EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 83 



which we are next to consider there are equations of condition due to 

 a cause of a different nature. 



Effect of a Diaphragm (Equilibrium 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 permeable to 

 some of the components and impermeable to others, we shall have the 

 equations of condition 



V+*f = 0, (72) 



<fo/ = 0, &/' = 0, (73) 



and for the components which cannot pass the diaphragm 



Sm a ' = 0, Sm a " = 0, <$ra; = 0, <$m 6 " = 0, etc., (74) 



and for those which can 



<**' + Sm h " = 0, SmS + Sm" = 0, etc. (75) 



With these equations of condition, the general condition of equilibrium 

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



*W, (76) 



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

 components of both masses, 



^' = /C, ti = tf, etc., ; ' (77) 



but not P'=P"> 



nor l*a' = Pa", Hb=Hb'> etc. 



Again, if the diaphragm is permeable to the components in certain 

 proportions only, or in proportions not entirely determined yet subject 

 to certain conditions, these conditions may be expressed by equations 

 of condition, which will be linear equations between Sm^, Sm 2 ', etc., 

 and if these be known the deduction of the particular conditions of 

 equilibrium will present no difficulties. We will however observe 

 that if the components S v S 2 , etc. (being actual components on each 

 side) can pass the diaphragm simultaneously in the proportions a 1} a 2 , 

 etc. (without other resistances than such as vanish with the velocity of 

 the current), values proportional to a v a 2 , etc. are possible for Sm^, 

 Sm 2 ', etc. in the general condition of equilibrium, Sm^', Sm 2 ", etc., 

 having the same values taken negatively, so that we shall have for 

 one particular condition of equilibrium 



a i fa' + a 2 fa' + etc - = a i A*i" + a 2 fa" + etc - (78) 



There will evidently be as many independent equations of this form 

 as there are independent combinations of the elements which can pass 

 the diaphragm. 



