352 On the Statistical Theory of Radiation. 



also small ; for that is needed in order to lead to this law of 

 approximately equable partition, in the form 



E r =it^ r =N f ir 1 T. 



In this special result the value of the element of energy 

 € r has become eliminated. Also N,-& -1 must be the gas con- 

 stant R ; and since N r € r must be a, another universal constant, 

 we have &e r =a/R. Hence in this case of simple gas-theory 

 the value of k should be inversely as the scale of magnitude 

 of the elements of energy chosen ; and the size of a standard 

 cell should be directly as that element. And this result 

 must be universal. 



Thus the conclusion is, briefly, that to render the entropy 

 independent of the scale of minuteness of sub-division of the 

 statistics, as is natural, we have only to define it as k log W, 

 where the value of k (if we decide to retain it in the formulas) 

 must vary directly as this amount of sub-division, or inversely 

 as the scale of sizes of the elements of energy that are 

 employed in the analysis. But, on the other hand, if k had 

 the same value whatever be the scale of the statistics that is 

 adopted, conclusions such as those of Prof. Wilson regarding 

 the magnitude of the ultimate element of energy would 

 necessarily follow. 



To connect formally the values of e, thus demanded by 

 experimental knowledge for gas-theory, with those that 

 obtain for the types of radiant en erg}-, would involve a rather 

 long argument. But the present type of theory works out 

 for the domain of radiation as above, and it is readily seen 

 that it works out for the domain of gas theory on the ordinary 

 lines as indicated in the paper referred to; while a bridge 

 can be constructed between the two, as there suggested, by 

 noting that both for translatory and rotatory motions in gas- 

 theory and for radiation of long wave-length, the principle 

 of equipartition is practically effective, so that we may take 

 advantage of Prof. Lorentz's train of ideas connecting these 

 equipartitions by a calculation of the amount of the natural 

 radiation from a thin metallic plate, considered as arising 

 from the collisions of the moving free electrons that are 

 required by its electric conductivity. 



The existence of another universal physical constant (a), 

 in addition to that of gas-theory, has been postulated without 

 any explanation as yet. But its existence is independent of 

 these statistical theories ; and it thus seems to have come to 

 stay in some form or other. In fact it was early pointed out 

 by Wien and by Thiesen that the value A™T, where \ m is the 



