ENSEMBLE OF SYSTEMS. 75 



posing that there were means of measuring these quantities 

 separately) had each separately uniform values.* Exceptions 

 might occur when for particular values of the modulus the 

 differential coefficient de q /d~e p takes a very large value. To 

 human observation the effect would be, that in ensembles in 

 which and e p had certain critical values, ~e q would be in- 

 determinate within certain limits, viz., the values which would 

 correspond to values of and e p slightly less and slightly 

 greater than the critical values. Such indeterminateness cor- 

 responds precisely to what we observe in experiments on the 

 bodies which nature presents to us.f 



To obtain general formulae for the average values of powers 

 of the energies, we may proceed as follows. If h is any posi- 

 tive whole number, we have identically 



phases phases 



t. e., by (108), 



_i ,, _i 



(215) 



Hence 



and 



* This implies that the kinetic and potential energies of individual systems 

 would each separately have values sensibly constant in time. 



t As an example, we may take a system consisting of a fluid in a cylinder 

 under a weighted piston, with a vacuum between the piston and the top of 

 the cylinder, which is closed. The weighted piston is to be regarded as a 

 part of the system. (This is formally necessary in order to satisfy the con- 

 dition of the invariability of the external coordinates.) It is evident that at 

 a certain temperature, viz., when the pressure of saturated vapor balances 

 the weight of the piston, there is an indeterminateness in the values of the 

 potential and total energies as functions of the temperature. 



