494 Mr. W. Sutherland on the 



blows and shield the water molecules from them. If then we 

 suppose a semipermeable membrane separating water and a 

 dilute solution of sugar in water, the sugar molecules are to 

 be regarded as replacing some water molecules, but their 

 collisions on the water in the membrane are rendered inopera- 

 tive by the shielding action of the framework, so that the 

 water molecules in the membrane receive more impacts on 

 the side of the pure water than on the side of the solution, 

 and therefore water flows through the membrane, until in the 

 solution there is enough excess of hydrostatic pressure estab- 

 lished to compensate for the inoperative impacts of the sugar 

 molecules ; this inequality of pressure which can be hydro- 

 statically balanced is the osmotic pressure. Next as to its 

 laws. Let us find the number of molecules that in a second 

 cross unit area of a plane in a collection of molecules. We 

 know that in the case of a gas with n molecules per unit 

 volume and mean velocity v the number is nv/Q. Next con- 

 sider a number arranged in cubical order, the edge of each 

 cube being d, and each molecule oscillating through a distance 

 a smaller than d in a fixed direction parallel to one set of 

 edges ; then a molecule crosses a plane within distance a of its 

 central position v/2a times a second in the same direction, 

 and a plane at greater distance than a no times a second. If 

 then a plane is placed at random at right angles to the 

 vibrating molecules its chance of being traversed by molecules 

 is c/d, therefore an area d® similarly placed at random will 

 on the average be crossed (v/2a)(a/d) times a second; so that 

 unit area is on the average crossed in the same direction vj2d z 

 times a second by molecules, that is to say, nv/2 times. If 

 the molecules were vibrating in all directions and we assumed 

 that a third of them vibrated in any direction, and the remain- 

 ing two thirds in directions at right angles to it, we should 

 then have the average number of times unit plane is crossed 

 as nv/6, the same as in the case of the perfect gas. We can 

 easily see after these two special cases that the result is 

 general, no matter how croweled the molecules may be, so 

 long as the time of collision of a molecule may be neglected ; 

 because as each molecule keeps moving on with the same 

 average velocity, if it is diverted by collision with another, 

 then on the average the only effect of collision is to alter the 

 actual path of each molecule, but not the average space 

 traversed : thus in any collection of molecules moving at 

 random, whether as compact as in a liquid or as free as in a 

 gas, the number of times a unit plane is crossed by molecules 

 per second is nv/6. 



If then our unit area is taken on the surface of the semi- 



