172 BUTLER 



ART. D 



In general the condition of the Hmit of stabiHty is represented 

 by substituting = for > in any of these equations. 



38. Critical Phases* Since a critical phase may be varied 

 without changing any of the quantities t, ni, n^, ... Mn, all the 

 expressions (245) may be equated to zero. The solution of the 

 equations so obtained is 



Rn+i = 0. (261) [203] 



(This also follows from the fact that a critical phase is at the 

 limit of stability with respect to continuous changes.) "To 

 obtain the second equation characteristic of critical phases, we 

 observe that as a phase which is critical cannot become unstable 

 when varied so that n of the quantities t, p, ni, )U2, ...Mn 

 remain constant, the differential of Rn+\ for constant volume, 

 viz., 



—j^ dv + -~- dmi ... + -J— ^ drrin (262) [204] 

 dri ami otw,, 



cannot become negative" when n of the quantities t, p, ni, m, 

 . . ./x„ remain constant. "Neither can it have a positive value, 

 for then its value might become negative by a change of 

 sign of dr], drrii, etc." Therefore the expression (262) has the 

 value zero, when n of the expressions (245) are equated to zero. 

 If *S is a determinant in which the constituents are the same as 

 in i^n+i except that the differential coefficients 



dr) ' drrii ' ' * ' dm,, 



are substituted in a single horizontal line, this condition is 

 expressed by the equation 



S = 0. (264) [205] 



This substitution may be made in any horizontal line of Rn + i- 



* Gibbs, I, 132-134. 



