ENSEMBLE OF SYSTEMS. 73 



(209) 



e-__ ? . 



~?~ "^~ + 5f p 



These equations show that when the number of degrees of 

 freedom of the systems is very great, the mean squares of the 

 anomalies of the energies (total, potential, and kinetic) are very 

 small in comparison with the mean square of the kinetic 

 energy, unless indeed the differential coefficient de q /de p is 

 of the same order of magnitude as n. Such values of de q jde p 

 can only occur within intervals (ej 1 e p f ) which are of the or- 

 der of magnitude of n~~\ unless it be in cases in which e g is in 

 general of an order of magnitude higher than e p . Postponing 

 for the moment the consideration of such cases, it will be in- 

 teresting to examine more closely the case of large values of 

 de q /de p within narrow limits. Let us suppose that for ej and 

 e p f the value of de q /de p is of the order of magnitude of unity, 

 but between these values of "e p very great values of the differ- 

 ential coefficient occur. Then in the ensemble having modulus 

 @" and average energies e p " and e s ", values of e q sensibly greater 

 than e q rl will be so rare that we may call them practically neg- 

 ligible. They will be still more rare in an ensemble of less 

 modulus. For if we differentiate the equation 



regarding e q as constant, but and therefore ^ as variable, 

 we get 



/drj q \ __1 dif/ q \It q q . 



\d) -~d ^~' 

 whence by (192) 



