7. The resistance on a cone moving in the direction of its axis : ditto on its base as 3:4; required the angle at the vertex of the

cone.

QUEEN'S COLLEGE, 1828.

1. THE weight of a vessel being given when empty, when filled with water of given specific gravity, and when filled with some other fluid; find the specific gravity of that fluid.

2. A rectangular flood-gate turns on a horizontal axis; find the position of the axis, that the water having risen to the top of the gate may just, by its pressure, be able to open it.

3. Compare the time of emptying a sphere with the time of emptying its circumscribing cube through a small orifice at the bottom.

4. A cylindrical tube of given dimensions, closed at one end and partly exhausted is immersed vertically to a given depth, when the height to which the water rises in it is observed. Hence find the quantity of air in the tube.

5. Construct the common pump, and find the height to which the water is raised by the two first ascents of the piston.

6. A tube of considerable length, but of small dimensions in other respects, is bent into the form of a circle and made to revolve about a vertical diameter. If 30° be occupied by water the rest being a vacuum; required the angular velocity that the lowest quadrant may be deserted by the water.

QUEEN'S COLLEGE, MAY 1829.

1. A CONE and a paraboloid of the same altitude, floating with their vertices downwards, have each 4th of the axis above water; compare their specific gravities.

2. Compare the resistance on the arc of a plane curve moving in a fluid, in the direction of its axis, with the resistance on the base; and apply the formula to the case of a semicircle.

3. A floodgate moving about a vertical axis, is divided by it into two parts, one of which is a rectangle, and the other an inverted

Given the base of the

right-angled triangle of the same altitude. triangle, find the width of the rectangle, so that the gate may just open by the pressure of the water, when it has risen to the top.

4. A parabolic vessel, terminated by the plane, at right angles to the axis, passing through the focus, empties itself by an orifice at the vertex. Compare the spaces described by the descending surface, in the first and last halves of the time.

5. A globe descends from rest by its gravity in a medium the resistance of which varies as (vel). Determine its velocity at any point, and the time of descending through a given space.

6. A small orifice is made in a vessel containing condensed air: determine the time of discharge till the density within equals that without.

7. A tube AB, open at A and closed at B, containing a piston of given weight and dimensions, between which and the closed end, air is included, is whirled about A in a horizontal plane. Given the whole length of the tube, also the length occupied by the air before and after the motion; to find the angular velocity.

8. What must be the dimensions of a balloon, the whole weight of which, with its appendages, is 800 pounds; that it may just rise two miles high; supposing air to have 10 times the specific gravity of gas under the same pressure; that 5 cubic feet of air at the earth's surface weighs 6 ounces; and that the density at the earth's surface is four times as great, as at the height of 7 miles?

QUEEN'S COLLEGE, MAY 1830.

1. A TRIANGLE is formed by drawing two straight lines from two given points in the diagonal of a square, to one of the opposite angles : the square is then immersed vertically to a given depth, so that one of the sides adjacent to the said angle is parallel to the surface; compare the pressures on the three sides of the triangle.

2. A flood-gate turns on a vertical axis, the area on one side of the axis being a right-angled triangle with the base lowest, and on the other side a rectangle, of the same altitude. Required the width of the rectangle, so that the gate may just open by the pressure of the water, when it has risen to the top.

3. Find the depth of a parabolic vessel which discharges that portion of the fluid that is above the horizontal plane passing through the focus, in the same time with that portion which is below the focus.

4. A cylindrical vessel of given dimensions and full of water is pierced at a given point; find another point in which if it be pierced, the fluid issuing from each orifice may fall on the horizontal plane on which the vessel stands, at the same point.

5. If an inverted hemisphere full of air be forced down, so as just to be immersed in mercury; construct the equation from which (x), the height to which the mercury rises within the hemisphere, is to be determined.

6. Explain the principle of the common pump, and find the altitude to which the water rises in consequence of the two first strokes.

7. Find the time in which the air rushing into an exhausted receiver through a given orifice will have acquired a given density within. The (velocity) varying as the pressure, and h being the height due to the initial velocity.

8. If water ascend and descend in the arms of a cylindrical canal, which are every where of the same diameter, and inclined to the horizon at any angles; determine the time of an oscillation.

ASTRONOM Y.

TRINITY COLLEGE, 1822.

1. GIVE proofs of the Earth's spherical form. What is its figure more accurately?

2. Define latitude and longitude on the Earth's surface, and explain some of the methods of determining the latitude.

3. Supposing a chronometer set at Greenwich, time not to have varied, and to indicate 7h 20′ P. M., when by observation the time of day at a given place is 11 35′ A. M. Required the longitude of the place from Greenwich.

4. By what phenomena are the periods of a day and a year ascertained? and what is the correct length of the latter, and what consequent adjustment of the civil year necessary?

5. What angle is the ecliptic inclined at to the equator? and what the magnitude of the different zones? Supposing the poles of the ecliptic and equinoctial to coincide, what effect would be produced upon the days and nights?

6. Required the Sun's place in the ecliptic on the 31st of May; and thence determine the Sun's declination and right ascension for that day.

7. Required the latitude of a place at which the Sun rises upon the N.N.E. point on the longest day.

8. Explain solar and lunar eclipses very briefly, and give the ecliptic limits for each, and mention the greatest number of each that can possibly occur in a year.

TRINITY COLLEGE, 1823.

1. WHAT are the direct proofs of the Earth's motion on its axis, and of its annual motion round the Sun?

2. Point out the necessity and sufficiency of the Gregorian correction of the Calendar.

3. The ecliptic polar distance of y Draconis, which is situated nearly in the solstitial colure, is about 15°. Supposing the diameter of the Earth's orbit to subtend an angle of 2′′ at the star, prove that its distance is at least 200,000 times greater than the distance of the Sun.

4. Project the diurnal path of a star orthographically upon the plane of the meridian; and prove upon the projection, that when the star is in the horizon,

rad: cos. azim. :: cos. lat. : sin. dec.

5. In a given latitude, on a given day, to find the time of Sunrise.

6. Having given the latitude of the place, and the declination of the Sun, investigate the equation to the locus of the extremity of the shadow which an upright style casts upon an horizontal plane. Prove that it is a conic section, and find its dimensions.

7. Find an expression for the angle which the Earth's disc subtends at a planet, and shew that some of the necessary observations do not require very great accuracy.

8. How did the ancients observe the angular distance of a planet from the Sun? Shew that this can be found from proper observations, without the intervention of a third object, and perform the computation.

9. Assuming the true system, demonstrate the retrograde, and stationary appearances of the planet Venus.

10. If the distance of Venus from the Sun were to the Sun's distance from the Earth ::1:√5; prove that her greatest brightness would be at her greatest elongation.

11. Give all the steps of the method by which Kepler found various distances of the planet Mars from the Sun, and the angles at the Sun contained by those distances, and was induced to conclude, that the orbit was not circular but oval.

12. Describe some of the more accurate methods of finding a Meridian Line.

13. Find the quantity of refraction by the circumpolar stars, (Boscovich's method); the refraction being supposed to vary as the tangent of the apparent zenith distance.

14. Transform the expression for the aberration in R. A. of a known star to the form m. cos.(L~ K) where L is the Sun's longi