390 Prof. L. Natanson on the 



Here, therefore, we may put R°=0 and ^U/d<? f =()V/d<7i ; 

 and the equation will be 



* 1 ^{8T-8U + 2P 4 ^ 4 + S / Q} = 0. . . (1) 



J*c 



We shall return to this case in § 13 below. 



§ 8. Reversible Thermodynamics — At present the imme- 

 diate object of the science called Thermodynamics is the study 

 of states of equilibrium. The modifications assumed in Ther- 

 modynamics to occur in a system are, for that reason, virtual 

 reversible transformations wdiich lead from one state of equi- 

 librium to another one. Let us admit the following assump- 

 tions : — first, that that part of the energy which we call T is 

 a constant quantity; secondly, that the variables are " normal" 

 variables ; thirdly, if a function of the variables 3 and q v called 

 the entropy, be denoted by S, that the term 8°Q is of the 

 form 



alfsa-t-asjrfy; (1) 



and, lastly, that the supposed transformation being reversible, 

 the term S'Q is equal to zero. Hence the laws of ordinary 

 Thermodynamics must be contained in 



and 



wh 



f 1 A{-2||8 ?j + 2P j g ? J=0, 



(3) 



ere 



av au as ... 



^r*ir% (4) 



Since the adopted variables S, g. are " normal " ones, we are 

 at liberty to define the quantity U — $S as representing what 

 in § 5 has been called the free energy of the system; hence 



-a? +p < =0; a-? +s=0 < ■ ■ ■ ^ 



and thus we are led to that well-known form of thermo- 

 dynamical equations which we have learned from MM. 

 Massieu, Gibbs, Duhem, Helmholtz, and others. 



§ 9. Irreversible Dynamics. — Let us now proceed to con- 

 sider cases of motion bearing perfect analogy with ordinary 



