322 Mr, G. U, Yule on the Passage of Oscillator 



and thus obtain again a geometrical series, the sum of which 

 is 



Ar) ' J '1 b 4 e^ Kd ~P^e~ li ^ 



This is rationalized by multiplying numerator and denominator 

 by l-bWf-PMeW ; and to further abbreviate the expression 

 we will write 



J je Kd-2p,t 2 _ £ -j 



^ = o>; j (41) 



thus obtaining for the sum of the second terms of (39), 



-AW 2 ^ <***+*-»» l-a'P*"* 



±v J ' b 2 ' 1- 2o)' 2 f cos 4^ + ft) 4 f 



Remembering that 



6 



and writing 



^-4t' + X = ^> .... (42) 



this becomes, retaining the real terms only, 



1 A 2.2/2 ^| COS0-ft) 2 g 2 COs((ft + 4^) 



4*7 7 * ^ 2 ' 1 - 2a> 2 | 2 cos 4^ + © 4 f * ' ' [ ' 



Adding (40) and (42) together, we get for I l9 the portion 

 of the transmitted intensity due to the square terms, 



, _ A2 £ 2 / 2 J jwf |_sec%_ cos0-&)Tcos(j) + 4f) '\ 

 1 4^ 2 Xl-cD 2 ? 2 " l-2o> 2 f cos4^ + ft, 4 f J ' V**' 



We have now to sum the product terms. We require 

 I 2 = 2S S \y m y n dt 



n = l ?n = J 



2n+ lt 2 



- T ' — T " 



— J-2 J-2 i 



say ; I 2 ' being the sum of the terms multiplied by sec %, and 

 I 2 '' the sum of the second terms. The integrated expression 

 is given in (38). Taking the sec% terms first, 



A 2 n 2-f'2 »=oo m=n— 1 



V= ' J So) 2}l + 2 6-^+ 2 ^sec% S co 2 " l cos2(?n-?i)S; 

 cos 2(m — n)8 may be treated as the real part of e 2i ^ m ~ n)S , or 



CO 



2m cos 2{m-n)B = e - 2inS (w 2 e 2lS ) m 



j 



