llieory of Radiation. 53 



third terms in (29) are not zero. There is no longer equi- 

 partition of the radiation, but the further question now arises 

 whether the distribution can be such as is at least empirically 

 represented in Planck's formula. Instead of (32), (29) leads 

 to the formula 



J£ x ^b7rmx-^ l ^ lu + ^ /l) )-- 1 (33) 



<& m = \ §m J (f), u{t -e) COS (c/C lH €)dedX, 



° r <£„= f(\,H)e- H '™dH, (34) 



Jo 



or y m =0 p 9 (X, H) e-*i™dH. (35) 



(33) suggests aformula of the same type as Planck's. 



Let ° f(\,H)=f(\H) (36) 



g(\,E)=XF(\H) (37) 



So that /(Xii) and ¥{\B) are functions only of the product 

 of wave-length and energy. The ordinary mechanical 

 theories in which radiation is explained as due to the acce- 

 lerated motion of electrons, make / (XH) contain some such 

 factor as e~ a)<H , so that the function has an essential singu- 

 larity and vanishes when ^diminishes to zero. 



I assume thnif(XH) and F (XH) both behave in this way 

 for the zero value of XR, and further that a series forF(Tu^) 

 exists such that 



*w=E ai 'd(?Hy J(xH) -- • • (38) 



Note that / {XH) and all its differential coefficients vanish 

 when XH vanishes. 



/"iCO 



% }l = Xd\ Y {\H)e- H ' lie dH, 

 Jo 



E x =87rKa- 1 {l+x/J(-l)% 1 p^y,} . • (39) 



