﻿470 Messrs. C. G. Darwin and R. H. Fowler on 



system A and B. The application of the multinomial 

 theorem then leads to the consideration o£ the expression 



[/(*)] M M*)f, 



and pursuing exactly the same course as in § 5, we find 



C=2-y^ 1 [/«] M [^)] N , • • • (9-4) 



Assume for the moment that we can choose a <\>(z) con- 

 forming to the requirements of § 6. The whole calculation 

 then goes on as before. The radius of the circle to be taken 

 as contour is given by the equation 



E = M^log/(3)+N^ log </(£). . . (9-6) 



This equation has one and only one root. We thus can 

 at once put down 



, *> \ . . . (9-7) 



=M3 J log /(d). ) 



In exactly the same way we have 



£=Mp,**/X»). (9-8) 



and we can also verify that in the case of an infinite bath 

 the fluctuations are again given by (8*4), (8*5), and that 

 equation (6'6) is still true. The exact forms of the fluctua- 

 tions {8'$)i (8*7) are also valid if we replace re by e r . 



We have now to examine whether <j>(z) can be properly 

 chosen. It is natural to take 



<f>(z)= Z -H/W Mm Um m - ■ ■ ■ (9-8 1 ) 

 By its definition it must satisfy (i.). For (ii.) to be true, 

 we must have 



E>Me + N>/o, 

 which is the trivial condition that there must be enough 

 energy to provide each system with the least amount Of 

 energy it is permitted to have. Condition (iv.) does not 

 appear at first sight inevitable, but must follow from Bohr's 

 Correspondence Principle *, for the convergence of the series 

 f(z) and g(z) depends on their later terms — that is, those of 

 * Bohr, loc. cit. 



