due to the Scattering of Light by Electrons. 715 



Let us now consider a plane wave travelling in the 

 positive direction of the axis of a?, let /(a*, t) be the value of 

 the electric force parallel to Z at the place x and the time t. 

 Consider now how the force at the point x + Sx, at the time' 

 t + hxjc, differs from that at x x and t v The force which is 

 at x at the time t will be found at # + d#,at the time t + Sxjc, 

 and in addition to this force there will be found at x + hx 

 the force due to the secondary waves which come from the 

 electrons in the slab 8x. This force, as we have just seen, 

 is equal to 



2tt (dz\ 5 

 - o^Kdtr' 



Hence we have 



f(x + fr, t + &r/«) =,/■(.!-, t) - 2 " t* (§) Bm. 



This equation is equivalent to 



df 1 df 2tt [dz\ /n . 



If the equation of motion of an electron in the slab is 



and if /varies as e lpt , then 



._ . ef 



m(n 2 —p 2 ) ' 

 and equation (2) becomes 



df ldf_ _2tt e 2 a df 

 dx c dt ~ c m(n 2 —p 2 ) dt 



f+-,f=0, (3) 



ax & dt 



(/ c c m ?i 2 — p 2 



The solution of (2) is 



f=<p{x-c't), 



representing a disturbance propagated with a velocity v. 

 1 i' yu is the refractive index, 



hence bv (4") 



1 2ire 2 a 



/*=! + 



m 



m ' (n 2 — p 2 ) 



3 A 2 



