the density at any point, and the whole pressure on the curved surface of the cylinder, neglecting the effect of gravity. 10. State and prove the general formula for finding the metacentre of a floating body; and apply it to discuss the stability of Noah's ark, which was 300 cubits long, 50 cubits broad, and 30 cubits high, and had three storeys; assuming with Bouguer that it was immersed to a depth of 10 cubits when loaded. MECHANICS. DR. TARLETON. I. A system of forces constant in magnitude and direction act at definite points of a rigid body. If the body be in equilibrium for two consecutive positions having a straight line in common, prove that the equilibrium is undisturbed by any angular displacement round this line as axis. 2. A uniform rod is placed at right angles to two fixed horizontal and parallel rough bars, which are at different heights from the ground. The rod passes over one bar and under the other; find the shortest length it can have consistently with equilibrium. 3. A uniform lamina, in the form of an isosceles triangle, is in equilibrium with its three angular points resting on the interior surface of a smooth hemisphere; determine the angle a vertical line makes with the plane of the triangle, in terms of one of its equal sides, its altitude, and the radius of the hemisphere. 4. A uniform elastic ring having its plane horizontal is stretched round a smooth surface of revolution, whose axis is vertical, and remains in equilibrium at whatever part of the surface it is placed. Determine the form of the surface. 5. A flexible string suspended from two points is in equilibrium under the action of gravity, and is such that its mass at each point of its length is proportional to the tension at that point. Determine the curve in which the string hangs, and prove that the tension varies as the radius of curvature. 6. Determine the amount of work which is required to scatter to an infinite distance from each other the particles composing a homogeneous sphere of mass M and radius a, the particles attracting each other with forces which vary inversely as the square of the distance. 7. An imperfectly elastic particle, projected with a velocity V in a direction making an angle a with the horizon, strikes a smooth vertical wall perpendicular to the plane of the trajectory and at a distance a from the point of projection. Find when and where the particle will meet the horizontal plane passing through the point of projection. 8. A number of particles are projected from a point with the same velocity, but in different directions, and are acted on by forces emanating from a fixed point and varying inversely as the square of the distance; prove that the centres of the orbits described are situated on the surface of a sphere. 9. A heavy particle moves on the interior surface of a paraboloid of revolution whose axis is vertical and vertex downwards. Prove that the velocity of the particle at the highest point of its path is that due to the height of the lowest point above the vertex of the paraboloid, and similarly for the lowest point. 10. A plane lamina receives a motion of translation in its own plane whose components parallel to rectangular axes are a and b, and a rotation round a point in itself which at the beginning of the motion coincides with the fixed origin. Determine the co-ordinates of the point, a rotation round which would bring the lamina from its initial to its final position. 11. A plane area is made to rotate with an angular velocity w round a fixed axis in its own plane by the expenditure of a given amount of work. When rotating it strikes a sphere of mass m, at a distance a from the fixed axis, whose velocity at the instant of impact is zero. Determine the moment of inertia of the plane area round the fixed axis, in order that the velocity imparted to the sphere should be a maximum. 