IVO J. W. Gibhs — EquiUhrium of Heterogeneous Substances. 



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 J into d in the first mem- 

 bers 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 dif- 

 ferential coefficients obtained from these by interchanging in any 

 way rj, m^, . . . m„, and simultaneously interchanging t, j.i^, . . . /J„ 

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

 Let us give to rj or t, to m^ or j.i^, .. . to m„_j or /y„_i, and to v, 

 the constant values indicated by these letters when accented. Then 



by (165) 



d^=iMu - l<)dm,. {Ill) 



Now 



""-"•'=(,17.) '('"•-'"•') (^'«> 



approximately, the differential coefficient being interpreted in accord- 

 ance with the above assignment of constant values to certain vari- 

 ables, and its value being determined for the phase to which the 

 accented letters refer. Therefore, 



and 



d^ = 1^^] {m„ - m„') dm,,, (179) 



^ = -m^y(m„-m„')^. (180) 



The quantities neglected in the last equation are evidently of the 

 same order as (v;?„ — w^„')^. Now this value of ^ will of course be 

 different (the differential coefficient having a different meaning) 

 according as we have made // or t constant, and according as we have 

 made m^ or /^^ constant, etc. ; but since, within the limits of stability, 

 the value of <?, for any constant values of «?„ and ?j, Avill be the least 

 when t^p, 1^1 . . . //„_i have the values indicated by accenting these 

 letters, the value of the differential coefficient will be at least as small 



* 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 152-156. 



