90 CERTAIN IMPORTANT FUNCTIONS 



the potential energy, either alone or with quantities which are 

 constant in the integration. 



We may often avoid the inconvenience occasioned by for- 

 mulae becoming illusory on account of discontinuities in the 

 values of V q as function of e q by substituting for the given 

 discontinuous function a continuous function which is practi- 

 cally equivalent to the given function for the purposes of the 

 evaluations desired. It only requires infinitesimal changes of 

 potential energy to destroy the finite extensions-in-configura- 

 tion of constant potential energy which are the cause of the 

 difficulty. 



In the case of an ensemble of systems canonically distributed 

 in configuration, when V q is, or may be regarded as, a continu- 

 ous function of e q (within the limits considered), the proba- 

 bility that the potential energy of an unspecified system lies 

 between the limits e q and e q ' is given by the integral 



where ^ may be determined by the condition that the value of 

 the integral is unity, when the limits include all possible 

 values of e q . In the same case, the average value in the en- 

 semble of any function of the potential energy is given by the 

 equation 



u = / ue d q . (275) 



V q =0 



When V q is not a continuous function of e ff , we may write d V q 

 for e* q de g in these formulae. 



In like manner also, for any given configuration, let us 

 denote by V p the extension-in-velocity below a certain limit of 

 kinetic energy specified by e p . That is, let 



V, = J. 



(276) 