12. The base of a smooth homogeneous circular semi-cylinder rests on a horizontal plane. A heavy particle is placed at a point on the surface of the semi-cylinder situated in a vertical plane containing its centre of inertia and perpendicular to its axis. Show that the particle will move in an ellipse. LOGICS. DR. STUBBS. 1. If an extreme be universal in the premiss, and particular in the conclusion, investigate the legitimate modes which can result. 2. Investigate (a) the circumstances under which the substitution of its contradictory for a premiss can give rise to a legitimate syllogism; and (b) the conditions which must be fulfilled by the retained and suppressed premises when the substitution of the conclusiou of a legitimate syllogism for one premiss produces new premises which are legitimate. 3. To what tendency of the human mind does Mill refer the doctrine of qualities as a peculiar species of entities? Show how this tendency : acts in the case referred to. 4. What does Condillac mean by the statement that Definition is an analysis? How is this put in Mill's language? What are the two kinds of imperfect definition? 5. Under what circumstances does it happen that the method of Differences cannot be made available at all, or not without a previous employment of the method of agreement? What example is given by Mill? 6. There is a characteristic imperfection of the method of Agreement from which the method of Differences is free: explain this, and show that it does not apply to the joint method. 7. What is Stewart's opinion concerning the foundation of Geometry? What remark does Bain make upon this statement ? and in what respect is the supposed refutation of Stewart's doctrine by Whewell defective? 8. Show that when a sequence of phenomena is resolved into other laws-(a) they are always laws more general than itself; and (b) they are more to be relied on. 9. Give cases from Locke's Essay in which he appears to recognize the intellect as a source of ideas. IO What evidence have we of the existence of spirit and of spirits, according to Locke ? II. How does he prove the necessary existence of pure space? In what words is the element of necessity introduced? 12. Why does he suspect that Natural Philosophy is not capable of being made a science? 13. Locke mentions an objection which may be brought against verbal truth? and how does he reply to it? 14. What reason does he assign for his opinion that the great ends of morality and religion are well enough secured without philosophical proofs of the soul's immateriality? He specifies two opposite errors in relation to this subject? 15. Berkeley distinguished between two kinds of magnitude? What kind of figure is the object of geometrical reasoning? How does Mansel disprove this theory? 16. What does Mansel mean by saying that the bodily organism is the debatable land between self and not self? 17. How does he distinguish between emotions and passions? 18. What subdivision of the genus representation is made by Kant? In what respect does Hamilton differ from him? 19. Berkeley pointed out Locke's error with respect to Abstraction? With what oversight may Berkeley be charged? 20. What four classes of judgments does Mansel enumerate, and upon what laws does each class depend? EXPERIMENTAL PHYSICS. MR. FITZGERALD. 1. Explain how to calculate the height to which a liquid will rise in a capillary tube, and hence how to measure the superficial energy of the liquid. 