70 The Law of Distribution [CH. v 



so that expression (162) can be written in the form 



B a e~ hm <'+*+) dudvdw (163), 



where B a is independent of u, v and w. It follows that the law of distribution 

 of translational velocities is of the form 



/. (u, v, w) = C a er hm (*+*+**) (164). 



In other words, the law of distribution of translational velocities is the 

 same as if the molecules were the smooth rigid spheres of Chapters II. 

 and III. 



79. In the only case of any importance, the potential energy can be 

 written in the form 



V=V i + x , 



where F,- is the potential energy of the molecule under its internal forces, 

 and % is its potential in the external field of force and depends only on x, y, z, 

 the coordinates of its centre of gravity. This covers the case of a gas acted 

 upon by a gravitational field of force for instance, the atmosphere. In this 

 case we have 



E a =V i + x + L, 



and % is the only part which involves x, y, z. 



By an argument similar to that of the last section, it will be seen that 

 the number of molecules for which x lies between x and x + dx, and similar 

 conditions are satisfied by y, z, is of the form 



D a e-^ dxdydz. 



In other words, if v a is the molecular density of gas of type a at the point 

 x, y, z we have 



Va = D a e-* h * (165). 



Also it appears that the number of molecules of type a which occur per 

 unit volume at x, y, z and of which the coordinates lie within the usual range 

 of limits d%! ... d^n^d^ . . . dij n is proportional to 



Va e-*(ri+UdZ l ...d n _ 3 d<r, l ...dr ]n ..., (166). 



This shews that the field of external force does not affect the law of 

 distribution of the coordinates of the molecule, other than x, y, z, or the law 

 of distribution of velocities, but that the distribution of density alone is 

 affected. 



The result expressed by equation (165) is simply a more general case of 

 that expressed by equation (94), namely 



v = constant, 



in the case of hard spherical molecules. The quantity v is the molecular- 

 density defined as in 11, being in fact the true density averaged through a 

 "cell" which is sufficiently large to contain a very great number of molecules. 



