CHAPTER V. 



AVERAGE VALUES IN A CANONICAL ENSEMBLE 

 OF SYSTEMS. 



IN the simple but important case of a system of material 

 points, if we use rectangular coordinates, we have for the 

 product of the differentials of the coordinates 



dxi dyi dzi . . . dx v dy v dz v , 



and for the product of the differentials of the momenta 

 m l dxi mi dyi m^ dz 1 . . . m v dx v m v dy v m v dz v . 



The product of these expressions, which represents an element 

 of extension-in-phase, may be briefly written 



mi dxi . . . m v dz v dxi . . . dz v ; 

 and the integral 



e @ mi dxi . . . m v dz v dxi . . . dz v (118) 



will represent the probability that a system taken at random 

 from an ensemble canonically distributed will fall within any 

 given limits of phase. 

 In this case 



(119) 

 and 



e 



=e & e 2> 20 . (120) 



The potential energy (e 3 ) is independent of the velocities, 

 and if the limits of integration for the coordinates are inde- 

 pendent of the velocities, and the limits of the several veloci- 

 ties are independent of each other as well as of the coordinates, 



