108 THE FUNCTION < AND 



falls within any given limits of energy (e' and e") is repre- 

 sented by 



f 



e^de. 



If we expand 77 and < in ascending powers of e e , without 

 going beyond the squares, the probability that the energy falls 

 within the given limits takes the form of the law of errors ' 



de. (353) 



i/ 



This gives 



We shall have a close approximation in general when the 

 quantities equated in (355) are very small, i. e., when 



is very great. Now when n is very great, d*$/de* is of the 

 same order of magnitude, and the condition that (356) shall 

 be very great does not restrict very much the nature of the 

 function 77. 



We may obtain other properties pertaining to average values 

 in a canonical ensemble by the method used for the average of 

 d<j>/de. Let u be any function of the energy, either alone or 

 with and the external coordinates. The average value of u 

 in the ensemble is determined by the equation 



6=00 4,-e 



/- - + 4> 

 ue e de. (357) 



F=0 



