716 Sir J. J. Thomson on some Optical Effects 



This is the well-known expression given by Lorentz. If 

 the electrons have not all the same frequency, then if 07 is 

 the density of those with the frequency n s , we can show 

 without difficulty that 



m n/—p z 



This expression has been used by Drude and others to- 

 determine the number of electrons in the atoms. In using 

 it for this purpose, however, it is necessary to take account 

 of some considerations which may be illustrated by the 

 following example. Let us consider the case of an electron 

 in an atom of such a character that the displacement of the- 

 electron b} r an electric force is not in general in the direction 

 of the electric force. There will be three directions fixed in 

 the atom such that a force along one of these directions 

 produces a displacement in the same direction. If these 

 directions are taken as the axes of f, 77, f, the equations of 

 motion of the electron may be written 



where F^, F^, F f are the components of the electric* force in 

 the directions f , r), f respectively. Let|the axes of f , r„ £* 

 cut a sphere at the points A, B, C, and the axis of % at the 

 point Z ; then, if Z. ZC = 0, and <£ is the angle between ZO 

 and CA, an electric force Z in the light wave will be 

 equivalent to the forces 



F^ = Z cos 0, F^Z sin 6 sin <f>, F^ = — Z sin 6 cos </>, 



and if Z varies as € pt , we get from the equations of motion 

 of the electron 



e Z sin cos c/> __ e Z sin 6 sin $ 



m n 2 —p 2 ' m n 2 2 —p 2 ' 



e Z cos 6 

 m n$—p 2 ' 

 The displacement resolved along the axis of z is thus 



/ cos 2 6 sin 2 6 sin 2 (f> sin 2 6 cos 2 <j>\ 

 W-p 2 + ~~n 2 2 -p 2 + n x 2 -p 2 ) ' 



