4. What is the object of a machine?

Find the relation between the power and the load in the system of pulleys in which each pulley hangs by a separate string, the pulleys being of the same weight and the strings parallel.

Show that the principle of "virtual velocities" holds in this

case.

5. Enunciate the proposition known as the triangle of forces.

Show that this proposition enables us to find, in certain cases, by a geometrical construction, the tensions and pressures in the several ties and struts of a jointed structure, such, for example, as a roof truss.

A triangle ABC is formed of three rods jointed at the angles, and B is fixed in such a way that the structure can turn freely about it. A given weight is suspended at A and a horizontal force applied at C so that AB remains horizontal. Neglecting the weight of the frame, draw a diagram representing the stresses in each of the rods and the forces at B and C.

6. Define the centre of gravity of a body.

Why does the centre of gravity of a body always coincide with its centre of mass (or inertia) ?

Find the centre of gravity of a thin conical shell of uniform thickness.

7. A shot of 1000 lbs. leaves a gun with a velocity of 1500 feet per second. How long must the shot have been under the action of the powder supposing the average pressure upon it to have been equal to the weight of 1200 tons ?

If the shot penetrate a target to the depth of two feet and then come to rest, what will be the average pressure exerted on the target?

8. Describe Atwood's machine, and explain how you would show experimentally by its means that the weight of a body is independent of its vertical velocity.

A smooth body and a rough body, each of one pound, are connected by an inextensible string 4 feet in length and placed side by side and close together on a plane inclined 45° to the horizon. If the coefficient of friction between the plane and the rough body be, discuss the motion completely and find the impulse on the string at the instant it becomes tight, and its tension subsequently.

9. A particle is projected with velocity v at an elevation a; find the latus rectum of its path.

A number of particles are projected from the same point in different directions, but so that the horizontal component of the velocity of each is the same. Find the locus of the foci of their paths.

10. Show that the time of oscillation of a heavy particle on a smooth cycloid whose axis is vertical and vertex downwards is independent of the extent of the arc.

The weight of 29 905 cub. ins. of mercury in London is equal to that of 29 898 cub. ins. in Manchester. How many seconds will a pendulum clock gain in a year in Manchester if properly regulated for London ?

MIXED MATHEMATICS (HIGHER).

W. D. NIVEN, Esq.

Friday, 5th July 1878. 10 A.M. to 1 P.M.

1. Define Acceleration, and find the tangential and normal accelerations of a point describing a plane curve.

A tangent rolls on a given circle with uniform angular velocity: and a point moves with relative velocity v along the tangent. Find the acceleration of the point.

1 under the action of a

x2 y/2 2. A particle describes the curve + az b2 force parallel to the minor axis. Find the law of the force, as also its absolute value if the time from the end of one latus rectum to the next be one second.

3. A particle moves freely under the action of a known central force. Find the differential equation of the path described, and show how to solve it.

4. A particle is constrained to move in a curved line under the action of given forces. Find the pressure on the curve.

The curve being a circle, and the only impressed force an attraction varying as the distance, and directed to a point outside the circle, a particle has a velocity V imparted to it causing it to move on the inside of the circle; find the least value of V that the particle may describe the complete circle.

5. Six forces act along the sides of a regular hexagon taken in order. Find the equation of the line of action of their resultant, the origin of co-ordinates being the centre of the hexagon.

What are the conditions that the forces may reduce to a couple?

6. Find the general conditions among a system of forces that they may hold a rigid body in equilibrium.

A heavy screw can turn about a vertical axis. If the friction be such that a weight P, in addition to its own weight, is required to make it fall, what force will be sufficient to raise it?

7. Find the general differential equations of equilibrium of a string under the action of given forces.

Solve these equations in the case of a uniform string suspended from two points in the same horizontal plane, determining in this way the equation of the curve in which the string hangs, and the tension at any point.

8. A body with a plane face is placed on the top of a fixed sphere. What must be the maximum of distance of the centre of gravity of the body from its plane face that it may rest on the sphere in stable equilibrium?

9. Show how to find the resultant pressure on any surface exposed to fluid pressure.

A circular cone, whose dimensions are known, is entirely under water, with its vertex at a known depth and its axis at a known inclination to the vertical. Determine the resultant pressure on its curved surface.

10. Find the centre of pressure of the flat end of the cone in question 9.

11. Explain why it is important to know the metacentre of a floating body.

A ship, in the form of a solid hemisphere, has a mast of uniform thickness erected at its centre, the weight of which is given. Find the greatest length of the mast consistent with the stability of the ship in small angular displacements.

12. Of what gaseous laws is the formula

the mathematical expression? Describe any experiments with which you are acquainted, confirming your statements.

A given quantity of air at a uniform temperature is confined in a sphere of given radius, and is acted on by a repulsion from the centre of the sphere, varying as the square of the distance. Find the pressure at any point of the surface of the sphere, the value of the repulsion at the surface being equal to the pressure of air of unit density, and at the same temperature.

