THE CANONICAL DISTRIBUTION. 105 



\/ 6 ^ Q 6 



> 



(339) 



Substituting this value in 



which expresses the probability that the energy of an unspeci- 

 fied system of the ensemble lies between the limits e' and e", 

 we get 



- 



**. (340) 



When the number of degrees of freedom is very great, and 

 e e in consequence very small, we may neglect the higher 

 powers and write* 



i . 



" (341) 



This shows that for a very great number of degrees of 

 freedom the probability of deviations of energy from the most 

 probable value (e ) approaches the form expressed by the 

 'law of errors.' With this approximate law, we get 



* If a higher degree of accuracy is desired than is afforded by this formula, 

 it may be multiplied by the series obtained from 



by the ordinary formula for the expansion in series of an exponential func- 

 tion. There would be no especial analytical difficulty in taking account of 

 a moderate number of terms of such a series, which would commence 



