﻿the Partition of Energy. 841 



advance the size of the units of energy in which the 

 expansion is to be made. 



The problem is exactly analogous to that solved by 

 Poincare * in his deduction of the necessity for quanta 

 from the fact that Planck's radiation formula agrees with 

 experiment. The machinery required has been examined 

 by one f of us in a recent paper. We give here a sketch 

 of how it may bo applied to the present problem : reference 

 must be made to the original works for further detail. If 

 we write t = 1/£T, we may suppose that the partition function 

 is known in terms of t. The relation of the partition 

 function to the weights and energies from which it is 

 generated may be put in the form of a Stieltjes' integral: — 



/(r) = 1 e- Te dw{e). 



Here dw{e) represents the weight corresponding to e, and 

 it is indifferent whether we are concerned with quantized 

 systems or mechanical ones with continuous distributions 

 of weight. The function w(e) can be determined by an 

 extension of the method of the Fourier integrals, which 

 (roughly speaking) leads to 



iv (e) 



1 r a+i " , s dr 



where a can have any positive value above a certain limit. 



This is a complete solution, but it requires that / should 

 be known for complex values of t, and in practice it would 

 be given in the form of a table, of course for real r only. 

 In general it would not be possible to find a simple analytic 

 expression to fit with the tabular values. This difficulty 

 can, however, be turned J, so that only the practical difficulty 

 of carrying out a large number of mechanical quadratures 

 would remain. For it is possible to associate with the real 

 function /(t) a complex function 



/ICO 



J(flO = j f(r)T*- l dr, 



where q may have complex values, and then 



W{ < ShriJ.^. r(q) q 

 This is the formal solution of the problem ; but it must be 

 doubtful whether it is really a practical method. 



* Poincare, Journal de Physique, ser. v. vol. ii. p. 5 (1912). 

 t Fowler, Proc. Roy. Soc. A. vol. xcix. p. 462 (1921). 

 % Fowler, loc. cit. § 5. 



Phil Mag. S. 6. Vol. 44. No. 263. Nor. 1922. 3 I 



