance is used as a filtering approximation, all the terms in the vorticity equation which involve both the rotational and the divergent part of the wind field should be retained, while, if the geostrophic equation is used, all of these terms in the vorticity equation should be omitted, if the equations are to possess suitable energy invariants. An n-layer model with the appropriate energy invariants is developed. The two-layer model may be the simplest possible model with variable static stability. The model appears to be suitable for theoretical studies of the general circulation.