Distribution e~ 2h x and van der Waals' Theorem. 407 



must be confined to such gases. But this separation of the 

 factors is the one thing in the proof that is not essential, as 

 will be shown later (art. 17). 



7. The Physical Interpretation of the Law. — Our law 

 expresses that the distribution of velocities among the mole- 

 cules is independent of the coordinates. This, however, as 

 above stated, is assumed but not proved, either at p. 134 of 

 the Vorlesungen or anywhere else. It comes out as a result 

 at the end of the mathematical process, only because it was 

 put in as an axiom at the beginning. 



The law determines a certain distribution of the molecules 

 in space, which depends on the nature of the forces acting. 



For external forces it expresses the permanent density in 

 different parts of space in stationary motion, as, for instance, 

 in a vertical column of air under the constant force g\ the 

 density at height s varies according to our law as e~ 2h 3 s . 



So also in case of intermolecular forces the law is a law of 

 density, in so far as 2% depends on the density, but in this 

 case it is the instantaneous density near any point that is 

 indicated. We must now distinguish between two cases. 

 Case I., the radius of action of a molecule is greater than 

 the distance between it and its nearest neighbours all the 

 molecules being uniformly distributed through space. Case 

 II., the radius of action is less than this distance. 



8. In case I. no single molecule, and no group of molecules, 

 is ever isolated. Sx is a function of the coordinates of all 

 the molecules, but cannot be divided into independent parts, 

 each relating to a separate group of molecules. 



About a point P conceive a sphere of radius c described, 

 and suppose at a given instant it contains n molecules, n being 

 a number which, however great, is small compared with N, 

 the number of molecules in the system. And similarly the 

 volume of the c sphere is small compared with the whole 

 space. The potential now consists of (1) 2%, the mutual 

 potential of the n molecules, (2) ^x', the potential of mutual 

 action between them and the molecules outside of the c sphere, 

 and (3) %", the potential of these external molecules inter se. 

 %x! i s *h en a function of [inter alia) the coordinates of the 

 n molecules within the c sphere. As the radius c becomes 

 very much greater than the radius of action, e~ 2h -x becomes 

 a much more important factor than €~ 2h ~X', but we cannot 

 generally assert that the chance of any configuration of the n 

 molecules depends on 2% only. 



i). Case II. is usually known as that of binary encounters. 

 In that case, of the n molecules within the sphere of radius c, 

 only very few pairs are at any instant in encounter. And we 



2 E 2 



