52 AVERAGE VALUES IN A CANONICAL 



the w's. The integrals may always be taken from a less to a 

 greater value of a u. 



The general integral which expresses the fractional part of 

 the ensemble which falls within any given limits of phase is 

 thus reduced to the form 



...<***&...%,. (134) 



For the average value of the part of the kinetic energy 

 which is represented by ^u^ whether the average is taken 

 for the whole ensemble, or for a given configuration, we have 

 therefore 



__ (135) 



--' 



I/ 



e 



00 



and for the average of the whole kinetic energy, JTI, as 

 before. 



The fractional part of the ensemble which lies within any 

 given limits of configuration, is found by integrating (184) 

 with respect to the w's from oo to + oo . This gives 



J f. 



da, 



which shows that the value of the Jacobian is independent of 

 the manner in which 2e p is divided into a sum of squares. 

 We may verify this directly, and at the same tune obtain a 

 more convenient expression for the Jacobian, as follows. 



It will be observed that since the M'S are linear functions of 

 the p's, and the jt?'s linear functions of the ^'s, the u's will be 

 linear functions of the <?'s, so that a differential coefficient of 

 the form du/dq will be independent of the q's, and function of 

 the <?'s alone. Let us write dp x jdu y for the general element 

 of the Jacobian determinant. We have 



