92 Dr. C. V. Burton on the /Scattering and 



therefore, when r is large in comparison with the wave-length, 

 we get 



h'= rot a' = &/*■-* sin (pt—vr—y)(y, -x, 0); . (43) 



while if d' is the electric vector, and c the velocity of light, 



6.' = c roth' = Cpvr~ z cos {pt — vr— y)( — xz, —yz, x 2 + y 2 )* 



Hence, since the disturbance which concerns us is periodic, 



d' = Ci/?'~ 3 sin (pt — vr— y)( — xz, -yz, x 2 +y 2 ). . (44) 



38. If the primary (plane-polarized) waves, which cause 

 the secondary disturbance (42) to be emitted, are repre- 

 sented by 



a = 0, 0, A cos {pt — vx), .... (45) 



and if the phase-lag y is known, C can be found in terms 

 of A. From (45), h and d for the primary waves are seen 

 to be 



h=0, -Avsm(pt— vx), 0; . . . (46) 



d=0, 0, Av sin (pt-vx) (47) 



We have now to introduce the condition that the secon- 

 dary vibrator merely scatters energy of frequency p/^ir, 

 without changing its total amount. To do this, we may 

 imagine a closed surface drawn enclosing the vibrator, and 

 write down the expression for the surface-integral of the 

 Poynting vector ; the condition then is that the time-average 

 of this surface-integral should vanish. More conveniently 

 the closed surface is replaced by two parallel planes, x= — I 

 and x=l, where l:\ is large. Thus of the vector-product 

 of d + d' and h + h' there is only the ^-component to be 

 considered, namely 



t> 2 {Asin {pt — vx)-\-Q,r~ z (x 2 + y 2 ) sin (pt — vr—y)}{Asin (p>t — vx) 



+ Cr~ 2 .£sin (pt — vr— y)} 



= v 2 [A 2 sin 2 i^pt — vx) + C 2 r~ 5 x(x 2 +y 2 ) sin 2 (pt — vr — y) 



+ AC{r~ z (x 2 + y 2 ) -\-r~ 2 x} sin (pt — vx) sin (pt — vr — y)]. 



The quantity which has to vanish is the surface-integral of 

 the last-written expression over the plane x = l minus the 

 surface-integral over the plane x=—l. Accordingly those 

 terms may be omitted from the expression which remain 

 unaltered when the sign of x is reversed. On averaging the 

 remaining terms with respect to time, and taking account of 



