Energy Distribution for Natural Radiation. 797 



the total energy represented by (iii.) will be 



§J Q {[%i(P)] 2 +toO>)] 2 }^. . . (xiii.) 

 From (iv.) and (v.), by squaring and adding, we obtain 



[%iO)] 2 + [% 2 (p)] 2 -f 2 Xi (p) x*(j>) cos (co 1 -<o 2 ) = [/(p)] 2 . 



.... (xiv.) 

 We may take o>i and w 2 as having all values between and 

 27r as being equally probable. Hence averaging (xiv.), we 

 may take 



Thus (xiii.) becomes 



Iffjo ^W*- 



We may also take it that the average magnetic energy in 

 the waves is equal to the average electrical energy. Thus 

 the total energy due to the waves represented by (ii.) is 





I£ and e depend onp, we must write instead o£ (xv.) 



J- Pcf [/(/>)]». «%», .... (xvi.) 



which then represents the amount of energy per volume OT 

 stored up in the radiation given by (ii.). It we consider the 

 effects of* Y and Z, we shall have to multiply expression (xvi.) 

 by three, for homogeneous radiation. 



§ 5. In the assumption represented by (iii.), we might 

 have introduced terms involving waves moving parallel to 

 0~, in addition to those moving parallel to Oy. The final 

 result is, however, the same, and we have not therefore 

 considered these terms. 



Another point to be noticed is this. The function f(p) is 

 a function of T, the time-interval in forming (ii.). Thus 

 [/(p)] 2 /T must be a function of the temperature, as (xvi.) 

 indicates, if there is a definite distribution of energy amongst 

 the wave-lengths. 



§ 6. It seems natural to ask what is the effect of intro- 

 ducing damping terms e~ apy/c in (vii.), e~ aQV /C in (viii.), where 

 a is a coefficient of absorption which is ultimately to be 

 taken as vanishing. The integral which arises corresponding 

 to (x. a) is interesting from the point of view of the pure 



