Page:Scientific Papers of Josiah Willard Gibbs.djvu/41

principle of continuity, as the whole figure is infinitely small, the ratio of the area of one of the small quadrilaterals into which the figure is divided to the work done in passing around it is approximately the same for all the different quadrilaterals. Therefore the area of the figure composed of all the complete quadrilaterals which fall within the given circuit has to the work done in circumscribing this figure the same ratio, which we will call $\gamma$. But the area of this figure is approximately the same as that of the given circuit, and the work done in describing the given circuit (eq. 5). Therefore the area of the given circuit has to the work done or heat received in that circuit this ratio $\gamma$, which is independent of the shape of the circuit.

Now if we imagine the systems of equidifferent isometrics and isopiestics, which have just been spoken of, extended over the whole diagram, the work done in circumscribing one of the small quadrilaterals, so that the increase of pressure directly precedes the increase of volume, will have in every part of the diagram a constant value, viz., the product of the differences of volume and pressure $(dv\times dp)$, as may easily be proved by applying equation (2) successively to its four sides. But the area of one of these quadrilaterals, which we could consider as constant within the limits of the infinitely small circuit, may vary for different parts of the diagram, and will indicate proportionally the value of $\gamma$, which is equal to the area divided by $dv\times dp$.

In like manner, if we imagine systems of isentropics and isothermals drawn throughout the diagram for equal differences $d\eta$ and $dt$, the heat received in passing around one of the small quadrilaterals, so that the increase of $t$ shall directly precede that of $\eta$, will be the constant product $d\eta \times dt$, as may be proved by equation (3), and the value of $\gamma$, which is equal to the area divided by the heat, will be indicated proportionally by the areas.^[1]

1. The indication of the value of $\gamma$ by systems of equidifferent isometrics and isopiestics, or isentropics and isothermals, is explained above, because it seems in accordance with the spirit of the graphical method, and because it avoids the extraneous consideration of the co-ordinates. If, however, it is desired to have analytical expressions for the value of $\gamma$ based upon the relations between the co-ordinates of the point and the state of the body, it is easy to deduce such expressions as the following, in which $x$ and $y$ are the rectangular co-ordinates, and it is supposed that the sign of an area is determined in accordance with the equation $A=\int ydx$:—

   ${\frac {1}{\gamma }}={\frac {dv}{dx}}\cdot {\frac {dp}{dy}}-{\frac {dp}{dx}}\cdot {\frac {dv}{dy}}={\frac {d\eta }{dx}}\cdot {\frac {dt}{dy}}-{\frac {dt}{dx}}\cdot {\frac {d\eta }{dy}},$

   where $x$ and $y$ are regarded as the independent variables;—or

   $\gamma ={\frac {dx}{dv}}\cdot {\frac {dy}{dp}}-{\frac {dy}{dv}}\cdot {\frac {dx}{dp}},$