One general consequence of the Nernst theorem is derived, i.e., the various
heat capacities of a thermodynamic system under different constraints approach
zero as the temperature approaches absolute zero. The temperature dependence of
the heat capacity of any thermodynamic system at ultra-low temperatures is
revealed through this consequence. Moreover, the general form and the simplest
expression of the heat capacities of thermodynamic systems at ultra-low
temperatures are deduced. Some significant discussion and results are given.
One new research method is provided by using this consequence. Finally, the
equivalence between the Nernst theorem and its consequence is rigorously
proved, so that this consequence may be referred to another description of the
third law of thermodynamics