Gate* and tJw Equation of Virial. 499 



From (20) we may fall back on (18) by the consideration 

 that in accordance with (10) fjJ and a can occur only in the 

 combination p/a*. 



It may be well to remark that the method of dimensions 

 does not tell us whether or no an available solution can be 

 deduced from particular assumptions. What it teaches us is 

 the form which an available solution must assume. For 

 example, equation (IF) gives the form of the term in the 

 virial proportional to r~ l . under the law of force (10) ; and 

 nothing has been said as to any restriction upon the value 

 of n. But it is easy to see that n must in fact be greater 

 than 4. Otherwise the integral representing the virial re- 

 lating to a given particle would not be convergent. We 

 have to consider 



§p<t>(p)p 2 dp 



with infinity for the upper limit, and this diverges unless n 

 exceed -1. 



It is not to be expected that any law included under (10) 

 could represent with completeness the mutual action of the 

 particles of a gas. Under it no provision can be made for 

 repulsion at small distances and attraction at greater ones. 

 And when >*>4, the aggregate virial depends too much 

 upon the encounters which take place at exceedingly small 

 distances. 



If, as for both the virial terms in Van der Waals' formula, 

 there be proportionality to r _1 , (20) becomes 



^)=4 T F(7). .... (21) 



or. if we prefer it. 



S^(ri = ";'F^), .... (22) 



F in both cases denoting an arbitrary function. According 

 to Van der Waals F in (21) is a linear function, the constant 

 part giving the collisional virial and the second term the 

 cohesional virial which is independent of T. Except for one 

 consideration to be mentioned presently, there would appear 

 to be good reason for supposing the virial of a rare gas to be 

 proportional to v 1 ; but on the other hand it is doubtful 

 whether the cohesional force- are altogether of the kind 

 supposed by Laplace and Van der Waals. We should ex- 

 pect the cohesional virial to be more directly influenced by 

 the approaches of molecules during an encounter ; and on 

 the experimental side I >. Berthelot has shown cause for pre- 

 ferring to that of Van der Waals the Rankine and Olausius 



