112 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



minors successively obtained, and R 1 for the last remaining con- 

 stituent. Then if dt, dfJL v ...djuL n _ v and dv all have the value zero, 



we have by (172) 



= R n+l dm n> (174) 



that is, - =i. (175) 



\<fr 



In like manner we obtain 



etc. 



Therefore, the conditions obtained by writing d for A in (166)-(169) 

 are equivalent to this, that the determinant given above with the n 

 minors obtained from it as above mentioned and the last remaining 



d 2 e 

 constituent -^ shall all be positive. Any phase for which this con- 



dition is satisfied will be stable, and no phase will be stable for 

 which any of these quantities has a negative value. But the con- 

 ditions (166)-(169) will remain valid, if we interchange in any way 

 TI, m p . . . m n (with corresponding interchange of t, fa, ... JUL^. Hence 

 the order in which we erase successive columns with the corresponding 

 rows in the determinant is immaterial. Therefore none of the minors 

 of the determinant (173) which are formed by erasing corresponding 

 rows and columns, and none of the constituents of the principal 

 diagonal, can be negative for a stable phase. 



We will now consider the conditions which characterize the'limifa 

 of stability (i.e., the limits which divide stable from unstable phases) 

 with respect to continuous changes.* Here, evidently, one of the 

 conditions (166)-(169) must cease to hold true. Therefore, one of 

 the differential coefficients formed by changing A into d in the first 

 members of these conditions must have the value zero. (That it is 

 the numerator and not the denominator in the differential coefficient 

 which vanishes at the limit appears from the consideration that the 

 denominator is in each case the differential of a quantity which is 

 necessarily capable of progressive variation, so long at least as the 

 phase is capable of variation at all under the conditions expressed 

 by the subscript letters.) The same will hold true of the set of 

 differential coefficients obtained from these by interchanging in any 

 way T], m v . . . m n , and simultaneously interchanging t, fJL v ... jm n in the 

 same way. But we may obtain a more definite result than this. 



* The limits of stability with respect to discontinuous changes are formed by phases 

 which are coexistent with other phases. Some of the properties of such phases have 

 already been considered. See pages 96-100. 



