﻿478 On the Partition of Energy. 



and let its total amount be Gr. The method now requires 

 the averaging process to be applied to expressions depending 

 on 



P! 



W 2 , 



where we now have not only 



but also 2, t c t fi t =Q, 



where fi t is the momentum in the given direction of a mole- 

 cule in the tth cell. To sum the appropriate expressions we 

 must take as our partition function 



With this function C will be the coefficient of z a G in 

 [h(z, #)] P , and this can be expressed as a double contour 

 integral. So can the other averages, and the usual asymptotic 

 expansions can be found. The correct distribution law 

 follows on replacing h(z, x) by the integral which is its limit 

 when the sizes of the cells tend to zero. This subject lies 

 rather outside the theme of the present paper and need not 

 be elaborated further. 



§ 16. Summary. 



The whole paper is concerned with a method of calculating 

 partitions of energy by replacing the usual calculation, which 

 obtains the most probable state, and is mathematically un- 

 satisfactory, by a calculation of the average state, which is 

 the quantity that is actually required and which can be found 

 with, rigour by the use of the multinomial theorem together 

 with a certain theorem in complex variable theory. 



After a review of principles and two preliminary examples 

 ihe real point of the method is illustrated in § 5. Here 

 there are two groups of interacting Planck vibrators of 

 different types. It is shown that the partition can be found 

 by evaluating the coefficient of a certain power of z in an 

 expression which is the product of power series in z. This 

 coefficient can be expressed as a contour integral and can be 

 evaluated by a well-known method, the " method of steepest 

 descents." The result expresses itself naturally in terms of 

 a parameter 3 which is identified with temperature measured 

 on a scale given by ^~e~ llk . 



The work is extended to cover the partition among more 



