128 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



aABb, a'A'B'b', the last two being different portions of the same 

 developable surface. 



But if, when the primitive surface is constructed for such a tem- 

 perature and pressure that it has three points of contact with the same 

 plane in the same straight line, the sheet (C) (which has the middle 

 position) at its point of contact with the triple tangent plane intersects 

 the developable surface formed upon the other sheets (A) and (B), the 

 surface of dissipated energy will not include this developable surface, 

 but will consist of portions of the three primitive sheets with two 

 developable surfaces formed on (A) and (C) and on (B) and (C). These 

 developable surfaces meet one another at the point of contact of (C) 

 with the triple tangent plane, dividing the portion of this sheet which 

 belongs to the surface of dissipated energy into two parts. If now 

 the temperature or pressure are varied so as to make the sheet (C) 

 sink relatively to the developable surface formed on (A) and (B), the 

 only alteration in the general features of the surface of dissipated 



energy will be that the developable 

 surfaces formed on (A) and (C) and 

 on (B) and (C) will separate from 

 one another, and the two parts of 

 the sheet (C) will be merged in 

 one. But a contrary variation of 

 temperature or pressure will give a 

 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 shee't (A) 

 on the left of the line a AAV, a portion 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 bBCy and b'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 

 maximum or minimum for constant temperature, when the com- 

 position of the three coexistent phases is such that one can be 

 formed by combining the other two. This result has been obtained 

 analytically on page 99. 



The preceding examples are amply sufficient to illustrate the use of 

 the m-f surfaces and curves. The physical properties indicated by the 

 nature of the surface of dissipated energy have been only occasionally 

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



