90] 



Maxwells Treatment of Equipartition 



81 



The ratio of this expression to (198), on replacing VTT by F(), is found 

 to be 



r f - 



This is the fraction for which r) n lies between rj. n and r) n + dr} n , of all the 

 systems for which q l} q z ...q n , E lie within the specified small ranges. 



Let us write K n = \c n T)J 



so that K n is the kinetic energy corresponding to the momentoid ij n ] then 



Cjdl) n = -f== , 



A 



and therefore expression (200) becomes 



(E VK n ] 



w-3 



_ ,, 



The mean value of K n averaged over all the systems for which 

 and E lie within the specified ranges, say K n , is therefore 



r 



J^=o 



so that from symmetry 



(202). 



In words, this result states that, averaged through all those parts of the 

 generalised space in which q 1} q 2 ... q n and E have specified value, the energies 

 of the various momentoids are equal. By addition, it follows that, averaged 

 through all parts of the generalised space for which E has a given value, 

 the energies of the various momentoids are equal. 



* Williamson, Integral Calculus, p. 161. 



J. 



