162 BUTLER 



ART. D 



cease to hold true. Therefore, one of the differential coefficients 

 like that in (234) must be zero. 

 The differential coefficients 



dt dni dfXn 



jri ^: ■■■i^: »36) [181] 



may be evaluated in a number of different ways, according to 

 whether the quantities which are to remain constant are chosen 

 from the numerators or the denominators of the other terms. 

 Gibbs shows that when the quantites which, together with 

 V, are to remain constant are taken from the numerators of the 

 others, their values will be at least as small as when one or more 

 of the constants are taken from the denominators. 



At least one of the coefficients determined in this way will 

 therefore be zero. But if one of these coefficients is zero it 

 can be shown that all the others, having their constants chosen 

 in the same way, will also be zero. Gibbs gives the following 

 proof of this proposition. "For if 



(dfin/dmn)t, V, ^,. . . . ^„_u (237) [182] 



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

 the component Sn without altering (if we disregard infinitesi- 

 mals of higher orders than the first) the temperature or the 

 potentials, and therefore, by [98], without altering the pres- 

 sure. That is, we may change the phase without altering 

 any of the quantities t, p, m, ...Hr,. 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 stabihty. And the relation which character- 

 izes the limit of stability may be expressed, in general, by setting 

 any one of these differential coefficients equal to zero." 

 We may write this condition in the form 



dfj.,,, 1 

 J( — 7-: = 0, (238) [183] 



