and find its value, having given that (x) has p(x) for its differential coefficient.

12. Find a formula for the volume of a solid of revolution.

revolves about the axis of x; find the volume generated.

MIXED MATHEMATICS.-PART I.

The Board of Examiners.

1. Shew how to find the magnitude and direction of the uniform acceleration of a particle moving in a plane, when the magnitude and direction of the velocity at two instants are known.

A body of mass 1 kilogramme is moving North with a velocity of 5 kilometres per hour at one instant, and an hour afterwards is moving East with a velocity of 6 kilometres per hour. Find the magnitude and direction of the force on it in dynes.

2. Shew how the ratio of the masses of two bodies may be found by allowing them to impinge on one another, and observing the alterations in their velocities.

A sphere of mass m overtakes and impinges directly on a sphere of mass m', the coefficient of impact being e. Shew that the former sphere cannot have the direction of its motion reversed by the impact unless m < em'.

3. Prove that the acceleration of a particle moving with uniform velocity v in a circle of radius r is v2/r.

Shew that a simple conical pendulum of length 1 metre must make one revolution in 1·7 seconds nearly if it makes an angle of 45° with the vertical.

4. Find the time of a small oscillation of a simple pendulum, and shew that a small percentage of increase in the length of the pendulum, or of decrease in gravity, increases the time of oscillation by half that percentage nearly.

5. Find the resultant of two parallel forces on a rigid body.

A circular cylindrical cup with a flat bottom is made of thin smooth sheet metal of weight w per unit area. It is placed on a horizontal table, and a uniform rod of weight Wis placed in it. Shew that the rod will upset the cup if its length is greater than

where h is the height of the cup and r its radius.

6. Prove that it is necessary and sufficient for the equilibrium of a rigid body acted on by forces in one plane, that the sums of the moments of the forces about three non-collinear points should vanish.

A uniform heavy rod rests with one end on a smooth horizontal plane, and the other against a smooth vertical wall. The lower end of the rod is connected with a point of the intersection of plane and wall by an elastic string of natural length 2/3 times that of the rod, and of modulus equal to the weight of the rod. Shew that there is equilibrium when the rod makes an angle of 45° with the vertical.

7. Two equal uniform rods AB, AC, each of weight w, are smoothly jointed to one another at A, and to fixed points B, C, such that BC is vertical and equal to AB. A weight W being hung at A, find the reactions at the joints.

8. Shew that, neglecting friction, the work done by a machine, working steadily, is equal to the work done on it.

Apply this principle to find the mechanical advantages of a smooth screw, and of that system of pulleys in which the same string passes over all the pulleys.

9. Define the intensity of fluid pressure at a point.

The safety-valve of a steam boiler is a circle of radius r, whose centre is at a distance a from the fulcrum, and the weight W on the valve is at a distance c from the fulcrum. Find the steam pressure when the valve is about to open.

10. Prove that the centre of pressure of a triangular area immersed in heavy liquid, with one side in the surface, is one-half the way down the median through the lowest point.

If the triangle is of height h, and its plane is vertical, shew that the addition of a depth k of liquid will raise the c.p. a height hk/(6k + 2h).

11. Investigate the pressure at any point of a mass of water rotating steadily as if rigid about a vertical axis.

If a particle of density 9 revolves with the water making 50 revolutions a second, and is at a distance of a decimetre from the axis, shew that there is a radial force on it about 113 times its weight.

12. Investigate an expression for the density of the air in an air-pump after n strokes, the clearance and leakage being neglected.

MIXED MATHEMATICS.-PART II.

The Board of Examiners.

1. Shew that the tension in a circular hoop rotating with angular velocity w is mw22 where m is the mass per unit length.

If the hoop is of iron (sp. g. 7-8) and breaks with a tension of 20 tons per square inch, shew that its velocity must not be greater than 650 feet per second.

2. Shew that if a particle describes an ellipse under a central force in a focus, the force varies as the inverse square of the distance.

Shew from the equation of energy and by normal resolution that a particle can describe an ellipse under two Newtonian central forces, one in each focus; the square of the velocity being the sum of the squares of the velocities at the same point when each force acts alone.

3. Prove that if a plane curve rolls, without slipping, on a fixed curve, the angular velocity and the velocity of the point of contact are connected by the equation w = v (k + k) where K, Kare the curvatures of the curves at the point of

contact.

A circle of radius a rolls on one of radius b, making n revolutions a second. Find the velocity of the point of the rolling circle furthest from the fixed circle.

4. Two heavy particles move in smooth vertical circles of different radii. Find how the particles must be started in order that their velocities may bear a constant ratio to each other when they are at equal angular distances from their lowest points, and shew that the reactions in corresponding positions will then be as the masses.

5. Investigate a graphic construction for the resultant of a system of forces in one plane, and obtain the graphical conditions of equilibrium.

6. Prove Pappus's theorem that the volume of the solid of revolution generated by a plane area