152 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



From (260), by (87) and (91), we obtain 



f = Em + mt( c H c log t + a log J + pv, 



and eliminating v by means of (263), we obtain the fundamental 

 equation 



=Em+mt(c+a-H~(c+a)logt+alog^. (265) 



From this, by differentiation and comparison with (92), we may 

 obtain the equations 



(266) 



(267) 

 P 



(268) 



The last is also a fundamental equation. It may be written in the 

 form 



p Hc a , c+a, , . u E /on\ 



' (269) 



or, if we denote by e the base of the Naperian system of logarithms, 



H-c-a c+a p-E 



p = ae ^~t~^e~^ r . (270) 



The fundamental equation between x> n> P> an d m may also be 

 easily obtained ; it is 



, (271) 



m 



which can be solved with respect to x- 



Any one of the fundamental equations (255), (260), (265), (270), 

 and (271), which are entirely equivalent to one another, may be 

 regarded as defining an ideal gas. It will be observed that most of 

 these equations might be abbreviated by the use of different con- 

 stants. In (270), for example, a single constant might be used for 



H-c-a C + d 



ae a , and another for - . The equations have been given in 



the above form, in order that the relations between the constants 

 occurring in the different equations might be most clearly exhibited. 

 The sum c + a is the specific heat for constant pressure, as appears if 

 we differentiate (266) regarding p and m as constant.* 



* We may easily obtain the equation between the temperature and pressure of a 

 saturated vapor, if we know the fundamental equations of the substance both in the 

 gaseous, and in the liquid or solid state. If we suppose that the density and the specific 

 heat at constant pressure of the liquid may be regarded as constant quantities (for such 



