Page:Scientific Papers of Josiah Willard Gibbs.djvu/267

Rh An equation, therefore, between may be called a fundamental equation for the surface of discontinuity. An equation between or between may also be called a fundamental equation in the same sense. For it is evident from (501) that an equation may be regarded as subsisting between the variables (510), and if this equation be known, since $$n+2$$ of the variables may be regarded as independent (viz., $$n+1$$ for the $$n+1$$ variations in the nature of the surface of discontinuity, and one for the area of the surface considered), we may obtain by differentiation and comparison with (501), $$n+2$$ additional equations between the $$2n+5$$ quantities occurring in (502). Equation (506) shows that equivalent relations can be deduced from an equation between the variables (511). It is moreover quite evident that an equation between the variables (510) must be reducible to the form of an equation between the ratios of these variables, and therefore to an equation between the variables (511).

The same designation may be applied to any equation from which, by differentiation and the aid only of general principles and relations, $$n+3$$ independent relations between the same $$2n+5$$ quantities may be obtained.

If we set we obtain by differentiation and comparison with (501) An equation, therefore, between $$\psi^S, t, s, m_{1}^S, m_{2}^S$$, etc., is a fundamental equation, and is to be regarded as entirely equivalent to either of the other fundamental equations which have been mentioned.

The reader will not fail to notice the analogy between these fundamental equations, which relate to surfaces of discontinuity, and those relating to homogeneous masses, which have been described on pages 85–89.

When all the substances which are found at a surface of discontinuity are components of one or the other of the homogeneous masses, the potentials $$\mu_{1}, \mu_{2}$$, etc., as well as the temperature, may be determined from these homogeneous masses. The tension $$\sigma$$ may