the Osmotic Tlieory of Solutions. 601 



Stratification of the solution by gravity. 



We will first consider the stratification as referred to 

 Pi, c 2 , u, s lm Take as independent variables the concentration c 2 

 at any point in the column of solution, and j), the hydrostatic 

 pressure at that point ; w and s l are both functions of c 2 and p ; 

 u is a function of p — P x ( = <7i), while T > 1 in turn is a function 

 of c 2 and p, so that w is a function of c 2 and p. 



dPi/ck 2 is the rate of change of the osmotic pressure 

 (measured against the pure solvent) with change of 

 concentration, the hydrostatic pressure on the solution 

 remaining constant. 



dPi/dp is the rate of change of the osmotic pressure with 

 change of hydrostatic pressure on the solution, the 

 concentration remaining constant. 



dp/dG is the rate of change of hydrostatic pressure in the 

 column as we pass to places of higher gravitational 

 potential (i. <?., as we pass downwards through the 

 column). 



dc 2 /dG is similarly the rate of change of concentration 

 with change of gravitational potential. 



When we pass from a level in the solution where the 

 .gravitational potential is G to a place where it is G + dG, 

 the hydrostatic pressure is different by 



dp = d P /dGxdG (1) 



Similarly, between the same levels, the increment in 

 concentration is 



dc 2 = dc 2 /dGxdG (2) 



Again, the osmotic pressure, P 1? when we pass from G to 

 G + </G, is different, because (1) the hydrostatic pressure has 

 changed by dp, and (2) the concentration has changed by 

 dc 2 , thus 



#±1= ^ dp+ ^ ac a 



-(^•&*VSK • • <» 



But, in the column AB 



dG=icdp; (4) 



similarly in the column CD we have 



dG = ud(p-P,) = udp-udF 1 . . . . (5) 

 (u — w)dp~udPi (6) 



