OF THE ENERGIES OF A SYSTEM. - 93 



substitute it in the general expressions (286), and obtain the 

 following values, which are perfectly general : 



~ * (289) 



It will be observed that the values of V p and <f> p for any 

 given e p are independent of the configuration, and even of the 

 nature of the system considered, except with respect to its 

 number of degrees of freedom. 



Returning to the canonical ensemble, we may express the 

 probability that the kinetic energy of a system of a given 

 configuration, but otherwise unspecified, falls within given 

 limits, by either member of the following equation 



Since this value is independent of the coordinates it also 

 represents the probability that the kinetic energy of an 

 unspecified system of a canonical ensemble falls within the 

 limits. The form of the last integral also shows that the prob- 

 ability that the ratio of the kinetic energy to the modulus 



* Very similar values for V q , <&*, V, and e* may be found in the same 

 way in the case discussed in the preceding foot-notes (see pages 54, 72, 77, and 

 79), in which e 3 is a quadratic function of the q's, and Aj independent of the q'a. 

 In this case we have 



(2 ')*( - 



P(Jn) 



+ i) 



