Star, and from that time the increment in longitude will vary as the versed sine of the Sun's longitude reckoned from that point. 15. If a Star be situated very near the pole of the ecliptic, the decrement of the apparent longitude varies as the versed sine of the Sun's longitude, reckoned from a point 90 degrees behind the Star's place in the ecliptic. TRINITY COLLEGE. 1820. ASTRONOMY. 1. EXPLAIN the construction and use of the Vernier. Within what limits may angles be read off by an instrument of which the arc is subdivided to 20', and 20 divisions of the Vernier are equal to 19 of the arc ? 2. Explain the mode of correcting a small error in the meridian plane by observations made with a transit-instrument on a circumpolar star. Supposing the time between the lower and upper transit T, and between the upper and lower = T +t, work out the proper correction. 3. Determine the obliquity of the ecliptic by meridian altitudes, taken on successive days before and after the solstice, and apply the proper corrections. 4. Explain the mode of determining the lengths of a sidereal and solar year. 5. Assuming the length of a solar year to be 365d. 5h. 48" 48", determine the correction of the civil year, in order that it may always nearly coincide with the solar. 6. Under what circumstances is a star said to rise or set cosmically, achronically, and heliacally? 7. "Hesiod says that sixty days after the winter solstice the star Arcturus rose at sun-set: from which it follows that Hesiod lived about 100 years after the death of Solomon." (Sir Isaac Newton's Chronology). Exhibit the calculations on which this conclusion is founded. 8. When the Sun is in either of the equinoctial points, determine the locus of the extremities of the shadow of a perpendicular style on a horizontal plane. 9. When the Sun is in either of the equinoctial points, and the style of the dial is perpendicular to the horizontal plane on which the hour-lines are drawn, determine the construction of the dial. 10. Upon what experiments does it appear that in the passage of a ray of light through a variable medium like our atmosphere, the sine of incidence is to the sine of refraction in a constant ratio? 11. Explain the mode by which Bradley obtained the following formula: Refraction = a 400 29.6 X tan. (-3) x 57" x 350 + h a altitude of the barometer in inches 29.6= mean altitude of do. z = zenith distance r57" x tan z h height of Fahrenheit's thermometer in inches. = 12. Define parallax, and determine the law of its variation for the same body at different altitudes. 13. Explain the mode by which (a) the effect of parallax in right ascension may be observed, and prove that the horizontal 15 xa x cos. dec". parallax= cos. lat. x sin. hour-angle * 14. If the velocity of the Earth be in a finite ratio to the velocity of light; (1) find the direction in which a telescope must be held, in order that a given heavenly object may appear in its axis: (2) On the same hypothesis shew that a ray of light proceeding from a heavenly body must strike the retina at a different point from what it would do if the eye of the spectator were at rest; and therefore, by the laws of vision, that the apparent place will differ from the true place of the body. Shew that the quantity and direction of aberration are the same on either of the two preceding considerations. 15. Write down the expressions for the aberration in latitude and longitude, and determine for any given star, when the corrections are positive, and when negative. 16. Bradley observed, (1) That each star was farthest north when it came to the meridian about six o'clock in the evening, and farthest south when it came about six in the morning." (2) "That in both stars the apparent difference of declination from the maxima, was always nearly proportional to the versed sine of the Sun's distance from the equinoctial points." Confirm these observations, and shew that they only apply to stars situated near the solstitial colure. 17. Prove that in elliptical orbits of small eccentricity, the greatest equation to the centre is twice the eccentricity. 18. Explain the mode of correctly determining the longitude of the Earth's apogee: and state at what era in the history of mankind the line of the apsides coincided with the line of the equinoxes. 19. Given the place of a planet as seen from the Earth, find its place as seen from the Sun, exhibiting the formulas of heliocentric latitude and longitude. 20. Account for the Moon's vibration in longitude. 21. Find the lunar and solar ecliptic limits; and thence determine the greatest and least number of eclipses, of either kind, that can happen in one year. 22. Suppose the Moon's right ascension to be exactly known, when the Sun is on the meridian; determine when the Moon's centre will be on the meridian. 23. Determine the difference of the longitudes of two places on the Earth's surface, by observations on the passage of the Moon's centre over the meridian. 24. If a small error be made in the assumed distance between two meridians, shew how that error may be corrected by observations on the occultation of a fixed star. TRINITY COLLEGE. 