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the circle, and produced to the circumference, of the two aros intercepted between these two lines, the one is 3 times the other.

3. Describe a circle which shall touch the base of a triangle and the two sides produced.

Show that if the centre of the circle so described, and the centre of the circle inscribed in the triangle be joined to either extremity of the base, the joining lines are at right angles.

4. Give Euclid's definition of proportion, and illustrate it by proving that triangles of the same altitude are to one another as their bases.

5. A point moves so that its distances from two fixed points are in a constant ratio to one another. Prove that its locus is a circle.

6. When is a straight line at right angles to a plane, and when are two planes perpendicular to one another?

If two straight lines are parallel, and one of them is at right angles to a plane, the other is also at right angles to the same plane.

7. Show that, in a parabola, tangents at the ends of a focal chord intersect at right angles in the directrix.

Show also that, if from the ends of a focal chord perpendiculars be let fall upon the directrix, the intercepted portion of the directrix subtends a right angle at the focus.

8. The rectangles contained by the segments of any two intersecting chords of a parabola are to one another as the parameters of the diameters which bisect the chords. Hence show that if a circle intersect a parabola, the common chords are equally inclined to the axis of the parabola.

9. If perpendiculars be let fall from the two foci upon any tangent to an ellipse, prove that the feet of these perpendiculars lie on the auxiliary circle, and the rectangle contained by the perpendiculars is equal to the square on the semi-axis minor.

10. Prove that in an ellipse the sum of the squares on two conjugate semi-diameters is constant.

If along the normal at any point of an ellipse a portion be measured equal to the conjugate semi-diameter, the locus of its extremity is a circle.

11. Define an asymptote of an hyperbola, and prove that the curve continually approaches its asymptote but never meets it.

If a directrix of the hyperbola cut an asymptote in E, prove CE = CA, where C is the centre, and A a vertex.

12. If a chord Qq of an hyperbola be produced to meet the asymptotes in R, r, prove QR = qr. Hence show that if the tangent at P meet one asymptote in T, and TQ, drawn parallel to the other asymptote, meet the curve in Q, and PQ be produced to meet the asymptotes in R,R', then RR' is trisected at the points P, Q.

TRIGONOMETRY AND MENSURATION.

Examiner-DR. H. W. M'CANN.

1. Explain what is meant by the "circular measure" of an angle, and prove that it is a correct measure.

If D, G, 0, be respectively the number of degrecs, grades, and units of circular measure in any angle, find the relation between D, G, 0.

2. Trase the variations in sign and magnitude of the secant of an angle as the angle increases from 0° to 360°.

Show how the trigonometrical ratios of all angles whatever, positive or negative, may be made to depend upon those of some positive angle less than 45°.

3. Write down the general expressions for all angles which have (i) the same sine, (ii) the same cosine, (iii) the same tangent, as a given angle a.

Solve the equations, obtaining the value of 0 in the most complete form,

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Assuming its truth when A, B, A + B, are each less than a right angle, prove it when A and B each lie between 90° and 180°, and (A + B) lies between 180° and 270°.

Express sin 2A, cos 2A, in terms of tan A, and express tan A in terms of cos 2A.

5. Given a, b, c, the three sides of a triangle, investigate a formula

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6. Show how to solve a triangle having given two sides and the angle opposite one of them, and explain completely the resulting ambiguity.

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7. A certain square pleasure-ground, containing 5 acres, has in its centre a circular sheet of water occupying 1 acre, 1 rood, 20 perches: find the lengths of the paths reaching from each of the angles to the water's edge.

8. Prove the rule for finding the area of the surface of a right circular cone. Also, given the expression for the volume of a right circular cone in terms of the radius of its base and its height, deduce that for the volume of the frustum of such a cone in terms of the radii of its two ends, and of its height.

9. The volume of a spherical orange is

99

cubic inches: the thick.

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ness of the rind is inch: after the rind is removed the orange is divided into equal parts by planes through a diameter making angles of 30° with one another: find the volume of each of these parts

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10. A rectangular tank is excavated whose sides are vertical, but whose ends slope at an angle of 45°, of length 383 feet, and of depth 5 feet the earth from the excavation is employed to fill up a pit 21 feet deep, of circular section, whose radius at the top is 8 feet and at the bottom 7 feet. How broad must the tank be so that the earth may just fill the pit?

STATICS AND DYNAMICS.

Examiner-Dr. H. W. M'CANN.

1. Define a "force" and show how it is possible to represent a force completely by a straight line.

Define and explain the principle of the transmissibility of a force to any point in its line of action.

2. If any number of forces acting on a particle be represented in magnitude and way of action by the sides of a polygon taken in order, they will keep the particle in equilibrium.

If the forces, instead of all acting at a point, actually act along the sides of the polygon, what is their resultant?

