J. W. Gibbs — Equilibrium of Heterogeneous Substances. 169 

 Let us write i?„+i for the determinant of the order n + 1 



(173) 



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

 equations (1*72), and i?„, B^_-^, etc. for the minors obtained by erasing 

 the hist column and row in the original determinant and in the minors 

 successively obtained, and R^ for the last remaining constituent. 

 Then if dt, dju^, . . . djn„-i, and dv all have the value zero, we have 

 by (172) 



Ji„ dj.i„ = i?„+i drji„, (174) 



that is. 



/d/Jr, \ 



\dinjt, v,/x,, 



/""— 1 





In like manner we obtain 



( d^„_i \ 



\dm„_Jt,v,fii, 



/"»-2> »»„ 



RZ. 



(175) 



(176) 



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^ £ 

 constituent -y— 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 condi- 

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

 77, w^i, . . . m„ (with corresponding interchange of t, ^t^, . . . /.i„). 

 Hence the order in which we erase successive columns with the cor- 

 responding 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 Avhich characterize the limits 

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



Trans. Conn. Acad., Vol. III. 22 January, 1876. 



