138 X W. Gibbs—Equilihriuin of Heterogeneous Substcmces. 



lem which we are next to consider there are eqnations of condition 

 due to a cause of a different nature. 



Eff'ect of a Diaphragm {EqniUbrmm of Osmotic Forces). 



If the given mass, enclosed as before, is divided into two parts, 

 each of which is homogeneous and fluid, by a diaphragm which is 

 capable of supporting an excess of pressure on either side, and is per- 

 meable to some of the components and impermeable to others, we 

 shall have the equations of condition 



6,f-\-6v"=% (72) 



(W=iO, 6v"=0, (73) 



and for the components which cannot pass the diaphragm 



6mJ=0, dmj'=0, Sm,,' = 0, Sm,," z=0, etc., (74) 



and for those which can 



dm,,' + d)j/,"= 0, Sm/ -f Stn/' = 0, etc. (75) 



With these equations of condition, the general condition of equilib- 

 rium (see (15)) will give the following particular conditions: 



t' = t", (76) 



and for the components which can pass the diaphragm, if actual com- 

 ponents of both masses, 



/'//=/'/', Mt'=^h", etc., (77) 



but not 2^' = p" ■> 



nor iA,lz=if.i^\ i.(f; = ii,'\ etc. 



Again, if the diaphragm is pei'meable to the components in certain 

 proportions only, or in proportions not entirely determined yet sub- 

 ject to certain conditions, these conditions may be expressed by 

 equations of condition, which will be linear equations between 6m^\ 

 Sm^'t etc., and if these be known the deduction of the i^articular con- 

 ditions of equilibrium will present no difficulties. We will however 

 observe that if the components aS',, S2, etc. (being actual components 

 on each side) can pass the diaphragm simultaneously in the propor- 

 tions a J, a^, etc. (without other resistances than such as vanish with 

 the velocity of the current), values proportional to a^, a^, etc. are 

 possible for dni^\ Sm^', etc. in the general condition of equilibrium, 

 6m ^", Sm^"^ etc. having the same values taken negatively, so that 

 we shall have for one particular condition of equilibrium 



^1 /'/+ "2 '"2' + ^^^- — '-^1 " 1" + ^h Ih" -^ etc. (78) 



There will evidently be as many independent equations of this form 



