2. Two weights connected by a stretched cord slide down a rough inclined plane; if their coefficients of friction be μ and μ', respectively, find an expression for their common acceleration.

3. A weight of 8 lbs. is suspended from the extremity of a string 20 ft. long; find the least velocity that should be given to it in order to break the string, if the greatest weight which it can sustain be 12 lbs.

4. Prove the formula for the time of a small oscillation in a simple pendulum.

5. A train moving at the rate of 40 miles an hour comes to the foot of an incline of 1 in 100; find how far it would ascend the plane before stopping, if the steam be shut off, assuming the resistances of the air and friction to amount to 8 lbs. per ton.

6. If a projectile describe a series of parabolic arcs by rebounding from a horizontal plane of elasticity e, prove that the times of flight form a geometrical progression whose common ratio is e, and the greatest distances from the plane form a geometrical progression whose ratio is e2. Find also the whole range on the plane before the body ceases to rebound.

DR. TRAILL.

7. If all the particles of a body attract an external point with a force varying directly as the distance, prove that the effect is the same as if the whole mass of the body were concentrated at its centre of gravity.

8. If forces be applied to the middle points of the sides of any closed polygon, proportional to those sides, they will be in equilibrium?

9. A uniform beam rests with one end in a fixed hemispherical bowl, having its base horizontal, and the other projects beyond the rim of the bowl; find the position of equilibrium.

10. Three forces act in equilibrium at the angles of a triangle, one bisecting the angle at which it acts, and the other two making equal angles with the side opposite to that angle; show that the forces are as the sides opposite to their points of application.

II. Find the time of vibration of a pendulum, which consists of a rod with n different weights attached to different points on it.

12. If a liquid be in equilibrium under the action of any forces, prove that the resultant force at any point is in the direction of the normal to the surface of equal pressure passing through the point.

MR. BURNSIDE.

13. Two spheres rest on two smooth inclined planes, and press against each other; to determine their position.

14. A ladder with twelve steps is placed with one end on the ground, and the other end resting against a smooth vertical wall; determine the coefficient of friction between the ladder and the ground which will just allow a man to ascend within two steps of the top of the ladder.

15. The edge of a square whose side is 10 ft. coincides with the foot of a plane inclined at 30° to the horizon; how high may a rectangle be constructed on this base without toppling over?

16. Two bodies are connected by a string which passes over a small smooth pulley fixed at the top of two inclined planes having a common height; determine the motion, and tension of the string, supposing one body to remain in contact with each plane.

17. Find the law of force by which a particle may describe a circle, the centre of force being in the circumference of the circle.

18. A weight P descending draws a weight Q up an inclined plane whose height is h, and length 7, by a chord passing over a pulley at the summit of the plane: prove that Q will ascend to the top, if the cord breaks when it has ascended the plane, a distance equal to

1. Prove that the form of each horizontal section of a vertical sail, acted on by a horizontal wind, is a catenary.

2. If the equation of a central orbit be rm = am cos me, the force being directed to the origin, prove that the force varies as

I

r2m+3

3. In the same case, prove that the velocity at each point is that due to motion from an infinite distance towards the centre of force.

4. Find, by D'Alembert's principle, the general equations of motion of a rigid body when acted on by any forces: apply to the case of a body turning round a fixed axis, gravity being the only force which acts.

5. A window sash is 4 ft. wide, and 18 in. high; find the greatest distance from the middle at which it can be raised by one hand, the coefficient of friction being o'6.

DR. TRAILL.

;

6. Round the edge of a smooth elliptical table there is a raised border a perfectly elastic ball is projected in any direction from one of the foci of the ellipse, and rebounds after impact on the border; find its direction after the nth impact, and show that it will eventually move along the axis major.

7. Find the condition that there should be no loss of vis viva after the impact of two bodies.

8. Prove that the cycloid is a tantochronous curve, i. e. that all bodies starting simultaneously from different points on the curve, will, under the influence of gravity, arrive at the lowest point at the same time.

9. A string which passes through a small heavy ring has its ends attached to the extremities of a lever which rests on a fulcrum; find the conditions of equilibrium—

(a) when the lever has weight;

(b) when the weight of the lever is neglected.

10. If a given volume of incompressible fluid be acted on by forces,

11. A board movable about a horizontal line in its own plane is supported by resting on a rough sphere which lies on a horizontal table; find the greatest inclination at which the board can rest.

