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

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6

GRAPHICAL METHODS IN THE

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:—^[1]

\delta W=\delta H={\frac {1}{\gamma }}\delta A.

(7)

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

where $v$ and $p$ are the independent variables:— or

\gamma ={\frac {dx}{d\eta }}\cdot {\frac {dy}{dt}}-{\frac {dy}{d\eta }}\cdot {\frac {dx}{dt}},

\eta

t

{\frac {1}{\gamma }}={\frac {-{\frac {d^{2}\epsilon }{dvd\eta }}}{{\frac {dx}{dv}}\cdot {\frac {dy}{d\eta }}-{\frac {dy}{dv}}\cdot {\frac {dx}{d\eta }}}},

v

\eta

These and similar expressions for ${\tfrac {1}{\gamma }}$ may be found by dividing the value of the work or heat for an infinitely small circuit by the area included. This operation can be most conveniently performed upon a circuit consisting of four lines, in each of which one of the independent variables is constant. E.g., the last formula can be most easily found from an infinitely small circuit formed of two isometrics and two isentropics.

↑ To avoid confusion, as $\delta W$ and $\delta H$ are generally used and are used elsewhere in this article to denote the work and heat of an infinite　short path, a slightly different notation, $\delta W$ and $\delta H$ , is here used to denote the work and heat of an infinitely small circuit. So $\delta A$ is used to denote an element of area which is infinitely small in all directions, as the letter $d$ would only imply that the element was infinitely small in one direction. So also below, the integration or summation which extends to all the elements written with $\delta$ is denoted by the character $\sum$ , as the character $\int$ naturally refers to elements written with $d$ .

[2] To avoid confusion, as $\delta W$ and $\delta H$ are generally used and are used elsewhere in this article to denote the work and heat of an infinite　short path, a slightly different notation, $\delta W$ and $\delta H$ , is here used to denote the work and heat of an infinitely small circuit. So $\delta A$ is used to denote an element of area which is infinitely small in all directions, as the letter $d$ would only imply that the element was infinitely small in one direction. So also below, the integration or summation which extends to all the elements written with $\delta$ is denoted by the character $\sum$ , as the character $\int$ naturally refers to elements written with $d$ .

[1]