ANN'S THEOREM OK THE AVERAGE DISTRIBUTION 



The quantity E- F. which occurs in this equation, is, by equation (1), 

 equal in magnitude to T, the kinetic energy of the system. The quantity T, 

 m defined explicitly in terms of the velocities or the momenta of the 

 whoffQM K-V does not involve these quantities explicitly, but is 

 M a function of the configuration. 



We t***H find it convenient, however, especially in the study of more 

 complicated problems, to remember that the number of systems in a given 

 j^lyifatfff" IB a function of the kinetic energy corresponding to that configuration. 

 If the kinetic energy is not expressed as a sum of squares, but in the 

 move general form, 



+ i[22]a,' + [23]a,a, + &c (42), 



where the quantities denoted by [11] &c. are functions of the co-ordinates, 

 which we may call the moments and products of mobility of the system ; 

 thm fjfusM the discriminant 



[n], [121 M 



A- ^J 2 ^;;;;:;;;. 12 ! 1 (> 



[ll [2J [nn] 



is an invariant, its value is the same when T is reduced to a sum of squares, 

 in which case all the elements except those in the principal diagonal of the 

 determinant vanish, and we have 



A =/*,/,.../* (44), 



and we may write the value of N (b), 



*' ^ (45). 



If the system consists of n' material particles, whose masses are m l ...m n -, 

 then the number of degrees of freedom is n = 3n' and 



/*I = / A I = / A = WI~ I , p t = fr = p- t = ini~ l and so on (46). 



e in this case we may write 



(r( 



db,...db n (47). 



