1917]
FROM A REGULARLY STRATIFIED MEDIUM
495
In that case there was no need to refer the vibrations to particular points, but for our purpose we refer the vibrations always to the points 0, 0', etc., bisecting the intervals between the plates. On this understanding the formal expressions are the same. <pm denotes the reflection from m plates, referred to the point 0 in front of the plates; ^rm the transmission referred to a point Om behind the last plate. " Consider a system of m + n plates, and imagine these grouped into two systems, of m and n plates respectively. The incident light being represented by unity, the light <f>m will be reflected from the first group, and <frm will be transmitted. Of the latter the fraction tyn will be transmitted by the second group, and <pn reflected. Of the latter the fraction tym will be transmitted by the first group, and </>m reflected, and so on. Hence we get for the light reflected by the whole system,
and for the light transmitted
which gives, by summing the two geometric series,
.(18)
The argument applies equally in our case, only (f>m> etc., now denote complex quantities by which the amplitudes of vibration are multiplied, instead of real positive quantities, less than unity, relating to intensities. By definition fa = r, ^ = t.
Before proceeding further, we may consider the comparatively simple cases of two or three plates. Putting m = n = l, we get from (17), (18)
l-r
l-r8'
.(19)
By (13), 1 - r2 + t3 = 1 - e^, and thus
l-r'
.(20)
It appears that <f>2 vanishes not only when r = 0, but also independently of r when cos 2p = 1. In this case i/ra = — 1.
When cos 2p = 1, r = ± sin 6, t = + i cos 6, so that r is real and t is a pure imaginary. From (9) we find that a real r requires that
or, as it may also be written,
tan i/c§. tan i&S' = ^——-J J 1 + ?)*
&'
.(22)d equidistant plates may follow the lines of Stokes5 work for a pile of plates, where intensities were alone regarded*. * Roy. Soc. Proc. 1862; Math, and Phys. Papers, Vol. iv. p. 145. supposing k real, we are postulating a transparent plate. The effect of absorption might be included by allowing k to be complex.