Distribution e~ 2h x and van der Waals* Theorem, 409 



does not produce complete uniformity, because, although the 

 more uniform distribution is more probable than the less, yet 

 (k being finite) it is more probable only in a finite ratio of 

 the form e~ 2h{x ~ x ''. (The case of infinite h is the case of 

 statical equilibrium.) Therefore in stationary motion the 

 density is not, and docs not tend to become, uniform, because 

 all the non-uniform distributions continue for all time to 

 occur, each with its appropriate frequency as given by the 

 law. 



11. In the general case where there are many molecules 

 within the sphere of action of any one molecule, the same 

 reasoning applies mutatis mutandis. As the difference nf—n^ 

 increases, the value of S^j + S^o for repulsive forces increases 

 on average, and therefore the probability 6~ lA(2x i +2 fo> dimin- 

 ishes. So that with repulsive forces, of two distributions of 

 2n molecules through two equal portions of space, that in 

 which each portion contains n molecules is more probable in 

 a finite ratio than the less uniform distribution. In this 

 general case, as in that of binary encounters, our law makes 

 (the forces being repulsive) for uniformity of density. But 

 except in the case mentioned below (art. 12), it is incom- 

 patible with complete uniformity, because all the non-uniform 

 distributions continue for all time to occur, each with its 

 approximate frequency, as given by the law. 



If, on the other hand, the intermolecular forces be wholly 

 attractive, the more uniform distribution is generally one of 

 higher potential than the less, and therefore the less probable 

 of the two ; and the completely uniform distribution, in 

 which the density is the same throughout, being one of 

 maximum potential, is according to our law less probable 

 than any other distribution. 



12. These inequalities of density must exist in all substances 

 to which the law e~ 2hx is applicable. If in any case they can 

 be proved not to exist, then for that case the law is not 

 applicable. And as a consequence of these inequalities the 

 mean potential has in case of repulsive forces a greater value, 

 in case of attractive forces a less value, than it would have if 

 the distribution were uniform throughout. The same is true 

 of the virial of the intermolecular forces. 



It will be said that experiment proves the density to be the 

 same for all measurable volumes of gas taken at the same 

 level of potential of the external forces. That result is quite 

 consistent with what I have said, indeed follows from it. 

 For if we compare two equal spaces IS and S', each of which 

 should, according to the average of the whole gas, contain 

 n molecules, the chance that at a given instant they shall 

 contain respectively nl + q and nl—q, where 0<'y<l, is 



