84 METEOROLOGICAL OPTICS 
point appears in the sun’s vertical at the elevation h, 
provided that in this direction 
Pi+5i=P+S. (4) 
For the sun at the horizon, P/P; = cos? h. The intensi- 
ties S; and S are normal to the direction h, and thus 
S, = 1, S = i, sin? h + 7, cos? h, and from (38) 
S/S: = (1 + 7 cos? h)/3. (5) 
In (5) the intensities can be expressed by the total in- 
tensities (P; + P, S: + 8), and then if the ratio R = 
(Si — S)/(Pi + P) is known, the elevation of the neu- 
tral point is determined from the equation 
sin? h/(1 + cos? h) 
= R(7 cos? h — 2)/(4 + 7 cos? h). (6) 
The most probable value of R les within the limits 
R= 0.1 and R = 0.2, which gives h = 16.7° and h = 
22.4°, in good agreement with observation. 
Ahlgrimm [7] extended Soret’s computation for arbi- 
trary solar elevation, neglecting the extinction and 
assuming only that the distribution of scattering par- 
ticles is the same in all azimuths. The unknown dis- 
tribution of scattering particles with height appeared 
in the integrand of mtegrals which could be evaluated 
by means of the transmission coefficients. Using the 
measured values by Abney,! Ahlerimm was able to 
compute the degree of polarization due to the primary 
and secondary scattering im any arbitrary direction. 
The values of maximum polarization as a function of 
solar elevation are reproduced in Fig. 1, values of the 
polarization in the zenith in Fig. 2. 
Consideration of secondary scattering thus brmgs a 
great improvement in the qualitative agreement of the 
theory with observations under normal conditions, but 
a quantitative agreement still cannot be reached. Ticha- 
nowski [66] extended Ahlgrimm’s computations, con- 
sidering the anisotropy of molecules and even the re- 
flection from the ground, but without any quantitative 
improvement. The effect of atmospheric extinction and 
refraction was taken into account by Link [89]. In such 
a case the integration for secondary scattermg cannot 
be performed in any analytic form and requires a tedi- 
ous quadruple numerical or graphical quadrature. Un- 
fortunately the computations were made for the de- 
pressions of the sun for which no observations are 
available yet. 
Comparison of the theoretical curve of the Ba-pomt 
with that obtamed during the period after volcanic 
eruption suggests the presence of another mechanism 
which may be even much stronger than the effect of 
secondary scattering. The appearance of Bishop’s ring 
and other twilight phenomena proved the presence of 
larger particles (according to Pernter [47] of a diameter 
from 3.2 \ to 6 \) than assumed in Rayleigh’s theory. 
The theoretical and experimental investigations of the 
scattering by particles of such a size, by Schirmann 
[55, 56) and later by Blumer [10], showed the possibil- 
ity of neutral points already in the primary scattered 
1. Abney’s values are reproduced in [5]. 
light, and led Mulch to the idea of using the presence 
of larger particles as an explanation of the deviation of 
the observed values from those given by Rayleigh’s 
theory. Neglecting the effect of secondary scattering, 
Milch developed the followmg expression for the de- 
gree of polarization: 
— mo Pp a of (hs , T) 
mo(p + ) + wo + v)f(hs, TL)’ 
where mp and po denote the number of molecules and 
large particles per unit volume at the ground; p and y, 
the intensity of the polarized part of the light scattered 
by molecules and by large particles respectively; p + n 
and wy + », the total intensity in both cases of scatter- 
ing; and finally, f(h;, T) is a function of the sun’s ele- 
vation and the turbidity coefficient computed from the 
extinction values for different turbidities. This expres- 
sion can explain all observed variation of the polariza- 
tion with turbidity, but its use for the computation of 
P is very limited by the lack of knowledge of the func- 
tions y and v. Assuming that at the point of maximum 
polarization y can be neglected with respect to p, 
Mulch determined » from the measured value of P for 
a given h,, and computed the variation of maximum 
polarization with h;. But the computed values showed 
a systematic deviation from the observed P. 
In continuation of Milch’s work, Blickhan [9] studied 
the correlation between the turbidity coefficient and 
the maximum polarization measured simultaneously. 
The values of P lie along a hyperbola P = A/(B + T;) 
ina (P, T;,) —diagram (7; is the turbidity coefficient for 
short-wave radiation, and A and B are constants), mm 
good agreement with the simplified formula (7). Ex- 
trapolating P by means of this empirical formula for 
an atmosphere without large particles (7 = 1), he ob- 
tained a value of P for h, = 50°, which is larger than 
that obtaimed for h, = 32.5°, in agreement with Ahl- 
erimm’s computation. From the difference between the 
extrapolated values and those computed by Ahlgrimm, 
assuming further that the light reflected from the 
ground is not polarized, Blickhan was able to compute 
the albedo a = 0.132, which is in good agreement with 
Dorno’s values. For the computation of the daily vari- 
ation of maximum polarization he used Milch’s pro- 
cedure, but with the difference that im (7) he inserted 
for p and n the values computed by Ahlgrimm. The 
values obtained in this way agreed with the measured 
ones much better than if the effect of the secondary 
scattering had been neglected. Other simultaneous 
measurements of polarization and turbidity were made 
(7) 
Tasie IJ. Pouartzation av ZenrrH (per cent of normal 
value) as A FUNCTION or TURBIDITY 
2.51-3.0 | 3.01-3.5 4.01-5.0 
112.0 | 104.0 93.2 
T 
iP 
>5.0 
68.3 
2.0-2.5 3.51-4.0 
114.4 102.8 
by Worner [70]. The polarization was measured this 
time at the zenith and was compared with the normal 
values computed by Jensen [2] (Fig. 2). The degree of 
polarization expressed in per cent of normal values 
