496 ON THE REFLECTION OF LIGHT [422
When i) is small we see that
In this case only the first and second components of the aggregate reflection are sensible.
If there are three plates we may suppose in (17) m = 2,n=l,
•
Thus . ^ = ^+l~^' ........................... (23>
<£o and ^3 being given by (19). If <j£>3 = 0>
<£2(l-r<£2) + rx/r22 = 0 ......................... (24)
In terms of p and 8
2 _ cos3#e2^
• ^9"~ .......... ( }
Using these in (24), we find that either sin 0, and therefore -r, is equal to zero,
or else that
Eico&0 + E(Z-E)(l-E)co&0 + (l-Ey = Q) ......... (26)
E being written for e2^. By solution of the quadratic
..... cosz0 = -(I-E)*IE or l-E~\
The second alternative is inadmissible, since it makes the denominators zero in (25). The first alternative gives
E = cos 2/o + i sin 2p = 1 - 1 cos2 6 ± i cos 9 \/(l - £ cos2 6)] whence cos0= + 2sm/> ............................... (27) ,
When 17, and therefore r, is small, cos 6= 1 nearly, and ^ in (15) may be omitted. Hence
8 + 8' = X(£orf) + sA, ........................ (28)
as might have been expected.
If we suppose ezip = 1, <£2 = 0, ^2 = — 1, and (23) gives <£3 = r. It is easy to recognize that for every odd number <jbm = r, and for every even number </>m = 0. . .
In his solution of the functional equations (17), (18)*, Stokes regards 0 and T/T as functions of continuous variables m and n, and he obtains it with the aid of a differential equation. The following process seems simpler, and has the advantage of not introducing other than integral values of m and n. If we make m = 1 in (17),
or if we write un = r$>n — 1,
n + t* = Q ...................... (30)
* Stirling has shown, Eoy. Soc. Proc. A, -Vol. xc. p. 237 (1914), that the two equations are not independent, (18) being derivable from (17).transparent plate. The effect of absorption might be included by allowing k to be complex.