On the Intensity of the Light reflected from a Pile of Plates. 485 

 and equations (13) now give 



&»*__$-*» a — or 1 ab m —a- l b~ m ' * ^ ' 



In the case of perfect transparency these expressions take a sim- 

 pler form. If r-\-t differ indefinitely little from 1, a and ft will 

 be indefinitely small. Making a and /3 indefinitely small in (13) 

 and (14), and putting 1 — r for t, we find 



<\>(m) ^p(m)_ 1 



mr 1 — r l-\-(m — l)r 



(19) 



In this case it is evident that each of the 2m reflecting surfaces 

 might be regarded as a separate plate reflecting light in the proportion 

 of p to 1, and therefore we ought also to have, writing 2m for m and 

 p for r in the denominators of the equations (19), 



<p(m) ^(m) _ 1 g , , 2() v 



2mp"\—p l + (2»2 — l)p '* 



It is easy to verify that when #=1 (4) reduces (19) to (20). 



The following Table gives the intensity of the light reflected from 

 or transmitted through a pile of m plates for the values 1, 2, 4, 8, 

 1C, 32, and co of m, for three degrees of transparency, and for certain 

 selected angles of incidence. The assumed refractive index /j. is 

 1*52. r) = l— e-<? T is the loss by absorption in a single transit of a 

 plate at a perpendicular incidence, so that c — corresponds to per- 

 fect transparency. The most interesting angles of incidence to select 

 appeared to be zero and the polarizing angle «r= tan -1 jx ; but in the 

 case of perfect transparency the result has also been calculated for an 

 angle of incidence a little (2°) greater than the polarizing angle. <j> 

 denotes the intensity of the reflected and \p that of the transmitted 

 light, the intensity of the incident light being taken at 1000. For 

 oblique incidences it was necessary to distinguish between light po- 

 larized in and light polarized perpendicularly to the plane of inci- 

 dence ; the suffixes 1, 2 refer to these two kinds respectively. For 

 oblique incidences a column is added giving the ratio of \p x to \p. 2t 

 which may be taken as a measure of the defect of polarization of the 

 transmitted light. No such column was required for £=0 and 8=«r, 

 because in this case i^ 2 =1000. 



* From a paper by M. Wild in Poggendorff's 'Annalen ' [vol. ix. (1S56) p. 240] 

 I find that the formula; for the particular case of perfect transparency have 

 already been given by M. Neumann. His demonstration does not appear to have 

 been published. 



