length, and rebounds from the horizontal plane; given the greatest elevation which it afterwards attains, determine its elasticity.

6. Three equal spheres are placed in contact upon a horizontal plane; if another equal sphere placed upon them just causes them to separate, what is the proportion of friction to pressure?

7. If two tangents to a parabola are at right angles to one another, and their point of intersection be in the axis, find the centre of gravity of the area contained between them and the arc; and determine the curve described by the centre of gravity, when the tangents sume all possible positions.

8. Find the equation to the path of a projectile; and shew that its length, between the body leaving the horizontal plane and returning to it, will be the greatest possible for a given velocity, when the angle of elevation (0) satisfies the equation

2sec.0=ecosec. + e-cosec.0

9. What segments of a sphere will oscillate about a tangent at the vertex in the shortest, and longest times possible? find the length of the isochronous simple pendulum in each case.

10. Determine the position of a weight supported on a smooth cycloidal lamina whose axis is vertical, by an elastic string which lies along the arc and is fastened at the vertex; supposing the length of the string when unstretched given, and its weight inconsiderable.

11. Two equal balls, resting on a smooth horizontal table, are fastened by two equal strings to the end of another string, which, passing over the edge of the table, sustains a given weight; determine the motion, supposing the initial direction of the latter string to bisect the angle between the two former.

12. An uniform beam rests in a vertical plane with its ends upon two curves; shew that if the angles of inclination to the horizon of the normals, at the points in contact with the beam, be respectively a and B, and be the inclination of the beam itself to the horizon, then 2 tan.0 = tan.a~ tan.B.

13. A lamina of matter in the form of an ellipse slides by gravity with its plane vertical, between two straight lines inclined at angles ά, and 90 a to the horizon, determine the motion. 14. An uniform and flexible string is suspended from two points

in the surface of a vertical cylinder, determine its equations; taking the origin where the axis of the cylinder meets a horizontal plane, whose distance from the lowest point of the string = length of a portion equivalent in weight to the tension at that point; and shew that the perpendicular upon the tangent from the origin is of a constant length.

ST. JOHN'S COLLEGE, MAY 1830.

1. IF three forces keep a point at rest, any one of them is proportional to the sine of the angle between the directions of the two others.

2. A stone dropped from a bridge, strikes the water in 2′′, what is the height of the bridge? Also, if the stone be projected downwards with a velocity of 3 feet per second, in what time will it strike the water?

3. Two weights sustain each other upon two inclined planes, having a common altitude, by means of a string which is parallel to the planes; find their position, taking into account the weight of the string which is supposed to be of uniform thickness.

4. If a body, whose elasticity is (e), impinge on a plane with a velocity (v), and be reflected with a velocity (v'); and if a, a, be the angles which the directions of its motion, in the two cases, make with a perpendicular to the plane, prove that tan.ae tan.a', v sin.a = v′ sin.ä.

5. The locus of the centres of gravity of the areas of all rightangled triangles on the same hypothenuse (2a), is a circle whose

a

radius= The locus of the centres of gravity of their perimeters 3*

is a spiral, whose equation is pa (sin.0 - sin.); the pole being

in the middle point of the hypothenuse, and measured from that line.

6. A hemispherical bowl is terminated by a cylindrical rim, having the same external and internal radius; what must be its breadth, in order that the vessel may rest upon any point of its spherical base?

D

7. A perfectly elastic ball is projected from the middle point of the base of a vertical square towards one of the angles, and after having been reflected at the sides containing that angle, fails at the opposite angle; find the velocity of projection.

8. Two weights P and Q, connected by a string of given length, are placed upon an inclined plane of the same length, P being at the highest point, and Q at the lowest; after what time will their distance be a minimum ?

9. A uniform rod whose length equals twice the diameter, passes through a hole in a spherical shell, and rests with one end against the internal surface; shew that if a be its inclination to the vertical, when in its position of stable equilibrium, (≈ + a) and (π — α)

α

will be its inclinations, in its positions of unstable equilibrium.

