Law of Molecular Force. 119 



for the potential energy increasing with the upper limit L, 

 which is opposed to the fundamental hypothesis of the theory 

 of molecular force, assumed in the definition of L. For values 

 of m or n greater than 3, the terms involving L become 

 negligible in comparison with those involving a, and the 

 potential energy is expressible as 2irp multiplied by a series 

 of inverse powers of a ; but a is a function of v 9 so that if the 

 series is finite, the potential energy cannot vary inversely 

 as v. 



If, however, the series reduces to the single term -3, then 



the potential energy becomes 



2wpAlog-, 



which, remembering that - is a very large number, we see 



to be independent of variations of a within the range of pre- 

 sent experimental possibilities. 



Thus, then, it is established that the only law of force ex- 

 pressible by a finite series of inverse powers of r which is 

 consistent with the variation of molecular potential energy of 

 a body inversely as its volume is the law of the inverse fourth 

 power. The same statement holds as regards variation of the 

 virial of the molecular force inversely as the volume. When 



and the value of the virial becomes 

 37TpA.log— . 



To show the relation between the general expressions for the 

 potential energy and virial, let us integrate the latter by parts, 

 remembering that 



f(r)dr= —d{<j))r, 

 irp 1 r B f(r)dr=—7rp 1 r s d(j)(r) 



Ja Ja 



Ja 



According to the treatment usual in the theory of capillary 

 action, the first term of the last expression would be shown to 

 vanish, because <f>(L) is negligible; and in the lower limit, when 

 a is replaced by 0, the term O 3 0(O) is assumed to vanish, as it 



