POLARIZATION OF SKYLIGHT 85 
shows a close correlation with the turbidity factor, as 
may be seen from Table II. Jensen’s normal values 
correspond thus to the turbidity factor 3.9. 
In connection with Milch’s and Blickhan’s work the 
recent investigation of Tousey and Hulburt [67] should 
be mentioned. The brightness and the polarization of 
the daylight sky were measured at different altitudes 
up to 10,000 ft. The curves of polarization with height 
showed clearly a much slower rise after a rapid increase 
within the first 2000 or 3000 ft, closely resembling the 
distribution of the turbidity coefficient with height [41]. 
They compared the measured values with theoretical 
values obtained on the assumption that the secondary 
scattered light is unpolarized, but taking full account 
of the extinction defined by a mean value of the theo- 
retical extinction coefficient 6 = 0.0126 km. They 
found that a somewhat better agreement for larger dis- 
tances from the sun can be obtained with 6 increased 
to 0.017, 0.018, or 0.021 km. The systematic devia- 
tions in the vicinity of the sun, expected by the au- 
thors, are caused by the assumption above, eliminating 
the neutral points around the sun. The increase of 8 
proves without doubt the presence of larger particles, 
sufficient to increase the theoretical scattering by about 
35 per cent. The increased values of 6 mentioned above 
correspond to the turbidity coefficient 7 = 1.35, 1.48, 
and 1.67, respectively. These values are in very good 
agreement with the measured values mentioned above 
[41], indicating the real presence of larger particles 
rather than a systematic error in taking too wide a 
spectral range, as suggested by van de Hulst [68]. In 
the theoretical computation the reflection by the ground 
was taken into consideration and the variation of the 
maximum and zenithal polarization due to the different 
values of the albedo is given in Table III. 
Tasie II]. Errecr or GrouND REFLECTION ON SKYLIGHT 
POLARIZATION 
Albedo a 0.0 0.1 0.2 0.4 0.6 
Tousey and Hulburt (67) 
P (maximum): h,; = 30°) 0.85 — | 0.755 | 0.667 | 0.612 
P (at zenith): h, = 0° 0.527 | — | 0.482 | 0.489 | 0.408 
Chandrasekhar [16] 
P (maximum): 
= 0.918 | 0.903 | 0.885 = 
ie = 130° 0.906 | 0.860 | 0.801 — —_ 
= 39.8° 0.906 | 0.796 | 0.673 = — 
iP Oe zenith) : 
«2 = 08 0.918 | 0.903 | 0.885 = = 
= 13.95 0.824 | 0.779 | 0.727 —_— _ 
hs = 39.8° 0.392 | 0.360 | 0.315 ~- 
The mean value of the measured albedo, a = 0.20, 
was taken for the computation, and for this value the 
theoretical degree of polarization at zenith, P = 0.482 
for h, = 30°, and P = 0.724 for h, = 15° may be com- 
pared with the values of P computed by Ahlgrimm 
(0.469, 0.738); and for h, = 25° the observed value 
P = 0.58 may be compared with the theoretical value 
P = 0.572 (Ablgrimm 0.558). 
From the discussion above it is quite evident that a 
better quantitative agreement between measurement 
and theory can be achieved when the original Rayleigh 
theory is extended by a consideration of (1) the effects 
of multiple (at least secondary) scattering, extinction, 
and reflection by the ground, and (2) the effect of the 
presence of large scattering particles. The effects men- 
tioned first can lower Rayleigh’s theoretical values to 
the observed values and explain the existence of neu- 
tral points in observed positions; but for the explana- 
tion of the great variety and magnitude of diurnal, in- 
terdiurnal, seasonal, and secular variations the highly 
variable content of larger particles in the atmosphere 
must be considered. This is more evident if the disper- 
sion of polarization is taken into account. The presence 
of larger particles can best be taken into account in 
the quantitative analysis, however, by separating and 
subtracting the effect of molecular scattering as a sim- 
pler and more nearly constant factor. For this purpose 
the recent theoretical investigations of similar problems 
in astrophysics offer excellent help. In a very elegant 
way, Chandrasekhar succeeded in reducing the exact 
solution of quite general multiple scattering to a solu- 
tion of two relatively simple integral equations of a 
form suitable for successive iteration. Once the solu- 
tion of these equations is known, the exact problem is 
solved. The great advantage of this method is not only 
that the extinction is very simply taken into account, 
but mainly that the effect of ground reflection can be 
included, as proposed by van de Hulst, and that the 
method can be extended for a more general law of scat- 
tering than in Rayleigh’s theory [13, 14, 15]. 
Chandrasekhar has recently accomplished the nu- 
merical computation of the effect of multiple scattering 
and of the ground reflection in the skylight polarization 
for a special value of the optical thickness + = 0.15, 
corresponding to \ = 450 wu under normal conditions 
[16]. The reflection on the ground affects the position 
of neutral points very little, in agreement with obser- 
vation (Neuberger [46]). A much larger effect is notice- 
able in the degree of polarization at zenith or at 90° 
from the sun. It is evident from Table III that the 
ground reflection is responsible for the daily variation 
of the maximum polarization, namely for the decrease 
of P with the sun’s elevation, in the sense of Fig. 1. 
The theoretical values for P are still much higher than 
the measured ones. 
The observed distances of the neutral points are also 
much higher than the theoretical values obtained by 
Chandrasekhar (for h, = 0°: A-point, 19.4° and Ba- 
point, 19.4°: for h, = 13.9°: A-point, 20.9° and Ba- 
point, 18.7°). However a remarkable agreement was 
found between the theoretical shape of the neutral lines 
and the shape of neutral lines observed by Dorno [22 
Larger scattering particles apparently affect primarily 
the magnitude of the polarization, and the position of 
the neutral points, but have only a slight effect on the 
position of the plane of polarization. This fact is an 
interesting aspect of the physics of scattering by larger 
particles and as such should be studied more exten- 
sively. 
Chandrasekhar’s method of exact evaluation of the 
molecular scattering makes possible a quantitative 
