﻿the Partition of Energy. 457 



■expanded by the multinominal theorem. The typical term is 



a ! a x ! a 2 \ ...~ 



where the as take any values consistent with (4*1). Then 

 if we pick out the coefficient of z F , we have the sum of all the 

 expressions for which the a's satisfy both (4*1) and (4*2). 

 Observe that we may take the whole infinite series because 

 the later terms are automatically excluded. 



Now this will be the coefficient of X s in (1—z) ~ M , and so 



(M + P-l)! a . 4 . 



°- (M_i)!P! ' ^** J 



which is the ordinary expression for the number of homo- 

 geneous products as formerly used by Planck *. 

 We next evaluate the average of a r ; 



M! 



a ! a l ! a 2 1 . . . 



-MS, ( M -^ ! 



a ! a l ! a 2 I . . . 



where X a > denotes summation over all values satisfying 

 «o' + «i / + « 2 / + «3 / --- =M — 1, 



a/ + 2a 2 / 4 3a 3 / ... = P — r. 

 The sum is thus 



M (MtP-r-2)! 

 (M-2)!(P-r)!' 

 and we have 



q-M(M n ( M + P-r-2)! P! 



This is exact, and holds for all values of r ; now r can have 

 any value up to P and the majority of the a r 's will be zero. 

 The ordinary method of proof applies Stirling's formula for 

 a r ! to these zero values. In the important case where both 

 M and P are large, it will be only necessary to consider 

 values of r which are small compared with P. Now, if r 2 is 

 small compared with P, P!/(P — r)! has the asymptotic 

 value P r . Using relations of this type and also disregarding 

 the difference between M and M — 1, we have 



M 2 P r 



"*= (M+ty+i (4 ' 5) 



* M. Planck in the earlier editions of his book on Radiation. 



