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r rrr r rr r rr 44 r rrr r rr r rr
r rrr r rr r r 45 r rrr r rr r rTHE THEORY OF CONIC SECTIONS
more represented-though, for the sake of sim-
plicity, in horizontal position. A line perpen- dicular to its plane at the center O must meet the point of the gnomon G. Hence CG is the extended gnomon. From figure 2 it follows that OCG = CGL = 6.
It is now easy to express the angle / = HGC
as function of a = COH. If r = OH is the radius of the parallel circle we have on the one hand (fig. 3)
CH = 2rsina CH = 2r sin-, 2
r Because CG = HG = we have on the
cos 6 '
other hand 2r ./
CH = s5sln- cos 6 2
FIG. 2.
of the ecliptic. Figure 2 represents the celestial
sphere with the shadow-casting point G of the
gnomon as center. ELW is the equator, RESWR' the horizon, RHCR' the daily orbit
of the sun, culminating at C. Hence CGL = 6 the declination, and CG the direction of the
gnomon. Assume the sun in H. The angle a,
measuring the distance from noon is given by a = COH, where O is the center of the parallel circle RHCR'. The length s of the shadow is
then given by s = tan ,, where / = HGC is the
angle between the ray HG and the direction of
the gnomon GC. All we have to do is to find / as function of a.
In figure 3 the parallel circle RHCR' is once
. C . a Thus sin = cos 6sin . Because s = tan /
2 2
we compute also tan . By a simple computa-
tion one finds iv
(1) tan =
2 1 - cos2 a sin2 -
This answers our question.
CONSEQUENCES
(A) Because cos 5 = cos (- 6) we obtain the
same shadow length for declinations symmetric to the equator, especially for 6 = e and 5 = - E. Our sundial shows equal shadow lengths for both
solstices.
(B) Because the geographical latitude has no
influence on 6, our sundial gives the same shadow
lengths for all localities on the earth.
(C) Because shadows are only cast on the
receiving plane when /3 < 90? we find for
161 = e the following limit for a. We substitute
in formula (1) the value tan = 1 and find 2
(2)
FIG. 3.
1 cos ao = 1 - o
cos" 2
Because 1/cos2 e > 1 we see that cos ao < 0 and therefore ao > 90?.
The angle c = ROC measures the half length of daylight for a solar declination 6. If c > ao the sundial does not operate for the interval
VOL. 92, NO. 3, 1948] 137
Crnos sn -
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