Theory of Illumination in a Fog. 445 
Some of the principles here enunciated have an acoustical 
as well as an optical application, and indeed first occurred to 
me some years ago in connection with Prof. Tyndall’s inves- 
tigations upon fog-signals. The effect of “acoustic clouds”’ 
analogous to fog (and unattended with absorption of energy), 
might be very different upon the report of a gun and upon 
the sustained sound of a siren, the latter being reinforced by 
reflection trom the acoustic fog. 
The theory presented in the present paper may be illus- 
trated by the known solution of the comparatively simple 
problem of a pile of transparent plates*. If p denote the 
proportion of the incident light reflected at a single surface, 
then the proportion reflected ¢(m), and transmitted y(m), by 
a pile of m plates is given by 
d(m) vn) _ 1 
2mp  1—p 1+(2m—1)p 
From these expressions it is evident that, however small p 
may be, 7. e. however feeble the reflection at a single surface, 
we have only to suppose m large enough in order that the 
reflection may be as complete, and the transmission as small, 
as we please. Such a pile may, under ordinary conditions, 
be regarded as impervious. 
But now suppose that after passing the pile of m plates, the 
light is incident upon a second pile of n plates, and consider 
the intensity between the two piles, the original intensity 
being unity, as before. For the intensity of the light tra- 
velling in the original direction we have 
+ap(m) .{P(nr) - (mm) fF +... 65 
or on summation of the geometric series, 
Wr(m) 
1—$(n) . o(m) 
If we introduce the values of @ and in terms of m, n, p, 
this becomes 
2np+1—p 
2(m-+n)p+1—p° 
In like manner, for the light going the other way we have 
p(m).b(m)_, 
1—$(m). $(n)’ 
* Stokes, Proc. Roy. Soc. vol. xi. p. 545 (1862). 
Phil. Mag. 8. 5. Vol. 19. No. 121. June 1885. 2H 
