CHAPTER XIV. 



DISCUSSION OF THERMODYNAMIC ANALOGIES. 



IF we wish to find in rational mechanics an a priori founda- 

 tion 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 



de = Td-q A l da l A 2 da z etc., (482) 



where e, T, and TJ denote the energy, temperature, and entropy 

 of the system considered, and A^da v etc., the mechanical work 

 (in the narrower sense in which the term is used in thermo- 

 dynamics, 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 hi 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 equa- 

 tion between e, ?/, and a v 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 de- 

 termined by the fundamental mechanical equation which 

 expresses the energy of the system as function of its mo- 

 menta and coordinates with those external coordinates (a v 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, 



