Page:Scientific Papers of Josiah Willard Gibbs.djvu/66

30 $$A$$ (fig. 13), let there be drawn the isometric $$vv'$$ and the isentropic $$\eta \eta^\prime$$, and let the positive sides of these lines be indicated as in the figure. The conditions $$\left( \frac{dp}{d\eta}\right)_{v}>0$$ and $$[dp:dv]_{\eta}<0$$ require that the pressure at $$v$$ and $$\eta$$ shall be greater than at $$A$$, and hence that the isopiestic shall fall as $$pp'$$ in the figure, and have its positive side turned as indicated. Again, the conditions $$\left( \frac{dp}{d\eta}\right)_{v}>0$$ and $$[dp:dv]_{\eta}<0$$ require that the temperature at $$\eta$$ and at $$p$$ shall be greater than at $$A$$, and hence, that the isothermal shall fall as $$tt'$$ and have its positive side turned as indicated. As it is not necessary that $$\left( \frac{dp}{d\eta}\right)_{v}>0$$, the lines $$pp'$$ and $$tt'$$ may be tangent to one another at $$A$$, provided that they cross one another, so as to have the same order about the point $$A$$ as is represented in the figure; i.e., they may have a contact of the second (or any even) order. But the condition that $$\left( \frac{dp}{d\eta}\right)_{v}>0$$, and hence $$\left( \frac{dt}{dv}\right)_{v\eta}<0$$, does not allow $$pp'$$ to be tangent to $$vv'$$, nor $$tt'$$ to $$\eta \eta^\prime$$.

If $$\left( \frac{dp}{d\eta}\right)_{v}$$ be still positive, but the equilibrium be neutral, it will be possible for the body to change its state without change either of temperature or of pressure; i.e., the isothermal and isopiestic will be identical. The lines will fall as in figure 13, except that the isothermal and isopiestic will be superimposed.

In like manner, if $$\left( \frac{dp}{d\eta}\right)_{v}<0$$, it may be proved that the lines will fall as in figure 14 for stable equilibrium, and in the same way for neutral equilibrium, except that $$pp'$$ and $$tt'$$ will be superposed.