8 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 139 



Let us derive an expression for B in terms of some of the other vari- 

 ables. If there are B s units of illumination striking the sea surface, 

 R S B 8 units will be lost owing to reflection, and B s (i—R s ) will be the 

 amount of light actually entering the water surface. At a depth of 

 {d+D) the light value will now be B s (i-R s )e- k ^+ D ^>. 



Since only R d of the light reaching the disc is reflected from it, 

 the light just leaving the disc would then have a value equal to 

 B s {i-R a )e- k U+ D ^R d , which is B oD . 



(5) B oD = B,(i-R 3 )e- Md+D) R d 



Traveling back upward, the light would be further attenuated over 

 the distance D, so that at the bottom of the hydroscope the brightness 

 value would now be equal to B s (i—R s )e~ k( - d + D '>R d e- kD . One more 

 reflective loss occurs at air-glass-water interface which may be as- 

 sumed to be equal percentagewise to the original surface reflective 

 loss so that the object brightness at the eye turns out to be : 



(6) B„ = B s (i — Rs)e- k(d + D) R d e-" D (i — R.) 



— B s R d {i—R,Ye- kia ^ D) 



Using the same methodology for calculation of the background 

 brightness, we get the following: 



(7) B b = B,Uw(i — R s ) z e-" D 



(8) B iD = B,U«(i — R.)e-* u + Di 



When (5), (6), (7), and (8) are substituted back in (4), the fol- 

 lowing is obtained : 



B s R d (i — R.ye- kii+2m — B.U«(i — R.ye-" D 



B s U w (i-Rs) 2 e' kD 



e- w B,R d {i — R s )e-™ +D) — B,U w (i — R a )e- kid+D > 



B s U w (i—R,)e- k(a+Dy 



which, upon simplification becomes : 



z >. R d e~ — Uv> ( R d — Uw\ _tn 



(9) — u: — =\nr^—) e 



Clearing fractions and transposing: 



--2fcD ( R d Uv> \ -kD Uv> __ _ 



\ Rd / Rd 



Letting -~=A, and simplifying, gives: 



e* kD — (i — A)e-* D — A = o 



or, multiplying by e 2kD to give positive exponents, we get : 



Ae 2kD + (i-A)e kD —i = o 

 which, when solved for e kD gives: 



Co *"=ji=vi 



