Page:Scientific Papers of Josiah Willard Gibbs.djvu/123

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Let then by differentiation and comparision with (86) we obtain If, then, $$\psi$$ is known as a function of $$t, v, m_{1}, m_{2}, ... m_{n}$$ we can find $$\eta, p, \mu_{1}, \mu_{2}, ... \mu_{n}$$ in terms of the same variables. If we then substitute for $$\psi$$ in our original equation its value taken from eq. (87), we shall have again $$n+3$$ independent relations between the same $$2n+5$$ variables as before.

Let then by (86), If, then, $$\chi$$ be known as a function of $$\eta, p, m_{1}, m_{2}, ... m_{n}$$, we can find $$t, v, \mu_{1}, \mu_{2}, ... \mu_{n}$$ in terms of the same variables. By eliminating $$\chi$$, we may obtain again $$n+3$$ independent relations between the same $$2n+5$$ variables as at first.

Let then, by (86), 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 (86), supposing the quantity of the compound substance considered to vary from zero to any finite value, its nature and state remaining unchanged, we obtain and by (87), (89), (91)    The last three equations may also be obtained directly by integrating (88), (90), and (92).