Statistical Relations of Radiant Energy. 659 



involved in the discussion have different extents or oppor- 

 tunities, then equi-partition can no longer exist in the steady 

 state of thermal equilibrium. 



5. Applications to the problem in radiation. 



We cau now apply the results just deduced to the funda- 

 mental problem of radiation, either in the method of Planck 

 or of Jeans and Rayleigh. 



The analysis given by Planck* still remains valid for 

 determining the energy distribution in an elementary con- 

 stituent of the radiation capable of being resolved by a 

 Fourier analysis, so that the energy in it corresponding to 

 the oscillations with wave-lengths between 



is, after him, given by 



8\s Sx s \ 

 T' Xs+ ^~) 



X, 4 1 



F-l 



The energy in this part of the total radiation is, as above 

 explained, to be obtained by the superposition of a large 

 number Nx« of such elementary amounts, and is, therefore, 



™ £. cb\* Nx,e 



which, in the limit, since N\ s = — is of the form 



E s cd\ s 7] 8 



X s oX s = 



which is Planck's formula for natural radiation if 



he 



a form which is in fact suggested by the considerations 



* See his book ■ Vorlesungeii iiber die Theorie der Warmestrahlung ' 

 (Leipzig, 1906). 



