158 J. W. Gihbs — Equilibrium of Heterogeneous Substances. 



homogeneous parts, we must regard the given condition as one of 

 neutral equilibiium. But in regard to these homogeneous parts, 

 which we may evidently consider to be all diiFerent phases, the fol- 

 lowing conditions must be satisfied. (The accents distinguish the 

 letters referring to the different parts, and the unaccented letters 

 refer to the whole mass.) 



Tf' + jf + etc. = ;^, 1 



v' -f v" -j- etc. =: y, I 



m/ + m/' + etc. = w,, 1^ (134) 



//?2' + ''*3 " + etc. = ^2, j 



etc. J 



Now the values of //, y, m^, m^, etc. are determined by the whole 



fluid mass in its given state, and the values of -„ —„ etc., —r, — ^, 

 etc — — -^ etc., etc., are determined by the phases of the various 



parts. But the phases of these parts are evidently determined by 

 the phase of the fluid as given. They form, in fact, the whole set of 

 coexistent phases of which the latter is one. Hence, we may regard 

 (134) as n + 2 linear equations between ?>', u", etc. (The values of 

 v' v" etc. are also subject to the condition that none of them can be 

 negative.) Now one solution of these equations must give us the 

 given condition of the fluid ; and it is not to be expected that they 

 will be capable of any other solution, unless the number of different 

 homogeneous parts, that is, the number of different coexistent phases, 

 is o-reater than w + 2. We have already seen (page 153) that it is 

 not probable that this is ever the case. 



We may, however, remark that in a certain sense an infinitely large 

 fluid mass will be in neutral equilibrium in regard to the formation 

 of the substances, if such there are, other than the given fluid, for 

 which the value of (133) is zero (when the constants are so deter- 

 mined that the value of the expression is zero for the given fluid, 

 and not negative for any substance) ; for the tendency of such a for- 

 mation to be reabsorbed will diminish indefinitely as the mass out of 

 which it is formed increases. 



When the substances aS'j, aS^, . . . S„ are all independently vari- 

 able components of the given mass, it is evident from (86) that the 

 conditions that the value of (133) shall be zero for the mass as given, 

 and shall not be negative for any phase of the same components, can 

 only be fulfilled when the constants T, P, M^, J/g, . . . M„ are equal 

 to the temperature, the pressure, and the several potentials in the given 



