1917] FROM A REGULARLY STRATIFIED MEDIUM 501
above limits, and then (ii) the reflection (and transmission) remain fluctuating however great m may be. When if is small, case (ii) usually obtains, though there are exceptions for specially related values of 8 and B'.
Particular cases, worthy of notice, occur when B' ± 8 = s\, where s is an integer. If 8' + 8 = s\,
sinh2 a = rf cos2 $kS - 1 , ........................ (57)
and is negative for all admissible values of 77, case (i). If &' — & = s\,
and we have case (i) or case (ii), according as if is greater or less than
cos2
When 77 is given, as would usually happen in calculations with an optical purpose, it may be convenient to express the limiting values of (56) in another form. We have
= tan PS . tan J&S', - = -coH&S. tan^S'. ...(59)
When the passage is perpendicular, Young's formula, viz. ?;=(//, — !)/(/* + 1), gives
(1T7?)/(1±7?) = ^1, ........................... (60)
/j, being the relative refractive index.
We will now consider more in detail some special cases of optical interest. We choose a value of 8 such as will give the maximum reflection from a single plate. From (5) or (9)
1 _ (I-??2)3 ,fin
|r|2 ^ 2^ (1>- cos AS)' ..................... ( }
so that \r\is greatest for a given t] when cos k& = — 1. And then
=i }
~
We may expect the greatest aggregate reflection when the components from the various plates co-operate. This occurs when e~ik(&+s"> = 1, so that in the notation of (50), Aa = A/2 = — 1. The introduction of these values into (54) yields
sinh2a = -l, .............................. (63)
coming under (i). The same result may be derived from (57), since here cos ^k8 = 0. In addition to ax = 0, sin /32 = 0, we now have by (63) sin «2 = + 1, cos «2 = 0, and (48) gives
\<f>m 2 = tanh2m&, |r |2 = tanh2& ................ (64)
We are now in a position to calculate the reflection for various values of m, since by (62)
tanh ft = ± r-^-- = ± tanh 2£, -written