Page:Scientific Papers of Josiah Willard Gibbs.djvu/85

The lines traced upon the primitive surface by the rolling double tangent plane, which have been called the limit of absolute stability, do not end at the vertices of the triangle which represents a mixture of those states. For when the plane is tangent to the primitive surface in these three points, it can commence to roll upon the surface as a double tangent plane not only by leaving the surface at one of these points, but also by a rotation in the opposite direction. In the latter case, however, the lines traced upon the primitive surface by the points of contact, although a continuation of the lines previously described, do not form any part of the limit of absolute stability. And the parts of the envelops of the rolling plane between these lines, although a continuation of the developable surfaces which have been described, and representing states of the body, of which some at least may be realized, are of minor interest, as they form no part of the surface of dissipated energy on the one hand, nor have the theoretical interest of the primitive surface on the other.

Problems relating to the Surface of Dissipated Energy.

The surface of dissipated energy has an important application to a certain class of problems which refer to the results which are theoretically possible with a given body or system of bodies in a given initial condition.

For example, let it be required to find the greatest amount of mechanical work which can be obtained from a given quantity of a certain substance in a given initial state, without increasing its total volume or allowing heat to pass to or from external bodies, except

such surfaces taken together will form a continuous sheet, which, if we reject the part, if any, for which $p<0$, forms the surface of dissipated energy and has the geometrical properties mentioned above. There will, however, be no such part in which $p<0$, if there is any assignable temperature $t'$ at which the substance has the properties of a perfect gas except when its volume is less than a certain quantity $v'$. For the equations of an isothermal line in the thermodynamic surface of a perfect gase are (see equations (b) and (e) on pages 12-13)

$\epsilon =C$

$\eta =a\log {v}+C'.$

The isothermal of $t'$ in the thermodynamic surface of the substance in question must therefore have the same equations in the part in which $v$ exceeds the constant $v'$. Now if at any point in this surface $p<0$ and $t>0$ the equation of the tangent plane for that point will be

$\epsilon =m\eta +nv+C'',$

where $m$ denotes the temperature and $-n$ the pressure for the point of contact, so that $m$ and $n$ are both positive. Now it is evidently possible to give so large a value to $v$ in the equations of the isothermal that the point thus determined shall fall below the tangent plane. Therefore, the tangent plane cuts the primitive surface, and the point of the thermodynamic surface for which $p<0$ cannot belong to the surfaces mentioned in the last paragraph as forming a continuous sheet.