114 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



constant the quantities occurring in the numerators of the others 

 together with v, will have the value zero. But if one such has the 

 value zero, all such will in general have the same value. For if 



for example, has the value zero, we may change the density of the 

 component S n without altering (if we disregard infinitesimals of 

 higher orders than the first) the temperature or the potentials, and 

 therefore, by (98), without altering the pressure. That is, we may 

 change the phase without altering any of the quantities t, p, fJ. v ... /J. n . 

 (In other words, the phases adjacent to the limits of stability exhibit 

 approximately the relations characteristic of neutral equilibrium.) 

 Now this change of phase, which changes the density of one of 

 the components, will in general change the density of the others 

 and the density of entropy. Therefore, all the other differential 

 coefficients formed after the analogy of (182), i.e., formed from the 

 fractions in (181) by taking as constants for each the quantities in 

 the numerators of the others together with v, will in general have 

 the value zero at the limit of stability. And the relation which 

 characterizes the limit of stability may be expressed, in general, by 

 setting any one of these differential coefficients equal to zero. Such 

 an equation, when the fundamental equation is known, may be 

 reduced to the form of an equation between the independent variables 

 of the fundamental equation. 



Again, as the determinant (173) is equal to the product of the 

 differential coefficients obtained by writing d for A in the first 

 members of (166)-(169), the equation of the limit of stability may be 

 expressed by setting this determinant equal to zero. The form of 

 the differential equation as thus expressed will not be altered by the 

 interchange of the expressions q, m l ,...in n> but it will be altered 

 by the substitution of v for any one of these expressions, which will 

 be allowable whenever the quantity for which it is substituted has 

 not the value zero in any of the phases to which the formula is to 

 be applied. 



The condition formed by setting the expression (182) equal to 

 zero is evidently equivalent to this, that 



I ^Mw I /\ /i oo\ 



I I =0, (loo) 



that is, that 



I I 



(184) 



