and supposing the radii of the two ends of the axis to be unequal, and the level to be incorrect, shew how the proper corrections may be made. 6. Explain by what observations Bradley detected the aberration of light, and how he distinguished its effects from those of a nutation of the Earth's axis. 7. If the fixed stars have an annual parallax, its effect on the place of a star may be fouud from the formulæ for aberration, by supposing the longitude of the Earth to be 90° greater than it really is. 8. If a be the right ascension, and ♪ the north polar distance of a star, ▲ its co-latitude, w the obliquity, sin.2 Δ 2 =sin. {(+)+ M} sin. {(a + d) — M }, where sin.2M = sin.w sin.♪ sin.o (90 — a). Apply these formula to a Arietis, w= 23° 27′ 46′′.3. log. sin.17° 52′ 12′′ 3 = 9.4869392. log. sin.27° 34′ 11′′ ·7 = 9·6654221. log. sin.30° 24′ 15′′·75 = 9·7042361. log. sin.63° 18′ 36′′ ·3 = 9·9510705. 9. Shew how we may find the time at which the aberration of a given star in declination is 0; and apply this to a Arietis, of which the latitude 9° 57' 37", = log. sin. = 9.23796. the angle of position = 20° 39′ 52", log. tan. = 19.57653. 65° 21′ 50′′, log. tan. = 10.33857. 10. Prove the formula for the effect of precession in R. A. 50" (cos. + cot.d sin.w sin.a). da How are the stars situated for which this correction is O? 11. A known star is observed to pass the prime vertical to the east and west, and the intervening time is observed; find the latitude. 12. In the last question supposing an error of 1 second of time in the observed interval; find the corresponding error in the latitude deduced. 13. Find the position of the ecliptic at any instant, the angle which it makes with the horizon, and the point where it cuts it. 14. Explain under what circumstances Venus will be direct and retrograde; and her distance being 723 the Earth's, find how long after conjunction she will be stationary. 15. In an orbit of which the eccentricity is e, the true anomaly v, the eccentric anomaly u; if sin. = e, the equation of the centre will be greatest when 16. If u be the distance of the Sun from the solstice in R. A., y the defect from the solstitial declination, w the obliquity, y = tan.u sin.2w tan.u sin.4w + &c. 17. Supposing S' to be the true Sun moving unequally in the ecliptic, S" a point moving equably in the ecliptic, S"" a point moving equably in the equator in the same time, explain the orders of arrangement which S, S", S"" will assume in the course of a year. 18. Shew that on any hypothesis of the density of the atmosphere, the refraction 19. A ship sails perpetually due N. E.: what will be the length of its course before it reaches the pole, and the area included? 20. What day of the week was April 27, 1769? What are the rules for this calculation? 21. The sphere being orthographically projected on the horizon of a given place, give rules for describing the meridians and parallels. 22. A ladder leaning against a south wall happens to be inclined to the horizon at an angle equal to the latitude of the place: shew that the shadow will occupy the same situation every day at the same hour; and determine the position of this shadow on the wall and on the ground. 23. Shew how the difference of longitude of two places may be determined by observing the transit of a star and of the Moon at the two places. How far does this method depend on the accuracy of the lunar tables? 24. Find the place of the node of a planet's orbit from observation. 25. Find the inclination of the Sun's equator to the ecliptic, and the time of the Sun's rotation. TRINITY COLLEGE, June 1829. 1. In what respects does the received system of the universe differ from those supposed by Ptolemy and by Tycho Brahe? 2. Give a brief account of the solar system, stating what is known concerning the relative distances of the planets from the Sun, and the satellites from their primaries. 3. Explain the distinction between solar and sidereal time. Which of them is kept by the Observatory clock, and for what reason? 4. What is the nature of the observations made at the Observatory, and what is the object of them? 5. At what time of the day, when the Sun's declination is 20° 16′, will the length of the shadow of a vertical gnomon be just equal to the height of the gnomon itself? 6. Explain the method of finding the latitude of a place from two equal altitudes of the Sun observed on the same day, including the correction for a change of declination during the interval. 7. What is meant by the first point of Aries? How does it happen to be in the constellation Pisces? How is its place at any time determined? 8. The right ascension of a star being 13h 14′ 18′′, its declination 10° 4′ 31′′, and the obliquity of the ecliptic 23° 28'; required the star's latitude and longitude. 9. If z represent the zenith distance of a star, t the excess of the height of the thermometer above 50°, 6 the height of the barometer, the refraction is 10. Explain the effect of parallax. Which of the heavenly bodies does it affect? How is the quantity of it determined? 11. Investigate the alteration caused in the right ascension and declination of a star by the precession of the equinoxes. 12. Explain the phenomenon of the aberration of light. 13. Explain the method used to find the time, magnitude, and duration of a lunar eclipse. 14. Explain the method of constructing a sun-dial to be affixed to a given vertical wall. 15. What kind of sun-dial should be used at the equator? 16. Explain the method of drawing a meridian line. 17. In latitude 39° 54′ N. longitude 35° 30 W. the altitude of the Sun's lower limb, on the 7th of May, at 5h 30′32′′ P. M. per watch, was observed to be 15° 40′ 57′′; required the error of the watch. 18. On the 15th of May, in altitude 33° 10′ N., and longitude 18° W., about 5 o'clock, A. M. the Sun was observed to rise E. by N. Required the variation of the compass. TRINITY COLLEGE, May 1830. 1. GIVE a brief sketch of the motions which are observed to take place among the heavenly bodies. 2. Define the terms latitude and longitude, as applied to places on the surface of the Earth, and enumerate the methods employed to determine them. 3. What is meant by an error in the line of collimation of a transit telescope? How is it detected? 4. Detail the method used to draw a meridian line, with the correction for the change in the declination of the Sun during the interval of the observations. 5. The altitude of the Sun's centre was observed one afternoon in London, (latitude 51° 32 ) to be 38° 19′, its declination being at that time 19° 39′ N. Required the hour of the day. 6. The latitude of a Star being 16° 3′ S. Its longitude 2s 25° 52′, and the obliquity of the ecliptic 32° 28′, required the sidereal time of its passage over the meridian. 7. It is required to calculate the time of the Sun's rising, in London, from the following data : = 17° 33′ Sun's semidiameter = 16′ Sun's declination Horizontal refraction = 32' Parallax = 9'. 8. Prove that for small zenith distances the refraction varies nearly as tan.z. 9. Explain the nature of parallax, and determine the amount by which it affects the apparent diameter of the Moon. 10. Explain what is known, as a matter of fact and observation with respect to the precession of the equinoxes and nutation of the Earth's axis. 11. Find the effect of aberration on the latitude of a Star. Show what it becomes when the Star is supposed to be at the pole of the ecliptic. 12. Explain the terms mean and true anomaly, and show how the latter is to be deduced from the former, in an elliptic orbit of small eccentricity. |