320 Mr. G-. U. Yule on the Passage of Oscillator 

 which emerges at time (2n + l)t 2 . Remembering that 



q= -^ = Wl cos g— 2 sin g J, 



this becomes 



where we have written for brevity 



2 + %)=^ 

 # i * 



Finally, if we take the initial form of the wave-train to be 

 given by the sine terms, we may w r rite this ray emerging 

 after 2n internal reflexions, y , as 



(33) 



y n = A . c/. yin^2n^\)Kd e - Pl t sm [^ + a _ 2n + 1 (p 2 £ 2 + i/r) ] . (34) 



However small yjr may be it should evidently be retained, 

 as it may become of importance owing to the multiple 

 reflexions. 



The wave-train y n emerges at time 2n + lt 2 . From 2n + lt 2 

 to 2n-\-ot 2 . 3/0 to y B inclusive are the only wave-trains in 

 existence. The total energy of any wave-train varies as the 

 time-integral of the square of the " displacement " 



We will call this time-integral the intensity of the wave-train 

 (or set of wave-trains forming the ray), though we are not 

 using intensity in its ordinary sense. 



To get the intensity of the ray transmitted through the 

 plate, we shall then have to sum up a series of integrals partly 

 of squares (y|j, yj, . . . y 2 ) and partly of products (2y m y n &c.) : 

 that is to say, if we call the intensity of the transmitted 

 ray I„ 



n=oo /*°° m—n—l n=<o /»°° 



I*= S \!/idt + 2 S S \y m y n dt. . . (35) 



ji=0 ,/ »i=0 n=l J 



2n+lt 2 2n + lt 2 



In the experimental case the deflexions of the electrometer at 

 E_, fig. 1, are proportional to I t . 



We have now to get out the integrals in (35) and sum 

 them. We will first multiply out the product y m y n from 

 the expression for y n given in (34). This can then be 

 integrated, and the integral of y\ obtained by putting m=n, 



