of Mixed Solutions. 289 



fi = thermodynamic potential per unit mass of component (t) of 

 the solution at pressure Pi. 



gi = thermodynamic potential per unit mass of pure substance (i) 

 at pressure P^. 



Pi is the density of the pure component (i), and pmi its mean value 

 between Pi and pi . 

 Partial derivation of (32) with respect to Cg gives 

 , dFi{c„ c^...Cr,Pi) ^ _ J^ /BttA 



8Cs PmiydCs^p.' 



Assuming that the pressures on the solution Pi are the same 

 for all osmotic systems and equal to p, the equation (33) will give 

 an expression in terms of osmotic pressure for all the thermo- 

 dynamic quantities appearing in equation (26). 



From the equation (27) we get the following relation 



(34) L (^A = 1. fijLA . 



Pmi \OCg/p Pms X^^i ■/ p 



This interesting reciprocal relation between the osmotic 

 pressures of the first order, which is a simple corollary from their 

 connections with the thermodynamic potential, would have been 

 very difficult to obtain from a separate consideration of the 

 osmotic pressures themselves. 



Part III. — The effect of gravity upon solutions found by 

 means of the osmotic pressures of the first order. 



§ 9. The osmotic pressure of the first order enables us to 

 generalise equations (1) to solutions in general. The method 

 used in the previous paper only needs to be slightly modified. 

 Instead of only one, we have to form (r) systems, which we get by 

 supposing the solution successively in osmotic equilibrium with 

 the r substances 1, 2...r, which, at the temperature considered, 

 all are supposed to be in the liquid state. Moreover, we must 

 assume that in the state of equilibrium the pressure in the solu- 

 tion is the same for all osmotic systems. 



Using the same way of reasoning as before*, we get 



^^^^ ?=Ad^s)pdx = V'-P^Pdp)dx' 



i= 1, 2 ... r. 



dU dc 



^— and -7-^ mean here the variation of U and c per unit length 



ax dx r & 



* Phil. Mag. [6] 13, pp. 607, 608. 



