Page:Scientific Papers of Josiah Willard Gibbs.djvu/155

for $n=3,$ or the $m\!-\!\zeta$ curve for $n=2$, regarded as a surface, or curve, which varies with the temperature and pressure.

As by (96) and (92)

$\zeta =\mu _{1}m_{1}+\mu _{2}m_{2}+\mu _{3}m_{3},$

and (for constant temperature and pressure)

$d\zeta =\mu _{1}dm_{1}+\mu _{2}dm_{2}+\mu _{3}dm_{3},$

if we imagine a tangent plane for the point to which these letters relate, and denote by $\zeta$ the ordinate for any point in the plane, and by $m'_{1},m'_{2},m'_{3}$, the distances of the foot of this ordinate from the three sides of the triangle $P_{1}P_{2}P_{3}$, we may easily obtain

$\zeta '=\mu _{1}'m'_{1}+\mu _{2}'m'_{2}+\mu _{3}'m'_{3},$

(199)

which we may regard as the equation of the tangent plane. Therefore the ordinates for this plane at $P_{1},P_{2}$ and $P_{3}$ are equal respectively to the potentials $\mu _{1},\mu _{2},\mu _{3}$. 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 $S_{1},S_{2},$ and $S_{3},$ 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 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 $\zeta$.

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 $\zeta$ and $m_{1},m_{2},m_{3}$ for homogeneous bodies when $t$ and $p$ have any constant values and $m_{1}+m_{2}+m_{3}=1$. 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