328 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



General Relations. For any constant state of strain of the surface 

 of the solid we may write 



de m = tdri S[l) +fadr i(l} +fJi 3 dr 8W +ete., (674) 



since this relation is implied in the definition of the quantities involved. 

 From this and (659) we obtain 



da-= ~-ijB(i)dt-~'r*(i)dt*2-~r 3(l) d[i 3 - etc., (675) 



which is subject, in strictness, to the same limitation that the state 

 of strain of the surface of the solid remains the same. But this 

 limitation may in most cases be neglected. (If the quantity <r repre- 

 sented the true tension of the surface, as in the case of a surface 

 between fluids, the limitation would be wholly unnecessary.) 



Another method and notation. We have so far supposed that 

 we have to do with a non-homogeneous film of matter between 

 two homogeneous (or very nearly homogeneous) masses, and that 

 the nature and state of this film is in all respects determined by the 

 nature and state of these masses together with the quantities of the 

 foreign substances which may be present in the film. (See page 314.) 

 Problems relating to processes of solidification and dissolution seem 

 hardly capable of a satisfactory solution, except on this supposition, 

 which appears in general allowable with respect to the surfaces 

 produced by these processes. But in considering the equilibrium of 

 fluids at the surface of an unchangeable solid, such a limitation is 

 neither necessary nor convenient. The following method of treating 

 the subject will be found more simple and at the same time more 

 general. 



Let us suppose the superficial density of energy to be determined 

 by the excess of energy in the vicinity of the surface over that which 

 would belong to the solid, if (with the same temperature and state 

 of strain) it were bounded by a vacuum in place of the fluid, and to 

 the fluid, if it extended with a uniform volume-density of energy just 

 up to the surface of the solid, or, if in any case this does not suffi- 

 ciently define a surface, to a surface determined in some definite way 

 by the exterior particles of the solid. Let us use the symbol (e s ) to 

 denote the superficial energy thus defined. Let us suppose a superficial 

 density of entropy to be determined in a manner entirely analogous, 

 and be denoted by (?/ s ). In like manner also, for all the components 

 of the fluid, and for all foreign fluid substances which may be present 

 at the surface, let the superficial densities be determined, and denoted 

 by (F 2 ), (F 3 ), etc. These superficial densities of the fluid components 

 relate solely to the matter which is fluid or movable. All matter 

 which is immovably attached to the solid mass is to be regarded as a 

 part of the same. Moreover, let y be defined by the equation 



9 = (B) - *(*s) - ft(r s ) - yU 3 (F 3 ) - etc. (676) 



