Page:Scientific Papers of Josiah Willard Gibbs.djvu/122

 exception of the sources of the work and heat expended, which must be used only as such sources.

We know, however, a priori, that if the quantity of any homogeneous mass containing $$n$$ independently variable components varies and not its nature or state, the quantities $$\epsilon, \eta, v, m_{1}, m_{2}, ... m_{n}$$ will all vary in the same proportion; therefore it is sufficient if we learn from experiment the relation between all but any one of these quantities for a given constant value of that one. Or, we may consider that we have to learn from experiment the relation subsisting between the $$n+2$$ ratios of the $$n+3$$ quantities $$\epsilon, \eta, v, m_{1}, m_{2}, ... m_{n}$$. To fix our ideas we may take for these ratios $$\frac{\epsilon}{v}, \frac{\eta}{v}, \frac{m_{1}}{v}, \frac{m_{2}}{v}, ... \frac{m_{n}}{v}$$, etc., that is, the separate densities of the components, and the ratios $$\frac{\epsilon}{v}$$ and $$\frac{\eta}{v}$$, which may be called the densities of energy and entropy. But when there is but one component, it may be more convenient to choose $$\frac{\epsilon}{m}, \frac{\eta}{m}, \frac{v}{m}$$, as the three variables. In any case, it is only a function of $$n+1$$ independent variables, of which the form is to be determined by experiment.

Now if $$\epsilon$$ is a known function of $$\eta, v, m_{1}, m_{2}, ... m_{n}$$, as by equation (12) $$\epsilon, \eta, v, m_{1}, m_{2}, ... m_{n}$$ $$t, p, \mu_{1}, \mu_{2}, ... \mu_{n}$$ are functions of the same variables, which may be derived from the original function by differentiation, and may therefore be considered as known functions. This will make $$n+3$$ independent known relations between the $$2n+5$$ variables, $$\epsilon, \eta, v, m_{1}, m_{2}, ... m_{n}, t, p, \mu_{1}, \mu_{2}, ... \mu_{n}$$. These are all that exist, for of these variables, $$n+2$$ are evidently independent. Now upon these relations depend a very large class of the properties of the compound considered,—we may say in general, all its thermal, mechanical, and chemical properties, so far as active tendencies are concerned, in cases in which the form of the mass does not require consideration. A single equation from which all these relations may be deduced we will call a fundamental equation for the substance in question. We shall hereafter consider a more general form of the fundamental equation for solids, in which the pressure at any point is not supposed to be the same in all directions. But for masses subject only to isotropic stresses an equation between $$\epsilon, \eta, v, m_{1}, m_{2}, ... m_{n}$$ is a fundamental equation. There are other equations which possess this same property.