10. A uniform rod of given length (a) is bent into the form of a cycloid, and oscillates about a horizontal line joining its extremi

ties; prove that the length of the isochronous pendulum =

a

5

11. A convoy moving uniformly along a road which runs east and west, is perceived at the instant it is due south of a battery; at what elevation, and towards what point of the compass, must a cannon, loaded with a given charge, be fired at the same instant, so as just to hit it?

12. A beam having one end on a vertical, and the other on a horizontal plane, is kept at rest by a string connecting its centre of gravity with the point of intersection of the planes; find the tension of the string; explain the result when the beam is of uniform thick

ness.

13. A rod whose length = a, and distance of its centre of gravity from one end =b, is placed with that end on the arc, and the other on the minor and vertical axis of an ellipse, whose semi-axes are (a) and (b); shew that it will rest in any position; but that if it be disturbed, its centre of gravity will describe the major axis, and its centre of instantaneous rotation, a circle radius (a — b) about the centre of the ellipse; also find the angular velocity of the rod in any position, and the velocity of its centre of gravity.

QUEEN'S COLLEGE, 1826.

1. IF two equal forces sustain each other by means of a string passing over a tack, shew that either force: the pressure upon the tack: 12 cosine of half the angle at which the forces act.

2. P and W being in equilibrio on an inclined plane, if the whole be put in motion, then

P's velocity: W's velocity :: W: P.

AD is horizontal, DC vertical, Q a weight connected with one extremity of a beam AB by a string passing over a pulley at C in such a manner that CB is vertical. Find Q when there is an equilibrium, and shew that the equilibrium will be maintained whatever be the position of the beam AB, CB remaining vertical.

4. A body is suspended from a given point in a horizontal plane, by a string of known length, which is thrust out of its vertical position by a rod (without weight) acting from a given point in the plane, against the body; shew that the tension of the string varies inversely as the tangent of the inclination of the rod to the horizon.

5. A quadrant being placed with one of its sides vertical, and a uniform chain equal in length to the arc being suffered to hang over it from a pin at the upper extremity, it is required to find the sustaining force.

6. P and Q are two weights connected by a string passing over a fixed pulley, whereof P is the greater, at the end of t" an additional weight (q) is annexed to Q. Find the velocity of P after any assigned time.

7. If be the angular distance of a body from the lowest point in a circular arc; shew that the force in the direction of the arc is

to that in the direction of the chord as 2 cos.

: 1. 2

8. Find the centre of gravity of a frustum of a pyramid.

9. An imperfectly elastic ball is projected vertically downwards from a given point with a given velocity, upon a hard horizontal plane; required the sum of the space described by it before the motion ceases, and the whole time of its motion.

10. A ball is projected in a given direction with a given velocity,

determine the time of flight in terms of the oscillations of a pendulum of given length.

11. If a body descends down any curve by the action of gravity, the velocity acquired at any point will be the same as if the body had descended down the same vertical space falling freely.

12. Required the length of a pendulum which vibrates as often in a minute as inches in length.

QUEEN'S COLLEGE, 1827.

2

1. THE uniform lever AC of given weight and length, and turning on the fulcrum C, has two given weights w, w suspended from the extremity A, and the middle point B, and is kept at rest by the given weight P acting at A by means of a string passing over the fixed pulley D, CD is horizontal and equal to CA: find the position of equilibrium.

A paraboloid of which the parameter is given, being cut off by a plane perpendicular to its axis, when prevented from sliding, just stands on an inclined plane of given inclination: required the length of the axis of the paraboloid.

3. If the power and weight in equilibrio on an inclined plane be put in motion, the velocity of the power that of the weight

weight the power.

the

4. If any number of forces act on a point, the sum of their moments referred to any other point is equal to the moment of their résultant referred to the same point.

5. Suppose (p) to be retained on the curve of a given ellipse the axis major of which is vertical, by a force (q) parallel to the horizon, find the position of equilibrium.

6. Find the centre of gravity of the eighth part of a sphere, i. e. of the solid generated by the revolution of a quadrant about one of its sides through 90°.

7. In the direct impact of two inelastic bodies, the difference between the sums of the products of each body into the square of its velocity before and after impact, is equal to the sum of the products of each body into the square of the velocity gained or lost by it.

8. Shew that a body projected in an oblique direction along an inclined plane describes a parabola, and find its latus rectum, having