92 METEOROLOGICAL OPTICS 
produced an adequate geometrical theory of scattering, 
serving as an extrapolation of the Mie theory. Many 
workers, notably Houghton and Radford [26] and Bri- 
card [6, 7], have measured the size distribution in 
natural fogs, obtaimmg unimodal curves with maxima 
in the region between 4 and 10 microns radius. Such 
fogs should be nearly nonselective. All these researches 
may be criticised on the basis of sampling, especially 
since Driving and his co-workers [18] have presented 
indirect evidence for the presence of large numbers of 
very small droplets. Such evidence is hard to obtain 
because, as Dessens [15] points out, the total optical 
effect of these small particles is not very important. 
The researches of Dessens should be repeated and an 
attempt made to extend them to natural fogs in order 
to decide this point. It is also possible that the use of 
the electron microscope in such work might settle the 
controversy about the nature of the nuclei. Another 
line of research which might be undertaken in some 
sparsely inhabited region concerns the explanation of 
Fie. 1.—Polar diagram of scattering by water droplets, nor- 
malized at 90°. The radii are in log units. The numbers on the 
curves refer to values of 27a/n. 
the surprisingly low selectivity shown by very clear 
air such as occasionally permits a visual range of 150 
or 200 km. Observations, chiefly in Europe, never 
seem to show anything lke the inverse fourth power 
of the wave length demanded by the theory for pure 
gases. As a hypothesis, one may assume the effect of a 
comparatively small number of relatively large particles. 
The Reduction of Contrast by the Atmosphere 
The common observation that the more distant an 
object, seen in daylight, the greater its luminance, was 
first reduced to mathematical form in 1924 by Kosch- 
mieder [29]. The simplest case is that of a black object 
of intrinsic luminance zero seen against a horizon sky 
of luminance B;. On the assumption of a uniform 
atmosphere having a scattering coefficient bo, and illu- 
mination by the sun and a uniform (overcast or cloud- 
less) sky, Koschmieder, showed that such an object 
seen at a distance r will have an apparent luminance 
By = Bl = ey, (2) 
He further showed that an object which is not black, 
but has an intrinsic luminance Bo, will at a distance r 
have an apparent luminance 
B= yew Se Bil = g”). (3) 
We cannot calculate Bj without knowledge of the 
distribution of light, even if we know the properties of 
the object. 
In these equations no mention is made of absorption, 
and while it is customary to write similar equations 
using o instead of 6, it is not immediately obvious how 
such an extension can be justified. Actually a more 
detailed analysis, using the concept of the space light 
B,, which is a function of the scattermg properties of 
the air and of its illumination, does lead to an equation 
B= Bye" se (ll = e"). (4) 
Such an analysis has been carried out by Duntley 
[19], who also threw off the restriction of horizontal 
vision and introduced a quantity R which he called the 
optical slant range, which “represents the horizontal 
distance in a homogeneous atmosphere for which the 
attenuation is the same as that actually encountered 
along the true path of length R” [19, p. 182]. The 
equation for downward oblique vision becomes 
BO) @ _ 
00 
B= 
in which &,(0) and oo are the values of B, and o cor- 
responding to the air near the ground. In the horizontal 
case it turns out that B; is to be identified with B,(0)/«o. 
A more useful statement of the law may be made in 
terms of contrast. If By and Bo are the inherent lumi- 
nances of two objects adjacent in the field of view and 
By and By, their apparent luminances, then defining 
contrast in the usual way, 
Y i? 
Co = Boe By Cr = Ba Be 5) (6) 
Bo Br 
we may write two equations such as (5) and simplify 
to obtain 
On = Co(Bo/ Bae, 7°". (7) 
This is completely general. If we confine ourselves to 
objects seen against the horizon sky, B’(= B;,) is mde- 
pendent of distance, and (7) reduces to 
C, = Ce”. Oo 
We have not space to expound the theory further in its 
applications to oblique vision, but it should perhaps be 
pointed out that practical situations generally require 
information which, at best, has to be estimated. 
Equation (8) has been tested more or less thoroughly 
by many workers, and there is no longer any doubt 
about its adequacy. A matter about which there is some 
disagreement, however, is the exact nature of the extinc- 
tion coefficient o. In deriving equations similar to (5), 
Duntley [19] has made use of a theory originally de- 
veloped by Schuster [41] which dealt with the distribu- 
tion of diffuse radiation in stellar atmospheres. The 
e 0F) a By e, (5) ‘ 
