118 Mr. W. Sutherland on the 



which it is permissible to take for L is negligible in com- 

 parison with the dimensions of experimental bodies. 



Noticing that nm is the mass of the whole body, which we 

 assume to be unity, we get for the potential energy of the n 

 molecules, 



f L 

 2 77-/91 r 2 (f>(r)dry 



*) a 



and for the virial of their forces, 



irpi r 3 f(r)dr. 



Now if (as Laplace does in his expression for the attraction 

 of a fluid on a column) we were to replace a and L in the two 

 integrals by and <x> , we should make the two integrals con- 

 stants, and we should have the potential energy of the mole- 

 cules of a body and the virial of their attractions both 

 proportional directly to its density, or inversely to its 

 volume, whatever the law of force might be, so long as it 

 satisfied the assumed conditions. But the highly artificial 

 character of these conditions becomes evident if we consider 

 how the true lower limit a was obtained, and how it must be 

 a function of the distance between a molecule and its nearest 

 neighbours, and therefore of v ; a natural assumption is that 

 a is approximately proportional to y/v. Now in the treat- 

 ment usual in capillary theory, the above integrals would be 

 supposed to vanish at their upper limits, so that their values 

 depend entirely on the values at the lower limit; before, then, 

 we can treat the integrals as constants, we have to give 4>(r) 

 and/(?*) a certain special property. 



It is easy now to see that (/>(r) cannot be of the form 



A B 



if it is to possess this property, unless when the series takes 



A 



the form -j, which corresponds to the law of the inverse 



fourth power of the distance; for, assuming cf>(r) to be expres- 

 sible as such a series, and integrating the expression for the 

 potential energy, we get 



27rpA( 1 _J_\ 2tt / oB/_1_ 1 > . 



It is obvious that we cannot admit any exponent such as m or 

 n to be less than 3, or we should have a part of the expression 



