﻿632 Dr. C. V. Burton on the Scattering and 



This will enable ns to make use of the formula (11), (12) 

 provided only that the value of v conforms to the restriction 

 already indicated ; the vibrators in the lamina dx' being 

 sensibly a plane distribution for which cr = vdx'. Writing 

 Xdat in place of k in (12) we have 



2tti/ sin 7 .,, N 



expressing, as to intensity and phase, the relation between 

 the resultant plane waves incident on the lamina dx' and the 

 plane waves emitted by the vibrators contained in that 

 lamina. 



17. But even if v is too great to allow of a gas-like distri- 

 bution of vibrators, so that (13) no longer holds good, the 

 homogeneousness of the swarm of vibrators still leads to the 

 conclusion that % is a (complex) constant ; or, in other 

 words, that the waves emitted by an elementary lamina dx 1 

 have an amplitude proportional jointly to dx' and to the 

 amplitude of the resultant incident plane waves, with a phase 

 differing from that of the incident waves by a constant. 



18. Understanding, then, that v is quite unrestricted, let 

 the waves originally incident on the slab 0<c2x:L be repre- 

 sented by (10;, and let the waves given oat by the lamina x' 

 to x' + dx' be 



B'expi{pt— v(x—a?)}dx\ B'ezp i{pt+v(x— w')}dx', . (14) 



where B' is a function of x', and is in general complex. At 

 any plane x — x'\ for which x" lies within the limits 0, L 

 the total disturbance arriving is 



&exipi(pt—vx")+\ B'exp i{pt— v(x"— rf)\da! 



j; 



B'exp i{pt + v(x' f -x J )}dx r 



= E"exp ipt, say; . . (15) 

 so that E" is a complex function of x" . 



19. Now by definition of % the waves emitted by the 

 lamina dx n will be 



-xE"exp i\pt - v(x-x")}, - x E"exp i{pt + v(x-x") } ; 



and these (on replacing single by double accents) must be 

 identical with (14) ; that is 



% " 1 B" + E" = 0, 



