Page:Scientific Papers of Josiah Willard Gibbs.djvu/102

66 of these quantities, which we may call the independent variables, is evidently $$(n+2)_{\nu}$$, $$\nu$$ denoting the number of homogeneous parts into which the whole mass is divided. All the quantities which occur in (19), (20), (21), are functions of these variables, and may be regarded as known functions, if the energy of each part is known as a function of its entropy, volume, and the quantities of its components. (See eq. (12).) Therefore, equations (19), (20), (21), may be regarded as $$(\nu - 1) (n + 2)$$ independent equations between the independent variables. The volume of the whole mass and the total quantities of the various substances being known afford $$n+1$$ additional equations. If we also know the total energy of the given mass, or its total entropy, we will have as many equations as there are independent variables.

But if any of the substances $$S_{1}, S_{2}, ... S_{n}$$ are only possible components of some parts of the given mass, the variation $$\delta m$$ of the quantity of such a substance in such a part cannot have a negative value, so that the general condition of equilibrium (15) does not require that the potential for that substance in that part should be equal to the potential for the same substance in the parts of which it is an actual component, but only that it shall not be less. In this case instead of (21) we may write

$$M_{1}, M_{2}$$, etc., denoting constants of which the value is only determined by these equations.

If we now suppose that the components (actual or possible) of the various homogeneous parts of the given mass are not the same, the result will be of the same character as before, provided that all the different components are independent (i.e., that no one can be made out of the others), so that the total quantity of each component is fixed. The general condition of equilibrium (15) and the equations of condition (16), (17), (18) will require no change, except that, if any of the substances $$S_{1}, S_{2},... S_{n}$$ is not a component (actual or possible) of any part, the term $$\mu \delta m$$ for that substance and part will be wanting in the former, and the $$\delta m$$ in the latter. This will require no change in