Page:Elementary Principles in Statistical Mechanics (1902).djvu/189



we wish to find in rational mechanics an a priori foundation for the principles of thermodynamics, we must seek mechanical definitions of temperature and entropy. The quantities thus defined must satisfy (under conditions and with limitations which again must be specified in the language of mechanics) the differential equation where $$\epsilon$$, $$T$$, and $$\eta$$ denote the energy, temperature, and entropy of the system considered, and $$A_1 da_1$$ etc., the mechanical work (in the narrower sense in which the term is used in thermodynamics, i. e., with exclusion of thermal action) done upon external bodies.

This implies that we are able to distinguish in mechanical terms the thermal action of one system on another from that which we call mechanical in the narrower sense, if not indeed in every case in which the two may be combined, at least so as to specify cases of thermal action and cases of mechanical action.

Such a differential equation moreover implies a finite equation between $$\epsilon$$, $$\eta$$, and $$a_1$$, $$a_2$$, etc., which may be regarded as fundamental in regard to those properties of the system which we call thermodynamic, or which may be called so from analogy. This fundamental thermodynamic equation is determined by the fundamental mechanical equation which expresses the energy of the system as function of its momenta and coördinates with those external coördinates ($$a_1$$, $$a_2$$, etc.) which appear in the differential expression of the work done on external bodies. We have to show the mathematical operations by which the fundamental thermodynamic equation,