10. At a given place, on a given day, at a given hour, let the great circle which joins two known stars be produced to cut the horizon find the points and angles of intersection. : 11. Find the change in the refraction at a given altitude, depending on the barometer and thermometer. 12. The mean right ascension of the star a Aquile for the year 1830 is 19h. 42m. 298 34, and its north polar distance 81°. 34′. 26" 7. Find whether its meridian altitude, at a given place, will be greater or less this day next year than it is to-day: shew how the magnitude of the variation may be computed. 13. On a given day find how much, and which way, the right ascension of a known star is affected by aberration; and on what days the effect is nothing or a maximum. 14 Given the mean anomaly of a planet; find its true anomaly. 15. The orbit of a planet being supposed a circle, whose radius is known nearly; approximate to the true radius. 16. Given the synodic time of a superior planet; find its periodic time. 17. A planet is a morning star, and stationary: when it begin to move again, find which way it will move : (1). Supposing it an inferior; (2). Supposing it a superior planet. 18. Find the eccentricity of the lunar orbit. 19. In the year 1823 there were seven eclipses, and the first was an eclipse of the Sun: state the others in the order in which they happened, and as near as you can the times of the year at which they happened. 20. Given the periodic times of the Earth and Venus, equal to 365 256 and 224-7 days, respectively, find the periods at which transits of Venus may be expected to recur. 21. Construct a horizontal dial for a place on the equator. SIDNEY SUSSEX COLLEGE, MAY 1831. 1. DEFINE the latitude of a place on the Earth's surface: by what angle within the Earth is it measured when the Earth is a sphere, and by what angle when the Earth is a spheroid? Find the latitude from two equal altitudes of the Sun observed before and after noon. 2. Supposing the Earth an oblate spheroid, find its radius, at any point, in terms of the latitude. 3. Investigate Flamstead's method of finding the right ascension of the Sun. 4. Investigate the reduction of the ecliptic to the equator in terms of the longitude. 5. Find the altitude and azimuth of a known star when that arc of the vertical circle passing through it which is intercepted between the star and the ecliptic, is a minimum. 6. Explain the differences between sidereal time, true solar time, and mean solar time: find the length of a sidereal day in mean solar time, and that of a mean solar day in sidereal time. 7. At a given place on a given day, find the time and place of sunrise, the length of the day and night, the Sun's meridian altitude and midnight depression. 8. On a given day at a given place, find the spherical area swept out by the arc ZS from sun-rise to sun-set. 9. On a given day at a given place, the true time is to be found from observing the Sun's altitude: find at what time in the morning the observation must be made, so that a small error in the observed altitude may have the least effect on the result. 10. On a given day at a given hour, let a globe of given radius be suspended in the air: having given the dimensions of its shadow cast by the Sun on the ground, find the latitude of the place, and the direction of the meridian line. Determine under what circumstances this problem admits of two answers. 11. Explain the construction and use of the transit instrument, and shew how to adjust its axis and line of collimation. 12. Investigate Brinkley's formula for the refraction, the atmosphere being homogeneous. 13. Explain the nature of parallax, and its effects on the hour angle and declination of a known body. 14. Find the aberration of a given star in declination: and trace the changes, arising from this cause, in the meridian altitude of the star, throughout the year. 15. Explain the terms 'Mean Anomaly' and 'True Anomaly :' and having given the mean anomaly, find the true. 16. Shew when that part of the equation of time which arises from the obliquity is additive or subtractive. 17. The orbits of the Earth and planets being circles, shew when an inferior or superior planet will appear stationary, when to be moving direct, and when retrograde. 18. Explain the Moon's librations in longitude and latitude, and account for them. 19. Compute the time of a solar eclipse at a given place. 20. Explain fully the process of finding the longitude by observing the distance between the Moon and the Sun or a fixed star. 21. Explain the stereographic projection of the sphere; and shew that the angle between any two circles on the surface of the sphere is the same as that between their stereographic projections. 22. An horizontal dial constructed for latitude l is fixed in a place whose latitude is 7+ 0, 0 being small when compared with 7: find the correction, which must be applied to the time shewn by the dial, in order to obtain the true apparent time. NEWTON, AND CONIC SECTIONS. TRINITY COLLEGE, 1821. 1. EXPLAIN the method of indivisibles, and compare the area of a parabola with that of the circumscribing parallelogram, by that method. 2. Prove that the ultimate ratio of the chord, arc, and tangent, is a ratio of equality. 3. Shew that the curvature of the semi-cubical parabola is infinitely greater, and that of the cubical parabola infinitely less, than that of the common parabola. 4. Compare the forces to two points within a circle, the periodic times being supposed different. 5. Find the variation of the force when the body moves in an ellipse, the force acting in a direction parallel to the ordinates. 6. Find the horizontal velocity in a cycloid, when the force acts parallel to the axis. 7. When is the paracentric velocity a maximum? In what point of all conic sections is it so? Does it admit of a maximum in a circle, the centre of force being in the circumference? 8. Shew fully that if a body move in a logarithmic spiral, the ; and compare the time of describing this spiral with that of describing a circle at distance SP. 9. Investigate the relation between the centripetal and centrifugal forces, the equation to the curve in which they are equal, and the law of force by which it will be described. 10. Shew that round different centres, the periodic times in all 11. Required the law of force in a parabola. 12. Determine that point in an ellipse, force in focus, in which the velocity is an arithmetic mean, and also the point in which it is a geometric mean, between the velocities at the greatest and least distances, 1 13. Force of gravity given the periodic time of the Moon, and her distance from the Earth; required how far a body falls in 1" at the Earth's surface. 14. The rectangle contained by two perpendiculars drawn from the foci of an ellipse to the tangent at any point, is equal to the of the semi-axis minor. square 15. To draw a tangent to any conic section from a given point without, which is not the centre of the hyperbola. 16. In an ellipse whose eccentricity is small, the increment of the radius vector, in moving from the extremity of the axis minor to that of the axis major, varies as the square of the sine of the angle through which it has passed. 17. To find the curve that cuts any number of similar concentric ellipses at right angles. TRINITY COLLEGE, 1821. 1. STATE the reasonings by which the following propositions are established: (1). That the planets with their satellites gravitate to the Sun; the satellites to their planets, and the Moon to the Earth. (2). That the force acting on the same body in different parts of its orbit, and on different bodies in different orbits, round the same centre of force, varies inversely as the square of the distance from that centre, that equal areas can only be described in equal times round a point, when the forces acting on the body tend to that point. |