1917]
FROM A REGULARLY STRATIFIED MEDIUM
503
since sin «2 = 0 in this case (ii). By (49 bis) we have now, setting m = 1,
J_ sinh2a1_(l +rf)-
| r |2 sin2 /32 47?2 '
as we see from (62). Comparing with (66), we find sin2 & = 1, /?2 = (s +|) TT. Thus sin2 m/32 is equal to 1 or 0, according as m is odd or even; and (49, bis) shows that when m is odd
and that when TO is even, | </>TO 2 = 0. The second plate neutralizes the reflection from the first plate, the fourth plate that from the third, and so on. The simplest case under this head is when 8 = £ X, &' X.
A variation of the latter supposition leads to a verification of the general formulse worth a moment's notice. We assume, as above, 8' = s'X, but leave 8 open. Since e^c&> =±l, (9) and (10) become
_,
~~ ~
and these are of the form (39), if we suppose a = 7;"1, b = e*m. The reflection (f>m from m plates is derived from r by merely writing bm for b, that is,
gitwtw for gi^ leaving least when 8' = 0.
equal to \r j*, as should evidently be the case, at
[* This statement does not hold in general, when S' = s'\, where s' is an integer and may be zero. We have
f_ 2isin£/t5
(ij"1 - 77) cos ^kd + i (f)~l + rf) sin i/cd'
p]2=4BTn2"p5 + 3"
so that
Hence consequently, if
where n is an integer, so that
5= - ^ .
This result may he verified for m=2 or 3 from (19), (23), and (68). It includes as a special case that dealt with in the preceding paragraph, if, when m is odd, we write » = (« + &) (mil), where s is an integer. When 5' = 0 the strata intervening between the plates disappear, but the theory is only applicable on the supposition that reflection and refraction continue to take place as before at each of the contiguous surfaces of the plates. W. F. S.]
Illition to ax = 0, sin /32 = 0, we now have by (63) sin «2 = + 1, cos «2 = 0, and (48) gives