Wave-trains through a Conducting Dielectric. 321 

 Writing, for brevity, 



p 2 t 2 + yjr = B, (36) 



y t y = A 2 c 2 / 2 5 2 («+» l )^-2p 1 < e 2(/»+»+i)/crf 



Xsin(^ + «— 2n + lS) sin Q? 2 ^ + a ~' 2m + 18) 



= - A V 2 pb^+^e—ZpitgZ (m+n+l)icd 



x |cos[2(m — w)8] — cos2[^ + a — (m + ;i+l)5] [. (37) 

 Integrating, 



) y m y n • ^ = iAV/*^***)^**-** 1 ^ 



2>l+l£ 2 



x |_ 2 z - + ^-cos{2(^+ a )+x-2(m + n + l)8}J. 



The expression in the square brackets vanishes at the upper 

 limit ; so we have, substituting rj cos % for pi in the 

 denominator of the first fraction, 



y m y n .dt=~ KWfh^+n^m+n-liyd 



2)i+ \t 2 



x e -2(2n+]) Px t 2 J v 1 cos [2(n — m)p 2 t 2 + 2a + % 



^ C ° S% -2(m + n+l)+].}.'(38) 



If we put m=n, we get 



J 



1 



4:77 



2»+i^2 x {sec %— cos [2a + % — (4n + 2)\|r] £. (39) 



Having now got the integrals, we must next perform the 

 summations indicated in (35). Let us take the y\ integrals 

 first : we have to sum the expression on the right-hand side 

 of (39) from w = to n=co . The two terms in the bracket 

 are best taken separately : the first forms a simple geometric 

 series whose sum is 



_A 2 c 2 f 2 X . . (40^ 



47} J * 1 — b 4 e^ Kd -P^ y V ' 



The second term requires different treatment, as the cosine 

 is a function of n. We may take the cosine to be the real 

 part of 



e i(2<x+ x -4n+2W 



