3. In the screw, P: W:: distance between two threads : circumference of the circle described by the power.

4. Prove the principle of virtual velocities in the wedge generally.

If a system be in equilibrium, the centre of gravity is at its highest or lowest point; required a proof.

6. If h k be the co-ordinates of the centre of gravity of a curvilinear area, shew that

Apply the expressions to a quadrant of an ellipse.

7. In an arch which is in equilibrium, the weights of the voussoirs. are as the differences of the tangents of the angles which their joints make with the vertical.

8. A solid is formed by the revolution of a semi-parabola through an angle of 60°. Find the centre of gravity.

9. Find the resultant of any number of forces acting in the same plane on a rigid body, and determine also the equation to the straight. line in which it acts.

10. A cone and hemisphere of equal bases and altitudes are placed on an horizontal plane, the extremities of their bases coinciding; find the position of equilibrium of a given rod (in length less than the diameter of the base) which shall rest between them.

11. Find the equation to the catenary between x and y beginning from the lowest point.

12. Explain and prove the first law of motion.

13. Two bodies, whose common elasticity is e, and moving with given velocities, impinge directly on each other; determine their velocities after impact.

and shew that the spaces described in the 1st, 2nd, 3rd, 4th, &c. seconds are as the numbers, 1, 3, 5, 7, &c.

15. Shew that the curve described by a projectile is a parabola, and find its latus rectum.

16. Two spherical bodies move uniformly in any two straight lines not in the same plane, if they meet, determine their concourse, if not, find when they approach nearest to one another.

17. A body perfectly elastic is projected up a plane whose angle of inclination equal B, at an angle a, and with a given velocity U, after reflexion it ascends vertically upwards, shew that its range on U2 cos.2 a and that the time of flight sin.ß. cos.28'

the inclined plane = from the point of

18.

g.sin.28

A body descends down an inverted cycloid; find the time of a whole oscillation.

19. Find the equations of motion of a body moving in a plane and acted on by any forces in that plane.

20. A body is acted on by a force tending to a centre; find the polar equation to the curve described.

21. P draws up W by means of the wheel and axle; find the accelerating force on P.

22. The strength of a string by which a body may be whirled round in a vertical circle, must be able to support six times the weight of the body.

23. A paraboloid resting on an horizontal plane has its axis drawn a little from the vertical, and begins to oscillate; find the time of one of its small oscillations.

24. Define the centre of oscillation, and shew that the centres of oscillation and suspension are reciprocal.

CAIUS COLLEGE, MAY 1830.

1. IF two weights balance each other on a straight lever, their distances from the fulcrum are inversely as their weights.

2. If any two forces act at the same point, they are equivalent to a single force whose magnitude is expressed by the diagonal of the parallelogram, of which the sides express the magnitude and direction of the two forces.

3. If any number of parallel forces P, P', P", &c. act on a system x, y, z, x, y, z, &c. being the co-ordinates of their points of application, and if x, y, z be the co-ordinates of the centre of forces; shew that

4. Find the centre of gravity of a pyramid whose base is a triangle.

5. Prove Guldin's theorems.

6. Given the resultant R of two forces P, Q, their sum, and their inclination; to find the forces.

7. P supports Q on a given inclined plane, P's direction making with the plane an angle ; to find the conditions of equilibrium.

8. Determine the same, taking into account friction.

9. In a system of pullies, each string is attached to the weight. Determine the equation of equilibrium, taking into account the weight of the blocks.

10. Find the equations to the common catenary, and if t = the tension at any point, p = the radius of curvature, and c = tension at

11. A beam (PQ) rests upon two inclined planes; to find the position of equilibrium.

12. Prove the principle of virtual velocities for the five mechanical powers.

13. Two bodies m, m' move in the same direction with velocities v, v′, and v is greater than v', determine their motions after impact, first when perfectly hard, and next when their common elasticity

is ε.

14. When a body is acted upon by a uniformly accelerating force shew that

15. A body is projected from a given point in a given direction with a given velocity; prove that

is to the equation to the curve, and shew the curve to be the common 2. a2 cos.2 a g

parabola of which the parameter =

16. The velocity acquired in falling down any curve = the velocity acquired by falling freely down the same vertical height.

17. Find the time of oscillation in a common cycloid, and in a circle.

HYDROSTATICS.

TRINITY COLLEGE, 1822.

1. WHEN a fluid is at rest its surface is horizontal.

2. Shew how the capacity of an irregular vessel may practically be found with accuracy.

3. Between every two positions of stable equilibrium of a floating body there is a position of unstable equilibrium.

4. Supposing that it requires one nth part the weight of a particle of water to overcome its adhesion: find what is the least slope down which water would flow.

5. It is found that on mixing 63 pints of sulphuric acid at 1.82 specific gravity, with 24 pints of water, one pint is lost by their mutual penetration: find the specific gravity of the compound.

6. Find the whole pressure on the side of a hollow octahedron filled with water,

(1). When suspended by an angle.

(2). When by two adjacent angles, with the intermediate edge horizontal.

(3). When by the three angles of a side, so that this side is horizontal;

the length of an edge being a inches: and the weight of a solid foot of water being 1000 ounces.

7. How far may a shaft 4X be bored perpendicularly into the vertical face of a canal-bank before the bank would be blown up; supposing that the tenacity of the earth would be accounted for by an addition to its specific gravity, and that the form of the bank is determined by the equation PM = ƒ(AM), (AM) being the vertical line from the point P on the bank, to M in the shaft.

Apply the result to the case where AB is 7 feet, BC is 2 feet, the bank is a plane of 30° elevation; the specific gravity of the earth being 2.25, and the addition for adhesion being of the weight.