122 EQUILIBBIUM OF HETEROGENEOUS SUBSTANCES. 



(B) are unstable, and a decrease of pressure will give a diagram 

 indicating two stable pairs of coexistent phases, in each of which 

 one of the phases is of the sort represented by the curve (B). When 

 the relation of the volumes is the reverse of that supposed, these 

 results will be produced by the opposite changes of pressure. 



When we have four coexistent phases of three component sub- 

 stances, there are two cases which must be distinguished. In the 

 first, one of the points of contact of the primitive surface with the 

 quadruple tangent plane lies within the triangle formed by joining 

 the other three; in the second, the four points may be joined so 

 as to form a quadrilateral without re-entrant angles. Figure 2 

 represents the projection upon the X-Y plane (in which m p m 2 , m 3 

 are measured) of a part of the surface of dissipated energy, when 

 one of the points of contact D falls within the triangle formed by 

 the other three A, B, C. This surface includes the triangle ABC 

 in the quadruple tangent plane, portions of the three sheets of the 

 primitive surface which touch the triangle at its vertices, EAF, GBH, 

 ICK, and portions of the three developable surfaces formed by a 

 tangent plane rolling upon each pair of these sheets. These develop- 

 able surfaces are represented in the figure by ruled surfaces, the lines 

 indicating the direction of their rectilinear elements. A point within 

 the triangle ABC represents a mass of which the matter is divided, 

 in general, between three or four different phases, in a manner not 

 entirely determined by the position of a point. (The quantities of 

 matter in these phases are such that if placed at the corresponding 

 points, A, B, C, D, their center of gravity would be at the point 

 representing the total mass.) Such a mass, if exposed to constant 

 temperature and pressure, would be in neutral equilibrium. A 

 point in the developable surfaces represents a mass of which the 

 matter is divided between two coexisting phases, which are repre- 

 sented by the extremities of the line in the figure passing through 

 that point. A point in the primitive surface represents of course a 

 homogeneous mass. 



To determine the effect of a change of temperature without change 

 of pressure upon the general features of the surface of dissipated 

 energy, we must know whether heat is absorbed or yielded by a 

 mass in passing from the phase represented by the point D in the 

 primitive surface to the composite state consisting of the phases A, 

 B, and C which is represented by the same point. If the first is the 

 case, an increase of temperature will cause the sheet (D) (i.e., the 

 sheet of the primitive surface to which the point D belongs) to 

 separate from the plane tangent to the three other sheets, so as to be 

 situated entirely above it, and a decrease of temperature, will cause 

 ja part of the sheet (D) to protrude through the plane tangent to 



