EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 115 



or by (98), if we regard t, JUL V ... // n as the independent variables, 



In like manner we may obtain 



z /1QC , 



' (186) 



Any one of these equations, (185), (186), may be regarded, in general, 

 as the equation of the limit of stability. We may be certain that 

 at every phase at that limit one at least of these equations will 

 hold true. 



Geometrical Illustrations. 



Surfaces in which the Composition of the Body represented is 



Constant. 



In the second paper of this volume (pp. 33-54) a method is 

 described of representing the thermodynamic properties of substances 

 of invariable composition by means of surfaces. The volume, entropy, 

 and energy of a constant quantity of a substance are represented 

 by rectangular co-ordinates. This method corresponds to the first 

 kind of fundamental equation described on pages 85-89. Any 

 other kind of fundamental equation for a substance of invariable 

 composition will suggest an analogous geometrical method. Thus, 

 if we make m constant, the variables in any one of the sets (99)-(103) 

 are reduced to three, which may be represented by rectangular 

 co-ordinates. This will, however, afford but four different methods, 

 for, as has already (page 94) been observed, the two last sets are 

 essentially equivalent when n \. 



The first of the above mentioned methods has certain advantages, 

 especially for the purposes of theoretical discussion, but it may 

 often be more advantageous to select a method in which the proper- 

 ties represented by two of the co-ordinates shall be such as best serve 

 to identify and describe the different states of the substance. This 

 condition is satisfied by temperature and pressure as well, perhaps, 

 as by any other properties. We may represent these by two of 

 the co-ordinates and the potential by the third. (See page 88.) 

 It will not be overlooked that there is the closest analogy between 

 these three quantities in respect to their parts in the general 

 theory of equilibrium. (A similar analogy exists between volume, 

 entropy, and energy.) If we give m the constant value unity, 

 the third co-ordinate will also represent f, which then becomes equal 

 to /UL. 



