﻿the Partition of Energy. 461 



The latter can also be written as 



{-Mjrgglog (!-<«) }(l-*«)-*(l. 



If these expressions are now expanded in powers of z by 

 the binomial theorem, they give a sum of products of factorials 

 which are, of course, the " homogeneous product " expressions 

 used by Planck. It is possible to approximate to these by a 

 legitimate use of Stirling's theorem and to replace the sums 

 by integrals without much difficulty. It would, indeed, have 

 been possibly to start from these expressions, but we have 

 not done so because in the general case to be discussed later 

 that method would not be available. To make further 

 progress* by a method of general utility, we discard Stirling's 

 theorem and express these coefficients of z® by contour 

 integrals taken round a circle y with centre at the origin and 

 radius less than unity. By well-known theorems on in- 

 tegration * we have at once 



c 



i r dz i . 



= 2mUz E+1 (l-z e ) M (l--z r i)K> ' ' (5 ' 71) 



d 



-Ms-T-logd-s 6 ) 



— 1 C d~ ~ dz & K J 



CEA= 2^-J y ^i-(1_^M (1 _^ '• ' ( 5 * 72 > 



AVe can no longer hope for the single-term formulae of 

 §§ 3 ; 4. But (5-71) (5-72) are exact, and when M, N, E are 

 all large in any definite fixed ratios, we can make use of the 

 method of steepest descents to obtain simple adequate approxi- 

 mations. The method is very powerful and can be applied 

 in a great number of cases without difficulty. Moreover, it 

 is comparatively easy to use it with mathematical rigour if 

 that is desired ; and thus the somewhat clumsy calculations 

 in the usual proofs of partition theorems are entirely avoided. 



In general terms the process is this. Consider the in- 

 tegrand on the real positive axis. It becomes infinite at z = 

 and again at z=l, and somewhere between at z = § there is a 

 minimum which is easily shown to be unique. Take as the 

 contour the circle with centre at the origin and radius S 

 passing through this minimum. Then we find that, for 



* For those not familiar with these theorems we may remark that 



5— -. 1 2 r <7z=0 when r is any integer other than — 1, while r> — ' 1 — =1*;. 



these equations at once give (5'71) and (5 - 72). 



