Page:Scientific Papers of Josiah Willard Gibbs.djvu/160

form a quadrilateral $ABCD$ (fig. 4) without re-entrant angles, the surface of dissipated energy will include this plane quadrilateral, portions of the four sheets of the primitive surface which are tangent to it, and portions of the four developable surfaces formed by double tangent planes rolling upon the four pairs of these sheets which correspond to the four sides of the quadrilateral. To determine the general effect of a variation of temperature upon the surface of dissipated energy, let us consider the composite states represented by the point $I$ at the intersection of the diagonals of the quadrilateral. Among these states (which all relate to the same kind and quantity of matter) there is one which is composed of the phases $A$ and $C,$ and another which is composed of the phases $B$ and $D.$ Now if the entropy of the first of these states is greater than that of the second (i.e., if heat is given out by a body in passing from the first to the second

state at constant temperature arid pressure), which we may suppose without loss of generality, an elevation of temperature while the pressure remains constant will cause the triple tangent planes to $(B),(D),$ and $(A),$ and to $(B),(D),$ and $(C),$ to rise above the triple tangent planes to $(A),(C),$ and $(B),$ and to $(A),(C),$ and $(D),$ in the vicinity of the point $I$. The surface of dissipated energy will therefore take the form indicated in figure 5, in which there are two plane triangles and five developable surfaces besides portions of the four primitive sheets. A diminution of temperature will give a different but entirely analogous form to the surface of dissipated energy. The quadrilateral $ABCD$ will in this case break into two triangles along the diameter BD. The effects produced by variation of the pressure while the temperature remains constant will of course be similar to those described. By considering the difference of volume instead of the difference of entropy of the two states represented by the point $I$ in the quadruple tangent plane, we may distinguish between the effects of increase and diminution of pressure.

It should be observed that the points of contact of the quadruple