Page:Scientific Papers of Josiah Willard Gibbs.djvu/42

6 This quantity $$\gamma$$, which is the ratio of the area of an infinitely small circuit to the work done or heat received in that circuit, and which we may call the scale on which work and heat are represented by areas, or more briefly, the scale of work and heat, may have a constant value throughout the diagram or it may have a varying value. The diagram in ordinary use affords an example of the first case, as the area of a circuit is everywhere proportional to the work or heat. There are other diagrams which have the same property, and we may call all such diagrams of constant scale.

In any case we may consider the scale of work and heat as known for every point of the diagram, so far as we are able to draw the isometrics and isopiestics or the isentropics and isothermals. If we write $$\delta W$$ and $$\delta H$$ for the work and heat of an infinitesimal circuit, and $$\delta A$$ for the area included, the relations of these quantities are thus expressed:— We may find the value of $$W$$ and $$H$$ for a circuit of finite dimensions by supposing the included area $$A$$ divided into areas $$\delta A$$ infinitely small in all directions, for which therefore the above equation will hold, and taking the sum of the values of $$\delta H$$ or $$\delta W$$ for the various areas $$\delta A$$. Writing $$W^C$$ and $$H^C$$ for the work and heat of the circuit $$C$$, and ∑$^C$ for a summation or integration performed within the limits of this circuit, we have