832 Mr. U. Doi on Scattering 



\f — =n , we obtain, for the general solution of the above 

 v m 



differential equation 



f = - -2 7 M 2, + Ae< "" t + Be ' i "" t > 



3 in ntf — rr — arie'*/ m 



in which A and B are arbitrary constants of integration. 



We have to observe at this point of discussion that there 

 is every reason to suppose that within a medium the radia- 

 tion and absorption due to free vibrations of the electrons 

 compensate each other, so that in consideration of the present 

 problem of scattering of light incident on them from an 

 external system, we may naturally put the terms of free 

 vibrations out of consideration, limiting our attention solely 

 to the part of forced vibration, 



z = e_ X e int 



m n 2 — n 2 — aXe 2 /m ' 

 Differentiating f twice as to time, 

 - e n 2 X e int 



mn 2 — n 2 — aNe 2 hn 

 The amount of radiation corresponding to this acceleration is 



e n 2 X ( - in ' 



677c 8 6-7rc 3 \ mn 2 — n 2 —aXe 2 jm) 



~~ ^irc z m 2 {n^—n 2 —aXe 2 jm) 2 ' 



This energy of radiation must be consumed from the 

 incident energy, which results in scattering after all. Take, 

 now, a thin lamina of the medium, of a thickness dx and 

 cross-section A, perpendicular to the incident ray of light. 

 The number of scattering electrons contained within this 

 elementary lamina will be NA . d%, so that the amount of 

 energy radiated from this lamina during a time-interval / is 



NA.^P-^P^ 

 Jo b7rc 



= NA.d# ; A .Q i e 2 ™tdt 



67rc 3 m 2 (n 2 — n 2 — aNe 2 lm) 2 J 



But the energy incident on this lamina during the same 

 interval of time is * 



I = A.C.CX 2 € 2int dt 



= A.cX 2 C e 2int dt. 

 * Cf. 0. W. Richardson, he. cit. p. 211. 



