Page:Scientific Papers of Josiah Willard Gibbs.djvu/398

changes. In fact, stable critical phases are situated at that limit. They are also situated at the limit of stability with respect to discontinuous changes. These limits are in general distinct, but touch each other at critical phases.

Geometrical illustrations.—In an earlier paper,^[1] the author has described a method of representing the thermodynamic properties of substances of invariable composition by means of surfaces. The volume, entropy, and energy of a constant quantity of the substance are represented by rectangular coordinates. This method corresponds to the first kind of fundamental equation described above. Any other kind of fundamental equation for a substance of invariable composition will suggest an analogous geometrical method. In the present paper, the method in which the coordinates represent temperature, pressure, and the potential, is briefly considered. But when the composition of the body is variable, the fundamental equation cannot be completely represented by any surface or finite number of surfaces. In the case of three components, if we regard the temperature and pressure as constant, as well as the total quantity of matter, the relations between $\zeta ,m_{1},m_{2},m_{3}$ may be represented by a surface in which the distances of a point from the three sides of a triangular prism represent the quantities $m_{1},m_{2},m_{3}$, and the distance of the point from the base of the prism represents the quantity $\zeta$. In the case of two components, analogous relations may be represented by a plane curve. Such methods are especially useful for illustrating the combinations and separations of the components, and the changes in states of aggregation, which take place when the substances are exposed in varying proportions to the temperature and pressure considered.

Fundamental equations of ideal gases and gas-mixtures.—From the physical properties which we attribute to ideal gases, it is easy to deduce their fundamental equations. The fundamental equation in $\epsilon ,\eta ,v$, and $m$ for an ideal gas is

$c\log {\frac {\epsilon -{\text{E}}m}{cm}}={\frac {\eta }{m}}-{\text{H}}+a\log {\frac {m}{v}};$

(17)

that in $\psi ,t,v$, and $m$ is

$\psi ={\text{E}}m+mt\left(c-{\text{H}}-c\log t+a\log {\frac {m}{v}}\right);$

(18)

that in $p,t$, and $\mu$ is

$p=ae^{\frac {{\text{H}}-c-a}{a}}t^{\frac {c+a}{a}}e^{\frac {\mu -{\text{E}}}{at}},$

(19)

where $e$ denotes the base of the Naperian system of logarithms. As for the other constants, $c$ denotes the specific heat of the gas at

1. [Page 33 of this volume.]