﻿the Partition of Energy. 469 



Finally it is of some interest to point out that we can 



obtain a formula for (Ea— "Ea) & of general validity. We 

 have in fact 



(E A -E A ) 2 * = 1.3...'(2s-l){(E A -E A ) 2 }*, . ($•$) 



where (E A — E A ) 2 is given by (8*7). We retain of course 

 only the highest order term *, which is thus 0(E A )*. 



§ 9. Generalization to any number of types of system, and 

 to systems of any quantized character. 



It is clear that the present method of treating partitions 

 is of a much more general character than has so far been 

 exhibited. Consider an assembly composed of two types, 

 A and B, of quantized systems more complicated than Planck 

 vibrators. We suppose generally that the systems of type A, 

 M in number, can take energies to the extents e , e 1} e 2 , ..., 

 and these states have weight factors po^Pi->I ) 2^ ...in conformity 

 with the discussion in § 2. Similarly, the B's, N in number, 

 can take energies rj , 77^ tj 2i ... with weights q , g^ q 2 , 

 We have to suppose that it is possible to determine a basal 

 unit of energy such that all the e's and n's can be expressed 

 as integers. Further, it simplifies the work if we suppose 

 that there is no factor common to all of them. Proceeding 

 exactly as before, we set down the weighted number of com- 

 plexions which correspond to the specification that, of the A's, 

 a r have energy e r ; of the B's, b s have energy ij s . This 

 number is 



a ,. , wv • • • • b , ; ?«v . . . ., • (9-i) 



a Q i a 1 i . . . o ! o 1 : . . . 



and the a's and 6's are able to take all values consistent with 

 \a=U, 2A = N, S r ^+S^A=E. . (9-2) 

 Now form the functions 



f(z)=p a z e « + /h ^+p.^ +...., . . . (9-3) 

 9 (z)=q z"° + q^+q,^+ (9-31) 



These will be called the partition functions f of the types of 



* Cf. Gibbs' < Statistical Mechanics,' p. 78. But (8*8) is generally 

 valid, while Gibbs' formulae really refer only to a group of systems in a 

 temperature bath. 



t They are practically the " Zustandsumme '' of Planck, ' Radiation 

 Theory/ p. 127. 