CHEMISTRY.

Professor G. D. LIVEING.

Wednesday, 10th July 1878. 10 A.M. to 1 P.M.

1. Define an element in Chemistry. What are the elements of marble, flint, steel, red-lead, ozone, brass?

2. Describe bromine, hydrochloric acid, and lime. Mention in each case substances which resemble them in chemical characters, pointing out both wherein they resemble them and wherein they differ from them.

3. Show how it is proved experimentally that steam contains its own volume of hydrogen, and that marsh gas contains twice its volume of hydrogen.

4. Calculate the density of olefiant gas as compared with air, having given that the density of hydrogen is ⚫069.

The density of acetylene is found to be that of air. Show what formula this leads to, having given that acetylene consists of carbon and hydrogen in atomic proportion. Send up the whole work.

5. Describe the process of preparing nitric acid.

What is the nature,

and what is the cause, of the red fu mes produced in the process?

How is nitric acid affected by admixture with hydrochloric acid?

6. What do you understand by a bi-basic acid, and what are the experimental tests by which the bi-basic character is recognised? Illustrate your answer by two cases of bi-basic acids.

7. Describe and explain the salt-cake process of preparing carbonate of soda from common salt.

8. What is the quantivalence of an element? Illustrate by the cases of arsenic, iron, and lead, pointing out, moreover, the variations of quantivalence which these elements exhibit.

9. Describe and explain a method of preparing pure silver nitrate from ordinary standard silver.

10. The analysis of a compound gave, of potassium 19.40, of magnesium 5.97, of SO4 47.76, of water 26 87. Construct a formula for the compound, and send up all the work. (K: Mg : S = 39: 24: 32.)

ELECTRICITY AND MAGNETISM.

Professor G. D. LIVEING.

Monday, 8th July 1878. 10 A.M. to 1 P.M.

1. Describe a gold leaf electroscope. An electroscope is taken on to the roof of a house, and when there the knob is for a moment touched with a wire; on now taking the electroscope indoors the leaves are found to diverge; show how to determine whether they diverge with positive or negative electricity. Supposing it to be negative, what inference can be drawn from the experiment?

2. Give reasons, founded on general principles, why a man standing under a wide-spreading tree at a distance from the trunk is in a safe position during a thunderstorm.

3. Show, by examples, that there is a loss of potential energy (or a conversion into kinetic energy) when electricity passes from a point of higher to one of lower potential; taking (1) negative electricity produced by friction and passing through air, (2) induced electricity passing through a metallic conductor, (3) voltaic electricity passing through an electrolyte.

4. Give an accurate definition of the capacity of a leyden jar. Show, by reference to a general law, whether more heat can be produced by the discharge of a leyden jar or by the direct discharge, into the ground from the conductor of the machine used to charge the jar, of a quantity of electricity equal to the charge of the jar.

H 667.

D

5. Define lines of magnetic force. How are the lines of force due to a bar magnet affected by placing a piece of soft iron at one end of it? Describe the lines of force due to the earth's magnetism at London. What are the means of telling how much closer they are at one place than at another?

6. Define the magnetic moment of a bar magnet; also a unit magnetic pole. Show how to find practically the ratio of the magnetic moments of two given bar magnets.

7. Describe a sine galvanometer, and prove the property from which it derives its name. If you require to measure powerful currents by such an instrument state what method you would employ in order to get accurate results, and the principle on which the method depends.

8. A single Bunsen's cell is found, when tested by a galvanometer, to give the same deflexion as two Daniell's cells joined up in the usual way; if now the positive pole of the Bunsen's cell be connected to the positive pole of the Daniell's battery, and the two negative poles be joined up to the galvanometer, will the needle necessarily remain at zero? Give reasons for your

answer.

9. Explain the use of a relay in a Morse telegraph, how it is employed, and how put in action.

10. An insulated wire is coiled round a bar of copper and a current passed for a short time through the wire; what effects will be produced in the bar? It is more difficult to draw the bar out of the coil while the current is passing than when it is not passing; state the general law to which this may be referred.

HEAT AND LIGHT.

Professor G. D. LIVEING.

Tuesday, 9th July 1878. 2 P.M. to 5 P.M.

1. The weight of mercury in a thermometer is 15 grams, and the length of a degree 2 millimeters; what is the section of the tube; the specific gravity of mercury being 13.5 and its coefficient of cubical expansion .00018?

2. How are the boiling point, the freezing point, and the point of maximum density, of water affected by the presence of common salt in solution in it? State general laws to which these effects, or any of them, may be referred.

3. How does radiation vary with the distance from the source? Show how this may be proved experimentally (1) for heat, (2) for light.

4. 1 lb. of iron at 100° C. is immersed in 40 lbs. of mercury at 10°, and the resulting temperature of the two substances is 17°; calculate the specific heat of iron, that of mercury being '03.

How does the specific heat of water change as the temperature rises; and how can this be proved?