Page:Scientific Papers of Josiah Willard Gibbs.djvu/338

302 surfaces, by $$\sigma$$ and $$s$$ the tension and area of one of these surfaces, and by $$E$$ the elasticity of the film when extended under the supposition that the total quantities of $$S_{1}$$ and $$S_{2}$$ in the part of the film extended are invariable, as also the temperature and the potentials of the other components. From the definition of $$E$$ we have and from the conditions of the extension of the film  Hence we obtain  and eliminating $$d\lambda$$,  If we set we have and With this equation we may eliminate $$d\sigma$$ from (643). We may also eliminate do- by the necessary relation (see (514)) This will give  or  where the differential coefficients are to be determined on the conditions that the temperature and all the potentials except $$\mu_{1}$$ and $$\mu_{2}$$ are constant, and that the pressure in the interior of the film shall remain equal to that in the contiguous gas-masses. The latter condition may be expressed by the equation in which $$\gamma_{1}'$$ and $$\gamma_{2}'$$ denote the densities of $$S_{2}$$ and $$S_{2}$$ in the contiguous gas-masses. (See (98).) When the tension of the surfaces of the film and the pressures in its interior and in the contiguous