290 Mr Vegard, On some general Properties, etc. 



from a point on the surface of the membrane and along some 

 direction in this surface, but as the direction of the surface is quite 

 arbitrary, the increment dx may have any direction whatever. 

 From equation (32) we get 



^ = 1-Piqi. 



Putting the vahie for ^ into (85) 



(36) *i'' - (^'^\ — ^ = n - ■) — 

 s=i pi^dcjpdx ^ ^^ dx 



^■ = l, 2...r. 



If we transform this system of (?') linear equations by means of 

 equation (33) we get the result expressed in equation (25), show- 

 ing that the osmotic method leads to the same result as that in 

 which we used the thermodynamic potentials. 



There is, however, one difference between the two solutions of 

 the problem. In order to apply the osmotic pressure method, we 

 had to assume that the r components were in a liquid state ; the 

 treatment with thermodynamic potentials only requires that the 

 solution is fluid at the temperature considered, and the components 

 may have any aggregate form whatever. 



§ 10. Summary of results. 



(1) Applying the general thermodynamic equilibrium condi- 

 tion, the variation of concentration caused by any field of gravity 

 is found for a solution containing any number of substances, and 

 the result is expressed in terms of thermodynamic potentials. 



(2) The conception of osmotic pressure is extended to a solu- 

 tion containing any number of substances. 



(3) Simple expressions in terms of thermodynamic potentials 

 are found for the partial osmotic pressure and for the osmotic 

 pressure of the first order. 



(4) By means of the osmotic pressure of the first order we ob- 

 tained a physical interpretation of the thermodynamic quantities 



f^. From their connection with the thermodynamic potentials a 



reciprocal relation was found for the osmotic pressure of the first 

 order. 



(5) The osmotic pressure of the first order gave us a simple 

 way of finding the influence of gravity upon a solution. 



