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

269 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).