Page:Scientific Papers of Josiah Willard Gibbs.djvu/80

44 entirely above it, except the single point of contact. A tangent plane at any point of the primitive surface within these lines will cut the surface. These lines, therefore, taken together may be called the limit of absolute stability, and the surface outside of them, the surface of absolute stability. That part of the envelop of the rolling plane, which lies between the pair of lines which the plane traces on the surface, is a part of the derived surface, and represents a mixture of two states of the substance.

The relations of these lines and surfaces are roughly represented in horizontal projection in figure 2, in which the full lines represent lines on the primitive surface, and the dotted lines those on the derived surface. $$S$$, $$L$$ and $$V$$ are the points which have a common tangent plane and represent the states of solid, liquid, and vapor which can exist in contact. The plane triangle $$SLV$$ is the derived surface representing compounds of these states. $$LL'$$ and $$VV'$$ are the pair of lines traced by the rolling double tangent plane, between which lies the derived surface representing compounds of liquid and vapor. $$VV$$ and $$SS$$ are another such pair, between which lies the derived surface representing compounds of vapor and solid. $$SS$$ and $$LL$$ are the third pair, between which lies the derived surface representing a compound of solid and liquid. $$LLL'$$, $$V'VV$$ and $$SSS'$$ are the boundaries of the surfaces which represent respectively the absolutely stable states of liquid, vapor, and solid.

The geometrical expression of the results which Dr. Andrews,