﻿the Partition of Energy. 459 



lowest terms. To avoid introducing new symbols, we may 

 suppose that the unit of energy is so chosen that e and r\ are 

 themselves integers without a common factor. 



We have already introduced the idea of weight, and seen 

 that we must assign the weight unity to every permissible 

 state of a linear vibrator. To calculate the number of com- 

 plexions of the assembly of any given sort, we have merely 

 to calculate the number of ways in which the energy may 

 be distributed among the vibrators, subject to the given 

 statistical specification. A simple example will make the 

 process clear. 



Let there be two A's and two B's; let ^ = 2e, E = 4e. 

 Then the possible complexions are : — 



{aaaa 

 r i r 

 a a a < 



( aaaa' 

 I aa'a'a 



aab 

 aab' 

 a'ab 

 Wa'b' 



job 



\aa!b 



\aa'V bb' 



Here, for example, aaaa' means that there is 3e of energy 

 on the first of the A's, e on the second, and none on the B's. 

 Each of the fourteen complexions is, by definition, of equal 

 weight, and is therefore to be reckoned as of equal probability 

 in the calculation of averages. Observe how with the small 

 amount of energy available a, good deal more goes into the 

 smaller than into the larger quanta ; for the pair of A's have 

 on the average ye, as against ye for the pair of B's. 



We pass to the general case. The statistical state of the 

 assembly is specified by sets of numbers a r , b s where a r is the 

 number of vibrators of type A which have energy re, and b s 

 the number of B's with energy stj. All weights are unity 

 and the number of complexions representing this statistical 

 state is the number of indistinguishable ways (combinations) 

 in which M vibrators can be divided into sets of a , a Y ... and 

 at the same time N into sets b , b ± ... . As illustrated by 

 the example, it is therefore given by the formula 



M! N! 



o !a l la 1 !...* a !6 l !W..." * * * ( ; 



In (5"1) a r and b s may have any zero or positive values 

 consistent with the conditions 



2 r a r = M, SA = N, 2 r r6a r + X s s V b s = E, . (5'2) 



where E is the total energy of the system — necessarily an 



