Page:Scientific Papers of Josiah Willard Gibbs.djvu/38

2 These are subject to the relations expressed by the following differential equations:—   where $$\alpha$$ and $$\beta$$ are constants depending upon the units by which $$v$$, $$p$$, $$W$$ and $$H$$ are measured. We may suppose our units so chosen that $$\alpha = 1$$ and $$\beta = 1$$, and write our equations in the simpler form,   Eliminating $$dW$$ and $$dH$$, we have  The quantities $$v$$, $$p$$, $$t$$, $$\epsilon$$ and $$\eta$$ are determined when the state of the body is given, and it may be permitted to call them functions of the state of the body. The state of a body, in the sense in which the term is used in the thermodynamics of fluids, is capable of two independent variations, so that between the five quantities $$v$$, $$p$$, $$t$$, $$\epsilon$$ and $$\eta$$ there exist relations expressible by three finite equations, different in general for different substances, but always such as to be in harmony with the differential equation (4). This equation evidently signifies that if $$\epsilon$$ be expressed as a function of $$v$$ and $$\eta$$, the partial differential co-efficients of this function taken with respect to $$v$$ and to $$\eta$$ will be equal to $$-p$$ and to $$t$$ respectively.