20S J. W. Gihhs — EqiiUihvii(m, of Heterogeneous Substances. 



We shall then have, for the general condition of equilihrium, 



fSDe + /AUt -jTSDm - fTADm ^ ; (239) 



and the equations of condition will be 



fSDi^ -\-fADt] — 0, (240) 



. . . ' . . . [ (241) 



fdl)m„ + fABrn^ = 0. ) 

 We may obtain a condition of equilibrium independent of these equa- 

 tions of condition, by subtracting these equations, multiplied each 

 by an indeterminate constant, from condition (239). If we denote 

 these indeterminate constants by T, 31^, . .. M„, we shall obtain 

 after arranging the terms 



/ 



SDs — T 6Dm — TdDtj - iHf, SDm^ . . . ^ M„ 6Dm„ 



fADe-TADm - TADi]-M^ ADm^ . . . -M„dI>m„^o^ (242) 



The variations, both infinitesimal and finite, in this condition are 

 independent of the equations of condition (240) and (241), and are 

 only subject to the condition that the varied values of J)e, i>//, 

 Dm^, . . . lJm„ for each element are determined by a certain change 

 of phase. But as we do not suppose the same element to experi- 

 ence both a finite and an infinitesimal change of phase, we must have 

 SJ)e~ FdDm - TdBi] - 31^ SBrn^ . . . - M„ SBm.„^0, (243) 



and 

 ADs — TADm - TAD?? — M^ A Dm, ... - 3/„ JX>w„^0. (244) 

 By equation (12), and in virtue of the necessary relation (222), the 

 first of these conditions reduces to 



{t — T) dDi] + (yu, - r— J^/,) SBm^ . . . 



+ (yu„ - r- M„) dDm„^0 ; (245) 

 for which it is necessary and suflicient that 



t = r, (246) 



V* (247) 



* The gravitation potential is here supposed to be defined in the usual way. But if 

 it were defined so as to decrease when a body falls, we would have the sign + instead 

 of — in these equations ; i. e., for each component, the sum of the gravitation and 

 intrinsic potentials would be constant throughout the whole mass. 



