132 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



We are not, however, absolutely certain that equation (200) will 

 always be satisfied by a critical phase. For it is possible that the 

 denominator in the fraction may vanish as well as the numerator for 

 an infinitesimal change of phase in which the quantities indicated 

 are constant. In such a case, we may suppose the subscript n to 

 refer to some different component substance, or use another differ- 

 ential coefficient of the same general form (such as are described on 

 page 114 as characterizing the limits of stability in respect to con- 

 tinuous changes), making the corresponding changes in (201) and 

 (202). We may be certain that some of the formulae thus formed 

 will not fail. But for a perfectly rigorous method there is an 

 advantage in the use of T], v, f m l , m 2 ,...m n as independent variables. 

 The condition that the phase may be varied without altering any of 

 the quantities t, fa, /i 2 , ... ju n will then be expressed by the equation 



7? ^20^ 



J-^n+i u > ^ziuo; 



in which R n+l denotes the same determinant as on page 111. 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, fa, fa, ... // n remain constant, the 

 differential of R n+i for constant volume, viz., 



dR dR dR 



T^-dn-\ T^dm, ... -\ ,-^cZm n , (204) 



rt vi fi IVY) * dfyv) ' 



(A//I \Jjlli/-i U/ 1 1 (/ft 



cannot become negative when n of the equations (172) are satisfied. 

 Neither can it have a positive value, for then its value might become 

 negative by a change of sign of dr\, dm^ etc. Therefore the expression 

 (204) has the value zero, if n of the equations (172) are ' satisfied. 

 This may be expressed by an equation 



S=0, (205) 



in which S denotes a determinant in which the constituents are the 

 same as in R n+ i, except in a single horizontal line, in which the 

 differential coefficients in (204) are to be substituted. In whatever 

 line this substitution is made, the equation (205), as well as (203), 

 will hold true of every critical phase without exception. 



If we choose t, p, m^, m 2 ,...m n as independent variables, and 

 write U for the determinant 



p? 



dX 



(206) 



