20 Prof. J. EL Jeans on the Interact! 



on 



scattered energy. By the result of § 3, F(q , q) mast be of 

 the form 



F ^ )= i*(fo'4 • • • • ( 8 > 



and so, by the conservation of energy, 



Jo Jo Wo J 9.0 



On integration this last is a function of c 2 only, so that 

 0(q ) does not depend on q , and may be replaced by 0. 

 The law of partition of the ^Yhole energy after unit time is 



(Wo0<M?o) + J^j <I>(%W%, Q)dq, 



or arranged according to frequency q, 



Ukw+i *(?,)%'/)%.]• • • (9) 



L J o -I 



If the radiation in the space is to be in temperature equi- 

 librium with the electron, the partition of energy must be 

 unaltered by the interaction between the electron and the 

 radiation. The final partition of energy (9) must accord- 

 ingly be identical with the initial partition of energy \ dqcj)(q). 

 Thus we must have 



/ICO 



00(2)+ I <K?o)F(2o, q)dq = cb{q), 

 Jo 



or by equation (8) 



$(q)[±-0] = f$(qo)®(^,c^\ . . (io) 



Jo V?o / c lo 



and the partition of energy required, $(q), is the solution of 

 this integral equation. 



The equation may be written 



or, if q = uq, 



v — 



so that the ratio of 4>(uq) to cj)(q) must be independent of q. 



