﻿the Partition of Energy. 825 



functions -which form the basis of our method. In § 8 

 the definition is considerably simplified mathematically by 

 replacing the "entropy" by the "characteristic function" 

 as the basal thermodynamic quantity. In § 9 we show 

 that for an assembly in a temperature bath our method 

 is exactly equivalent to Gibbs' use of his "canonical 

 ensemble," and in § 10 we consider briefly the possibility 

 of inverting the argument so as to obtain information 

 about the elementary states from thermodynamic data. 



§ 2. Summary of our previous paper. 



It was shown in the previous paper how the partition 

 of energy could be evaluated for various types of assembly. 

 Those chiefly treated were quantized systems, for which the 

 energy was sole variable ; but it was also shown how to 

 apply the method in the case of a perfect gas, where both 

 energy and volume are variable. It is easy to see that the 

 method is applicable in considerably more general cases. 

 The partition of energy could be evaluated when any of the 

 types enumerated were mixed together, the essential point 

 of the method being the existence of a certain function, the 

 partition function, for each type of system. By means of 

 these functions all the rather tiresome combinatory ex- 

 pressions can be very easily dealt with so as to obtain mean 

 values, and also the fluctuations about those mean values. 



The partition function of a system — which with a different 

 notation is the " Zustandsumme " of Planck — is constructed 

 as follows. The possible states of the system may be divided 

 into cells ; these cells are fixed and finite for quantized 

 systems, but for the systems of classical mechanics must 

 ultimately tend to zero in all their dimensions. Associated 

 with each cell is a weight factor, determined by the usmil 

 statistical principles. The weight of any cell of a system 

 obeving Hamiltonian equations is proportional to its ex- 

 tension. The relative weights of the cells of a quantized 

 system are determined by Bohr's Correspondence principle, 

 and the weights are all assigned definite magnitudes by 

 the convention that a simple quantized cell shall have unit 

 weight. For consistency in physical dimensions the cells 

 for Hamiltonian systems are divided by the appropriate 

 power of h to give their weight. Associated with any cell 

 there is a definite energy, depending on the cell and on 

 certain external parameters .t u x 2 , . . . ; this last is a slight 

 extension of our previous paper, which must be made so as 

 to deal with questions of external work. Then, if p r is the 



Phil. Mag. S. 6. Vol.44. No. 263. Nov. 1922. 3 H 



