372 On the Proof of the Second Law of Thermodynamics. 



and temperature that v may be expressed as/(T), r ~ must in 



any case be a complete differential. This latter relation is 

 assumed by M. Szily when he puts r)=f(i;) or /'(f), taking 

 for granted that the path of the state must be capable of expres- 

 sion by some such equation. In a paper " On the Moving 

 Force of Heat/' printed in the Philosophical Magazine in 1851 

 ([IV.] vol. ii. p. 1), M. Clausius pointed out that under such 

 circumstances dQ also is a complete differential. 



The hypothesis of Boltzmann gives •-= j— = — T-^ —=-, 



but is not necessarily the only hypothesis which will satisfy 

 this condition and make -^ a complete differential indepen- 

 dently of any constant relation between v and T. 



The equations above obtained require some modification to 

 meet the case of a gas supposed to consist of molecules the 

 elements of which possess internal energy. 



There are two ways of applying the formula of Clausius 

 (which in any case holds good) to such a gas. (1) We may 

 consider T to represent the kinetic energy of the molecules 

 only, and 2S(JR?') the virials of the intermolecular forces 

 only. T then corresponds to temperature, and we may, in the 

 application of this formula, disregard the internal motions and 

 forces of the elements of the molecules. 



Or (2) we may regard T as the total energy, and S2(i R?0 

 as the sum of the virials of all forces whatever. But then T 

 ceases to correspond in the same manner to temperature, and 

 that part of T which represents the internal energy of the mo- 

 lecules is equal to that part of 22(-§-Rr) which represents 

 the virials of the internal forces taken negatively. 



These latter do not vary with the change of v } or affect 



d\J w 2 Ti + T 2 dW 2 T 2 , 



■*' Hereof— — , ^= -_-}_, and 



^7~' < 8 > 



where T x is the kinetic energy of the molecules, and T 2 their 



dW 

 internal energy, and —j— dv is the hypothetical change of po- 

 tential of the intermolecular forces only — the same result as 

 we should have obtained by neglecting altogether the effect 

 of the latter portion of T ; and also the internal forces of the 

 molecules. 



