Page:Scientific Papers of Josiah Willard Gibbs.djvu/40

the integration being carried on from the beginning to the end of the path. If the direction of the path is reversed, $W$ and $H$ change their signs, remaining the same in absolute value.

If the changes of state of the body form a cycle, i.e., if the final state is the same as the initial, the path becomes a circuit, and the work done and heat received are equal, as may be seen from equation (1), which when integrated for this case becomes $0=H-W$.

The circuit will enclose a certain area, which we may consider as positive or negative according to the direction of the circuit which circumscribes it. The direction in which areas must be circumscribed in order that their value may be positive, is of course arbitrary. In other words, if $x$ and $y$ are the rectangular co-ordinates, we may define an area either as $\int ydx$, or as $\int xdy$.

If an area be divided into any number of parts, the work done in the circuit bounding the whole area is equal to the sum of the work done in all the circuits bounding the partial areas. This is evident from the consideration, that the work done in each of the lines which separate the partial areas appears twice and with contrary signs in the sum of the work done in the circuits bounding the partial areas. Also the heat received in the circuit bounding the whole area is equal to the sum of the heat received in all the circuits bounding the partial areas^[1]

If all the dimensions of a circuit are infinitely small, the ratio of the included area to the work or heat of the circuit is independent of the shape of the circuit and the direction in which it is described, and varies only with its position in the diagram. That this ratio is independent of the direction in which the circuit is described, is evident from the consideration that a reversal of this direction simply changes the sign of both terms of the ratio. To prove that the ratio is independent of the shape of the circuit, let us suppose the area ABCDE (fig. 1) divided up by an infinite number of isometrics $v_{1}v_{2},v_{2}v_{2},$, etc., with equal differences of volume $dv$, and an infinite number of isopiestics $p_{1}p_{1},p_{2}p_{2}$, etc., with equal differences of pressure $dp$. Now from the

1. The conception of areas as positive or negative renders it unnecessary in propositions of this kind to state explicitly the direction in which the circuits are to be described. For the directions of the circuits are determined by the signs of the areas, and the signs of the partial areas must be the same as that of the area out of which they were formed.