सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

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may further note that at the north pole, the altitude of
the Sun is always δ so that the length of the shadow cast is
always equal to cot δ and this will be infinite when δ = 0.
If now φ = 90°, and δ ≠ 0, though A = cos φ / sin δ be
comes zero, A tan φ will not be zero because A tan φ =
cos φ / sin δ × tan φ = sin φ / sin δ = 1 / sin δ (∵ φ = 90°). On the
other hand A² tan φ = A × A tan φ = / sin δ = 0 because
A = 0. Thus the term containing y in eqn. (3) vanishes.
∴ The equation reduces to
x² + y² = A² tan² φ - 1 = cosec² δ - 1 = cot² δ (ie. +ve)
ie. at the north pole the locus will be a circle with radius
cot δ. When
A = ∞ the locus √(x² + y² + 1) / A = y + tan φ becomes
y = - tan φ which means that the hyperbola degenerates
into the straight line which is parallel to the east-west
line and is in the north at a distance of tan φ ie. s, the
equinoctial shadow since the length of the gnomon
is taken to be unity. In particular when A tan φ = 1 ie.
φ = δ, the constant in (3) is zero, so that the locus
passes through the foot of the gnomon as is also evident
from the fact that the Sun passes through the zenith.
Taking a northern latitude say 17°, the loci of the
extremity of the shadow are shown in fig. 48A (page 270)
on important days when δ=ω, when δ=0, when δ = - ω,
when δ = φ. The maximum mid-day shadow is
tan (φ + ω), when δ = - ω taking the gnomon's length
to be unity; this shadow is cast north of the gnomon along
the south-north line through the gnomon, on Dec. 23rd of
the year. The minimum length of the mid-day shadow
occurs when δ = φ, the shadow being zero and being at the
foot of the gnomon, the Sun being then just overhead. The
maximum shadow cast south of the gnomon at mid-day
is tan (ω - φ).