Page:Scientific Papers of Josiah Willard Gibbs.djvu/43

Rh We have thus an expression for the value of the work and heat of a circuit involving an integration extending over an area instead of one extending over a line, as in equations (5) and (6).

Similar expressions may be found for the work and the heat of a path which is not a circuit. For this case may be reduced to the preceeding by the consideration that $$W = 0$$ for a path on an isometric or on the line of no pressure (eq. 2), and $$H = 0$$ for a path on an isentropic or on the line of absolute cold. Hence the work of any path $$S$$ is equal to that of the circuit formed of $$S$$, the isometric of the final state, the line of no pressure and the isometric of the initial state, which circuit may be represented by the notation $$[S, v, p^0, v'].$$ And the heat of the same path is the same as that of the circuit $$[S, \eta , t^0, \eta '].$$ Therefore using $$W^S$$ and $$H^S$$ to denote the work and heat of any path $$S$$, we have  where as before the limits of the integration are denoted by the expression occupying the place of an index to the sign ∑. These equations evidently include equation (8) as a particular case.

It is easy to form a material conception of these relations. If we imagine, for example, mass inherent in the plane of the diagram with a varying (superficial) density represented by $$\tfrac {1}{\gamma},$$ then ∑$$\tfrac {1}{\gamma} \delta A$$ will