1911.] Note on Surface Electric Charges of Living Cells. 223 



for the superficial film. So long as the depth of the effective film is greater 

 than the range of the molecular forces the attractive forces across the inter- 

 face will be constant for films of the same composition and at the same 

 temperature. When the thickness of the film is less than this range the 

 Laplacian pressure at the interface, and therefore 7, must diminish and the 

 velocity of the water under unit electric field increase. 



The most probable assumption is that 7 varies directly with the intrinsic 

 pressure at the interface. 



Let the attraction of a molecule of water upon a molecule of the film be 

 mm'<f)(f), where / is the distance between them. Then, if z be the depth of the 

 film and dz an infinitely thin plate, the attraction of the whole mass of the 

 water on the film is 



27rmp^(f)fdf, where tt (/) = j (/) df. 



The density of the water may be taken as uniform. The density of the 

 film will vary rapidly. Call its density p and let 



The pressure at the interface will now be 



27rp^p'+(z)dz. 



Leaving out of account for the moment the variation of density p {dz), and 

 putting p equal to unity, we have 



p = 2tt jV (z) dz, which is equal to 2 tt [a j tt (/) fdf+ |"tt (J) f 2 dfj . 



Putting tt (J) = Kfi'h'M as an analytically simple hypothesis, this integral 

 reduces to 



^ = 2K7r/3- 4 [2-^(z/3+2)], 



where p is the pressure at the interface. 



Kiieker* gives as the range of molecular attraction 50 p,fj,. The estimate is 

 based upon measurements of the thickness of soap films made by himself in 

 association with Eeinold, and upon a critical analysis of measurements by 

 Quincke and others. If /3 be put equal to 10 6 the force is approximately 4- at 

 Tib" at 50 /a/a, and vanishingly small at 100 /a/a. Thus /3 = 10° 

 approximates closely to Eucker's estimate. 



* ' Journ. Chem. Soc.,' 1888, p. 222. 



