Page:Scientific Papers of Josiah Willard Gibbs.djvu/153

The sheet which gives the least values of $\zeta$ is in each case that which represents the stable states of the substance. From this it is evident that in passing around the projection of the triple point we pass through lines representing alternately coexistent stable and coexistent unstable states. But the states represented by the intermediate values of $\zeta$ may be called stable relatively to the states represented by the highest. The differences $\zeta ^{(L)}-\zeta ^{(V)},$ etc. represent the amount of work obtained in bringing the substance by a reversible process from one to the other of the states to which these quantities relate, in a medium having the temperature and pressure common to the two states. To illustrate such a process, we may suppose a plane perpendicular to the axis of temperature to pass through the points representing the two states. This will in general cut the double line formed by the two sheets to which the symbols $(L)$ and $(V)$ refer. The intersections of the plane with the two sheets will connect the double point thus determined with the points representing the initial and final states of the process, and thus form a reversible path for the body between those states.

The geometrical relations which indicate the stability of any state may be easily obtained by applying the principles stated on pp. 100 ff. to the case in which there is but a single component. The expression (133) as a test of stability will reduce to

$\epsilon -t'\eta +p'v-\mu 'm,$

(197)

the accented letters referring to the state of which the stability is in question, and the unaccented letters to any other state. If we consider the quantity of matter in each state to be unity, this expression may be reduced by equations (91) and (96) to the form

$\zeta -\zeta '+(t'-t)\eta -(p-p')v,$

(198)

which evidently denotes the distance of the point $(t',p',\zeta ')$ below the tangent plane for the point $(t,p,\zeta )$, measured parallel to the axis of $\zeta$. Hence if the tangent plane for every other state passes above the point representing any given state, the latter will be stable. If any of the tangent planes pass below the point representing the given state, that state will be unstable. Yet it is not always necessary to consider these tangent planes. For, as has been observed on page 103, we may assume that (in the case of any real substance) there will be at least one not unstable state for any given temperature and pressure, except when the latter is negative. Therefore the state represented by a point in the surface on the positive side of the plane $p=0$ will be unstable only when there is a point in the surface for which $t$ and $p$ have the same values and $\zeta$ a less value. It follows from what has been stated, that where the surface is doubly convex