2. Describe Bunsen's ice calorimeter, and how to use it. 3. Describe Andrews' experiments on the continuity of the liquid and gaseous states. 4. Describe how the laws of the absorption of radiant heat have been studied. 5. How has it been shown that all reversible engines working between the same limits of temperature have the same efficiency? (a) How has an absolute scale of temperature been founded upon this? 6. Explain Faraday's method of mapping out the electric forces in space by means of tubes of force; and assuming that the intensity of the force varies as the number of tubes per unit area, show how to calculate the amount of the force near a charged sphere. 7. How are the changes in the horizontal intensity of the terrestrial magnetic force measured? 8. 5 cells, each with an el. m. f. of 1.1 volts, and a resistance of .6 ohms, are used to drive a current through a resistance of 2 ohms, what will the current be in the following cases: (a) initially; (b) after half the liquid had been spilled out of three of the cells; (c) when the liquid left in these three cells had been so much used up that their el. m. f. had fallen to .8 volts ? 9. How are resistances compared by means of Wheatstone's Bridge? 10. Explain how Ruhmkorff's coil works, and why the secondary coil is made so long and of such thin wire. Translate: Classical Scholarships. DR. INGRAM. I. Beginning, Ὁ δὲ ἀμείβετο λέγων· κ.τ.λ. 2. Beginning, καὶ εἴ τῳ ἄρα παρέστηκε τὸν μὲν Συρακόσιον, κ τ. λ. Ending, ἔργῳ δὲ τὴν αὑτοῦ σωτηρίαν. HERODOTUS, lib. vii. cap. 49, 50. THUCYDIDES, lib. vi. cap. 78. Ibid., lib. iv. cap. 86. PLATO, Phaed., xlvii. 3. Beginning, καὶ εἴ τις ἰδίᾳ τινὰ δεδιὼς ἄρα, μὴ ἐγώ, κ. τ. λ. Ending, παρέχεται ὡς καὶ ξυμφέρει ὁμοίως ὡς εἶπον. 4. Beginning, εἰ δέ τις λέγοι ὅτι ἄνευ τοῦ τὰ τοιαῦτα ἔχειν, κ. τ. λ. Ending, ἔφη, ἐπίδειξιν ποιήσωμαι, ὦ Κέβης ; 5. Beginning, Οὐ τοίνυν εὔκολος ὁμοίως γίγνοιτ', κ.τ.λ. Ending, πάντων τελεώτατον πρὸς ἀρετὴν ἀνδρῶν. ID., De Leg., lib. iv. P. 708. 6. Beginning, ὡς μὲν τοίνυν οὐχὶ καλῶς οὗτος ἔχει, κ.τ.λ. Ending, νόμον εἰσφέρειν ἢ ὃν νῦν ἀφ ̓ αὑτοῦ τίθησιν. DEMOSTHENES, Adv. Lept., P. 487. 1 DR. MAGUIRE. [You are requested to begin each passage on a new page.] Translate : (α) Beginning, ὡς δ ̓ ὅτ ̓ ἂν ἀΐξῃ νόος ἀνέρος, ὅστ ̓ ἐπὶ πολλὴν, κ.τ.λ. Ending, "Ηρη, τίπτε βέβηκας, ἀτυζομένῃ δὲ ἔοικας; HOMER, I., xv. 80-90. (6) Beginning, αὐτόν μιν πληγῇσιν ἀεικελίῃσι δαμάσσας, κ. τ. λ. Ending, πρίν γε τὸν ἐς νῆάς τε θοὰς κλισίας τ ̓ ἀφικέσθαι· ID., Od., iv. 244-255. (c) Beginning, οἳ δ ̓ ἄρ ̓ ἐν ἐλλεδανοῖσι δέον καὶ ἔπιτνον ἀλωήν. κ. τ. λ. Ending, οἵ γε μὲν ἐτράπεον, τοὶ δ ̓ ἤρυον. HESIOD, Heracl., 290-301. (α) Beginning, ΗΛ. ἐῶ, κελεύω, θῦε· μηδ ̓ ἐπαιτιῶ, κ. τ. λ. Ending, σπείρῃ ματαίαν βάξιν ἐς πᾶσαν πόλιν. SOPHOCLES, Electr., 632-642. (e) Beginning, καὶ τοὺς μὲν ηὐχένιζε, τοὺς δ ̓ ἄνω τρέπων, κ. τ. λ. Ending, κόμην ἀπρὶξ ὄνυξι συλλαβὼν χερί. ID., Ajax, 298-310. (f) Beginning, ᾧ γὰρ τὸν ἐνδυτῆρα πέπλον ἀρτίως, κ. τ. λ. Ending, χαλκῆς ὄπως δύσνιπτον ἐκ δέλτου γραφὴν. ID., Trach., 674-683. (9) Beginning, 10. ὡς μὴ μενοῦντα, τἄλλα σοι λέγειν πάρα. κ. τ. λ. Ending, τοὺς μὲν μάχεσθαι, τοὺς δὲ δειλίᾳ μένειν. EURIPIDES, Heracl., 693-701. (h) Beginning, tθ ̓ ὡς πόσις μοι πλησίον γαμηλίους, κ. τ. λ. Ending, ὅστις τὰ σιγῶντ ̓ ὀνόματ ̓ οἶδε δαιμόνων. ID., Phaeth. ID., Rhesus, 780-791. (i) Beginning, καί μοι καθ ̓ ὕπνον δόξα τις παρίσταται· κ. τ. λ. Ending, βάλλει με δυσθνήσκοντος αἵματος νέου. MR. GRAY. Translate : I. Beginning, Quum mihi dixisset Caecilius quaestor puerum, CICERO, Epist. ad Att., ii. 9. 2. Beginning, Quae vero, cum de facto non ambigitur, Ending, et, prope dicam, decantata abere debeant. .... ID., De Oratore, lib. ii, cap. 32. |