Diurnal Aberration. 11. Find the diurnal aberration of right ascension and declination of Polaris for Jan. 1, 1839, and latitude 45°, when the hour angle is 0' 30". R-8".04 0.53 0.9050 D=0".03 8.4564 12. Find the diurnal aberration of & Ursa Minoris in right ascension and declination for Jan. 1, 1839, and latitude 0°, when the star is upon the meridian. Dec. of Ursa Minoris = 86° 35'. Ans. R— 08.35 & D = 0. Refraction of a star. CHAPTER X. REFRACTION. 120. Light proceeds in exactly straight lines, only in the void spaces of the heavens; but when it enters the atmosphere of a planet, it is sensibly bent from its original direction according to known optical laws, and its path becomes curved. This change of direction is called refraction; and the corresponding change in the position of each star is the refraction of that star. 121. Problem. To find the refraction of a star. Solution. Let O (fig. 50) be the earth's centre, A the position of the observer, AOK the section of the surface formed by a vertical plane passing through the star. It is then a law of optics, that Astronomical Refraction takes place in vertical planes, so as to increase the altitude of each star without affecting its azimuth. Let, now, ZIH be the section of the upper surface of the upper atmosphere formed by the vertical plane, SI the direction of the ray of light which comes to the eye of the observer. This ray begins to be bent at I, and describes Ratio of sines in the law of refraction. the curve 14, which is such, that the direction AC is that at which it enters the eye. Let, now, 4 = ZAC the ✶'s apparent zenith distance, r the refraction, the diff. of directions of AC and IS, Again, it is a law of optics that the ratio of the sines of the two angles LIS and ZAS' is constant for all heights, and dependent upon the refractive power of the air at the observer. Denote this ratio by n, and we have sin. (ur) =n, sin. P (704) and if U and R = the values of u and r at the horizon, sin. φ sin. (¶—u+r) = n = cos. (U — R). (705) 1 cos. (UR) (706) simp-sing = 2 cos{ (p+q) sin ([p-q) Approximate refraction. and since (ur) is small, ≥ (u—r) = N tang. [4 — § (u — r)]. Again, to find u, the triangle COA gives (708) (709) Now the point Cis at different heights for different zenith distances of the star; but this difference in the values of OC is small, and may be neglected in this approximation; so that (708), which, compared with this rough value of (ur) from and the values of m and p must be determined by observation; their mean values, as found by Bradley, and adopted in the Navigator, are m = 57".035, p=3, by which Table XII is calculated. (720) 122. The variation in the values of m and p for different altitudes of the star, can only be determined from a knowledge of the curve which the ray of light describes. But this curve depends upon the law of the refractive power of the air at different heights; and this law is not known, so that the variations of m and p must be determined by observation. At altitudes greater than 12 degrees, the mean values of m and p are found to be nearly constant, and observations at lower altitudes are rarely to be used. 123. The mean values of m and p, which are given in (720), correspond to the height of the barometer = 29.6 inches, (721) Now the refraction is proportional to the density of the air; but, at the same temperature, the density of the air is proportional to its elastic power, that is, to the height of the barometer. If then |