Ꮓ Z A T S 142. An approximation to the law of refraction in terms of the apparent distance of the celestial body from the zenith of the observer is thus investigated: supposing the atmosphere to be of uniform density, the earth to be a plane, and the upper surface of the atmosphere parallel to that plane. Let o be the place of the observer, s the true place of a star, SA the incident ray, and AO the course of the ray in the atmosphere: produce OA to T; then T may represent the apparent place of the star. Let zo be a vertical line passing through the spectator, and z'A be parallel to it; then z'As is the angle of incidence, ZAT, or its equal ZOT, the angle of refraction and also the apparent zenith distance of the star. The angle TAS is called the refraction. Let the angle zor be represented by z, and TAS by r; then, ZAS Z+r. Now, by Optics, the sine of the angle of incidence is to the sine of the angle of refraction in a constant sin. (z+r) ratio, the medium remaining the same; therefore is constant: let it be represented by a. Then α= sin. (z+r) or (Pl. Tr. art. 32.) a = sin. z sin. z cos. r+cos. z sin. r " sin. z whence a sin. zsin. z cos. r+cos. z sin. r, or (a-cos. r) sin. z COS. Z. = sin z. sin. r; or again, (a-cos. r) tan. z=sin. r. But since r is a very small angle, let radius (=1) be put for cos. r and r for sin. r; then, (a-1) tan. z=r, or the refraction varies as the tangent of the apparent zenith distance of the celestial body. Delambre, on developing the value of a in terms of tan. r, found that tan. r may be represented, on the above hypothesis, by the series A tan. zB tan. 3z+C tan. 5z+&c.; and he shows that all the formula which had before his time been employed, on the supposition that the surface of the earth and the strata of the atmosphere are spherical, might be brought to that form. The coefficients A, B, C, &c., are supposed to be determined by comparisons of observations. 143. By direct experiments on the refractive power of the air we have a=1.00028 (Barom. 30 in., Fahrenheit's Thermom.=50°): therefore, = and the formula r=57".817 tan. z is very near the truth when the zenith distance does not exceed forty-five degrees. 144. Dr. Bradley, guided probably by a comparison of observations with the above formula, was led to adopt one which may be represented by r=m tan. (z-nr); and he found that the observations were represented by making m=56′′.9, n=3. Subsequently MM. Biot and Arago, from numerous observations found m=60".666 and n=3.25, the latitude of the observer being 45°, the temperature of the air =32° Fahr., and the pressure of the atmosphere being represented by the weight of a column of mercury 30 in. Eng. And, lastly, Mr. Groombridge (Phil. Trans. 1824) by a mean of the refractions obtained from the observed meridian altitudes of sixteen circumpolar stars, found m (commonly called the constant of refraction) 58".133 and n=3.634. In a paper on "Astronomical Refractions" in the Memoirs of the Astronomical Society (vol. ii. part 1.) Mr. Atkinson, by an investigation conducted on optical principles, found for the value of refraction the expression 57".754 tan. (z-3.5937 r), the barometer being at 29.6 inches and Fahrenheit's thermometer at 50°. The table of refractions in the Nautical Almanacs from 1822 to 1833 was computed from a formula given by Dr. Young in the "Philosophical Transactions" for 1819. S 145. Refractions deduced from formula, without regard to the density and temperature of the air, are called mean refractions, and these differ from that which depends merely on tan. z, chiefly on account of the curvature of the earth and of the atmospherical strata: in order to obtain their values from observations in latitudes corresponding to those of the greater part of Europe, the circumpolar stars may be employed in the following manner: let ZHO, supposed to be in the celestial sphere, represent part of the meridian of the observer at C; let z be his zenith, and P the pole of the equator; and let s, s' be the apparent places of any star at the lower and upper culmination. Also, let zs=z, zs'=', and let r and r' (=ss and s's') be the corresponding refractions; then, z+r and rare the true zenith distances at the two times of culminating; and since at those times, after the corrections for refraction have been applied, the star is equally distant from the pole, 1⁄2 (z + r + z + r) or 1 (z+z′+r+r') will be equal to PZ, the colatitude of the station; therefore 2PZ- (z+z′) =r+r'. Hence, PZ being accurately known, also z and z' being given by the observations, and the value of r' (the refraction at the H upper culmination) being computed from the formula r=57′′.817 tan. z', which will be sufficiently near the truth, the value of r may be obtained. The columns entitled " Alt." contain the apparent altitudes of a celestial body as they are obtained from the observations after being corrected for the index error; and immediately on their right hand are the columns containing the refractions corresponding to the altitudes, the barometer being at 30 inches, and Fahrenheit's thermometer at 50°. The columns headed "Var." contain the variations of the refraction for one minute of altitude; and the number opposite the nearest degree or minute in the column of altitudes is to be multiplied by the difference between the given number of minutes and such nearest degree or minute: the product is to be subtracted from, or added to the refraction corresponding to the nearest degree or minute, according as the latter is less or greater than the given altitude. The columns entitled "Bar." contain the corrections for the actual state of the barometer; and the number in the column being multiplied by the inches, or decimals of an inch by which the height of the mercury exceeds or falls short of 30 inches, the product is, in the former case, to be added to, and in the latter to be subtracted from, the refraction corrected, as in the preceding step. The columns headed "Therm." contain the corrections for temperature; and the number being multiplied by the number of degrees by which the mercury in the thermometer exceeds or falls short of 50°, the product is, in the former case, to be subtracted from, and in the latter to be added to, the refraction corrected, as in the preceding steps. 146. The refractions in the tropical regions appear to differ but little from those which are given in the tables computed for Europe, on making due allowance for the temperature of the air. According to the observations of Colonel Sabine (Pendulum Experiments, p. 505.) the difference does not exceed half a minute when the celestial body is in the horizon: but in the arctic regions the case is far otherwise; for according to observations made between 1822 and 1830, at Igloolik (Lat. 69° 21' N., Long. 81° 42' w.) at an altitude equal to 8' 40" the refraction was found to be 1° 21′ 19′′, the temperature being 28 degrees below the zero of Fahrenheit's scale, while from the existing tables, for that altitude and a like temperature, the refraction would be but 41' 57". In those regions the amount of refraction may be obtained by comparing the declinations of fixed stars, deduced from their observed altitudes with the apparent declinations given in the Nautical Almanac. 147. The effects of refraction on the polar distance and right ascension of a star when not on the meridian, being sometimes required for the adjustments of astronomical instruments, the following process for obtaining those effects sufficiently near the truth is here introduced. P Let P be the pole of the equator, z the zenith of the station, s the true, and s' the apparent place of the celestial body: then PZ the colatitude of the station, zs the true zenith distance of the star, and ZPS the star's hour angle, being given, we have (art. 61.) sin. zs sin. PZ :: sin. ZPS: sin. PSZ. But, letting fall st perpendicularly on PS' produced, the angle PSZ may be considered equal to ss't: and ss't being considered as a plane triangle, also ss' being taken S H H S' Q from tables of refractions or computed from the formula |