THROUai-l A STKATIFIED MEDIUM, ETC. 77
In the .small terms containing s'a we may substitute the approximate values of H and K from (34). For the problem of reflection we suppose Hm + Kw = 0. Hence
1 J_ rr A -f
m 11 + -111 (37), Si — iu,, (,'ft, - ajj, and so on, HO that
tfa3, 2-*-=* * I tr <ta, .................(38)
cr (6., ./ '
the. integration extending over the layer of transition.
One conclusion may be drawn at once. To this degree of approximation {.he. refled.ion is independent of the order of the strata. It will be noted that the- Hums in (37) are pure imaginaries. In what follows we shall suppose! that ain, is real.
As the final result for the reflection, we find ft, 7f, 4- 1L
A "A--// R('is>................................(39)
XI j ./V J ^^ J. i l
/<> _ ^'"l/0"] ~" ((»1 ('l f4l\\
M = , , , .................................(40)
^m/O-j + <*«/«! «wi
tan S = 2 ~
To UUH order of approximation thc^ intensity of tho reflection is unchanged by the pawnee, of tlvo intc'.rmediaUi layers, unlesH, indeed, the circumstances are midi that, (40) w iLwjlf nmall. If Om/eri = «•„,/(/•! absolutely, we have
(42)
.O'w.' «mJ O" J
and S sa |TT, This case is important in Optics, as representing the reflection at the polarising angle from a contaminated surface.
The two important optical cases: (i) where cr is constant, leading (when there is no transitional layer) to Freanel's formula (11), and (ii) where & sin9 0 is constant, leading to (10), are now easily treated as special examples. Introducing the refractive index //,, we find after reduction for case (i)
where Xu pi relate to the first medium, /xm is the index for the last medium, and the integration is over the layer of transition. The application, of (43)while the 's's are retained as quantities of the first order. Adding togetl the column of equations constituting the first members of (32), (33), etc., find