12. Find the ellipse of least eccentricity in order that it may be capable of resting in equilibrium upon a perfectly rough inclined plane.

13. A square board is supported in a horizontal position by three vertical strings; if one be attached to a corner, where must the other two strings be attached in order that the weight which can be placed on any part of the board without overturning it may be the greatest possible?

14. A rod having been placed in a given position, with one end resting on a smooth horizontal plane, and the other leaning against a smooth vertical wall, descends in a vertical plane under the action of gravity; determine where the rod will detach itself from the vertical wall.

15. A body symmetrical with regard to a plane through the centre of gravity revolves round a horizontal axis perpendicular to this plane; if gravity be the only force acting, and if the centre of gravity lies initially in the horizontal plane through the axis, determine the pressures on the axis.

C.

DIFFERENTIAL AND INTEGRAL CALCULUS.

MR. WILLIAMSON.

1. Prove that the points of inflexion on the curve

into another in which z is the independent variable, being given x =

3. Integrate the expressions

dx 1

dx

I

(x+a) √T-x2' (1+r2x2) √1-x2

4. Find the whole area of the loop of the curve

y3 - zaxy + x3 = 0.

DR. TRAILL.

5. Find, by Maclaurin's theorem, six terms of the development of in ascending powers of x.

ex

6. A man in a boat, 3 miles from shore, wants to reach a town in the shortest time, the town being 5 miles from the nearest point on the shore. He can sail 4 miles an hour, and walk 5 miles an hour. At what point should he land?

9.

MR. BURNSIDE.

Show that the maximum and minimum values of the function
ax3 + 3bx2 + 3cx + d = U

are determined by the equation

a2 U2-2GU+A = 0,

when Ga2d-3abc + 2b3 and ▲ is the discriminant of U.

10. If u be a function of the rectangular co-ordinates of a point, show that the function

is unaltered when the axes are turned through any angle.

11. Determine the value of the integral

Classics.

MR. MAHAFFY.

Translate the following passages :—

1. Beginning, ΦΙΛ. ὦ πταναὶ θῆραι, χαροπῶν τ', κ.τ.λ. Ending, ἔχειν μυρίον ἄχθος, ᾧ ξυνοικεῖ.

SOPHOCLES, Philoct., 1123-44.

2. Beginning, ἑπτὰ λοχαγοὶ γὰρ ἐφ ̓ ἑπτὰ πύλαις, κ. τ. λ. Ending, Βάκχιος ἄρχοι.

Ibid., Antig., 141–54.

3. Beginning, ἄναξ, βροτοῖσιν οὐδέν ἐστ ̓ ἀπώματον, κ. τ. λ. Ending, ἀλλ' ἔστ' ἐμὸν θοὔρμαιον, οὐκ ἄλλου τόδε.

Ibid., 386-95.

4. Beginning, φωνῆς δ' ἄπωθεν ὀρθίων κωκυμάτων, κ.τ.λ. Ending, φθόγγον συνίημ', θεοῖσι κλέπτομαι.

Ibid., 1185-97.

5. Beginning, ΗΑ. κἀμοὶ προσέστη καρδίας κλυδώνιον, κ. τ. λ. Ending, βροτῶν Ὀρέστου· σαίνομαι δ ̓ ὑπ ̓ ἐλπίδος.

ESCHYLUS, 183-94.

1. Explain accurately what is meant by the growth of language. 2. In what sense are languages mixed? Is Max Müller's statement too universal?

3. Sketch the history of grammatical studies among the Greeks, and at Rome.

4. What did the Greeks mean by barbarous languages?

5. Classify the languages of Europe.

6. What is the chief importance of the Lithuanian language?

7. Explain the origin of the term genitive case, and give the IndoEuropean forms for the Latin and Greek genitives in ov and i.

MR. PALMER.

Translate the following passages into English :

1. Beginning, Ph. Dorio, obsecro, parumper audi. De. Quin omitte

me...

Ending, Ut phaleratis dictis ducas me, et meam ductes gratiis? TERENCE, Phormio, act iii. sc. 2,

2. Beginning, Plus triginta natus annis sum, quom interea loci... Ending, Sublatum est convivium! edepol venio advorsum temperi ! PLAUTUS, Menæchmus, act iii. sc. 1.