326 Mr. C. Gr. Darwin on the 



must be replaced by 



- i'2mrcc r - ->— - 2 (2»*r) 2 (1\ 



r ' 



This applies to both monochromatic and white radiation. 



11. The Scattering of a Single Atom. 



We must now discuss the form of the function /. 

 When a light-wave in which the electric vector is along x 

 falls on a system capable of vibration, a wave is scattered 

 which is greatest in amplitude in the yz plane, vanishes in 

 the line of x, and in any intermediate direction is propor- 

 tional to the cosine of the angle between that direction and 

 the yz plane. In this way we find that in f 2 (20, h) there 



l_j_ cos 2 20 

 will be a factor ~ — — due to the two polarized com- 

 ponents of the incident beam. This gets rid of the polariza- 

 tion, and we need only consider the form of / in a plane 

 perpendicular to the electric vector. 



The atom consists of a positive charge and of electrons, 

 but the former is much too heavy to scatter radiation and 

 may be neglected. Though there can be little doubt that it 

 does not represent the reality of the case, we shall proceed 

 according to the ordinary electromagnetic theory, as applied 

 to dispersion. In optics this gives satisfactory results, and 

 it should do so here as well. Let e, m be charge and mass 

 of an electron, and let the forces which hold it in equilibrium 

 have a " stiffness " mk 2 O 2 , so that the emission wave-length 

 is 2ir/k Q . Under the action of an electric force X the electron 

 moves according to the equation 



Then if X=e ik <- Ct - X ) we have 



t. <?X 



m(k s -k*)C 2 + %e*k 3 e 



At a great distance r in the plane of yz this gives a wave of 

 amplitude 



e*Xe-* r 7c 2 



m(k 2 -l<?)C 2 +%e 2 kHr 



(8) 



Tf we take this expression and add together the terms for 

 each electron in the atom, and substitute in (2) for the 



