﻿Resonators exposed to Primary Plane Waves. 217 



comparison with the case of a single resonator, for which 

 b=1 and the X's vanish. We fall back on (24) by merely 

 putting n = 2, and dropping the signs of summation, as there 

 is then only one 11. 



If the tuning is such as to make the offect of the aggregate 

 of resonators a maximum, the cosines in (29) are to be 

 dropped, and we have 



. . . (30) 



s = 



n\ 2 1 



it ^ sin kR ' 



1 + z TkT 



As an example of (29), we may take 4 resonators at the 

 angular points of a square whose side is b. There are 

 then 3 R's to be included in the summation, of which two 

 are equal to b and one to by/2, so that (28) becomes 



A ,{ 1+ ,^ + «^}. . . . {n) 



A similar result may be arrived at from the value of -ty at 

 an infinite distance, by use of the definite integral * 



»|t 



sm x 



JoO sin 6) sin 6 d6 = ^~ (32) 



Jo x 



As an example where the company of resonators extends 

 to infinity, we may suppose that there is a row of them, 

 equally spaced at distance R. By (18) 



1 , p -ikTL p -2ikR p -3ikR > 



The series may be summed. If we write 



lie~ 2ix We' 



'Mx 



S = ,-*« + ^_ + -^l_ +..;., . . . (34) 



where h is real and less than unity, we have 

 dX ie~ ix 



dx l-he-^ 



and 



2 = — jhog(l—/i«- & ) (35) 



no constant of integration being required, since 

 X = -] r i] a(l-h) when x = 0. 



* Enc. Brit. 1. c. equation (43) ; Scientific Papers, iii. p. 98. 



