EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 119 



for n = 3, or the m-f curve for n = 2, regarded as a surface, or curve, 

 which varies with the temperature and pressure. 

 As by (96) and (92) 



and (for constant temperature and pressure) 



if we imagine a tangent plane for the point to which these letters 

 relate, and denote by f the ordinate for any point in the plane, and 

 by w/, ra 2 7 , w 8 7 , the distances of the foot of this ordinate from the 

 three sides of the triangle PjPgPg, we may easily obtain 



which we may regard as the equation of the tangent plane. Therefore 

 the ordinates for this plane at P p P 2 , and P 3 are equal respectively 

 to the potentials JJL V yu 2 , fa- And 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 93.) Among 

 the bodies which may be formed of S v S 2 , and S B , there may be some 

 which are incapable of variation in composition, 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 deter- 

 mine 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 f 



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 

 surface, 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 

 coexistent phases. 



The surface which we have considered represents the relation 

 between f and m v w 2 , m 8 for homogeneous bodies when t and p 

 have any constant values and m 1 +m 2 +m 3 =l. 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 



