Page:Scientific Papers of Josiah Willard Gibbs.djvu/394

358 then, by (5), If, then, $$\zeta$$ is known as a function of $$t, p, m_{1}, m_{2},...m_{n}$$, we can find $$\eta, v, \mu_{1}, \mu_{2},...\mu_{n}$$ in terms of the same variables. By eliminating $$\zeta$$, we may obtain again $$n+3$$ independent relations between the same $$2n+5$$ variables as at first. If we integrate (5), (6) and (8), supposing the quantity of the compound substance considered to vary from zero to any finite value, its nature and state remaining unchanged, we obtain   If we differentiate (9) in the most general manner, and compare the result with (5), we obtain  or  Hence, there is a relation between the $$n+2$$ quantities $$t, p, \mu_{1}, \mu_{2},... \mu_{n}$$, which, if known, will enable us to find in terms of these quantities all the ratios of the $$n+2$$ quantities $$\eta, v, m_{1}, m_{2},...m_{n}$$. With (9), this will make $$n+3$$ independent relations between the same $$2n+5$$ variables as at first.

Any equation, therefore, between the quantities

is a fundamental equation, and any such is entirely equivalent to any other.

Coexistent phases.—In considering the different homogeneous bodies which can be formed out of any set of component substances, it is convenient to have a term which shall refer solely to the composition