BY SMALL OBSTACLES OF CYLINDRICAL AND SPHERICAL FORM. 267 



resultant at O of all the secondary vibrations coming from the stratum dx is given by 



-2im<fo;(iA fc- !l + A 1 fc- 8 )e' < *" ........ (2), 



where the time factor e l9t has been restored. 



Now, from the results obtained in 7 for A! and A , we find 



2 + A,/r 3 = W {3 V/2X-V*- 1 



Except for very minute obstacles, it will be sufficient to write 



Substituting this last expression in (2), we obtain for the resultant of all the 

 secondary vibrations coming from the stratum dx 



- I + GX- 8 - 2 + J/iV) + 1 (f 

 of which the real part is 



cos 



- 1 a- 1 )sin(/a; + <r)} (3). 

 To this is to be added the corresponding expression for the primary wave 



The coefficient of cos (hx+a-t) is thus altered by the obstacles in the layer dx from 

 unity to 



1 - mra 2 { G + 3 ^o- 1 V 2 /c + 1 o- V/c 4 1 t/a;. 

 L cct J 



Thus, if E be the energy in the incident wave, we have 



dE/E = -WTra 2 J6 +3 v /2 - 1 V' 2 /c + |o-V/c 4 l^. 

 I ca J 



Integrating this, we obtain 



E = Eoe- 



where E is the energy in the primary waves at incidence, and a is given by 



a = mra 2 ( 6 + 3 x /2a 1/ V 2 /c+ lo-V/c 4 ) . (4). 



\ ea / 



2 M 2 



