﻿454 Messrs. C. G. Darwin and R. H. Fowler on 



total number of all possible weighted complexions. This 

 part of the problem depends on the nature of the particular 

 assembly considered, and so must be treated separately in 

 each case. 



We start in § 3 with a problem which concerns not the 

 partition of energy, but the distribution of molecules in a 

 volume. It illustrates the method in its simplest aspect and 

 has the advantage of being purely algebraic. Next, in § 4, 

 we take the distribution of energy among a set of similar 

 Planck vibrators, which is again a purely algebraic process, 

 and then proceed in § 5 to introduce the main theme of this 

 paper by dealing with the partition of energy between two 

 sets of Planck vibrators of different type. This is most 

 conveniently treated by using the complex variable, and in 

 § 6 there is a discussion of the required theorem. The par- 

 tition of itself introduces the temperature, and in § 7 the 

 special scale is compared with the absolute. In §§ 9, 10, 11 

 the partition law is generalized to more complicated types of 

 system, such as the quantized rotations of molecules. In 

 §§ 12, 13 the method is extended so as to deal with the free 

 motion of monatomic molecules, intermixed with vibrators. 

 The work leads to a rather neat method of establishing the 

 Maxwell distribution law. 



§ 3. The Distribution of Molecules in Space. 



The first example we shall take is not one of a partition of 

 energy, but of the distribution of small molecules in a vessel. 

 It illustrates in its simplest form the averaging process, 

 and has the advantage of depending only on elementary 

 algebra. 



Let there be M molecules, and divide the vessel into m 

 cells of equal or unequal volumes r l5 v 2 ... v m , which may 

 each be as large or as small as we like. Then 



Vi + V 2 + ... +v m = V (3-1) 



By well-known arguments which we need not consider, it 

 follows that any one molecule is as likely to be in any element 

 of volume as in any other equal one. So by a slight ex- 

 tension of the idea of weight we attach weights v h v 2 , ... v m to 

 the cells. To specify the statistical state we say that the 

 first cells has a x molecules, the second a 2 , and so on. Then 



a 1 + a 2 -\- ... +a m = M (3*2) 



By the theory of permutations the number of complexions 



