402 PROCEEDINGS OF THE AMERICAN ACADEMY § II 



nest possible film (in a soap-bubble for instance), as in 

 the surface of deep water. The contractile force depends, 

 accordingly, upon the breadth of the film, and not on its 

 thickness; and in Everett's Units and Physical Con- 

 stants (page 42, § 46) we find the tension in dynes of a 

 surface a centimetre broad. Since a film has two surfaces, 

 each with a tension independent, nearly, of the thickness 

 as assigned, to produce a given film would require a 

 perfectly measurable quantity of work, which, in the thin- 

 nest possible films, must be a considerable fraction of 

 that necessary to convert the liquid into vapor. 



This fraction is easily determined theoretically. For 

 the latent heat, instead of the series, 



• i2(i + (i)' + a)= + a)' + etc.) 



which represents the equivalent of the total number of 

 atoms from which a given atom has to be separated, we 

 now have only 



6(i+(l)'+G)^+(i)' + etc.) 



the molecules all 13'ing in one plane. From each of these 

 must be subtracted, as before, the equivalent of the num- 

 ber of atoms which remain clinging to a given atom in 

 the state of vapor. The value of the last series, which is 

 the more convergent of the two, is easily found to be 

 7.206, which, subtracted from the first series, or 19.740, 

 leaves 12.534 to represent the work done in stretching the 

 film. 



Plence the fraction of the work spent in this way is 



^^ ■ '^^^^ — , , or for ordinary liquids about two-thirds of 

 ^(19.740) 



that required for complete vaporization. 



It is therefore easy, if we know the principal ratio, K^ 



to find the thickness of the thinnest possible film. To 



generate each square centimetre of such a film, the tension 



