Wave-trains through a Conducting Dielectric, 329 

 and integrating from t — 2nt 2 to £=°o , 



f y m y n olt = i. A 2 c 2 / 2 6 2 ^+«" 1 > ^'■W 



x e -in Pl t 2 ( cos2(m-n)8 _ ^ ^ , _ v + 2 « + v] - 2 (n + m) f X (60) 

 l cos^ u J 



Putting m = n, 



J 



x{sec^— cos (2a 4-^-4^)}. . . (61) 



The summation of (61) from n=l to n = co is performed 

 in exactly the same way as that of the analogous expression 

 for l t in equations (39) to (44). The result is 



S f n dt = 



n=0 J 



AV/ 2 



F» 2 f 



2 £2 



4t;6 2 



2n# 2 



f sec% _ cos (ft - 2ifr) - 6) 2 f cos (ft + 2^) 1 ( „ 



\ 1-«T l-2o) 2 g 2 COs4i|r + ft) 4 f J * ^ 



The summation of the product integrals in (60) is also 

 analogous to that of the product integrals in l t given in equa- 

 tions (44) to (46) . Taking the integral in two parts as before, 

 we have for the sum of the terms containing sec % : — 



27}h 2 *>£ sec X (i_ ft) 2£^ 1 _ 2 f 2 cos26>£ 4 ) ' * {K)0) 



and for the sum of the second terms, those not containing 

 sec^ : — 



AV/ 2 c^? 2 



2rjb 2 1 - 2a> 2 £ 2 cos 4^ + a> 4 f 4 



[ + 0> 2 £ 6 COS(2i/r + ft) 1.(64) 



g 2 cos (28— 6>fr + ^)— g 4 cos (ft - 2f ) - a> 2 g 4 cos (23 - 2ijr + ft) \ 

 x l-2£ 2 cos2S+g 4 j 



Adding together (Q2) (63) and (64), bringing the second 

 half of (62) and (64) to a common denominator and simplifying, 

 we get the expression for that portion of I r which is a function 

 of the uniform series of ?/'s only. Calling it T r , and dividing 

 out by I the " intensity " of the incident ray, we get :— 



