On the Statistical Theory of Radiation. 351 



Thus, finally, for the distribution of energy among the parts 

 of the system we have the formula (Planck's) 



The argument of Prof. Wilson is that E 1 (=n 1 e l ) as thus 

 determined cannot be independent of the size of the energy- 

 element e l5 because e l is the only variable that enters except 

 N l5 which measures the extent of the system, so that any 

 change of e L must change the value of E ]? even though e^i 

 is kept constant : for example, if e x is taken very small, the 

 formula becomes 



E^N^T, 



which represents the law of equipartition. But this un- 

 welcome conclusion is evaded simply by recognizing that the 

 value of k must be some function of the size of the energy- 

 element which is taken as the basis of the statistics ; it would 

 indeed be strange if it were otherwise. If key remains finite 

 as ei diminishes, the equipartition is not attained unless T is 

 very great. We shall find that it is ke± that is to be taken 

 as constant when e l9 the statistical element for any given type 

 of energy, is changed. 



The two independent constants in the formula are in fact 

 N^, and ke x . Their ratio N^ -1 is equal to the gas-constant. 

 That universal quantity, and N^ (say a) which is the ratio 

 of the energy-element to the extent of a cell, are what affect 

 the distribution and are thus of pre-determined values ; but 

 there seems to be nothing that demands a definite magnitude 

 of the energy-element itself. 



On the Boltzmann form of the theory of probability of 

 distributions of energy among the molecules of gases, k 

 turned out indeed to be the gas constant. On the present 

 form of theory, which involves distribution of elements of 

 disturbance with their appropriate energies in the containing 

 system as mapped out into cells, instead of mere collocation 

 of elements with regard to one another, this conclusion need 

 not hold. We may probe this point further. It is known 

 as a fact that, under ideal conditions, equable partition is 

 very nearly attained as regards the translatory and rotatory 

 parts of the energy of the molecules of a gas. This requires 

 that, if € r is the value of e corresponding to each of the 

 translatory or rotatory types of freedom, it must prove to be 

 so small compared with e 1? e 2 , ... that the exponent ke r \T is 



