166 



BUTLER 



ART. D 



continuous and to discontinuous changes. Therefore, Hn is an 

 increasing function of m„ when t, v, ni, H2, . . .At„_i have con- 

 stant values* determined by any critical phase." If 



((Ptj.n/dmJ)t. V. 



Ml- 



• Mn-1 



had either a positive or a negative value, ^n would be a maxi- 

 mum or a minimum with respect to m„, at the critical point, 

 when (242) is satisfied. Thus, since Hn is an increasing function 

 of nin, we have 



(j^) = 0, (243) [201] 



\am„ /t, v,^i, Hi, . . . ,i„_, 



but one of the higher differentials must be positive, i.e., 



( -J — 3 ) ^ 0, etc. (244) [202] 



XII. Generalized Conditions of Stabilityf 



37. The Conditions. A single phase of n components has n + 1 

 degrees of freedom. Therefore, if n of the quantities t, p, ni, 

 . . -Hn are given constant values, the phase is only capable of 

 one independent variation. If we take rj, wi, Wi, . . .w„ as the 

 independent variables, we may write (when dv = 0) 



dt dt 



at = — di] -{- - — dm\ . 



(17) dmi 



dfi\ dfjLi 



dfii = —r- d-n + - — dmi . 

 dr] ami 



dt 

 + T"" dnin, 



+ 



dm„ 



dm 

 dnin 



dnin, 



> (245) [172] 



dUn dfJLn dfXn 



dun = ~r dv -f - — dm.1 . . . + "; — dm„. 

 arj ami dm„ 



When dt = 0, dm = 0, . . . dun-i = 0, we have 



dlJ,n\ Rn + l 



(: 



dmn/t, v,^i,...fin-l 



Rn 



(246) [175] 



* t; is included to insure that a change in the amount of the critical 

 phase is excluded, 

 t Gibbs, I, 111-112. 



