CHAPTER VII. 



FARTHER DISCUSSION OF AVERAGES IN A CANONICAL 

 ENSEMBLE OF SYSTEMS. 



RETURNING to the case of a canonical distribution, we have 

 for the index of probability of configuration 



as appears on comparison of formulae (142) and (161). It 

 follows immediately from (142) that the average value in the 

 ensemble of any quantity u which depends on the configura- 

 tion alone is given by the formula 



r au ^ *<r-*g 

 =J...Jue " ^d qi ...dq n} (179) 



u 



conflg. 



where the integrations cover all possible configurations. The 

 value of i|r g is evidently determined by the equation 



r ^ r _!? 



=J . . .J e %*dfc . . . dq n . (180) 



e 



config. 



By differentiating the last equation we may obtain results 

 analogous to those obtained in Chapter IV from the equation 



- *" ~ f 







J * J & &dPl ' ' ' 



e 



. 



phases 



As the process is identical, it is sufficient to give the results : 

 dfa = rj q d J^i J^da^ etc., (181) 



