87 TIME TO ELAPSE BEFORE OR ELAPSED SINCE MIDDAY Therefore, the Rsine of the corresponding arc of the equator, i. e., Rsin 2 x Rsin € Rsin m X. r so that Therefore, = Rsin 2 x Rsin € day-radius X₁ Rsin (90⁰ m) 1 R² (Rsin m)². 5400' -m= Rsini [Rsin (90° — m)], which gives the required time in asus, because 5400'- m is the number of minutes in the arc of the equator lying between the Sun's hour circle and the local meridian. Rcos Rsin An alternative rule: 40. Or, multiply the Rsine of the Sun's prime vertical zenith distance by the radius and divide by the day-radius. Then applying the method of finding out the arc (correspond. ing to a given Rsine), convert the resulting Rsine into (the corresponding) arc. Then reduce the asus (thus obtained) to nadis, etc. These lie between the (prime vertical) Sun and the meridian. SA = Rsin z, SB = LSAB = 90°, and LSBA= the angle between the Sun's day-radius, Rcos Rsin In Fig. 9, let S be the Sun on the prime vertical, B the centre of the Sun's diurnal circle, and A the point where the plane of the Sun's diurnal circle intersects the zenith-nadir line. Then in the plane triangle SAB, we have Rsin LSBA= hour circle and the local meridian, z being the Sun's zenith distance. Rsin z X R day-radius A Fig. 9 = arc of the equator intervening between the Sun's hour circle the local meridian. Therefore, we have B
