Page:Scientific Papers of Josiah Willard Gibbs.djvu/49

Rh  The constant of integration becomes 0, if we call the entropy 0 for the state of which the volume and energy are both unity.

Any other equations which subsist between $$v, p, t, \epsilon$$ and $$\eta$$ may be derived from the three independent equations ), and . If we eliminate $$\epsilon$$ from  and, we have  Eliminating $$v$$ from  and , we have  Eliminating $$t$$ from  and , we have  If $$v$$ is constant, equation  becomes  i.e, the isometrics in the entropy-temperature diagram are logarithmic curves identical with one another in form,—a change in the value of $$v$$ having only the effect of moving the curve parallel to the axis of $$\eta$$. If $$p$$ is constant, equation  becomes  so that the isopiestics in this diagram have similar properties. This identity in form diminishes greatly the labour of drawing any considerable number of these curves. For if a card or thin board be cut in the form of one of them, it may be used as a pattern or ruler to draw all of the same system.

The isodynamics are straight in this diagram (eq. ).

To find the form of the isothermals and isentropics in the volume-pressure diagram, we may make $$t$$ and $$\eta$$ constant in equations and  respectively, which will then reduce to the well-known equations of these curves:—