44 The Law of Distribution [CH. in 



48. We must now determine the number of mass-centres i.e., points at 

 which all the Jt's have integral values which occur within this region. The 

 density of distribution of these centres is of course constant throughout the 

 (?i l)-dimensional space in which equation (67) is satisfied. If a system of 

 lines is drawn through these centres parallel to one of the axes of coordinates, 

 these lines will form a network of parallel lines at unit distance apart ; their 

 intersections with any orthogonal (n 1 )-dimensional region will therefore be 

 at the rate of unity per unit volume of this latter region. The region specified 

 by equation (67) is not orthogonal to this system of lines, but since it has 

 direction cosines 



n~i, n~$, n~^ . . . 



it will make an angle cos" 1 ^"*) with any orthogonal region. Hence the 

 number of intersections of the system of lines with the region specified by 

 equation (67) is n~* per unit volume. 



The number of mass-centres in this region for which K lies between K 

 and K + dK is accordingly equal to expression (82) multiplied by n~~*. In 

 accordance with equation (65) we place at each of these centres a mass 



and we therefore have, for the total mass in the region in which K lies 

 between K and K + dK, 



n-l n-l 



dK\ (84). 



rf P ' 



11 ( 2 ) 

 After simplification this expression reduces to 



AT~JT -3 



:....(85). 



49. Remembering the significance of the masses placed in the new 

 generalised space, it appears that in the original generalised space of 6N 

 dimensions, that fraction of the whole space which represents systems for 

 which K lies between K and K + dK is given by expression (85). By 

 integration, the fraction for which K Ties between K l and K 2 is given by 



rr f e~ NK K 2 dK (86). 



(!Lt*\ J *-x 



\ 2 ) 



This result, it must be remembered, has been obtained on the assumption 

 that the e's are small compared with N and is therefore, by equation (77), 



