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LXVIII. On the Statistical Relations of Radiant Energy, 

 By G. H. Livens*. 



1. Introduction. 



THERE still appears to be some uncertainty as to the 

 principles on which the statistical and thermodynamical 

 relations o£ radiant energy are to be based, and as to the 

 final expression for the law o£ radiation which follows from 

 these principles. The following considerations are offered in 

 an attempt to elucidate the difficulties involved in the more 

 usual modes of formulation of the problem. 



Planck's method involves the consideration of an enclosure 

 filled with radiation involving an entirely arbitrary succession 

 of phases and polarizations along each ray, and also contain- 

 ing a system of fixed linear electric oscillators of the Hertzian 

 type, which are taken to represent the transforming action 

 of radiating and absorbing matter. The radiation contained 

 in the enclosure will be passed through these oscillators over 

 and over again, now absorbed, now radiated, and each con- 

 stituent will then settle down in a unilateral or irreversible 

 manner towards some definite intensity and composition, 

 which is determined by the density of the energy in the 

 sether and its distribution among the various wave-lengths. 

 Denoting now by E A <i\ this energy density as far as it refers 

 to the radiation with wave-lengths between X and \-\-d\ 

 then we know from the laws of Kirchhoff, Boltzmann, and 

 Wien, that E A is of the form 



T being the absolute temperature of the matter which is in 

 equilibrium of exchanges of energy with the radiation. The 

 actual form of the function <j> can only be determined by 

 probability considerations of the nature of those introduced 

 by Boltzmann in gas theory. In a system of Hertzian 

 vibrators, such as is discussed by Planck, with known pro- 

 perties and in given circumstances, there is a definite pro- 

 bability of the occurrence of each statistical distribution of 

 energy that is finally possible when all velocities consistent 

 with given total energy are considered to be equally likely 

 as regards each vibrator. The distribution of greatest pos- 

 sible probability is then, after Boltzmann, assumed to be the 

 state of thermal equilibrium for the system, and the proba- 

 bility of any other state is the usual function of the entropy 

 * Communicated by the Author. 



