EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 125 



tangent plane with the primitive surface may be at isolated points or 

 curves belonging to the latter. So also, in the case of two component 

 substances, the points of contact of the triple tangent line may be at 

 isolated points belonging to the primitive curve. Such cases need 

 not be separately treated, as the necessary modifications in the pre- 

 ceding statements, when applied to such cases, are quite evident. 

 And in the remaining discussion of this geometrical method, it will 

 generally be left to the reader to make the necessary limitations or 

 modifications in analogous cases. 



The necessary condition in regard to simultaneous variations of 

 temperature and pressure, in order that four coexistent phases of 

 three components, or three coexistent phases of two components, shall 

 remain possible, has already been deduced by purely analytical pro- 

 cesses. (See equation (129).) 



We will next consider the case of two coexistent phases of identi- 

 cal composition, and first, when the number of components is two. 

 The coexistent phases, if each is variable in composition, will be 

 represented by the point of contact of two curves. One of the curves 

 will in general lie above the other except at the point of contact; 

 therefore, when the temperature and pressure remain constant, one 

 phase cannot be varied in composition without becoming unstable, 

 while the other phase will be stable if the proportion of either 

 component is increased. By varying the temperature or pressure, we 

 may cause the upper curve to protrude below the other, or to rise 

 (relatively) entirely above it. (By comparing the volumes or the 

 entropies of the two coexistent phases, we may easily determine 

 which result would be produced by an increase of temperature or 

 of pressure.) Hence, the temperatures and pressures for which two 

 coexistent phases have the same composition form the limit to the 

 temperatures and pressures for which such coexistent phases are 

 possible. It will be observed that as we pass 

 this limit of temperature and pressure, the pair 

 of coexistent phases does not simply become 

 unstable, like pairs and triads of coexistent 

 phases which we have considered before, but 

 there ceases to be any such pair of coexistent 



phases. The same result has already been p . . 



obtained analytically on page 99. But on 



that side of the limit on which the coexistent phases are possible, 

 there will be two pairs of coexistent phases for the same values 

 of t and p, as seen in figure 6. If the curve AA' represents vapor, 

 and the curve BB' liquid, a liquid (represented by) B may exist 

 in contact with a vapor A, and (at the same temperature and 

 pressure) a liquid B' in contact with a vapor A'. If we compare 



