The paper introduces new developments of the original AREA method. A rigorous mathematical proof that the equilibrium points are the only ones which satisfy the maximum AREA criterion in the case of a two-component, two-phase system is given for the first time. A rigorous proof that the maximum AREA criterion is a necessary but not a sufficient condition for equilibrium in the case of an N-component, two-phase system is given also for the first time in the paper. Two test examples which reinforce the validity of the theoretical results obtained are presented and discussed.