J. W. Gibbs — Equilibrium of Heterogeneous Substances. 139 



as there are independent combinations of the elements which can 

 pass the diaj^hragra. 



These conditions of equilibrium do not of course depend in any 

 way upon the supposition that the volume of each fluid mass is kept 

 constant, if the diaphragm is in any case supposed immovable. In 

 fact, we may easily obtain the same conditions of equilibrium, if we 

 suppose the volumes variable. In this case, as the equilibrium must 

 be preserved by forces acting upon the external surfaces of the fluids, 

 the variation of the energy of the sources of these forces must appear 

 in the general condition of equilibrium, which will be 



6t'-\-6e" -^P' dv'-ifP" SV'^O, (79) 



JP and P" denoting the external forces per unit of area. (Compare 

 (14).) From this condition we may evidently derive the same 

 internal conditions of equilibrium as before, and in addition the 

 external conditions 



p' — P\ p"z=P". (80) 



In the preceding paragraphs it is assumed that the permeability of 

 the diajjhragm is perfect, and its impermeability absolute, i. e., that it 

 offers no resistance to the passage of the components of the fluids in 

 certain proportions, except such as vanishes with the velocity, and 

 that in other proportions the components cannot pass at all. How 

 far these conditions are satisfied in any particular case is of course to 

 be determined by experiment. 



If the diaphragm is permeable to all the n components without 

 restriction, the temperature and the potentials for all the components 

 must be the same on both sides. Now, as one may easily convince 

 himself, amass having n components is capable of only /i + 1 inde- 

 pendent variations in nature and state. Hence, if the fluid on one 

 side of the diaphragm remains without change, that on the other side 

 cannot (in general) vary in nature or state. Yet the pressure will 

 not necessarily be the same on both sides. For, although the pres- 

 sure is a function of the temperature and the n potentials, it may be 

 a many-valued function (or any one of several functions) of these 

 variables. But when the pi-essures are different on the two sides, 

 the fluid which has the less pressure will be practically unstable, in 

 the sense in which the term has been used on page 133. For 



£"_?;" 7/' +/>"?/'—/.</'»*," — /1 2" ??^2" . . . — //„"/>/„" = 0, (SI) 

 as appears from equation (12) if integrated on the supposition that 

 the nature and state of the mass remain unchanged. Therefore, if 

 p<}j" while t'-t'\ ;t,'= ;:,", etc. 



