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1912] THROUGH A STRATIFIED MEDIUM, ETC. 75
pure imaginary. Thus a^ is real, and a, is imaginary; cl is real always, and ,Vj i.s imaginary as before; the /9's arc- always real. Than, if wo separate tho real and imaginary parts of the numerator and denominator of (24), we get
of which the modulus is unity. In this ca.se, accordingly, the reflection back in the first medium in literally total, whatever may be the thickness of the intermediate, layer, as was to be expected.
The. separation of real and imaginary parks follows the Maine- rule when «.. is imaginary, as well as «:!. For then a, is imaginary, while <y»t ,v, are real. Thus #,0^/3, remains real, and c, «,<*.,, «,(*,#., remain imaginary. Tim reflection back in the. first medium is total in this case. also.
Tho only other case, requiring consideration occurs when rta is imaginary and (t,.t real. The reflection is then total if the middle layer be thick enough, but if this thickness be reduced, the. reflection cannot remain total, an in evident if we suppoHo the thickness to vanish. The ratios aj, a. are now both imaginary, while .v, is real. The separation of real and imaginary parts stands as in (24), and tho intensity of reflection is still expressed by (25). If we take </a — — if//, we may write in place of (25),
(A#, l- cx.aJ'^^Hh3 f/,; (^ - ,r,) ~> (au/y, h of,/i, When .r., — ,r, is extre.nusly nmall, thin reduces to
in accordance with (!)).
When on the other hand a?a — /P, exceeds a few wave-lengtliH, (29) approaches unity, as we we from a form, equivalent to (2(J), vis?.,
It IH to be remembered that in (80), «ts. #/• aia« Have negative values.
The form awwrnod when the third medium is Himilar to tho first may be noted. In thin case a,aa » 1, /i?,/^* 1, and we get from (21))
" a'/>)' Binh* f'a/ (<7<a ~ 'ri)
' --«- 4
In thin cane, of course, the reflection vanishes when x^ — x^ in Hufficiently reduc«(d.
Efjuationa (21), etc., may bcj regarded as constituting the solution of the general problem. If there are m media, we suppose //mas~l, Kin**l,ference may be made to Theory of Sound (loc. cit). In the case of the simple gases the compressibilities are made a year ago gave promising results. Ten lantern-slides were prepared from a portrait negative. The exposure (to gas-light) was for about 3 seconds through the negative and for 30 seconds bare, i.e. with negative removed, and the development was rather light. On single plates the picture was but just visible. Some rough photometry indicated that each plate transmitted about one-third of the incident light. In carrying out the exposures suitable stops, cemented to the negative, must be provided to guide the lantern-plates into position, and thus to ensure their subsequent exact superposition by simple mechanical means.