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9: Oblique ascensions of the ecliptic for latitude +40◦ . 10: Oblique ascensions of the ecliptic for latitude +50◦ . 11: Oblique ascensions of the ecliptic for latitude +55◦ . 12: Oblique ascensions of the ecliptic for latitude +60◦ . 13: Oblique ascensions of the ecliptic for latitude +65◦ . 14: Oblique ascensions of the ecliptic for latitude +70◦ . 15: Oblique ascensions of the ecliptic for latitude +75◦ .

Here, a and A are the altitude and azimuth, respectively, of this point in the sky. 51) is a tangent to the altitude circle passing through point Y in the direction of increasing altitude. It follows from the definition of parallactic angle, and elementary vector algebra, that cos µ = ∂r ∂r · ∂λ ∂a = − sin λ cos A v × e · r + sin λ sin A v × n · r + cos λ cos A s × e · r − cos λ sin A s × n · r. 52) However, according to Eqs. 55) s × n · r = cos λ (cos L cos ǫ + sin L sin ǫ sin α0). 56) The previous five equations can be combined to give cos µ = − cos A sin ǫ cos(α − t) − sin A [cos L cos ǫ + sin L sin ǫ sin(α − t)] .

Note that the value of µ at t = 0 represents the inclination of the ecliptic to the vertical as point Y culminates. Furthermore, the values of µ at t = th (corresponding to a = 0◦ ) represent the inclination of the ecliptic to the vertical as point Y rises and sets. 26 show the altitudes of twelve equally spaced points on the ecliptic, as well as the parallactic angle at these points, as functions of time, calculated for a series of observation sites in the earth’s northern hemisphere with equally spaced terrestrial latitudes.

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