Page:Scientific Papers of Josiah Willard Gibbs.djvu/59

Dividing by the area $dv\,d\eta$, and writing $\gamma _{v,\eta }$ for the scale of work and heat in a diagram of this kind, we have

${\frac {1}{\gamma _{v,\eta }}}=-{\frac {d^{2}\epsilon }{dv\,d\eta }}={\frac {dp}{d\eta }}=-{\frac {dt}{dv}}$

(15)

The two last expressions for the value of $1\div \gamma _{v,\eta }$ indicate that the value of $\gamma _{v,\eta }$ in different parts of the diagram will be indicated proportionally by the segments into which vertical lines are divided by a system of equidifferent isopiestics, and also by the segments of equidifferent isothermals. These results might also be derived directly from the propositions on page 5.

As, in almost all cases, the pressure of a body is increased when it receives heat without change of volume, ${\frac {dp}{d\eta }}$ is in general positive, and the same will be true of $\gamma _{v,\eta }$ under the assumptions which we have made in regard to the directions of the axes (page 21) and the definition of a positive area (page 22).

In the estimation of work and heat it may often be of use to consider the deformation necessary to reduce the diagram to one of constant scale for work and heat. Now if the diagram be so deformed that each point remains in the same vertical line, but moves in this line so that all isopiestics become straight and horizontal lines at distances proportional to their differences of pressure, it will evidently become a volume-pressure diagram. Again, if the diagram be so deformed that each point remains in the same horizontal line, but moves in it so that isothermals become straight and vertical lines at distances proportional to their differences of temperature, it will become an entropy-temperature diagram. These considerations will enable us to compute numerically the work or heat of any path which is given in a volume-entropy diagram, when the pressure and temperature are known for all points of the path, in a manner analogous to that explained on page 19.

The ratio of any element of area in the volume-pressure or the entropy-temperature diagram, or in any other in which the scale of work and heat is unity, to the corresponding element in the volume-entropy diagram is represented by ${\frac {1}{\gamma _{v,\eta }}}$ or $-{\frac {d^{2}\epsilon }{dv\,d\eta }}\cdot$ The cases in which this ratio is 0, or changes its sign, demand especial attention, as in such cases the diagrams of constant scale fail to give a satisfactory representation of the properties of the body, while no difficulty or inconvenience arises in the use of the volume-entropy diagram.

As $-{\frac {d^{2}\epsilon }{dv\,d\eta }}={\frac {dp}{d\eta }}$, its value is evidently zero in that part of the diagram which represents the body when in part solid, in part liquid,