upon its sides in directions perpendicular to them: prove that if the forces are in equilibrium, they are proportional to the sides on which they act.

4. Three forces, represented by 3, 4, and 5 pounds respectively, keep a particle in equilibrium: determine the angles at which they act.

5. Forces act on a triangle which are represented in magnitude and direction by the straight lines drawn from the vertices to the points of bisection of the opposite sides; show that they will maintain the triangle in equilibrium.

6. The resultant of two forces is 100 lbs., and the angles which it makes with the directions of the forces are 20° and 30°; find the component forces.

7. A heavy body is drawn up an inclined plane by means of a string, which being parallel to the plane, and passing over a pulley, has another heavy body hanging vertically at the end of it; find the accelerating force. If the plane be inclined at an angle of 30°, and the accelerating force beg, find the ratio of the masses of the two bodies.

8. A weight Q is drawn along a smooth horizontal table by a weight hanging vertically: find (1) the accelerating force on P, (2) the accelerating force on the centre of gravity of P and Q.

9. A body is projected with a velocity v in a direction making an angle of 30° with the horizon; find its velocity and direction of motion when it has reached a height 2 g. What is the horizontal space described in this time?

10. A body is projected horizontally with a velocity 4 g from a point whose height above the ground is 16g; find the direction of motion (1) when it has fallen half way to the ground, (2) when half the time of flight has elapsed.

11. Two particles are projected simultaneously from a given point with different velocities and in different directions, but in the same vertical plane; find the path described by their centre of gravity.

12. A lump of metal weighs 59 oz. in water and 61 oz. in alcohol, whose s. g. is 8; find the weight and s. g. of the metal.

II.

1. Three forces act upon a particle in a plane; show how to find their resultant.

2. Three forces of 3, 4, 5 lbs. act on a particle in the centre of a square in directions towards three of the angles of the square: find the magnitude and direction of the force which will keep the particle at rest.

3. Can the forces 3, 5, 11 acting on a particle keep it at rest?

4. A force of 40 lbs. acting on a point between two forces of 20 lbs. and 20√3 lbs. respectively, makes an angle of 60° with the former and 30° with the latter; find the magnitude and direction of the force which will keep the point at rest.

5. Two forces, P and Q, act upon a point; find the greatest and the least forces which can keep the point at rest. How will the directions of the forces in each case be adjusted?

6. At what angle must two forces, P and Q, act so that a single force which keeps them at rest shall be a mean proportional between the greatest and least values, as determined in Quest. 5?

7. A ball a impinges directly upon a ball в, in motion: show that the velocity of their centre of gravity is not affected by the impact.

8. Show that the time of a projectile's describing any arc of its path is the same as would be occupied in moving uniformly along the chord with the velocity which it has when its motion is parallel to the chord.

9. Find the direction in which a body must be projected with a given velocity to hit a given mark, and show generally that there are two directions that will satisfy the conditions.

10. A body oscillates once in 2.5"; find the distance between its centres of suspension and oscillation.

11. A cube of wood floating on water descends in. when a weight of 15 oz. is placed upon it; find the size of the cube, supposing a cubic foot of water to weigh 1000 oz.

12. A B C D is a parallelogram whose diagonals A C, BD intersect in E, and A B is in the surface of a fluid; show that when the fluid is homogeneous the pressures on the triangles A E B, B E C, C E D are as 1, 3, 5. Compare the pressures when the parallelogram is a square and the density of the fluid varies as the depth.

III.

1. Show how forces may be represented geometrically by straight lines. Define a moment, and show how it may be represented geometrically, in accordance with the above representation of a force.

2. Deduce the principle of equality of moments from that of the parallelogram of pressures.

3. There is a uniform bar, weighing 10 lbs., and 5 ft. long: weights of 9 lbs. and 5 lbs. are suspended from its extremities: on what point will it balance?

4. A uniform beam A B, whose weight is w, rests with one end on a smooth horizontal plane A C, and the other end B on a smooth plane CB inclined to the horizon at 60°. Find the tension of a string CA=CB which keeps the beam from slipping.

5. Two planes of equal altitude are inclined at angles of 60° and 45° to the horizon; required what weight resting on the latter will balance 20 lbs. on the former, the weights being connected by means of a string passing over the common vertex parallel to the planes.

6. The mouth of a harbour is a mi. wide, and the tide is running out of it at the rate of 21 mi. an hour; in what direction must a man, who can row in still water at the rate

of 5 mi. an hour, point the head of the boat in order to make for a point directly opposite to him?

7. In Qu. 6, find the length of the man's course when he keeps the head of the boat pointed to the opposite shore, and determine the length of time employed in making this course, and compare it with that employed in making a course direct across the harbour.

8. A body has fallen from rest through a feet at the moment when a second body begins to fall, b feet below the point which the first then occupies; find the distance traversed by the second before it is overtaken by the first, the force of gravity being constant, but unknown in amount.

9. Find the inclination of a gradient in a railway that a carriage descending down it, by its own weight, may move through a mile in 30"; find the space it will describe in the next 30".

10. Describe the force-pump, and explain its action.

11. The specific gravity of gold being taken as 19.3 and of copper as 8-6, find the specific gravity of an alloy containing 2 oz. of copper to 23 oz. of gold.

12. Find the depth at which a hemisphere floats with its base downwards, when the specific gravities of the fluid and solid are as 16: 11.

IV.

1. Show that if be the angle between the directions of two given forces, their resultant is greatest when 0=0; least when 0=7; and intermediate for intermediate values of A.

2. Find the resultant of two equal forces acting on a point.

3. Three forces, 99, 100, 101 lbs. respectively act upon a point in directions making an angle of 120° with each other, successively; find the magnitude of their resultant, and the angle which it makes with the force 100.

4 An endless band A B C D passes over three pulleys at

A, B, C, and a weight of 100 lbs. is attached to D. Supposing A B, C B each to make angles of 45°, and A D, C D angles of 30°, with the vertical, and a and to be in the same horizontal line, find the magnitude and direction of the pressures on B, A, and c.

5. Four bodies weighing 1, 2, 4, 7 lbs., respectively, are placed with their centres of gravity in a straight line at the respective distances of 3, 5, 7, 9 ft., from a given point in that line; find the common centre of gravity.

6. Find the conditions of equilibrium of any system of forces acting in one plane. Account for only one of these conditions being necessary, in the case of a lever attached to a fulcrum.

A uniform beam A B, whose weight is w, rests in equilibrium between a vertical wall B C and the horizontal plane A C, both smooth; CE is a string without weight attached to a point E in the beam. If B A C=a, and A C E=3, show

W cos a

that the tension of the string=2 sin (a-3)

7. If a bullet be fired from a level ground in a direction making an angle i with the horizon, and the horizontal range be R, find the time of flight.

8. If a body be projected from A in the direction A B C, and from c any point in that line a vertical line CD be drawn, meeting the curve described by the projectile in D, and if B, the middle point of the finite line A B C, be joined to D, show that BD will be the direction of the motion at D, and that the velocity at D will be to that at A as BD; A B.

9. How long will 10 men be in pumping dry the hold of a ship which contains 30,000 cubic ft. of water, the centre of gravity of the water being 14 ft. below the point of discharge, and each man yielding 1500 units of effective work per minute?

10. A uniform rod of length and weight w resting vertically upon one extremity on a horizontal plane, is allowed to fall over. On what point must it strike a nail in