Wave-trains through a Conducting Dielectric. 323 

 which is a geometric series whose sum from to n — 1 is 



■2inS 



e 



_ 1-(o>V*)* 



1— ©V* ' 



Multiplying numerator and denominator by 1 — co 2 e~ 2iS to 

 rationalize, and retaining only the real quantities in the 

 numerator, this becomes 



cos 2nB — co 2n — w 2 cos (2n+2)S + ft) 2n+2 cos 28 

 1- 2ft> 2 cos 28 + to 4 . 



Therefore 



V = A8 *y 1 8 S6C * 4 T(cos 2nS-co 2 » 



2 2yb 2 1 — 2w 2 cos 28 + <w 4 n==1 v 



-ft) 2 cos (2n + 2)8+ (o 2n+2 cos 28)<i^V-< 4 * + *>*»* i . 



The first and third terms in the bracket may be summed 

 together, as also the second and fourth. Carrying out the 

 summation as before, we get 



A 2 c 2 / 2 sec x-">$ f P(cos28-P) - to*? (cos 48 - f 2 cos 28) 





12 ~ 2 v b* l-2a> 2 cos2S + a) 4 (. l-2£ 2 cos2S + £ 4 



-(l-^cos2S) r ^}. 



Bringing the two fractions to a common denominator, 

 (1 — 2ft) 2 cos 28 + ft) 4 ) divides out, and 



AV/ 2 sec % . ft>g f(cos28-P) 

 2 " 2t7& 2 1-ft) 2 ? 2 l-2£ 2 cos28 + £ 4 ' * ' [ * 0) 



Returning now to the second half of L, the term not 



containing sec %, we have 



A 2 J2 /2 »=oo m=n— 1 



I/=i±-4f o,.f. S S P l ft) 2 ™cos{2(^-m)8-4^ + ^>}, 



^ t> »=1 m=0 



where we have rewritten the expressions (38) in terms of 

 the abbreviated symbols defined by equations (36), (41), 

 and (42). Carrying out the summation first with regard 

 to m, 



1 „^ A 2 .c 2 / 2 ft>g 



2 2rjb 2 1- 2ft) 2 cos 28 + ft) 4 



X S p l [cos(2nS-47n/r + <£)-~ a> 2 » cos {4nf-<h) 



— G) 2 cos (2n + 28 — 4ni/r + 0) + o) 2tt+2 cos (4ni/r — <£ — 28) |. 



