42 Prof. Thomson, On the vibrations of atoms 



The changes in the periods of this oscillation are the same as 

 those of the single particle investigated by Lorentz. 



The effects on the other principal coordinates are more complex. 



Let p» + qi = -fz, p s + r 1 = (f> 2 , q 3 + r 2 = d 1 , 



p 2 -q 1 = -v|r 3 ', p 3 -r x = tfy 2 f , q 3 -r 2 = 0/, 

 Pi + q-2 + r 3 = s, 

 p 1 -q 2 = \ 1 , 

 p 1 -r 3 = \ 2 . 



Then the equation 



m 



d 2 yjr 3 

 alt 2 



= - ^1^3 



becomes when there is a magnetic field 



+^|.(fx+^-(i3+^))-^^(r 1 +? 2 -(?3.+r 4 )), 



or ™^=-kifs + ePhj t (h~ e O + W^- ea ij t (<l>*- &')• 



Now if the vibrations are only slightly disturbed we need only 

 retain on the right-hand side the variables whose free periods 

 are the same as that of yjr 3 , i.e. we need only retain the terms 

 in #! and </> 2 , hence we have 



d 2 yjr 3 7 , , n dOj 1 d6 2 



«Tp--**+W»- a ^i«-£. 



«■ -i -i d 2 <b 2 7 , i dylr 3 1 dd x 



Similarly m -^ = - k<f> 3 + \e* -^ - \ ey -^ , 



m dp ^ + ^y dt s*p dt • 



From the form of these equations we see that they differ from 

 those for a single particle in that the terms involving the magnetic 

 force are multiplied by the factor \, hence the effect of the 

 magnetic force on this vibration is only \ the normal effect, while 

 the effect on the vibration corresponding to p, q, r is as we have 

 seen normal. The magnetic field has, to a first approximation, no 

 effect on the vibrations of the coordinates s, X 1} \ 2 . 



