286 MOREY 



ART. G 



It will be observed that these two expressions are identical 

 except for the first row of the determinants, which in (C) con- 

 tains the composition terms of phase P' , and in (D) contains 

 the composition terms of phase P". Hence it is evident that 

 the numerical values of expressions (A) and (B) will be the 

 same, i.e., the two curves will be stable in the same direction 

 from the invariant point, when (C) and (D) have opposite 

 signs (since (A) and (B) have opposite signs). But (C) and (D) 

 will have opposite signs only when phases P' and P" lie on 

 opposite sides of the onefold P'", P'^, . . . P"+\ P"+\ In a 

 two-component system this onefold is a point; in a three- 

 component system, a line; in a four-component system, a plane, 

 etc. 



The above may be summarized as follows: When two adjoin- 

 ing p-t curves (which represent the relation between the 

 variations in pressure and temperature in two different uni- 

 variant equilibria between 7i -\- 1 phases in a system of n com- 

 ponents) coincide, owing to a linear relation being possible 

 between the compositions of the n phases common to both 

 equilibria, i.e., to these n phases lying on the onefold n, whose 

 position is determined by the above Hnear relation, these 

 equilibria are stable in the same direction from the invariant 

 point, i.e., their stable portions coincide, when the other two 

 phases lie on opposite sides of the onefold n. By "the other 

 two phases" is meant the phases, one in each of the univariant 

 equilibria, which do not lie on the onefold n. In a two-com- 

 ponent system, the onefold n is a point; in a three-component 

 system, a line; in a four-component system, a plane, etc. This 

 has been proved for the case that a linear relation exists 

 between the compositions of n of the (n + 2) phases that 

 coexist at the invariant point. The cases where a linear relation 

 exists between the composition of (n — 1), {n — 2), ... {n — a), 

 phases may be regarded as special cases. 



3S. Generalizations Concerning p-t Curves. Before illus- 

 trating the application of the above principles to actual cases, 

 certain generalizations will be made concerning the p-t curves 

 from the state of aggregation of the phases. The actual 

 value of dp/dt for any univariant equilibrium is given by 



