Page:Scientific Papers of Josiah Willard Gibbs.djvu/51

We have first,

$dv=0$,

then $dW=0$,

and $d\epsilon =td\eta$.

If we add $dH=td\eta$,

these four equations will evidently be equivalent to the three independent equations (1), (2) and (3), combined with the assumption which we have just made. For a liquid, then, $\epsilon$, instead of being a function of two quantities $v$ and $\eta$, is a function of $\eta$ alone,—$t$ is also a function of $\eta$ alone, being equal to the differential co-efficient of the function $\epsilon$; that is, the value of one of the three quantities $t$, $\epsilon$ and $\eta$, is sufficient to determine the other two. The value of $v$, moreover, is fixed without reference to the values of $t$, $\epsilon$ and $\eta$ (so long as these do not pass the limits of values possible for liquidity); while $p$ does not enter the equations, i.e., $p$ may have any value (within certain limits) without affectiong the values of $t$, $\epsilon$, $\eta$ or $v$. If the body change its state, continuing always liquid, the value of $W$ for such a change is 0, and that of $H$ is determined by the values of any one of the three quantities $t$, $\epsilon$ and $\eta$. It is, therefore, the relations between $t$, $\epsilon$, $\eta$ and $H$, for which a graphical expression is to be sought; a method, therefore, in which the co-ordinates of the diagram are made equal to the volume and pressure, is totally inapplicable to this particular case; $v$ and $p$ are indeed the only two of the five functions of the state of the body, $v$, $p$, $t$, $\epsilon$ and $\eta$, which have no relations either to each other, or to the other three, or to the quantities $W$ and $H$, to be expressed.^[1] The values of $v$ and $p$ do not really determine the state of an incompressible fluid,—the values of $t$, $\epsilon$ and $\eta$ are still left undetermined, so that through every point in the volume-pressure diagram which represents the liquid there must pass (in general) an infinite number of isothermals, isodynamics and isentropics. The character of this part of the diagram is as follows:—the states of liquidity are represented by the points of a line parallel to the axis of pressures, and the isothermals, isodynamics and isentropics, which cross the field of partial vaporization and meet this line, turn upward and follow its course.^[2]

In the entropy-temperature diagram the relations of $t$, $\epsilon$ and $\eta$ are

1. That is, $v$ and $p$ hve no such relations to the other quantities, as are expressible by equations; $p$, however, cannot be less than a certain function of $t$.
2. All these difficulties are of course removed when the differences of volume of the liquid at different temperatures are rendered appreciable on the volume-pressure diagram. This can be done in various ways,—among others, by choosing as the body to which $v$, etc., refer, a sufficiently large quantity of the fluid. But, however we do it, we must evidently give up the possibility of representing the body in the state of vapor in the same diagram without making its dimensions enormous.