118 J. W. Gihhs — Equilibrium of Heterogeneous Substances. 



all substances measured by weight oi' mass. Yet in special cases, it 

 may be more convenient to adopt chemical equivalents as the units 

 of the component substances. 



It may be observed that it is not necessary for the validity of 

 equation (12) that the variations of nature and state of the mass to 

 which the equation refers should be such as do not disturb its homo- 

 geneity, provided that in all parts of the mass the variations of 

 nature and state are infinitely small. For, if this last condition be 

 not violated, an equation like (12) is certainly valid for all the infin- 

 itesimal parts of the (initially) homogeneous mass; i. e., if we write 

 2>f, Z>//, etc., for the energy, entropy, etc., of any infinitesimal part, 



dDe = t dDt] - p dBv -\- /< ^ dDm ^-\- fi.^ dJDni^ ...-{-/'« dDm„^ (13) 



whence we may derive equation (12) by integrating for the whole 

 initially homogeneous mass. 



We will now suppose that the whole mass is divided into parts so 

 that each part is homogeneous, and consider such variations in the 

 energy of the system as are due to variations in the composition and 

 state of the several parts i*emaining (at least approximately) homoge- 

 neous, and together occupying the whole space within the envelop. 

 We will at first suppose the case to be such that the component sub- 

 stances are the same for each of the parts, each of the substances 

 aSj, *S'2, . . . Sn being an actual component of each part. If we 

 distinguish the letters referring to the different parts by accents, 

 the variation in the energy of the system may be expressed by 

 Se' -\- Se" -\- etc., and the general condition of equilibrium requires 



that 



(Jt'+.f^f" -h etc, ^ (14) 



for all variations which do not conflict with the equations of condi- 

 tion. These equations must express that the entropy of the whole 

 given mass does not vary, nor its volume, nor the total quantities of 

 any of the substances ^Sj, aS^j, . . . Sn- We will suppose that there 

 are no other equations of condition. It will then be necessary for 

 equilibrium that 



i; 8r,' ^ p' 6v' H-///(Jm,' -\- i.i„' dm^J . 

 J^t" 67}" - p" 6v" -\- 1.1 ^" dm ^" + lA^" dm.J' . 

 -I- etc. ^ 

 for any values of the variations for which 



6if -f 67]" + 67/" 4- etc. = 0, 

 Sv' -j- 6v" + 6v"' + etc. = 0, 



