J. TF. Gibhs — Equilibrmm of Heterogeneoxis Substances. 183 



variation of the pressure wliile the temperature remains constant will 

 of course be similar to those described. By considering the diiference 

 of volume instead of the difference of entropy of the two states repi-e- 

 sented by the point I in the quadruple tangent plane, we may distin- 

 guish between the effects of increase and diminution of pressure. 



It should be observed that the points of contact of the quadruple 

 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 

 modificatioiis in analogoiis 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 con- 

 tact ; 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 com- 

 ponent 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 pos- 

 sible. 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 obtained analytically on 



