238 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



remain uniform, and on account of its infinitely greater size to be 

 infinitely less altered in its nature than the first part. Let Ae 8 denote 

 the increment of the superficial energy of this first part, A^ 8 , Am 8 , 

 Am?, etc., Am 8 , Am 8 , etc., the increments of its superficial entropy 

 and of the quantities of the components which we regard as belonging 

 to the surface. The increments of entropy and of the various com- 

 ponents which the rest of the system receive will be expressed by 



A;/ 8 , Am 8 , Am 8 , etc., Am 8 , Am?, etc., 

 and the consequent increment of energy will be by (12) and (501) 



- 1 A;/ 8 - yu a Am 8 - fjL b Am? - etc. - fi g Am 8 - jm h Amf - etc. 

 Hence the total increment of energy in the whole system will be 

 Ae 8 - 1 A*? 8 - fJL a Am 8 - [j. b Am? - etc." 



If the value of this expression is necessarily positive, for finite 

 changes as well as infinitesimal in the nature of the part of the film 

 to which Ae 8 , etc. relate,* the increment of energy of the whole 

 system will be positive for any possible changes in the nature of the 

 film, and the film will be stable, at least with respect to changes in 

 its nature, as distinguished from its position. For, if we write 



De 8 , D^ 8 , Dm 8 , Dm?, etc., Dm 8 , Dmf, etc., 



for the energy, etc. of any element of the surface of discontinuity, we 

 have from the supposition just made 



A De 8 t A Dif fji a A Dm 8 , fi b ADm? etc. 



ju. g ADm 8 fji h A Dm 8 etc. > ; (517) 



and integrating for the whole surface, since 



A/Dm 8 = 0, A/Dm 8 = 0, etc., 

 we have 



A/De 8 - 1 A/D^ 8 - fji a A/Dm 8 - fa A/Dm? - etc. > 0. (518) 

 Now A/Djy 8 is the increment of the entropy of the whole surface, 

 and A/D?/ 8 is therefore the increment of the entropy of the two 

 homogeneous masses. In like manner, A/Dm 8 , A/Dm?, etc., 

 are the increments of the, quantities of the components in these 

 masses. The expression 



- 1 A/D>? 8 - fi a A/Dm 8 - p b A/Dm? - etc. 



denotes therefore, according to equation (12), the increment of energy 

 of the two homogeneous masses, and since A/De 8 denotes the 



*In the case of infinitesimal changes in the nature of the film, the sign A must be 

 interpreted, as elsewhere in this paper, without neglect of infinitesimals of the higher 

 orders. Otherwise, by equation (501), the above expression would have the value zero. 



