THE CANONICAL DISTRIBUTION. 103 



Moreover, if we eliminate dty from (326) by the equation 



d^ = 0^ + ^d + de, (331) 



obtained by differentiating (325), we get 



de = -dv-!Jr-da l - < Q-da 2 - etc., (332) 



Cia-l OLa^, 



or by (321), 



. _^ = ^e + ^^ + ^^ + etc. (333) 



de da, aa 2 



Except for the signs of average, the second member of this 

 equation is the same as that of the identity 



ty = ^de + ?da l + ^da 2 + etc. (334) 



de da l da 2 



For the more precise comparison of these equations, we may 

 suppose that the energy in the last equation is some definite 

 and fairly representative energy in the ensemble. For this 

 purpose we might choose the average energy. It will per- 

 haps be more convenient to choose the most common energy, 

 which we shall denote by e . The same suffix will be applied 

 to functions of the energy determined for this value. Our 

 identity then becomes 



= de + da, + da, + etc. (335) 



\de J \dajo \da 2 J 



It has been shown that 



? = (^ = l, (336) 



de \de) ' 



when n > 2. Moreover, since the external coordinates have 

 constant values throughout the ensemble, the values of 

 d(p/da v d(f>Jda^ etc. vary in the ensemble only on account 

 of the variations of the energy (e), which, as we have seen, 

 may be regarded as sensibly constant throughout the en- 

 semble, when n is very great. In this case, therefore, we may 

 regard the average values 



<25 ~d4 



-5-S -=-S etc., 



