302 EQUILIBKIUM OF HETEROGENEOUS SUBSTANCES. 



surfaces, by <r 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 $ x and $ 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 



da 



2do- = E> (643) 



8 



and from the conditions of the extension of the film 



ds = 



s 



Hence we obtain 







ds 

 + 2r 2(1) ) = - 



o 



and eliminating d\, 



ds 



2yir 2(1) = -Xy 1 c?y 2 + Xy 2 c?y 1 -2y 1 ^r 2(1) . (645) 



o 



If we set r = *a, (646) 



we have dr = ~*, (647) 



Vi 



d* 



and 2r 2(1) =-Xy 1 dr-2dr 2{1) . (648) 



s 



With this equation we may eliminate ds from (643). We may also 

 eliminate do- by the necessary relation (see (514)) 



d(T= 1^2 



This will give 



4r 2(1) 2 dft = E(\ 7l dr + 2 cT 2(1) ), (649) 



or 



where the differential coefficients are to be determined on the con- 

 ditions that the temperature and all the potentials except // 1 and /z 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 



(ri - y/)^ + (y 2 - y 2 ')^ 2 = o, (651 ) 



in which y^ and y 4 / denote the densities of 8 l and $ 2 in the con- 

 tiguous gas-masses. (See (98).) When the tension of the surfaces 

 of the film and the pressures in its interior and in the contiguous 



