Surface Forces in Fluids. 343 



imagine that we describe about it as centre a sphere of radius 

 equal to the radius of its molecular action. The molecules 

 which exercise on this surface-molecule any attractive action 

 are contained in the hemisphere at whose centre it is situated. 

 From the symmetry of their distribution it is evident that 

 their total resultant action is vertically downwards, also that 

 the resultant action of each successive layer of molecules on 

 the central one which we are considering is vertically down- 

 wards. We do not know the law according to which the 

 attraction between two molecules varies with the distance. 

 The only assumption we shall make is, that within the limits 

 of variation with which we shall deal the attraction does not 

 increase with the distance. 



Let Fj represent the vertical component due to the action 

 of the first adjacent layer on the molecule A, F 2 that due to 

 the second layer, F 3 that due to the third, and so on. 



The last term to be considered is the nth or F n , where n is 

 the number of times that the distance between consecutive 

 layers is contained in the radius of molecular action. 



Now let us introduce between the molecules such repulsive 

 forces as shall be consistent with the maintenance of equili- 

 brium at the uniform distance apart which we have assigned, 

 and let us proceed to examine what this repulsive force 

 must be. 



The total action downward on the surface-molecule A which 

 we are considering is 



F. + F2 + F3 + + F W =2F. 



If its equilibrium is to be maintained, it must experience a 

 repulsive pressure upwards equal to 2F, and every molecule 

 in the surface-layer will be similarly acted on. 



Also, since we cannot but suppose that to the action there is 

 an equal and opposite reaction, the repulsive pressure on the 

 first layer upwards is equal to that on the second layer down- 

 wards ; and since the uniform distribution with which we have 

 started implies that there are the same number of molecules 

 in each layer, the repulsive pressure downwards on each mole- 

 cule (B) of the second layer is equal to the upward pressure 

 on each molecule A of the first. 



Now let us consider the attractive actions oe a molecule (B) 

 of the second layer. These are 2}F downwards and F 1 upwards; 

 and therefore, taking into account the repulsive pressure from 

 above, the total resultant action downwards on (B) is 



22F-F 1 = F 1 + 2F 2 + 2F 3 + + 2F-, 



and to maintain equilibrium it is necessary that this should 



