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2. Find the whole area of the curve

a2 y2 = x3 (2α- x).

3. A number of heavy particles slide from rest down the chords of a vertical circle, starting from the highest point, and afterwards move freely; prove that the locus of the foci of their paths is a circle, whose radius is half that of the given circle.

4. Find the time of a small oscillation in a simple pendulum, and show how to determine the corrections due to a change of place, and of length, respectively.

5. A square board ABCD, whose plane is vertical, rests with its side AD in contact with a rough vertical wall, perpendicular to the plane of the board, the side AB resting on a rough peg, indefinitely near to B ; find the least value of the coefficient of friction, supposing it the same for the wall and for the peg.

MR. BURNSIDE.

6. Find the maximum and minimum values of r where

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7. Find the value of the radius of curvature at any point of the curve determined by the equations

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9. Determine the equations of equilibrium of a flexible string, under the action of given forces; and find the condition that the tension be constant throughout.

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

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12. A weight P, after falling freely through h feet, begins to pull up a heavier one Q by means of a cord over a pulley, as in Atwood's machine; find the height through which it will lift it.

13. The Moon's orbit being supposed circular, and her periodic time P, prove that if her motion was suddenly stopped, the time of falling into

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14. Find the position of equilibrium of a rod resting in a smooth vertical parabola.

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3. Show how to determine the superficial area of an ellipsoid of revolution, distinguishing between the cases as the surface is prolate or oblate.

4. Two bodies connected by a string slide down a rough inclined plane; if the coefficients of friction be μ and u' respectively, find the acceleration, the string being supposed parallel to the plane.

5. A spherical ball impinges on another at rest; show that if the coefficient of elasticity be equal to the ratio of their masses, the balls will move in directions at right angles to each other, whatever be the direction of the impact.

MR. BURNSIDE.

6. A uniform rod passes through a hole in a spherical shell, and rests with one end against the internal surface, the length of the rod being equal to twice that of the diameter; determine its positions of equilibrium.

7. A molecule is placed at a point within a triangle formed of three uniform rods of densities, A, u, v, which attract inversely as the square of the distance; find the conditions of equilibrium.

8. A particle acted on by two central forces, each attracting according to the law of the inverse square of distance, is projected along the line joining the centres; find the velocity of projection so that the particle may just arrive at the neutral point of attraction, and remain at rest there.

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11. Prove that a parabola can be described under the rectangular forces,

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12. Find the least coefficient of friction that will keep up a pyramid of four equal spheres on a horizontal plane.

13. Given two sides of a triangle, find the third so that if the triangle revolve about it the solid formed may have the maximum volume.

14. What is the curve in which the sine of the angle between radius vector and tangent varies inversely as the square of the radius vector ? 15. A particle revolves in a central ellipse; if t be the time of describing an angle a show that

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Translate the following passages :—

1. Beginning, Proximo senatus die multa in luxum...

Ending, delenimentis curarum et periculorum carendum esse.

Annal., lib. ii. cap. 33.

2. Beginning, Possessa ipso transitu Vicetia; ..
Ending, terrarum spatiis consilia post res adferebantur.

Hist., lib. iii cap. 8.

3. Beginning, Nomen sedemque Caesaris Domitianus acceperat, Ending, nec perinde prosperis socius quam adversis abstractus.

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Ibid., lib. iv. cap. 2.

(c). Raptus penatium hauserunt animo.

(d). Agitatum num et Piso proficisceretur maiore praetextu.
(e). Experti invicem sumus ego et fortuna.

(f). Ipso Vespasiano inter initia imperii ad obtinendas iniquitates haud perinde obstinante.

(9). Quid enim profuturas cohortes alasque si unus alterque praesenti facinore paratum ex diverso praemium petat.

1. Write notes on the preceding passages where necessary.

2. What are our resources for the history of the Empire besides Tacitus ?

3. Give some account of Agrippa Postumus.

4. Describe the contents of the Orbis Pictus of Agrippa.

5. What light has been thrown by A. Zumpt on the history of Quirinius?

MR. GRAY.

