external to the system; show, again by any method, that the several shells thus constructed have a common centre of inertia, common central principal axes, and confocal ellipsoids of gyration with the system itself. KINETICS. I. A material particle, constrained to move without friction in a vertical circle under the action of gravity, being supposed retarded throughout its entire motion by a tangential resistance of constant intensity; determine, given all particulars, the law of diminution of its several successive arcs of vibration, and the entire number of oscillations subsequent to the commencement of any arc. 2. The first arc of vibration of a material particle, constrained to move on a uniformly rough vertical circle under the action of gravity, being supposed an exact semi-circle; determine the limiting angle of friction consistent with the possibility of an amplitude of that magnitude, and show that the velocity of its description, when possible, follows precisely the same law as for the same amplitude on a perfectly smooth circle. 3. A material particle, constrained to vibrate on a uniformly rough inclined plane under the combined action of gravity and the elasticity of an extensible cord of evanescent mass connecting it with a fixed point directly above it on the plane, being supposed in any oscillation to have no velocity when the cord has no extension; discuss the different circumstances of the termination of its motion, and determine the corresponding tensions of the cord. 4. A system of material particles, restricted only by connexions and acted on only by forces which are definite functions of the co-ordinates determining their positions, being supposed to reach with exhausted energy a position of unstable equilibrium under the action of the forces; show that, in general, such a position could, under the circumstances, be attained to only at the expiration of an infinite time. 5. The forces acting on the particles of a system, circumstanced as in the preceding question, being supposed to have a potential; show, from the general equation of energy applied to the motion of the system consequent on a slight disturbance in any position of its equilibrium under the action of the forces, that the positions of stable equilibrium correspond to the maxima values of the potential, and conversely. 6. The several particles of a material system, circumstanced again in all other respects as aforesaid, but resting in addition on rough surfaces at once fixed and rigid, being supposed to receive slight disturbances in any position of their equilibrium under the action of the forces; show again, from the equation of energy applied to the resulting motion, how to determine in general whether the equilibrium of the system in the position is necessary or only possible. 7. A material particle, connected with a fixed point by a rigid rod of evanescent mass capable of free rotation round the point, and resting in addition against a fixed vertical plane of uniform roughness not passing through the point, being supposed in a position of equilibrium under the action of gravity; discuss the nature of its equilibrium in different positions, whether stable or unstable, and whether necessary or only possible. 8. A rigid body of any form, capable of free rotation round a fixed point, being supposed struck simultaneously by a number of impulsive forces; investigate completely, and interpret geometrically, the several conditions requisite to be fulfilled in order that the change of motion resulting from their joint action may take place without strain on the point. 9. Two rigid bodies of any form, connected with each other at a universal joint round which each may turn freely in all directions, being supposed struck simultaneously by two impulsive forces in directions passing through their respective centres of inertia; show, given all particulars, how to represent by geometrical construction the magnitude and line of action of the resulting strain at the joint. 10. A rigid bar, revolving round a fixed axis intersecting it at right angles, being supposed to come in direct collision with a uniform sphere moving perpendicularly to its length in the plane of its motion; determine, given all other particulars, the point of its length at which it will strike the sphere with the greatest force, and also the actual magnitude of the percussion corresponding to the point. II. A continuous mass of fluid of any nature, moving in its space according to any law, being supposed divided into thin strata by an arbitrary system of non-intersecting surfaces moving with and containing the same particles throughout the motion; investigate, by any method, the equation of condition throughout the mass that the several strata should contain exactly the same particles throughout the motion. 12. The motion of a continuous mass of fluid of any nature, homogeneous throughout its entire extent in its state of equilibrium, being supposed irrotational at any instant under the action of any forms which have a potential; show, by any method, that it will continue irrotational throughout the motion; and investigate, for the same case, the simplified equations for the determination at any time of the velocity and pressure at any point. DYNAMICAL PROBLEMS. 1. A heavy flexible chain, of uniform thickness, being supposed to occupy longitudinally just half the interior of a slender circular tube, of uniform roughness, bounded radially by coaxal horizontal cylinders, and laterally by parallel vertical planes; determine, given all particulars, its two extreme positions of equilibrium under the action of gravity. 2. A circular arc being supposed described through the central and through both terminal points of the curve of strained equilibrium of a uniform elastic straight rod, fixed horizontally at both extremities, and bent slightly by its own weight; shew, by any method, that the vertical depression at any point of the rod is a third proportional to the central depression and the corresponding ordinate to the circle. 