OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 725 



These expressions give the number of systems in a given configuration 

 only when E V is positive for that configuration, for since the kinetic energy 

 is necessarily positive, the potential energy cannot exceed the total energy. 

 For configurations specified in such a way that if they existed V would be 

 greater than E, the value of N(b) is zero. 



The value of N(b) is also zero for configurations which, though they make 

 V less than E, cannot be reached by a continuous path from the original 

 configuration without passing through configurations which make V greater than E. 



We shall return to this expression for the number of systems in a com- 

 pletely specified configuration, but in the mean time it will be useful to 

 consider how many of these systems have one of their momenta, p n , between 

 given limits. In this way we shall be able to determine completely the 

 average distribution of momentum among the variables without making any 

 assumptions about the nature of the system which might limit the generality 

 of our results. 



In order to find the number of systems in the configuration (b) for which 

 one of the momenta, say p n , lies between a n and a n + da n , we must stop 

 before the last integration. Putting r = n 2 in equation (40) 



2 / 



The whole number of systems in configuration (b) is given by (45). Hence 

 the proportion of these systems for which a n lies between a n and a n + da n is 



(49). 



If we write 



then k n denotes the part of the kinetic energy arising from the momentum a n . 

 The proportion of the systems in configuration (6) for which k n is between k n 

 and k a + dk n is 



