J. W. Gihbs — Equilibrium oj' Heterogeneous /Substances. 177 



and by >«,', wig'j "^a'j ^^^^ distances of the foot of this ordinate from 

 the three sides of the triangle Pj P3 Pg, we may easily obtain 



C' = /(,mj' + /-/o Wo' + Ms "'3', (199) 



which we may regard as the equation of the tangent plane. There- 

 fore the ordinates for this plane at P^, P,, and P3 are equal respect- 

 ively to the potentials yu,, yUg? 'i^^*^ /'s- -"^nd in general, the ordinate 

 for any point in the tangent plane is equal to the potential (in the 

 phase represented by the point of contact) for a substance of which 

 the composition is indicated by the position of the ordinate. (See 

 page 149.) Among the bodies which may be formed of S^, aS^, and 

 -83, there may be some which are incapable of variation in composi- 

 tion, or which are capable only of a single kind of variation. These 

 will be represented by single points and curves in vertical planes. 

 Of the tangent plane to one of these curves only a single line will be 

 fixed, which will determine a series of potentials of which only two 

 will be independent. The phase represented by a separate point will 

 determine only a single potential, viz., the potential for the substance 

 of the body itself, which will be equal to 'C. 



The points representing a set of coexistent phases have in general 

 a common tangent plane. But when one of these points is situated 

 on the edge where a sheet of the surface terminates, it is sufficient if 

 the plane is tangent to the edge and passes below the surface. Or, 

 when the point is at the end of a separate line belonging to the sur- 

 face, or at an angle in the edge of a sheet, it is sufficient if the plane 

 pass through the point and below the line or sheet. If no part of the 

 surface lies below the tangent plane, the points where it meets the 

 plane will represent a stable (or at least not unstable) set of co- 

 existent phases. 



The surface which we have considered represents the relation 

 between 'C, and m^, m^, m„ for homogeneous bodies when t and jo 

 have any constant values and ni^ -|- m^ -f-^s = 1- It will often be 

 useful to consider the surface which represents the relation between 

 the same variables for bodies which consist of parts in different but 

 coexistent phases. We may suppose that these are stable, at least in 

 regard to adjacent phases, as otherwise the case would be devoid of 

 interest. The point which represents the state of the composite 

 body will evidently be at the center of gravity of masses equal to 

 the parts of the body placed at the points representing the phases of 

 these parts. Hence from the surface representing the properties of 

 homogeneous bodies, which may be called the primitive surface, we 



Trans. Conn. Acad., Vol. III. 23 January, 1876. 



