380 Lord Rayleigh on the Transmission of Light through 



order to determine the attenuation by this process it would be 

 necessary to supplement (1) by a term involving 



but this is of higher order of smallness. We could, however, 

 reverse the process and determine the small term in question 

 a posteriori by means of the value of the attenuation obtained 

 indirectly from (1), at least as far as concerns the secondary 

 light emitted in the direction of the primary ray. 



The theory of these effects may be illustrated by a com- 

 pletely worked out case, such as that of a small rigid and 

 fixed spherical obstacle (radius c) upon which plane waves of 

 sound impinge *. It would take too much space to give full 

 details here, but a few indications may be of use to a reader 

 desirous of pursuing the matter further. 



The expressions for the terms of orders and 1 in spherical 

 harmonics of the velocity-potential of the secondary disturbance 

 are given in equations (16), (17), § 334. With introduction 

 of approximate values of 7 and 7 b viz. 



we get 



[f »] + DM = - % 3 (! + t) cos *(*-') 



+ ^( 1 ~t) sin *( a *-'-)> • • ( 15 ) 



in which c is the radius of the sphere, and k = 2ir/\. This 

 corresponds to the primary wave 



1$] = cos k(at + w), (16) 



and includes the most important terms from all sources in the 

 multipliers of cos k(at—r), sin k(at -r). Along the course of 

 the primary ray (fi= — 1) it reduces to 



k 2 c z lk b c 6 



[^o] + [fi]=-^ cos ^-^)+ -gg^ sin *(o*-r)- • ( 17 ) 



We have now to calculate by the method of FresnePs zones 

 the effect of a distribution of n spheres per unit volume. 

 We find, corresponding to (6). for the effect of a layer of 

 thickness dx, 



1irndx{\ke % sin k{at + «)- 3 7 ^c 6 cos h{at + x)}. . (18) 



To ihis is to be added the expression (16) for the primary 

 wave. The coefficient of cos k\at-\-x) is thus altered by the 

 * ' Theory of Sound,' 2nd ed. § 334. 