1820.. NEWTON AND CONICS. 1. EXPLAIN by short examples, the method of exhaustions, of indivisibles, and of prime and ultimate ratios. 2. Prove that if a radius vector be drawn bisecting any arc, it must ultimately bisect the chord. 3. If a straight line EDA make with the curve CBA a given angle at the point A, and the ordinates CE, BD be drawn; the triangles ACE ABD, are ultimately in the duplicate ratio of the sides. 4. Let AB be the subtense of the arc, AD the tangent, BD the subtense of the angle of contact perpendicular to the tangent, as in the 11th lemma: then let a series of curves be drawn in which DB ∞ AD1, AD", AD", &c., the angle of contact in each succeeding case will be infinitely less than in the preceding. 5. If the areas described by the radius vector are not propor tional to the times, the revolving body is not acted on solely by a force towards a fixed centre. 6. If a body be acted on by a given force and revolve in a circle, the arc described in any given time is a mean proportional between the diameter of the circle and the space through which a body would descend in the same time from rest if acted on by the same force. 7. The velocity at any point of a curve described round a centre of force the velocity which a body, acted on by the given force at that point, would acquire by descending through part of the chord of curvature. 8. Given the force of gravity earth 4000 miles; deduce a numerical comparison between the 32 feet, and the radius of the force of gravity and the centrifugal force at the equator. 9. If a heavy body be whirled round in a vertical plane, and the centrifugal force at the top just keep the string extended; what will be the tension of the string at the lowest point of rotation? 10. In any orbit, let x = dist. p = perpendicular on the tangent: centripetal force o dp p3 dx Apply this expression to determine the law of the force in an ellipse round the centre, and in a circle with the centre of force in the circumference. 11. Deduce expressions for the chord of curvature passing through the focus, and the diameter of the curvature at any point of an ellipse. 12. All parallelograms described about any conjugate diameters of a given ellipse or hyperbola are of equal area. 13. Compare the centripetal and centrifugal forces at any point of an orbit; prove that in an ellipse round the centre, there are four points where these forces are equal. 14. Prove (Newton, Prop. XI.) that GvxvP : Qu2::CP2 : CDa. 15. The perpendicular from the focus of a parabola upon the tangent is a mean proportional between the focal distances of the point of contact and the vertex. 16. Prove that the force tending to the focus of a para bola c 1 D2 17. The velocity of a body revolving in a parabola round the focus the velocity of a body revolving in a circle at half the distance. 18. If two bodies revolve in an ellipse in the same periodic time; one about the focus, and the other about the centre; compare the forces towards these centres at the extremities of the major axis, and find the distance from the centres at which the forces are equal. 1 D2 19. If the force and a body be projected in any direc tion, except directly to or from the centre of force; prove that it will describe a conic section, and point out the relation between the velocity of projection and the particular curve described. APPENDIX. TRINITY COLLEGE. 1820. NEWTON'S PRINCIPIA. Book I. 1. (1) THE centripetal force (F) in any curve Q. dp p❜dx being the perpendicular from the centre of force on the tangent, at distance (x). Determine Q. (2) Find the value of (F) in the ellipse-the force tending to the centre. 2. If a body be acted on by two forces tending to two fixed centres, it will describe, about the straight line joining those centres, equal solids in equal times. 3. A body describes a parabola about a centre of force situated in the focus: (1) Find its position at any assigned time. (2) Given two distances from the focus, and the difference of anomalies. Find the true anomaly. 4. The time of a body's descent, in a right line, towards a given 1 centre of force varies as (dist.) law of the variation of the force. from that centre. Required the 5. A body at P is urged by an uniformly-accelerating force in the direction PS, and at the same time is impelled in the opposite 1 direction by a force varying as (dist.) from S. Find its velocity at any point N. 6. In the logarithmic spiral find an expression for the time of a body's descent from a given point to the centre, and prove that the times of successive revolutions are in geometrical progression. from the centre, 7. A body acted on by a force varying as 1 (dist.)" is projected from a given point, in a given direction, and with a given velocity. (1) Find the equation to the trajectory described. (2) Determine in what cases the body will fall into the centre, or go off to infinity. (dist.)5 shew under what restrictions of the velocity of projection, the body's approach towards the |