Scholium. Having given the mean results of observations on the terrestrial refraction, it may not be amiss, though we cannot enter at large into the investigation, to present here a correct table of mean astronomical refractions. The table which has been most commonly given in books of astronomy is Dr. Bradley's, computed from the rule = 57" x cot (a + 3r), r where a is the altitude, r the refraction, and r = 2′35′′ when a = 20°. But it has been found by numerous observations, that the refractions thus computed are rather too small.Laplace, in his Mecanique Celeste (tome iv pa. 27) deduces a formula which is strictly similar to Bradley's; for it is r=mx tan (z-nr), where ≈ is the zenith distance, and m and h are two constant quantities to be determined from observation. The only advantage of the formula given by the French philosopher, over that given by the English astronomer, is, that Laplace and his colleagues have found more correct coefficients than Bradley had. KR n Now, if R = 57°-2957795, the arc equal to the radius, if we make m = = (where k is a constant coefficient which, as well as n, is an abstract number,) the preceding equation will become =kx tan (z-nr). Here, as the refraction ris R always very small, as well as the correction nr, the trigonometrical tangent of the arc nr may be substituted for we shall have tan nr = k. tan (z— nr). But nr=42-(1z — nr) 2 = - nr) tan ( 마음 Hence, sin (z- 2nr) = This formula is easy to use, when the coefficients n and 1-k 1+k are known: and it has been ascertained, by a mean of many observations, that these are 4 and 99765175 respectively. Thus Laplace's equation becomes sin (z-8r) = 99765175 sin z: and from this the following table has been computed. Besides the refractions, the differences of refraction, for every 10 minutes of altitude, are given; an addition which will render the table more extensively useful in all cases where great accuracy is required, Table PROBLEM XI. To find the Angle made by a Given Line with the 1. The easiest method of finding the angular distance of a given line from the meridian, is to measure the greatest and the least angular distance of the vertical plane in which is the star marked a in Ursa minor (commonly called the pole star), from the said line: for half the sum of these two measures will manifestly be the angle required. 2. Another method is to observe when the sun is on the given line; to measure the altitude of his centre at that time, and correct it for refraction and parallax. Then, in the spherical triangle zps, where z is the zenith of the place of observation, P the elevated pole, and s the centre of the sun, there are supposed given zs the zenith distance, or co-altitude of the sun, PS the co-declination of that lu minary, Pz the co-latitude of the place of observation, and ZPS the hour angle, measured at the rate of 15° to an hour, to find the angle szp between the meridian PZ and the vertical zs, on which the sun is at the given time. And here, as three sides and one angle are known, the required angle is readily found, by saying, as sine zs: sine ZFS :: sine PS : sine pzs; that is, as the cosine of the sun's altitude, is to the sine of the hour angle from noon; so is the cosine of the sun's declination, to the sine of the angle made by the given vertical and the meridian. Note. Many other methods are given in books of Astronomy; but the above are sufficient for our present purpose. The first is independent of the latitude of the place; the second requires it. PROBLEM XII. To find the Latitude of a Place. The latitude of a place may be found by observing the greatest and least altitude of a circumpolar star, and then applying to each the correction for refraction; so shall half the sum of the altitudes, thus corrected, be the altitude of, the pole, or the latitude. For This method is obviously independent of the declination of the star: it is therefore most commonly adopted in trigonometrical surveys, in which the telescopes employed are of such power as to enable the observer to see stars in the day-time: the pole-star being here also made use of. Numerous other methods of solving this problem likewise are given in books of Astronomy; but they need not be de tailed here. Corol. If the mean altitude of a circumpolar star be thus measured, at the two extremities of any arc of a meridian, the difference of the altitudes will be the measure of that arc : and if it be a small arc, one for example not exceeding a degree of the terrestrial meridian, since such small arcs differ extremely little from arcs of the circle of curvature at their middle points, we may, by a simple proportion, infer the length of a degree whose middle point is the middle of that arc. Scholium. = = Though it is not consistent with the purpose of this chapter to enter largely into the doctrine of astronomical spherical problems; yet it may be here added, for the sake of the young student, that if a right ascension, d declination, altitude, λ= longitude, p= angle of position (or, the angle at a heavenly body formed by two great circles, one passing through the pole of the equator and the other through the pole of the ecliptic), i inclination or obliquity of the ecliptic, then the following equations, most of which are new, obtain generally, for all the stars and heavenly bodies. = 1. tan a = tan λ. cos i — tan l. sec a. sin i. 2. sin d = sin λ. cos l. sin i + sin l. cos i. 3. tan sin i tan d. sec a + tan a. cos i. sin d. COS i sin a. cos d. sin i. 4. sin 5. cotan p = cos d. sec a. cot i 6. cotan pcos . sec A. cot i 7. cos a COS d = cos l. cos λ. 8. sin p. cos d = sin i. cos λ. 9. sin p. cos λ= sini. cos a. 10. tan tan λ. cos i. Į when 11. cos λ = cos a cos d. sin d. tan a. = sin . tan λ. 70, as is always the case with the sun. The The investigation of these equations, which is omitted for the sake of brevity, depends on the resolution of the spherical triangle whose angles are the poles of the ecliptic and equator, and the given star, or luminary. PROBLEM XIII. To determine the Ratio of the Earth's Axes, and their Actual Magnitude, from the Measure of a Degree or Smaller Portion of a Meridian in Two Given Latitudes; the earth being supposed a spheroid generated by the rotation of an ellipse upon its minor axis. E Let ADBE represent a meridian of the earth, DE its minor axis, AB a diameter of the equator, M, m, arcs of the same number of degrees, or the same parts of a degree, of which the lengths are measured, and which are so small, compared with the magnitude of the earth, that they may be considered as coinciding with arcs of the osculatory circles at their respective middle points; let Mo, mo, the radii of curvature of those middle points, be = R and r respectively; MP, mp, ordinates perpendicular to AB: suppose further CD=C; CB = d; d2. c2 = e2; CP = x; cpu; the radius or sine total = 1; the known angle вSM, or the latitude of the middle point M, L; the known angle Bsm, or the latitude of the point m, = 7; the measured lengths of the arcs м and m being denoted by those letters respectively. Now the similar sectors whose arcs are м, m, and radii of curvature R, r, give R:r:: M: m; and consequently Rm= The central equation to the ellipse investigated at p. 29 of this volume gives PM = ¿√(d2 — x2); pm = ́√ (d2 — u2); ΤΜ. c2x c'u C also SP sp(by th. 17 Ellipse). And the method = of finding the radius of curvature (Flux. art. 74, 75), applied to the central equations above, gives On the other hand, c4d the triangle SPM gives SP: PM :: cos L: sin L; · : √√ (d2 — x2): : cos L : sin L; whence a2 = And from a like process there results, u2 = that is, d4 cos2 L d-e sin L' d4 cos2 l d2-e2 sin2 l* Substituting in the equation Rmrм, for R, and r their values, |