3. Point out the omission in Poisson's investigation, which led him to the erroneous conclusion that the moment of torsion is constant throughout the entire length of an elastic rod of any form in strained equilibrium under the action of any forces; and, by its supply, determine the true rate of variation of that moment from point to point of the rod. 4. A thin uniform circular ring, capable of free rotation in a vertical plane about a fixed point of its mass, being supposed to support a thin uniform rectilinear bar in frictionless contact at both extremities with its inner circumference; investigate, given all particulars, the quadratic determinant whose roots give the periods of the harmonic vibrations of the system about its position of stable equilibrium under the action of gravity. 5. A solid ellipsoid of uniform density being supposed to revolve round its least axis of figure, and to carry with it a surrounding envelope of homogeneous incompressible fluid of different density, the entire mass attracting according to the ordinary law of the inverse square of the distance; investigate completely the conditions requisite to the possibility of the free surface also of the fluid assuming the ellipsoidal form. 6. A solid circular cylinder, of uniform density and infinite length, being supposed to attract, according to the law of the inverse sixth power of the distance, a material particle projected, with the velocity from infinity under its action, from any point external to its mass, in any direction perpendicular to its axis; show that the particle will describe freely, under the action, a circular arc orthogonal to the surface of the cylinder. 7. A rigid body of any form, in unconstrained equilibrium in free space, being supposed set in motion by a single impulsive force applied to a definite point of its mass; show that all axes of initial pure rotation, corresponding to different directions of the percussion, envelope a quadric cone, diverging from the centre of inertia, and touching the three central principal planes of the body. 8. A uniform flexible cord, strained tightly between two fixed terminal points, being supposed slightly deformed, by uniform longitudinal extension, into a circular arc of small curvature, and left, without initial velocity, to vibrate by virtue of its elasticity; investigate, given all particulars, the general formula at any time of its motion for the transverse deflection at any point of its length. 9. Two co-axal segments, of equal length, of a uniform cylindrical elastic rod, moving from opposite directions, with equal velocities, along the common line of their axes, being supposed to come simultaneously into longitudinal collision with a third of the same length, resting midway between them; assuming all three to be free from longitudinal vibration before the collision, determine, given all particulars, their respective states of vibration at the mean epoch of its duration. 1. Investigate the differential equation of the moon's radius vector, namely. 2. Find the values of the forces P, T, S, true to the second order. 3. Compute the moon's latitude to the second order, and give a geometrical construction for the result. 4. Find the terms in u the reciprocal of the moon's radius vector which do not involve e' and k sufficient to approximate to the second order. 5. Calculate c to the third order, assuming prove the following formulas determining the eccentricity of the moon's orbit, and correcting the longitude of the apse, on the supposition of uniform progression, 7. Find the difference of the forces by which bodies are retained in the fixed and revolving orbit, and justify Newton's construction. 8. Give Newton's account of the changes of inclination of the moon's orbit, and the motion of the line of nodes produced by the disturbing action of the sun. 9. Give Newton's construction for the sun's disturbing force, and hence deduce approximate expressions for P, T making Newton's abstractions. 10. Explain generally the difference between the geometrical and analytical methods of approximating: and as an illustration, show geometrically and analytically that the transverse disturbing force cannot affect the motion of the apse, unless the square of the disturbing force is taken into account. Moderatorships in Classics. DR. INGRAM. Translate the following: 1. Beginning, Τὸν δὲ χολωσάμενος προσεφώνεεν Ἶρος ἀλήτης· κ. τ. λ. Ending, οἵην ἐκ ῥακέων ὁ γέρων ἐπιγουνίδα φαίνει. HOMER, Odyss., xviii. 25-33, 72-74. 2. Beginning, εἰ δὲ τύχῃ τις ἔρδων, μελίφρον ̓ αἰτίαν, κ. τ. λ. Ending, εὐώνυμον ἐς δίκαν. PINDAR, Nem., vii. 11-48. 3. Beginning, νῦν παραιτουμένᾳ μοι, πάτερ Ζεῦ θεῶν Ὀλυμπίων, κ. τ. λ. Ending, τα δ' ἀποστατεῖ φίλων. ESCHYLUS, Choeph., 770-810. 4. Beginning, ἀκτις ἀελίου, τὸ κάλλιστον ἑπταπύλῳ φανὲν, κ. τ. λ. Ending, πάταγος ̓́Αρεος ἀντιπάλῳ δυσχείρωμα δράκοντι. SOPHOCLES, Αntig., 100-126. 5. Beginning, εἴπ ̓, ὦ νεᾶνι, τῷ σ' ἐχεγγύῳ λόγῳ, κ. τ. λ. Ending, τῷ σῷ προσίζειν ἀνδρὶ δειμαίνουσ ̓ ἐᾷς. EURIPIDES, Androm., 192–228. 6. Beginning, ΑΛ. ἔγωγε νὴ τοὺς κονδύλους, οὓς πολλὰ δὴ ἐπὶ πολ λοῖς, κ. τ. λ. Ending, ΧΟ. ἀνὴρ ἂν ἡδέως λάβοι. τοὺς τερθρίους παρίει. ARISTOPHANES, Equit., 411-440. MR. PALMER. Translate the following passages:— I. Beginning, Αἰγινητέων δὲ οἱ παχέες, ἐπαναστάντος, κ. τ. λ. HERODOTUS, vi. 91, 92. 2. Beginning, Λακεδαιμονίοις οὐκ ἂν ἀντὶ πόνων χάρις καθίσταιτο, κ. τ. λ. Ending, γῆν δὲ τὴν ὑμετέραν δῃῶν πειράσομαι βιάζεσθαι. THUCYDIDES, iv. 86, 87. 3. Beginning, Πάλιν ἄρα, ἦν δ ̓ ἐγώ, ὦ παῖδες, οὓς τὸ πρῶτον, κ. τ. λ. Ending, ηττηθέντες οὖν αὐτῶν διελύσαμεν τὴν συνουσίαν. PLATO, Lysis, fin. 4. Beginning, "Οπου δ ̓ αὐτὰ τὰ πράγματα ἐφ ̓ αὑτῶν ἔστιν, κ. τ. λ. Ending, οὗτοι δὲ φωραθεῖεν τὰ ψευδή μεμαρτυρηκότες. DEMOSTHENES, Cont. Steph., i. 5. Beginning, Πάντα δὲ τὰ τοιαῦτα τὴν μὲν θεωρίαν ἐλεύθερον, κ. τ. λ. Ending, ἀτέχνων καὶ τῷ σώματι μόνῳ χρησίμων. ARISTOTLE, Pol., i. 11. |