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that the kinetic energy of the system is diminished in the ratio : 1, e being the coefficient of elasticity.
11. The upper half of an inclined plane is smooth, the lower rough: determine the coefficient of friction if a weight started from the top just reaches the bottom.
12. Two particles are projected from the same point with the same velocity of projection V, and in the same vertical plane, the interval of time between their projections being T; show that the condition they shall meet is
sin (a – B) gT
cos } (a +B) a, ß being the angles of projection.
MR. W. R. ROBERTS.
13. Find the inclination of a smooth plane, 8 ft. in length, being given that a heavy particle slides down it in one second.
14. Describe Newton's proof of the parallelogram of forces.
15. Two perfectly elastic spheres meet directly with equal velocities : find the relation between their magnitudes if one remain at rest after the collision.
16. Find the rough line of quickest descent from a point to a circle.
17. Two weights, P and Q, connected by a string, rest on the convex side of a smooth vertical circle : find the position of equilibrium.
18. A heavy ring is placed on a rough vertical circle: find the limits of its position consistent with equilibrium.
1. A string coiled round a cylinder, placed on a rough inclined plane, is attached to a heavy particle which also rests on the plane, friction being the same for the cylinder and the particle. If the inclination of the plane be gradually increased, determine the nature of the initial motion and the angle of inclination at which this takes place.
2. Investigate the nature of the orbit described by a body revolving round a force attracting directly as the distance ; and determine the magnitude and position of the orbit for given initial values of the distance, velocity, and direction of projection.
3. A particle acted on by gravity only descends along a curved line in a vertical plane, find the form of the curve that the same pressure may be exerted at every point.
4. Prove Steiner's theorem that the locus of a point with respect to which the pedal of a given closed curve has a constant area is a circle.
5. Apply the integral calculus to determine the curres in which the radius of curvature is equal to the normal.
6. Find the centre of gravity of the portion of the area of an ellipse contained between two conjugate semidiameters and the curve.
MR. F. PURSER.
7. A chain 15 ft. long rests on a rough table; at the edge of the table, and in the same vertical plane with the chain, is erected a post 2 ft, high supporting a smooth pulley. If one end of the chain be now lifted up, and thrown over the pulley, determine the least length of chain which must hang vertically over the pulley in order to lift the remainder from the plane, the coefficient of friction being .
8. The attraction of a given centre of natural force being represented by u (distance)-?, determine the change produced in the numerical value of u when the unit of length is altered in the ratio m: 1, and that of time in the ratio n :
:1. 9. A particle is projected in a given direction from a given point P, and is acted on by a central force attracting according to the law of nature emanating from a given centre 0. Show that the eccentricity will be a minimum when the point of projection is at the extremity of the axis minor in the resulting orbit, and determine the velocity of projection in order that this should be the case.
10. Find the equation of the evolute of a parabola. Draw a diagram of the form of the curve.
u. Determine the position of the inflexions of the conchoid of Nicomedes, r = a sec 8+, and show that their locus for a family of such curves obtained by varying the constant b is a semicubical parabola,
12. Calculate the definite integral
13. Two uniform heavy, bars are freely jointed at a common extremity, and are fixed at their other extremities to two smooth joints in a vertical line : find the stresses at the joints.
14. Two equal smooth spheres are placed inside a smooth hollow cylinder open at both ends, which rests on a horizontal plane : find the least weight of the cylinder consistent with equilibrium.
15. A heavy particle is placed on the exterior of a vertical circle at its highest point 0, and moves from rest leaving the circle at Pa; it is again placed on the circle at P1, and moves from rest, leaving the circle at P2; and this is repeated n times. Find the sum of the cosines of the arcs OPI, OP2,
18. Divide an elliptic quadrant into two parts whose difference shall be equal to the difference of the semiaxes.
Translate : 1. Beginning, νομίζω δε δύο τα εναντιώτατα ευβουλία είναι, κ. τ.λ. Ending, και αυτός προσάγεσθαι το πλήθος.
THUCYDIDES, lib. iii. c. 42.
2. Beginning, & χρή γνόντας, και ιδιώτην ιδιώτη, κ. τ. λ. Euding, ελθόντες ευλόγως άπρακτοι απίασι.
Ibid., lib. iv. c. 61,
3. Beginning, κεραίαν μεγάλην δίχα πρίσαντες εκοίλαναν, κ. τ. λ. Ending, ναι και το τείχισμα τούτο το τρόπο αλώναι.
Ibid., lib. iv. c. 100.
4. Beginning, το δε Νικιου στράτευμα απείχεν εν τω πρόσθεν, κ. τ.λ. Ending, τη ιδέα καταδαμασάμενοι λήψεσθαι αυτούς.
Ibid., lib. vii. c. 81.
1. What information does Thucydides give as to the races by whom Sicily had been successively inhabited ? and of the several Greek settlements ?
