Page:Scientific Papers of Josiah Willard Gibbs.djvu/62

being that of the hands of a watch, the work and heat of the circuit will be positive. But if we draw any circuit $EFGH$ in the same direction on the other side of the line $LL$, the work and heat will be negative. For

$W=H=\sum {\frac {1}{\gamma _{v,\eta }}}\delta A=\sum {\frac {dp}{d\eta }}\delta A,$

and the direction of the circuits makes the areas positive in both cases. Now if we should change this diagram into any diagram of constant scale, the areas of the circuits, as representing proportionally the work done in each case, must necessarily have opposite signs, i.e., the direction of the circuits must be opposite. We will suppose that the work done is positive in the diagram of constant scale, when the direction of the circuit is that of the hands of a watch.

Then, in that diagram, the circuit $ABCD$ would have that direction, and the circuit $EFGH$ the contrary direction, as in figure 11. Now if we imagine an indefinite number of circuits on each side of $LL$ in the volume-entropy diagram, it will be evident that to transform such a diagram into one of constant scale, so as to change the direction of all the circuits on one side of $LL$, and of none on the other the diagram must be folded over along tht line; so that the points on one side of $LL$ in a diagram of constant scale do not represent any states of the body, while on the other side of this line, each point, for a certain distance at least, represents two different states of the body, which in the volume-entropy diagram are represented by points on opposite sides of the line $LL$. We have thus in a part of the field two diagrams superposed, which must be carefully distinguished. If this be done, as by the help of different colors, or of continuous and dotted lines, or otherwise and it is remembered that there is no continuity between these superposed diagrams, except along the bounding line $LL$, all the general theorems which have been developed in this article can be readily applied to the diagram. But to the eye or to the imagination, the figure will necessarily be much more confusing than a volume-entropy diagram.

If ${\frac {dp}{d\eta }}=0$ for the line $LL$, there will be another inconvenience in the use of any diagram of constante scale, viz: in the vicinity of the line $LL$, i.e., $1\div \gamma _{v,\eta }$, will have a very small value, so that areas will be greatly reduced in the diagram of constant scale, as com-