﻿the Partition of Energy. 8)U 



this probability we are indifferent about the distribution 

 among the B's, so we sum the complexions involving all 

 values of the &'s consistent with the selected values of 

 the a's. Then 



V -L, '•: t. :<«'■ ■■■)(2.v,?T-. «'■-)/<'■ 



. . . (4-0) 



where the a's may have any values (which do not involve a 

 greater total energy than that of the whole assembly), while 

 Xb denotes summation over all different values of the b's 

 such that 



XsVsh = E — Z r €,.a r , 



and of course, as always, S,a r = M, ^.6 5 = N. Now, provided 

 that N is much larger than M, the factor 



(2' 6 !m'.... «°1* ■ • •)/ C 



will be practically independent of the <x r 's and of the energy 

 of the group of A's — that is to say, it may be taken as 

 constant and omitted from the calculation, and we are left 

 with the "thermodynamic probability" as the only variable 

 part. 



It is only in this sense that a strict meaning can be 

 assigned to Boltzmann's Hypothesis; and it is of the 

 greatest interest that the conditions under which it has 

 meaning correspond exactly to the conditions of the 

 "canonical ensemble" of Gibbs, as will be shown later. 

 But, even so, it is not a very convenient expression, for 

 we must always suppose that the assembly is a part of 

 some much larger one, whereas the expression for the 

 entropy is purely a function of the group and the tempe- 

 rature. It is therefore more convenient to abandon the use 

 of the principles of probability and to define the entropy 

 as k times the logarithm of the number of complexions 

 (weighted if necessary). We shall call this the kinetic 

 entropy. This number of complexions has the multiplicative 

 property (4'1), but now in virtue of its own combinatory 

 formula and not of an appeal to an inapplicable probability 

 theorem. The new definition does not appear to have the 

 same simplicity as the old, but that is only because in 

 the old the necessity for a detailed definition of what is 

 meant by probability was concealed. It would appear that 



