BY SMALL OBSTACLES OF CYLINDRICAL AND SPHERICAL FORM. 255 



Also by a well known result we have 



r'* t x I TJ. \',2 



cos (hv 2 ) dv = sin (hv*) dv = - 

 Jo Jo 



Consequently 



Substituting tbis result in (2) and restoring the time factor, we obtain for the 

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



A-V (ta+< " ) ........ (3). 



When Xa is small, A t is great compared with A ; neglecting A and using the 

 expression for A, given in (15), 3, we obtain instead of (3) 



2n . dx . (log 



of which the real part is 



2ndx {(\og%\ + y)cos(hx + a-t) + Trsm(hx + at)} {(log\a + y) 2 + ^Tr-} (4). 



c 



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



<f> = cos (hx + crt). 



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

 unity to 



dx ~ (log i 



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



dE/E = 4u.r/.r^(log 

 c 



Integrating this, we obtain 



E = Eoe-", 



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



- 1 V 2 } ..... (5). 



The coefficient of sin (hx + a-t) in (4) gives the refractivity of the medium as 

 modified by the wires. If 8 be the retardation due to the wires of the stratum dx, 



