

Statistical Relations of Radiant Energy. 657 



chances, each weighted, however, in a manner corresponding 

 to the particular set of coordinates to which it refers, is 

 proportional to the number of ways X, in which this statis- 

 tical distribution can be realized, which is of the form * 



X=f(n, p) ./,(«!, p{) f r (tt r , p r ), 



wherein each separate number is weighted properly, so that 

 /. (n„ p>) = [/(«•, Ps) ] (#= 1, 2, 3, ... r) f. 



In the natural distribution this must be a maximum subject 

 to the restriction 



E 



p+pi+p*+ ••• +iv=7' 



which requires 



f (n,p) __ /.' (n„p 8 ) _ 6 B/; 



ffap) "/("•> P.) ~ d /S ~^' 



for all values of 5, where is an undetermined quantity, the 

 same for all sets of coordinates. The usual argument de- 

 rived from ordinary considerations of gas theory shows that 

 if the first type of coordinate is the ordinary molecular type 

 then $ = kT, T being the absolute temperature and h the 

 usual absolute constant. 



The value of f(n, p) is the number of modes of distribu- 

 tion of p like objects in n like compartments and, following 

 Planck, this is 



(w + p-1) ! 

 (n-1)! pV 



which when n and p are very large numbers, by application 

 of Stirling's approximation, assumes the form 



J \ > PJ n npp ' 



so that , n + p . n 8 + p 8 e 



log — = a .log— =Z j, 



or ne n s € 



P € = ~ » Ps£= — • 



* See J. Larmor, Proc. Eoy. Soc. A. Ixxxiii. p. 82 (1909). 



t In applications where the number r is infinite no meaning can be 

 attached to the expression for X unless the infinite product on the 

 right is in some manner convergent. This implies convergence of the 

 series 2«s which leads to the conclusions adopted above. 



