﻿the Partition of Energy. 475 



equal. Then by the principles of § 2 the weight factor for 

 the tth cell is 



8t= («** •» <fo\ .... ( i2-l) 



provided of course that the cell is relevant to our assembly. 

 Only those cells have a weight for which the q } s lie in the 

 vessel; but the //s may range over all values from -co to 

 + co, for the method of summation will automatically 

 exclude values which could not be allowed. Associated 

 with the tth cell there is energy given by 



r ( =^0'i 2 +^ 2 +ps 2 ). . • • (12-iD 



The state of the molecules in the assembly is specified by 



the numbers c l5 c 2 , ... of molecules in cells 1, 2, The 



specification of the A's is as before. The number of weighted 

 complexions corresponding to the specification is then 



Mf Pi 



" '— i>o> gl ... > t V^V., . (12-12) 



. a ! fll !../ u ^ •••c 1 U' 2 ! 

 where Z r a r = M, $ t c t = F, 2,.a r e r + W* = E - . (12-13) 



In proceeding thus we are constructing an artificial 

 assembly in which the energy is taken to have the same 

 value f f in all parts of the tth cell, and in which all the f's 

 and all the e's can be expressed as multiples of some basal 

 unit, without a factor common to thern all. 



This assembly can be made to resemble the real one to 

 any standard of approximation required. For such an 

 artificial assembly w r e can make use of the whole of our 

 machinery. The results all depend on integrals such as 





■*T [««)]*[*«]*, • • a 2 " 2 ) 

 where the partition function of the artificial molecules is 



fcO)=2A/<, (1*21) 



and the formulae of § 8 follow at once for E c , (E c — E c ) 2 , c t 

 and (<' t — c t ) 2 . These results give completely the exact partition 

 laws for any artificial assembly of the type considered. To 

 obtain the actual distribution law for the real assembly, we 

 must make all the dimensions of all the cells tend to zero, 

 and obtain the limit of the partition function. Now, by the 