Forces represented in magnitude and way of action by the sides AB, BC, CD, AD of a quadrilateral act on a particle: find their resultant. If these same forces do not act on a particle, but act along the sides of the quadrilateral, find their resultant.

3. Define the centre of gravity of a body or a system of bodies, and find that of a plane triangular lamina.

From a plane triangular lamina the triangle obtained by joining the middle points of its sides is cut away; find the centre of gravity of the remainder.

4. Explain what is meant by the "tension" of a string at any point.

Find the ratio of Power to Weight in the 3rd system of pullies.

If the strings, instead of being fastened to a weight, are fastened to a scale-pan in which a man, weight W, stands, find with what force he must pull down the free end of the string passing over the lowest pully in order to support himself, the strings being all vertical.

5. State the three laws of friction.

A uniform rod rests with one end against a rough vertical wall, the other end being supported by a string of equal length fastened to a point in the wall; prove that the least angle 0 which the string can make with the wall is given by the formula μ tan is the coefficient of friction between the rod and the wall. 6. State and discuss Newton's Three Laws of Motion.

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7. A body starts with velocity u and is acted on by a uniform force in the direction of the velocity during time t; if ƒ be the acceleration due to the uniform force, and s the space described in time t, then prove s = ut + ft2.

If a body be projected vertically upwards with velocity u, find for what time it will rise, what height it will reach, and after what time it will return to the point of projection, neglecting the resistance of the air.

8. Prove that a body projected in any direction not vertical and acted on by gravity will describe a parabola.

From two points A, B, not in the same vertical line, two particles are projected at the same instant towards one another: prove that the line joining the particles is always parallel to AB, and that the particles will meet; and find the time of meeting.

9. Two bodies, masses m, m', moving with velocities v, v', in the same direction impinge; find expressions for their velocities after impact. Show that the momentum of the system is unaltered, but the vis viva diminished, unless the bodies be perfectly elastic.

An engine weighing 40 tons, and 3 coal-trucks, each weighing 15 tons are at rest on a horizontal line; there is an interval of one foot between the engine and the first truck, and between each truck and the next. The engine starts off and strikes the first truck which then strikes the second, and so on. Supposing the engine to be constantly impelled by a force equal to the weight of one ton, and the bodies to be inelastic, find the velecity with which the last truck starts, and the whole time occupied in starting the train, neglecting friction, and taking g - 32.

10. Explain what is meant by a "simple pendulum", write down the time of a small oscillation, and show how it may be used to determine the force of gravity at any place.

A pendulum which oscillates in a second at one place gains 5 beats an hour if carried to another place. Compare the weights of the same substance at the two places.

HYDROSTATICS AND OPTICS.

Examiner-DR. W. H. M'CANN.

1. Distinguish carefully between pressure at a point, and pressure on a point in a fluid.

Prove that the pressure at any point within a heavy inelastic homogeneous fluid at rest, not subject to external pressure, is equal to the weight of a column of fluid whose base is a unit of area and whose height is equal to the depth of the point below a horizontal plane through the highest point of the fluid. How is this result modified if the atmospheric pressure be taken into account?

2. State the rules for determining the vertical and horizontal components of the pressure of a fluid at rest under the action of gravity on any surface in contact with it, and hence deduce the resultant pressure of a fluid upon the surface of a solid either wholly or partially immersed in it.

A body floating on an inelastic fluid is observed to have volumes V1, V2, respectively above the surface at times when the density of the surrounding air is P1, P2; find the density of the inelastic fluid in terms of V1, V2, P1, P2.

3. Find the centre of pressure of a triangular lamina immersed in liquid with its base in the surface (i) neglecting the atmospheric pressure (ii) taking it into account.

If a quadrilateral lamina ABCD in which AB is parallel to CD be immersed in liquid with the side AB in the surface, the centre of pressure will be at the point of intersection of AC and BD if AB2 3 CD2.

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4. Explain the construction and use of the common hydrometer. The hydrometer being graduated upwards, its readings for two different fluids are x1, x2, and for a mixture of equal parts of these ; show that the volume of a unit of length of the stem is to the volume of the whole instrument below the zero point as x1 + x2 - 2x: xx1 + xx 2 2x1 x2.

5. Describe the Forcing Pump, and explain why the air-vessel is necessary for its efficiency.

6. Explain the terms geometrical focus, principal focus, applied to a pencil of rays reflected or refracted at a spherical surface.

In direct reflection at a spherical concave or convex surface,

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7. Rays of light diverging from a point are refracted directly at a spherical surface; write down the formulæ

(i) connecting the distances of the conjugate foci from the centre of the surface.

(ii.) connecting the distances of the conjugate foci from the centre of the sphere.

An eye is placed close to the surface of a sphere of glass (μ = 3) which is silvered at the back; shew that the image which the eye sees of itself is of its natural size.

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