13. In an imperfect barometer tube, of which the length is 33 inches, the mercury stands at 29, when in a perfect one it is at 30: at what height will it stand in the imperfect one, when it is at 20 in the perfect barometer?

14. A rectangular parallelopiped of which the specific gravity is one half, floats with a side horizontal; what is its stability? In what case will the equilibrium of indifference subsist?

15. A heavy piston descends by its weight in a tube filled with air; find its velocity at any point, and the depth to which it will descend (neglecting friction.)

16. Explain the common pump. What are the conditions under which it will not work when the piston does not descend to the fixed valve?

17. If the elasticity of the air, including the effect of temperature, vary as the (density)1+m, shew that the pressure (p) at any altitude z, will be given by the formula

A being the height of a homogeneous atmosphere, of density D, and r the radius of the Earth.

Also m being; find the whole height of the atmosphere.

TRINITY COLLEGE, MAY 1828.

1. IF two fluids communicate in a bent tube, their perpendicular altitudes above the plane where they meet are inversely as their specific gravities.

2. To find the specific gravity of a body lighter than the fluid in which it is weighed.

the

3. A hollow iron sphere just floats in water: find its thickness. If the exterior and interior surfaces are not concentric, how may fact be ascertained?

4. Compare geometrically the resistance on a semicircle moving

in a fluid perpendicularly to its diameter, with the resistance on its diameter.

5. A close paraboloid containing common atmospheric air is let down in water to a certain depth, and a small orifice being opened at its vertex, which is downwards, the water rises up to the middle ‹of its axis. Determine the depth, and the velocity with which the water first rushed in.

6. A cylindrical bucket, containing a given quantity of water, is whirled round its axis with a given angular velocity, not great enough to throw any water over the top. A small orifice being opened at the centre of the base, find the quantity of water which can escape by it, and the time in which a given quantity will flow

out.

7. A sphere full of water is placed on a horizontal plane. Find the point at which a hole being bored through it perpendicularly to its surface, the water shall spout to the greatest distance on the horizontal plane.

8. Determine the position of equilibrium of a homogeneous semiparabola in a fluid, supposing the axis and the extreme ordinate wholly

extant.

9. A vessel full of water has a side loose, whose shape is a given triangle, with the base horizontal, and vertex downwards. Find the magnitude and point of application of a force which shall keep it in its place.

10. Define the Metacentre, and show how it may be found. Explain how its position determines the stability of a floating body. 11. The height of a homogeneous atmosphere is the same for all distances above the Earth's surface.

12. Describe the construction and action of the common airpump, and determine the quantity of air remaining in the receiver after t turns.

13. Determine the curve traced out by the extremity of the horizontal diameter of a small spherical air-bubble ascending in a fluid.

14. Describe the air-gun, and having given the quantity of air and the space occupied by it at first, find the velocity of the bullet at any given point of the tube.

15. The force varies as the distance, and the resistance of the medium as the square of the velocity. Find the velocity of a body descending towards the centre, and shew from your result that when the resistance vanishes, the velocity coincides with that in a nonresisting medium.

TRINITY COLLEGE, JUNE 1829.

1. FIND the equation to the surface of revolution, such that when filled with fluid, and placed with its vertex upwards, the pressure on any horizontal section equals the pressure on the curve surface above it.

2. If two fluids communicate in a bent tube, their perpendicular altitudes above the plane where they meet are inversely as their specific gravities.

3. If a lighter fluid rest upon a heavier, and their specific gravities be as a b, and a body, whose specific gravity is c, rest with one part P in the upper fluid, and the other part Q in the lower, then P: Q: b

cc-a.

4. Define the centre of pressure, and shew how its position may be determined.

5. In the side of a vertical flood-gate is a rectangular aperture, whose sides are vertical. To this aperture is fitted a door moveable, in a direction perpendicular to its plane, about a horizontal axis passing through its centre of gravity. Given the height at which the water stands on one side of the gate; find the height at which it must stand on the other, in order that the door may remain at rest. If the water rise on the deeper side, which way will the door be pushed open?

6. A hemisphere, with a flat lid, filled with fluid, is held with a point in its edge uppermost. Find its position when the sum of the pressures on the curve and plane surfaces is a maximum.

7. It is found that when a small horizontal orifice is opened in the side of a cone filled with fluid at a height = of the cone's height, the fluid spouts to the greatest distance on the horizontal

plane on which the cone is standing. Compare the height of the cone with the diameter of its base.

8. Determine the greatest angular velocity, with which a sphere filled with water may be whirled round its vertical axis, so as to allow the whole of the water to escape by a small orifice at the lowest point.

9. A cone full of water is pierced with innumerable holes perpendicular to its surface. Find the boundary of the issuing fluid.

10. A cylinder whose height is a, and base b, open at the top, and filled with fluid, is placed by the side of a paraboloid of equal height and base, also open at the top. A communication being opened between the vessels by means of a small orifice whose area is Aπ2, shew that the time which elapses before the fluid stands at the same height in each

=

bea A2Ng

11. Over two pullies in the same horizontal line passes a string of inconsiderable weight, to which are attached two cylindrical vessels, equal in every respect, containing unequal quantities of water. A small orifice of the same size being simultaneously opened in the base of each, and the vessels being at the same moment allowed to move, find (1) the quantity of water remaining in one vessel when the other is empty, and (2) the time in which this remaining quantity will run out. (N. B. It is supposed that the ascending vessel does not reach the pully towards which it is moving before both are emptied.)

12. Two equal cylinders being placed as in the last problem, but containing equal weights of fluids of different specific gravities; state the effects which will result from simultaneously opening a small orifice of the same size in the base of each.

13. Define the Metacentre, and shew how its position determines the stability of a floating body.

14. In an imperfectly exhausted barometer, the depression below the standard altitude is to the standard altitude as the space which the air left in the tube occupied before immersion is to the space which it occupies after.

15. Describe the construction and action of the common Pump.

1.

TRINITY COLLEGE, MAY 1830.

THE equality of the pressure of fluid in all directions may be deduced from the principle, that when a mass of fluid is in equilibrium, the state of rest is not altered by supposing any portion of the mass to become solid.

2. The pressure at any point of a fluid mass of uniform density contained in any vessel and acted upon by gravity, is proportional to the perpendicular depth of the point below the surface of the fluid. In what manner is the pressure estimated, when it is said to be equal to gph, g being the measure of the accelerative force of gravity, h the depth below the surface, and p the density ?

3. The pressure of a fluid against any surface in a direction perpendicular to it, varies as the area of the surface multiplied into the depth of its centre of gravity below the surface of the fluid.

4. The pressure of a fluid downwards against the sides and bottom of any vessel whatever, is the weight of the fluid contained in it.

5. The weight of P in water was 10 grains, of Q in air 14 grains, of P and Q connected together, the weight in water was 7 grains; the specific gravity of water being 1, and of air 0013, shew that the specific gravity of Q was 8237, and that it was as large as 17-023 grains of water.

6. A tube, the upper part of which was bent like a retort, and contained a quantity of powder, was plunged vertically into mercury to a certain depth, after having been partially exhausted of air; the mercury stood in it at the height 1, and after the tube was depressed through an additional space a, at the height -f: the powder having been taken out, the tube was exhausted again so that when plunged to the same depths as before, the mercury stood at the heights / and / g; hence shew that if k2 the transverse section of the tube, and h = the height of the mercury barometer, the solid content of the powder was