Law of Molecular Force. 117 



energy of m and the shell is 



4z7rmpr 2 <j>(r)dr, 



and the potential energy of the whole continuous mass and 

 the molecule m is 



4c7rmpr 2 (f)(r)dr, 



neglecting, in accordance with the definition of L, the action 

 on m of all molecules beyond the surface of the sphere of 

 radius L. 



For the virial of the forces acting between m and all the 

 other molecules, we have 



ir. 



ij 4;7rpr d f(r)di 



J a 



Summing these two expressions for the n molecules and 

 dividing each by 2, we get results which do more.than repre- 

 sent the sums ^22m 2 0(r) and -J- . ^22mV/(r), as each in- 

 cludes the action of the n molecules on a layer of molecules of 

 thickness L added over the whole surface S. If, then, v is the 

 volume occupied by the n molecules, the number of molecules 



T S 

 in the layer is — . n. If, then, we imagine a layer of thick- 

 ness L removed from the surface of the actual body, and sum 



(T S\ 

 1 1 molecules 



remaining, we shall obtain two sums too small to represent the 

 desired sums, seeing that the mutual actions of the molecules 

 in the layer removed will not be taken into account. Thus, 

 the whole potential energy of the molecules lies between 



■l 

 r*tf>(r)dr, 



2irmnp\ 



J a 



2irn (l ) mp \ r 2 <p{r)dr, 



and may be represented by 



/ LS\ f L 



27rnll — 6 — \mp\ r 2 <f>(r)dr, 



where 6 is a proper fraction. 



The term involving S plays an important part in the theory 

 of capillary action; but for our present purpose we may 

 neglect it in comparison with the other term, if we make the 

 assumption usual in capillary theory that the smallest value 



