394 Sir J. J. Thomson on the Scattering of 



can be displaced along any one of them without causing 

 displacements along the other two. 



Suppose that when the electrons are displaced by 

 distances £, 77, f parallel to OA, OB, OC respectively, the 

 restoring forces are respectively A£, B?7, Cf ; then, if 

 F l5 F 2 , F 3 are the forces acting on an electron in thes^ 

 directions, the equations for £, 77, f are respectively 



&£ A y t? 



m 



d 2 7) 



} + B v = F 2 , 



"dt 2 





If the applied forces vary as e lpt the solutions are 



*- -A- «- F *- r = Fs - 

 * A-mp 2i ' B-mp*' * C-mp 2 ' 



When A, B, C are small compared with mp 2 , 



-E\ _-F 2 „_ -F 3 



wi/r mp z mp z 



and 



i. jl_! 



F, _ F 2 - F 3 - 



The electron is thus displaced along the direction of the 

 force and the usual theory will apply. Thus, as long as 

 the frequency of the incident vibrations is large compared 

 with the free frequencies of the electrons, the system of 

 electrons will behave as if it were quite symmetrical and 

 the light will be scattered in accordance with the usual 

 theory. When however A, B, C cannot be neglected in 

 comparison with )np 2 , we see that unless A = B = C the 

 direction of the displacement will not be that of the force 

 and the theory requires modification. 



Let A, B, be the points where the axes OA, OB, OC 

 of an atom cut a sphere of unit radius ; X, Y, Z the points 

 where this sphere is cut by three fixed axes OX, OY, OZ : 

 let us calculate the displacements along these axes of an 

 electron acted on by a force Z parallel to OZ. 



