74 AVERAGE VALUES IN A CANONICAL 



That is, a diminution of the modulus will diminish the proba- 

 bility of all configurations for which the potential energy exceeds 

 its average value in the ensemble. Again, in the ensemble 

 having modulus ' and average energies e p ' and e^, values of 

 e q sensibly less than e g ' will be so rare as to be practically neg- 

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

 modulus, since by the same equation an increase of the 

 modulus will diminish the probability of configurations for 

 which the potential energy is less than its average value in 

 the ensemble. Therefore, for values of O between ' and ", 

 and of e p between e p ' and e p /; , the individual values of e q will 

 be practically limited to the interval between e/ and e g ' r . 



In the cases which remain to be considered, viz., when 

 de q /de p has very large values not confined to narrow limits, 

 and consequently the differences of the mean potential ener- 

 gies in ensembles of different moduli are in general very large 

 compared with the differences of the mean kinetic energies, it 

 appears by (210) that the anomalies of mean square of poten- 

 tial energy, if not small in comparison with the mean kinetic 

 energy, will yet in general be very small in comparison with 

 differences of mean potential energy in ensembles having 

 moderate differences of mean kinetic energy, the exceptions 

 being of the same character as described for the case when 

 de q /de p is not in general large. 



It follows that to human experience and observation with 

 respect to such an ensemble as we are considering, or with 

 respect to systems which may be regarded as taken at random 

 from such an ensemble, when the number of degrees of free- 

 dom is of such order of magnitude as the number of molecules 

 in the bodies subject to our observation and experiment, e e, 

 P pi *q % would be in general vanishing quantities, 

 since such experience would not be wide enough to embrace 

 the more considerable divergencies from the mean values, and 

 such observation not nice enough to distinguish the ordinary 

 divergencies. In other words, such ensembles would appear 

 to human observation as ensembles of systems of uniform 

 energy, and in which the potential and kinetic energies (sup- 



