Page:Scientific Papers of Josiah Willard Gibbs.djvu/70

we will call the thermodynamic surface of the body for which it is formed.^[1]

To fix our ideas, let the axes of $v,\eta$ and $\epsilon$ have the directions usually given to the axes of $X,Y$, and $Z$ ($v$ increasing to the right, $\eta$ forward, and $\epsilon$ upward). Then the pressure and temperature of the state represented by any point of the surface are equal to the tangents of the inclinations of the surface to the horizon at that point, as measured in planes perpendicular to the axes of $\eta$ and of $v$ respectively. (Eqs. 2 and 3.) It must be observed, however, that in the first case the angle of inclination is measured upward from the direction of decreasing $v$, and in the second, upward from the direction of increasing $\eta$. Hence, the tangent plane at any point indicates the temperature and pressure of the state represented. It will be convenient to speak of a plane as representing a certain pressure and temperature, when the tangents of its inclinations to the horizon, measured as above, are equal to that pressure and temperature.

Before procedding farther, it may be worth while to distinguish between what is essential and what is arbitrary in a surface thus formed. The position of the plane $v=0$ in the surface is evidently fixed, but the position of the planes $\eta =0,\epsilon =0$ is arbitrary, provided the direction of the axes of $\eta$ and $\epsilon$ be not altered. This results from the nature of the definitions of entropy and energy, which involve each an arbitrary constant. As we may make $\eta =0$ and $\epsilon =0$ for any state of the body which we may choose, we may place the origin of co-ordinates at any point in the plane $v=0$. Again, it is evident from the form of equation (1) that whatever changes we may make in the units in which volume, entropy, and energy are measured, it will always be possible to make such changes in the units of temperature and pressure, that the equation will hold true in its present form, without the introduction of constants. It is easy to see how a change of the units of volume, entropy, and energy would affect the surface. The projections parallel to any one of the axes of distances between points of the surface would be changed in the ratio inverse to that in which the corresponding unit had been changed. These considerations enable us to foresee to a certain extent the nature of the general properties of the surface which we are to investigate. They

1. Professor J. Thomson has proposed and used a surface in which the co-ordinates are proportional to the volume, pressure, and temperature of the body. (Proc. Roy. Soc., Nov. 16, 1871, vol. xx, p. 1; and Phil. Mag., vol. xliii, p. 227.) It is evident, however, that the relation between the volume, pressure, and temperature affords a less complete knowledge of the properties of the body than the relation between the volume, entropy, and energy. For, while the former relation is entirely determined by the latter, and can be derived from it by differentiation, the latter relation is by no means determined by the former.