366 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



of discontinuity. Now if we compare the actual state of things with 

 the supposed state, there will be in the former in the vicinity of the 

 surface a certain (positive or negative) excess of energy, of entropy, 

 and of each of the component substances. These quantities are 

 denoted by e 8 , if, m?, mf, etc., and are treated as belonging to the 

 surface. The s is used simply as a distinguishing mark, and must not 

 be taken for an algebraic exponent. 



It is shown that the conditions of equilibrium already obtained 

 relating to the temperature and the potentials of the homogeneous 

 masses, are not affected by the surfaces of discontinuity, and that the 

 complete value of Se a is given by the equation 



S e s = t STJ S + cr Ss + yu^m? + /* 2 <$mf + etc., (23) 



in which s denotes the area of the surface considered, t the tempera- 

 ture, fi lt /x 2 , etc., the potentials for the various components in the 

 adjacent masses. It may be, however, that some of the components 

 are found only at the surface of discontinuity, in which case the letter 

 IUL with the suffix relating to such a substance denotes, as the equation 

 shows, the rate of increase of energy at the surface per unit of the 

 substance added, when the entropy, the area of the surface, and the 

 quantities of the other components are unchanged. The quantity & 

 we may regard as defined by the equation itself, or by the following, 

 which is obtained by integration : 



e 8 tq s + o-s + //! m? + // 2 m f + etc. (24) 



There are terms relating to variations of the curvatures of the 

 surface which might be added, but it is shown that we can give the 

 dividing surface such a position as to make these terms vanish, and it 

 is found convenient to regard its position as thus determined. It is 

 always sensibly coincident with the physical surface of discontinuity. 

 (Yet in treating of plane surfaces, this supposition in regard to the 

 position of the dividing surface is unnecessary, and it is sometimes 

 convenient to suppose that its position is determined by other con- 

 siderations.) 



With the aid of (23), the remaining condition of equilibrium for 

 contiguous homogeneous masses is found, viz., 



<r( Cl +c 2 ) -p'-p', (25) 



where p', p" denote the pressures in the two masses, and c lf c 2 the 

 principal curvatures of the surface. Since this equation has the same 

 form as if a tension equal to a- resided at the surface, the quantity or 

 is called (as is usual) the superficial tension, and the dividing surface 

 in the particular position above mentioned is called the surface of 

 tension. 



By differentiation of (24) and comparison with (23), we obtain 



etc., (26) 