2. To what epochs does Thucydides resort in fixing the date of the occupation of Platæa by the Thebans ? How does he give the date of its destruction by the Spartans ? What date does he give of Aminocles the Corinthian ship-builder, of the first recorded Greek naval action, and of the existing Spartan constitution ?
3. The commencement of the Peloponnesian war was an epoch of marked deterioration in the Greek character. By what facts does Mure prove this? Thucydides himself is no exception ?
4. Obscurity, the chief defect of Thucydides as a writer, has been accounted for in two ways by most critics ? How does Mr. Mahaffy reply?
5. Sketch the life of Antiphon the Orator. Quote the character given of him by Thucydides.
6. Under what circumstances did Critias meet his death ? 7. Quote “the violent sentencc” in Thucydides about Hyperbolus. How does it affect Grote's theory of Ostracism?
8. State exactly the position of the following places :—Naupactus, Dium, Eniadæ, Stagirus, Anactorium.
Translate :1. Beginning, Sustinuit labantem aciem Antonius accitis, Ending, set Flavianis aequior a tergo.
Tacitus, Hist. iii. 23. 2. Beginning, Classis quoque faces intulit, vacuo mari eludens, Ending, indiscretum et innoxium est.
Ibid., iii. 47. 3. Beginning, Hic exitus viri haud sane spernendi: .... Ending, partibus Vitellii amolitus videbatur.
Ibid., iii. 75. 4. Beginning, Undecimo kalendas Iulias serena luce spatium, Ending, templi magnificentiae defuisse credebatur.
Ibid., iv. 53. 5. Beginning, Ceterum si omisso optimo illo et perfectissimo, Ending, histriones diserte saltare dicantur.
Ibid., De Orat., 26.
1. Write notes on Tacitus's use of pagani, vernile, clavarium, sinus, aversari, invidia.. 2. Describe the state of Rome in the
6 A.D. 3. Discuss the causes of Ovid's banishment. 4. What accounts have we of the last moments of Augustus ?
5. Describe the writing materials used at this time. Have any specimens survived ?
6. What was the usual rate of travelling by sea and land ? Compare the highways of that age with those of modern Europe before railways.
7. Translate : saltuariorum testamenta, quibus cum clogio exheraedebatur.
What is the context ?
MR. L. C. PURSER.
Translate into Greek Prose :Hermokratês impresses upon his hearers that the aggressive schemes of Athens, now the greatest power in Greece, were directed against all Sicily, and threatened all cities alike, Ionians not less than Dorians. If they enfeebled one another by internal quarrels, and then invited the Athenians as arbitrators, the result would be ruin and slavery to all. The Athenians were but too ready to encroach everywhere, even without invitation : they had now come, with a zeal outrunning all obligation, under pretence of aiding the Chalkidic cities who had never aided them
- but in the real hope of achieving conquest for themselves. The Chalkidic cities must not rely upon their Ionic kindred for security against evil designs on the part of Athens : as Sicilians, they had a paramount interest in upholding the independence of the island. If possible, they ought to maintain undisturbed peace; but if that were impossible, it was essential at least to confine the war to Sicily, apart from any foreign intruders. Complaints should be exchanged, and injuries redressed, by all, in a spirit of mutual forbearance; of which Syracuse—the first city in the island and best able to sustain the brunt of war-was prepared to set the example ; without that foolish over-valuation of favourable chances so ruinous even to first-rate powers, and with full sense of the uncertainty of the future. Let them all feel that they were neighbours, inhabitants of the same island, and called by the common name of Sikeliots; and let them all with one accord repel the intrusion of aliens in their affairs, whether as open assailants or as treacherous mediators. — GROTE,
Translate into Latin Prose :
The gates of the city were thrown open, and the new emperor of the Romans, encompassed on every side by the Gothic arms, was conducted, in tumultuous procession, to the palace of Augustus and Trajan. After he had distributed the civil and military dignities among his favourites and followers, Attalus convened an assembly of the senate; before whom, in a formal and florid speech, he asserted his resolution of restoring the majesty of the republic, and of uniting to the empire the provinces of Egypt and the east, which had once acknowledged the sovereignty of Rome. Such extravagant promises inspired every reasonable citizen with a just contempt for the character of an unwarlike usurper, whose elevation was the deepest and most ignominious wound which the republic had yet sustained from the insolence of the barbarians. But the populace, with their usual levity, applauded the change of masters. The first days of the reign of Attalus were fair and prosperous. An officer of confidence was sent with an inconsiderable body of troops to secure the obedience of Africa; the greatest part of Italy submitted to the terror of the Gothic powers; and though the city of Bologna made a vigorous and effectual resistance, the people of Milan, dissatisfied perhaps with the absence of Honorius, accepted, with loud acclamations, the choice of the Roman senate.-GIBBON.