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XLIV. On the Partition of Energy. By C. Gr. Darwin, M.A., 

 L.M.S., Fellow and Lecturer in Christ's College, Camb., 

 and R. H. Fowler, M.A., Fellow and Lecturer in Trinity 

 College, Camb.* 



§ 1. Introduction. 



AN important branch of atomic theory is the study of 

 the way in which energy is partitioned among an 

 assembly of a large number of systems — molecules, Planck 

 vibrators, etc. This study is based on the use of the principles 

 of probability which show that one type of arrangement is 

 much more common than any other. The most usual method 

 is to obtain an expression for the probability of any state 

 described statistically and then to make this probability a 

 maximum. This always involves a use of Stirling's approxi- 

 mation for factorials, which in many cases is illegitimate at 

 first sight, and though it is possible to justify it subsequently, 

 this justification is quite troublesome. It is also usually 

 required to find the relation of the partition to the temperature 

 rather than to the total energy of the assembly, and this is 

 done by means of Boltzmann's theorem relating entropy to 

 probability- — a process entailing the same unjustified approxi- 

 mations. 



The object of the present paper is to show that these 

 calculations can all be much simplified by examining the 

 average state of the assembly instead of its most probable 

 state. The two are actually the same, but whereas the most 

 probable state is only found by the use of Stirling's formula, 

 the average state can be found rigorously by the help of the 

 multinomial theorem, together with a certain not very 

 difficult theorem in the theory of the complex variable. By 

 this process it is possible to evaluate the average energy of 

 any group in the assembly, and hence to deduce the relation 

 of the partition to temperature, without the intermediary of 

 entropy. The temperature here is measured on a special 

 scale, which can be most simply related to the absolute scale 

 by the use of the theorem of equipartition, and we shall also 

 establish the same relationship directly by connecting it with 

 the scale of a gas thermometer. Throughout the paper the 

 analysis is presented with some attempt at rigour, but it will 

 be found that apart from this rigour it is exceedingly easy to 

 apply the method of calculation. Most of the results are not 



* Communicated by the Authors. 



