9. In what cases do the verbs with radical u [acudir, tussir, &c.] change this u into o? 10. Write sentences exhibiting the peculiar usage which permits the suffixing of personal endings to the infinitive in Portuguese. Translate these phrases : (a) um relogio que regula bem. (b) elle pegou na sua espada, e elles tomarão as de Villadiogo. (c) não tem duvida. (d) apanhei uma constipação. (e) viver aos dias. (f) reparaste a que horas o navio sahiu ? UNDERGRADUATE PRIZE EXAMINATION PAPERS. Michaelmas Term. JUNIOR SOPHISTERS. Mathematics. A. MR. WILLIAMSON. I. Give Newton's proof that a body describing an orbit round a centre of force describes equal areas in equal times, round the centre of force. 2. A ladder rests on a horizontal plane, and against a vertical wall; being given the coefficients of friction, find the greatest weight that can be attached at a given point of the ladder without disturbing the equilibrium. 3. Find the radius of the least circle of aberration in the case of reflexion at a spherical surface. 4. If the atmosphere be supposed to consist of a dry mixture of oxygen and hydrogen, and if the atmospheric pressure be 30 inches, calculate how much of this pressure is sustained by the nitrogen, and how much by the oxygen: assuming the specific gravity of the oxygen to be 1.106, and of the nitrogen 0.972. 5. Show how the latitude of a place can be determined from the altitude of a star observed near the meridian. 6. Determine the aberration of a star in right ascension, and in declination. MR. BURNSIDE. 7. Determine the duration of twilight at the Equator. 8. Determine the geometrical focus of rays after direct refraction through a sphere. 9. A hollow cylinder, open at one end, is partly immersed in water: determine the height of the surface of the water within the cylinder. 10. A spherical envelope contains gas at a given pressure: it is required to find the tension at any point. 11. A heavy body is to be conveyed to the top of a rough inclined plane determine whether a greater force must be exerted to lift the body or drag it along the plane. 12. A particle starting from rest descends down the convex side of a circle from a given point: find where it will leave the curve. MR. M'CAY. 13. A body slides down a rough incline AB, then down another BC; prove that the velocity at C is independent of the position of B, the points AC being fixed." 14. How does Godfray show by a figure that the equation of time vanishes four times a year? 15. Give a general explanation of the phenomenon of the tides. 16. Prove that the free surface of a rotating liquid is a paraboloid, and determine the height to which the liquid will rise in a cylindrical vessel with a given velocity of rotation. 17. A particle A is thrown straight at a particle B at the instant that B commences to fall freely prove that A and B will meet. 18. Determine the relation between the distances of conjugate foci of a thin lens, and deduce Mr. Townsend's construction. B. MR. WILLIAMSON. 1. Give Newton's proof that if a central attractive force vary directly as the distance, the orbit is an ellipse, having the centre of force for its centre. Show also, by his method, that the periodic times are the same in all orbits described round the same centre of force. 2. Define a principal axis, and show that at any point of a rigid body there are three principal axes. 3. Prove that the time of describing any arc of a focal orbit can be expressed in terms of the chord of the arc, and of the focal distances of its extremities. 4. Being given the errors in the tabular Right Ascension and Declination of a star, find the corresponding errors in its latitude and its longitude. 5. Find the position of the metacentre in the case of a solid cone floating with its axis vertical and its vertex downwards; and determine whether the equilibrium is stable or unstable. 6. Determine the dispersion of a ray of light produced by transmission through two prisms placed in any manner with respect to each other. MR. BURNSIDE. 7. A particle is projected along a smooth groove from a point equidistant from two centres of force of equal intensity varying as the dis tance: determine the form of the groove that the particle may move uniformly. 8. A cylinder descends down a perfectly rough inclined plane by the action of gravity, its axis being horizontal: to determine the motion of the cylinder, and the friction at any point of its descent. 9. Determine the force to the pole under which the pedal of the curve r = f(p) can be described. (a). Apply the method to the curve given by the equation 10. If a lamina moves in its plane so that two fixed points in it describe right lines, with accelerations u, v, prove that the aceleration of the centre of instantaneous rotation is √u2 + v2 2uv cos 0 O being the angle between the lines. 11. Determine the depth of the centre of pressure of a triangle, the depths of the angular points below the surface of the fluidbeing given. 12. Prove that the caustic by refraction of a circle is the evolute of a cartesian oval. MR. M'CAY. 13. Prove that the velocity acquired by a particle running down a rough plane curve, from one fixed point to another, is independent of the nature of the curve. 14. How does Newton determine the centre of force, being given the velocity and direction of motion at three points of the orbit? 15. Show how to determine the position of the ecliptic in the heavens at any given instant. 16. Given the ratios of the lengths of the shadows of a vertical pole at noon on the days of the solstices, determine the latitude. 17. Prove that the image of a right line in a spherical mirror is a conic. 18. Prove that the caustic by refraction of a plane surface is the evolute of a conic. Classics. MR. ABBOTT. Translate the following passages :— 1. Beginning, Praeterea, omnis odos, fumus, vapor, atque aliae res.... Ending, Quam quae tenuia sunt, hiscendi est nulla potestas. LUCRETIUS, lib. iv. 88-102. 2. Beginning, Sed quod amara vides eadem, quae fluvida constant,... Ending, Aspera, quo magis in terris haerescere possint. Ibid., lib. ii. 463–476. 3. Beginning, Lubrica nascentes implent conchylia lunae:. Ending, Sunt, quorum ingenium nova tantum crustula promit. HORACE, Sat. iv. 30–47. 4. Beginning, At vos incertam mortales funeris horam.... Ending, Concessum nulla lege redibit iter. PROPERTIUS, lib. ii. El. 27. 1. Quote Cicero's remark on the poems of Lucretius, and explain it. 2. On what grounds is it argued that Cicero himself edited the work of Lucretius? 3. Write a note on Lucretius' use of the words: caula, decello, protelum, videlicet. 4 Give some account of Trajan's expedition into Armenia. 5. Mention the principal incidents in Jewish history during the reign of Hadrian. 6. State briefly the symptoms of decline of the empire observable under M. Aurelius Antoninus. MR. GRAY. Translate into Greek Prose : First of all, a man should always consider how much he has more than he wants. I am wonderfully pleased with the reply which Aristippus made to one who condoled him upon the loss of a farm: "Why," said he, "I have three farms still, and you have but one; so that I ought rather to be afflicted for you, than you for me." On the contrary, foolish men are more apt to consider what they have lost than what they possess; and to fix their eyes upon those who are richer than themselves, rather than on those who are under greater difficulties. All the real pleasures and conveniences of life lie in a narrow compass; but it is the humour of mankind to be always looking forward, and straining after one who has got the start of them in wealth and honour. For this reason, as want: there are few rich men in any of the politer nations but among |