328 Mr. Gr. U. Yule on the Passage of Oscillator 



V. The Intensity of the Reflected Ray. 



We will now proceed to calculate the intensity of the 

 reflected ray. As before, let the incident wave be 



A portion of this incident wave is at once reflected, namely 

 Y^A.b.ePV+^e-V (55) 



A second portion enters the electrolyte and reaches the second 

 surface in the form 



is there reflected, reaches the first surface again and emerges 

 at time 2t 2 , 



?/!= — A . hcf. e -2/>?d£P(*+^&*) e Km-w-w. 



The next wave emerges after three internal reflexions at 

 time 4^ 2 , 



and so on ; the expression for the wave emergent at time 2nt 2 

 being 



y =- — A . l 2n " ■ cf. e~ 2n Pi d e v{t+ s/ ^ e l ^ 2 ~ 2 ^ e^ 2n ~ l) ^. . (56) 



^=7 7 v / ^risin^+%^-cos^+%)]. 

 Inserting this value in (56), and using our previous abbre- 



viations, 



*=Wft/>cos(g+ X ), «=(§-2^ + ^), . (33) 



8=iV2 + V r , (36) 



and retaining only the sine terms, we have 



y n = —Kcfb 2n - l e 2nKd e-^Hm (p 2 t+a— 2n8). . (57) 



It is to be noted that the first reflected wave, Y, does not 

 fall into this series so the general expression for the reflected 

 intensity will be 



n=oo-/ °° m=n— 1 Jt=oo /» °° 



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



« = 1 J2»i# 2 m=l « = 2 J2m* 2 



+ j Y 2 ^ + 2lT I Y-yitft. . . (58) 



Jo «=1 J2nt 2 



Multiplying out y m y n we get 



y m y n - J A 2 c 2 / 2 & 2 (™+ w -D ^(m+n^-ap,* 

 cos 2(m -n)S— cos[2(p 2 £ + a) — 2(m + n)8]} ; (59) 