Translate the following passages:—

1. Beginning, τῶνδε δὲ οὔτε πλούτου τις τὴν ἔτι, κ. τ. λ. Ending, μᾶλλον ἢ τοῦ δέους ἀπηλλάγησαν.”

THUCYDIDES, lib. ii. cap. 42.

2. Beginning, “Η μὲν γενομένη ναυμαχία, ὦ ἄνδρες, κ. τ. λ. Ending, προβαλλομένους εἰκότως ἂν ἔν τινι κακοὺς γενέσθαι.” Ibid., lib. ii. cap. 87.

3. Beginning, Οἱ γὰρ Κερκυραῖοι ἐστασίαζον, κ. τ. λ. Ending, πείθει ὥστε τῷ νόμῳ χρήσασθαι.

Ibid., lib. iii. cap. 70.

4. Beginning, τέλος δὲ τραυματιζομένων ἤδη πολλῶν, κ. τ. λ. Ending, χαλεπῶς τε καὶ μόλις περιελθὼν ἔλαθε.

Ibid., lib. iv. caps. 35, 36.

5. Beginning, “ Χρῆν μὲν, ὦ ἄνδρες Βοιωτοί, κ. τ. λ. Ending, πολέμου ἄρχοντα ἧσσον ἑτοίμως κατέχειν.”

"Ibid., lib. iv. cap. 92.

1. Arnold, by an illustration taken from Roman history, enables us to understand the real meaning of the proceedings of Peithias given in the third passage?

2.

In what important particulars did the Spartan military system differ from the Athenian ?

3. On what occasions does Thucydides first introduce the names of Brasidas, Cleon, and Nicias? On two occasions Cleon influenced the Athenians towards the war?

4. State the view Thucydides seems to have taken of the proceedings of the Athenians towards Milos ?

5. Draw a map showing the position of the Dorian and Ionic cities in Italy and Sicily. The quarrel between Corinth and Corcyra brought the Sicilian parties nearer to co-operation in the Peloponnesian quarrel in two ways?

6. Write a comparative view of the state of society in Greece in the different portions of the 5th century, as depicted by Herodotus and Thucydides. Illustrate by examples.

7. How is the fact accounted for that in the reported political discussions during the Peloponnesian war, Athens is assumed to be the usurper, and Sparta the champion of Greek constitutional liberty, notwithstanding the known political tendency of the one to democracy, and of the other to oligarchy ?

8. Give an account of the probable sources and materials of the history of Thucydides.

9. Write a note on the speeches of Thucydides. On what ground is it suspected that their introduction into the text may have been an afterthought of the historian ?

10. Quote as accurately as you can the character of Pericles given by Thucydides Contrast with this that given by Plutarch.

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It is this

And it is well that nature imposes upon us in this manner. deception which rouses and keeps in continual motion the industry of mankind. It is this which first prompted them to cultivate the ground, to build houses, to found cities and commonwealths, and to invent and improve all the sciences and arts, which ennoble and embellish human life; which have entirely changed the whole face of the globe, have turned the rude forests of nature into agreeable and fertile plains, and made the trackless and barren ocean a new fund of subsistence, and the great high road of communication to the different nations of the earth. The earth, by these labours of mankind, has been obliged to redouble her natural fertility, and to maintain a greater multitude of inhabitants. is to no purpose that the proud and unfeeling landlord views his extensive fields, and without a thought for the wants of his brethren, in imagination consumes himself the whole harvest that grows upon them. The homely and vulgar proverb, that the eye is larger than the belly, never was more fully verified than with regard to him. The capacity of his stomach bears no proportion to the immensity of his desires, and will receive no more than that of the meanest peasant.

It

GREEK PROSE.

Among savages and barbarians it is quite otherwise. Every savage undergoes a sort of Spartan discipline, and, by the necessity of his situa

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