./. W. Gihbs — Eqailibrlum of Hetet'oyeneoufi /Substances. 187 



surface of dissipated energy such as is represented in figure (9), con- 

 taining two plane triangles ABC, A'B'C belonging to triple tangent 

 planes, a portion of the sheet (A) on the left of the line aA A'a', a por- 

 tion of the sheet (B) on the right of the line bBB'b', two separate 

 portions cCy and c'C'y' of the sheet (C), two separate portions aACc 

 and a'A'C'c' of the developable surface formed on (A) and (C), two 

 separate portions bBC;/ and h'B'C'y' of the developable surface 

 formed on (B) and (C), and the portion A'ABB' of the developable 

 surface formed on (A) and (B). 



From these geometrical relations it appears that (in general) the 

 temperature of three coexistent phases is a maximum or minimum for 

 constant pressure, and the pressure of three coexistent phases a maxi- 

 mum or mininuim for constant temperature, when the composition of 

 the three coexistent phases is such that one can be formed by com- 

 bining the other two. This result has been obtained analytically 

 on page 156. 



The preceding examples are amply sufficient to illustrate the use 

 of the m-'C, surfaces and curves. The physical properties indicated by 

 the nature of the siirface of dissipated energy have been only occa- 

 sionally mentioned, as they are often far more distinctly indicated by 

 the diagrams than they could be in words. It will be observed that 

 a knowledge of the lines which divide the various different portions 

 of the surface of dissipated energy and of the direction of the recti- 

 linear elements of the developable surfaces, as projected upon the 

 JC-Y' plane, without a knowledge of the form of the m-'Q surface in 

 space, is sufficient for the determination (in respect to the quantity 

 and composition of the resulting masses) of the combinations and 

 separations of the substances, and of the changes in their states of 

 aggregation, which take place when the substances are exposed to 

 the temperature and pressure to which the projected lines relate, 

 except so far as such transformations are prevented by passive re- 

 sistances to change. 



CRITICAL PHASES. 



It has been ascertained by experiment that the variations of two 

 •coexistent states of the same substance are in some cases limited in 

 one direction by a terminal state at which the distinction of the 

 coexistent states vanishes.* This state has been called the critical 

 state. Analogous properties may doubtless be exhibited by com- 

 pounds of variable composition without change of tempei-ature or 



* See Dr. Andrews " On the continuity of the gaseous and liquid states of matter." 

 Phil. Trans., vol. 159, p. 575. 



