EQUILTBEIUM OF HETEROGENEOUS SUBSTANCES. Ill 



true of any two infinitesimally differing phases within the limits of 

 stability. Combining these two conditions we have 



(170) 



which may be written more briefly 



AtfA;/ Ap Av+A/^Amj ... + A/z n Am n >0. (171) 



This must hold true of any two infinitesimally differing phases within 

 the limits of stability. If, then, we give the value zero to one of 

 the differences in every term except one, but not so as to make the 

 phases completely identical, the values of the two differences in the 

 remaining term will have the same sign, except in the case of A/> 

 and A-y, which will have opposite signs. (If both states are stable 

 this will hold true even on the limits of stability.) Therefore, within 

 the limits of stability, either of the two quantities occurring (after the 

 sign A) in any term of (171) is an increasing function of the other, 

 except p and v, of which the opposite is true, when we regard as 

 constant one of the quantities occurring in each of the other terms, 

 but not such as to make the phases identical. 



If we write d for A in (166)-(169), we obtain conditions which 

 are always sufficient for stability. If we also substitute ^ for >, we 

 obtain conditions which are necessary for stability. Let us consider 

 the form which these conditions will take when rj, v, m v . . . ra n are 

 regarded as independent variables. When dv Q, we shall have 



dt 



dt 



dt , 

 j dm n 

 dm 



(172) 



Let us write R n+1 for the determinant of the order n + 1 



^dri ' dm n drj 



dr] d 



(173) 



dt]dm n dm l dm n ' 



of which the constituents are by (86) the same as the coefficients in 

 equations (172), and R n , R n _ v etc., for the minors obtained by erasing 

 the last column and row in the original determinant and in the 



