180 J. W. Gibbs — Equilibruim of Heterogeneous Substances. 



eating two stable pairs of coexistent phases, in each of which one of 

 the pliases 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 substances, 

 there are two cases which must be distinguished. In the iirst, one of 

 the points of contact of the primitive surface with the qiaadruple 

 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 repi-esents the 

 projection upon the A'^ Y plane (in which ni^, m^, m^ are measured) 

 of a part of the snrftice of dissipated energy, when one of the points 

 of contact D falls within the triangle formed by the other thi-ee A, B, 

 0. 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 developable surfaces are repre- 



FlG. 2. 



sented in the figure by ruled surfaces, the lines indicating the direc- 

 tion 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 cori-esponding points. A, B, C, D, their 

 center of gravity would be at the point representing the total mass.) 



