362 EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 



changes. In fact, stable critical phases are situated at that limit. 

 They are also situated at the limit of stability with respect to dis- 

 continuous changes. These limits are in general distinct, but touch 

 each other at critical phases. 



Geometrical illustrations. In an earlier paper,* the author has 

 described a method of representing the thermodynamic properties 

 of substances of invariable composition by means of surfaces. The 

 volume, entropy, and energy of a constant quantity of the substance 

 are represented by rectangular coordinates. This method corresponds 

 to the first kind of fundamental equation described above. Any 

 other kind of fundamental equation for a substance of invariable 

 composition will suggest an analogous geometrical method. In the 

 present paper, the method in which the coordinates represent tem- 

 perature, pressure, and the potential, is briefly considered. But 

 when the composition of the body is variable, the fundamental 

 equation cannot be completely represented by any surface or finite 

 number of surfaces. In the case of three components, if we regard 

 the temperature and pressure as constant, as well as the total quantity 

 of matter, the relations between f, m 1} m 2 , m 3 may be represented 

 by a surface in which the distances of a point from the three sides 

 of a triangular prism represent the quantities m x , m 2 , m 3 , and the 

 distance of the point from the base of the prism represents the 

 quantity In the case of two components, analogous relations may 

 be represented by a plane curve. Such methods are especially useful 

 for illustrating the combinations and separations of the components, 

 and the changes in states of aggregation, which take place when the 

 substances are exposed in varying proportions to the temperature 

 and pressure considered. 



Fundamental equations of ideal gases and gas-mixtures. From 

 the physical properties which we attribute to ideal gases, it is easy 

 to deduce their fundamental equations. The fundamental equation 

 in e, 77, v, and m for an ideal gas is 



n e Em r\ , m /1f _x 



clog = H+aW ; (17) 



cm m ' v 



that in i/r, t, v, and m is 



^ = Em+m*(c-H-clog+alog-); (18) 



that in p, t, and JUL is 



H-c-a c+a /u.-E 



p = ae a t~^e~ r t (19) 



where e denotes the base of the Naperian system of logarithms. As 

 for the other constants, c denotes the specific heat of the gas at 



* [Page 33 of this volume.] 



