﻿458 Messrs. C. Gr. Darwin and R. H. Fowler on 



The same methods give the fluctuations of a r . For 



(a r — a r ) 2 — a r a r — 1) -f a r — (a r ) 2 , 

 and a process similar to the above gives 



0-^?^=T) = M(M - X) 



The exact expression for the fluctuation can be at once put 

 down. When M and P are taken large the leading term 

 cuts out, and so it is necessary to carr j the approximation to 

 the next order. If we substitute 



P!/(P-r)I--P^-ir(r-I)F- 1 , 



we find that 



\(Xf (Jby J — 



£ { l ~ (M + Pr* [2P 2 -(2?-l)MP + r*W] j . (4-6) 



The formula for a r can be put into a more familiar form by 

 the substitution P = M/(^ a — 1), which gives 



~a r = Me- ra (l — e- a ), . . . . (4 ! 7) 



and leads to a corresponding but more complicated ex- 

 pression for the fluctuation. Here, as we shall see later, ol 

 can be identified with the familiar e/kT., Equation (4* 6) 

 establishes at once that the statistical state specified by (4*5) 

 or (4* 7) is a normal property of the assembly. 



§ 5. The Partition of Energy among two Sets 

 of Planck Vibrators. 



After these preliminary examples we now apply our method 

 to a problem which will bring out its distinctive character, 

 that of the partition of energy in an assembly composed of 

 different types of system. We shall consider first the simplest 

 of such cases — an assembly consisting of a large number of 

 Planck vibrators of two types A and B. The number of A's- 

 is M, and the energy unit of an A is e as before. There are 

 now also N B's with energy unit tj. To make exchanges of 

 energy possible we have to suppose, say, that there are 

 present a few gas molecules, but that the latter never possess 

 any sensible amount of energy. (Later on in § 12 we develop 

 a method by which we shall be able to include any number 

 of such molecules in our assembly.) We also require for 

 the purposes of the proof to assume that e and rj are com- 

 mensurable, but it does not matter how large the numbers 

 "may be which are required to express the ratio e/rj in its 



