﻿the Partition of Energy. 4G5 



in such a way that M/E and N/E are constant. This func- 

 tion satisfies all the conditions of this section — the fact that 

 it is in general many-valued is irrelevant, for we are only 

 concerned with that particular branch which is real when z 

 is real and 0<^<1 and this branch is one-valued in the unit 

 circle. The unique minimum S is determined by the equa- 

 tion cj>'=0 or 



Me N^ 



E =s=zzi + $=v-l ( fi "4) 



The value of the integral (5*71) is then by {6'2) (omitting 

 the second term the form of which is only required in 

 calculating fluctuations), 



s-B(i-y)-M(i-s*)-g 



If, contrary to hypothesis, we had taken e and rj as having 

 a common factor /3, condition (v.) would have been violated. 

 In this case C would be /3 times as great as before, but so 

 would CE A , so that E A and all other averages would be 

 unaffected. The use of (6'2) to evaluate CE A (5 - 72) etc. 

 leads at once to expressions similar to (6*5), and so to the 

 results given formally in the last section and to others to be 

 given later. 



As we shall see, ■& is the temperature measured on a special 

 scale, and there is great advantage in regarding 3, rather 

 than E, as the independent variable which determines the 

 state of the assembty. If this is done the expression 

 ~E$ 2 (f>"/(f) can be put into simpler form. For it is easily 

 verified that 



Eayv<ft= ^* * + v 



(a- e -i) 2 ' (&-"-i) a ' 

 =*SIM»>=*§' ■ • ( 6 " 6 ) 



if E is regarded as a function of S, given by (5'S). 



It may be remarked that the constant occurrence of the 

 operator $d/d§ suggests a change of variable to log 3. 

 Though this has some advantages we have not adopted it, 

 partly because it makes the initial argument about the 

 multinomial theorem a little harder to follow and complicates 

 the contour of the integration, and also because log $ is not 

 itself the absolute temperature — if it had been, the physical 

 simplicity might have outweighed the other objections — but 

 only a quantity proportional to its reciprocal. 



Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 H 



