﻿460 Messrs. C. G. Darwin and It. H. Fowler on 



integer in the units we employ. The total number C of all 

 complexions is therefore 



M ! N ! 



G = ^ b aJa'JaTj::. b l b ± l b 2 \ ... ' * * (5 ' 3) 



where the summation X a ,b is to be carried out over all positive 

 or zero values of a r and b s which satisfy (5*2). 



By using (5*1) and (5*3) we can at once obtain an expression 

 for the average value, taken over all complexions, of any 

 quantity in which we are interested. We have already 

 studied ~a T in § 4. The main interest centres in Ea, the 

 average energy on the A's. We have at once 



PF -? (2,.r £ q r )M!N! 



^-^A — A*, b i 1 1 7 I 7 I 7 I • • • \p ^) 



The following process leads to simple integrals to express 

 the quantities C, CEa, etc. Consider the infinite series 



(l + z e + z 2e + ) M 



expanded in powers of z by the multinominal theorem. The 

 general term is 



'Ml z ? r rea„ m 



a Q ! % ! ..." 



It follows that if we select from the expansion of 



(l + * e + ^ e +...) M (l + s*^* + ...)* . (5-5) 



the coefficient of z E , we shall obtain the sum of all possible 

 terms such as (5*1) subject to the conditions (5*2), that is to 

 say C Similarly, if we form the expression 



j^(l+s e H-2 2e ...) M 1(1-^ + ^...)*, . (5-6) 

 the general term in the first bracket must be 



{% r T€a r )M\ 2 r rea r 

 — Z , 





\^\ax\ 



and by the same reasoning the coefficient of z E in (5*6) must 



be CE A . 



Expressions (5*5) and (5*6) are easily simplified— they are 

 respectively 



