Page:Scientific Papers of Josiah Willard Gibbs.djvu/158

$(B)$ 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 substances, 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 $m_{1},m_{2},m_{3}$ 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 developable 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 represented 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 a part of the sheet $(D)$ to protrude through the plane tangent to