Page:Scientific Papers of Josiah Willard Gibbs.djvu/67

The case that $\left({\frac {dp}{d\eta }}\right)_{v}=0$ includes a considerable number of conceivable cases, which would require to be distinguished. It will be sufficient to mention those most likely to occur.

In a field of stable equilibrium it may occur that $\left({\frac {dp}{d\eta }}\right)_{v}=0$ along a line, on one side of which $\left({\frac {dp}{d\eta }}\right)_{v}>0$, and on the other side $\left({\frac {dp}{d\eta }}\right)_{v}<0$. At any point in such a line the isopiestics will be tangent to the isometrics and the isothermals to the isentropics. (See, however, note on page 29.)

In a field of neutral equilibrium representing a mixture of two different states of the substance, where the isothermals and isopiestics are identical, a line may occur which has the threefold character of an isometric, an isothermal and an isopiestic. For such a line $\left({\frac {dp}{d\eta }}\right)_{v}=0$. If $\left({\frac {dp}{d\eta }}\right)_{v}$ has opposite signs on opposite sides of this line, it will be an isothermal of maximum or minimum temperature.^[1]

The case in which the body is partly solid, partly liquid and party vapor has already been sufficiently discussed. (See pages 23, 24.)

The arrangement of the isometric, isopiestic, etc., as given in figure 13, will indicate directly the sign of any differential co-efficient of the form $\left({\frac {du}{dw}}\right)_{z}$, where $u,w$ and $z$ may be any of the quantities $v,p,t,\eta$ (and $\epsilon$, if the isodynamic be added in the figure). The value of such a differential co-efficient will be indicated, when the rates of increase of $v,p$, etc., are indicated, as by isometrics, etc., drawn both for the values of $v$, etc., at the point $A$, and for values differing from these by a small quantity. For example, the value of $\left({\frac {dp}{dv}}\right)_{\eta }$ will be indicated by the ratio of the segments intercepted upon an isentropic by a pair of isometrics and a pair of isopiestics, of which the differences of volume and pressure have the same numerical value. The case in which $W$ or $H$ appears in the numerator or denominator instead of a

directions of the axes be such as we have assumed, the agreement with figure 13 will be without inversion, and the agreement with fig. 14 will also be without inversion for volume-entropy diagrams, but with inversion for volume-pressure or entropy-temperature diagrams, or those in which $x=\log v$ and $y=\log p$, or $x=\eta$ and $y=\log t$.

1. As some liquids expand and others contract solidifying, it is possible that there are some which will solidify either with expansion, or without change of volume, or with contraction, according to the pressure. If any such there are, they afford examples of the case mentioned above.