Surface Forces in Fluids. 347 



And we may write the whole action 



Similarly, 



expresses the action between the substance (ft) and a molecule 

 C of the third layer; and if there be, in the substance a, n 2 

 molecules per unit of area, the total action of (ft) on (a) per 

 unit of area may be written n 2 \^(j) ; and since the second 

 substance (ft) is in equilibrium with the uppermost layer of 

 (a), this must be equal to the repulsive reaction downwards 

 on each unit of area of the uppermost layer of (a). Therefore 

 the pressure on this layer is n 2 J £</> per unit of area, or j £<£ 

 per molecule; and the total action on a molecule A of the first 

 layer is therefore made up of j 2^> (downwards) + XF (down- 

 wards) — £(/> A (upward); and this must therefore be the value 

 of the repulsive pressure between the second layer of the 

 liquid (a) and every molecule A of the first layer ; or, using 

 previous notation, 



[(l)(2)] = fS* + 2F-2* A . 



Again, the total action on every molecule (B) of the second 

 layer of (a) will be 



(J24>+£F-2tf> A )+2F~(F 1+ 2tf> B ), 



which is therefore the value of the elastic repulsive reaction 

 between the third layer and the molecule B, — or 



[(2)(3)]=j^ + 22F-2^. A -2^ B -F 1 . 

 Similarly, 



[(3)(4) J =j2<£ + 32F-2£ A -24> B -2</> c -2F 1 -F 2 , 



and 



[(4)(5)]=J24> + 42F-2<£ A -2tf> B -^ c -2tf> D -3F 1 



-2F 2 -F 3 , 

 and 



[0.)0- + l)]=j"2^ + ,-2F-2^ A -2</» B -...2^-(r-l)F 1 



-(r-2)F 2 ...-F,_,; 

 from which we find 



[(2)(3)]-[(l)(2)] = 2F-2^ B -F 1 



= F, + F, + F, + ...-(* B .+* B , + * Bi + ...) 



[(3) (4)] - [(2) (3)] =2F-2* -F 1 -F, 



=F 3 + F 4 + ...-(<£ +</> +<£ C3 + ...); 



2A2 



