pended. (a) Show that the tensions of the several portions of the string re as the secants of the angles those parts make with the horizon; and (b) that the successive weights are as the sums of the tangents of the angles which the conterminous parts of the string make with the horizon. 12. A heavy particle is sustained on a smooth inclined plane, successively, by different forces. From the particle draw right lines representing the sustaining forces in magnitude and direction. Show that the locus of the free extremities of these lines is a right line perpendicular to the inclined plane. MR. LESLIE. 13. Two weights connected by an inextensible cord are placed on the surface of a horizontal cylinder; find, by the principle of virtual velocities, the position of equilibrium, and prove that it is unstable. 14. Two forces balance each other on the wheel and axle; calculate the pressure at the two points of support. 15. Two bodies, whose weights in tons are T and T", rest on inclined planes, and are joined by a rope passing over a pulley at the common vertex of the planes; find, by the principle of work, the velocity acquired in descending a given portion of the incline, the resistance from friction being plbs. per ton. 16. Two perfectly elastic spheres, M and M', meet directly with equal velocities; find the relation between their masses, so that, after collision, one of them may remain at rest. 17. A body is placed within a rough circular hoop, which is made to revolve round a vertical axis; find the least velocity of rotation, so that the body may not fall from the hoop. 18. A pendulum which beats seconds is taken to the top of a mountain one mile high; it is required to find the number of seconds which it will lose in the day, allowing the radius of the earth to be 4000 miles. B. MR. SALMON. 1. Find the centre of gravity of an arc of a cycloid. 2. A body falls from a considerable height h to the ground; ifr be the Earth's radius, T the time of fall, on the supposition that gravity is constant; and 7" the time, taking into account the variation of gravity: prove that, approximately, 5h T' = T 1 + 6r 3. A double ladder rests on a rough horizontal plane; find the greatest inclination of its sides to the vertical, consistent with equilibrium. 4. A particle is placed on the line joining the centres of two attractive forces varying as the distance; find the time of its oscillations. 5. What is the time of the small vibration of a sphere suspended from a point on the surface? с DR. SHAW. : 6. The velocity of a body is measured by the number of space-unit which it describes in one time-unit its mass or inertia by the number o force-units which, acting on it for one time-unit, generate in it one velo city-unit: an (a) Give analogous measures of a body's slowness and mobility; (6) designating the four quantities thus measured by the symbols v, m σ, and μ; prove the following relations: vσ = 1, mu = I. 7. From a particle which rests on a rough inclined plane draw right lines representing in magnitude and direction the forces which, acting singly, just suffice to move the particle up the plane; and also the forces which, acting singly, just suffice to sustain the particle on the plane. Show that the locus of the free extremities of each set of lines is a right line inclined at an angle to the perpendicular to the plane: tan being the coefficient of friction. 8. By means of the above locus lines, or otherwise, prove that— When we calculate the force which, acting in a given direction, will keep at rest a particle on an inclined plane, the error occasioned by neglecting friction must lie between the limits w being the particle's weight; ✪ the angle between direction of force, and the perpendicular to the plane; and & the elevation of the plane. ε 9. At a given point on the upper surface of a vertical pillar, of uniform horizontal section, is applied a pressure P, the direction of which passes through the axis of the pillar, and makes an angle with the horizon. Prove that the locus of the point in which each horizontal section is pierced by the resultant of P and the weight of the superincumbent part of the pillar, is a hyperbola. 10. A particle moves from rest along a right line towards a centre of force, the force varying inversely as the square of the distance. Prove that the velocity at any instant may be geometrically represented as follows:— On the original distance of the particle from centre of force, as diameter, construct a semicircle; and at the position of the particle corresponding to t, raise an ordinate to the diameter; at the point where this ordinate cuts the semicircle, draw a tangent; the intercept which this tangent makes on the fixed tangent drawn at the centre of force is inversely proportional to the velocity. MR. LESLIE. II. Two spheres in contact rest one on each of two inclined planes; find the position of equilibrium. 12. Find the condition of equilibrium in the screw, taking friction into account. 13. A ball of known elasticity falls from a given height on a horizontal plane; find the whole space described by the ball before it arrives at rest, 14. A body is projected up the concave side of a vertical circle, from its lowest point; find the point where it leaves the curve. 15. A particle moves in a plane acted on by a force varying as the distance directed to a centre which moves uniformly along a right line in the piane; determine its motion. Experimental and Natural Science. HEAT. DR. APJOHN. 1. If a metallic bar, whose coefficient of expansion for 1° Fahrenheit is k, have at temperatures t' and t" the respective lengths l' and l', what is the value of k? 2. In a U-tube containing mercury, one of whose legs is raised to t°, while the other is maintained at 32°, the height k of the cold column,—and the difference of heights (h'-h) of the two mercurial columns is ascertained by direct measurement. Deduce from these data the coefficient of the absolute expansion of mercury. 3. If a thermometer, in a medium whose temperature is t, be exposed to a constant heat, and that, when the thermometer becomes stationary, for example, at t', t'-t does not exceed about 40°, this rise of temperature is an exact measurer of the heat that it receives in a given time; why is this the case? 4 If a weight thermometer, containing at 32° 3760 grains of mercury, after being heated, is found to retain but 3724 grains; to what temperature Fahrenheit was it raised? 5. Explain the requisite experiments for determining with accuracy the specific gravity of a gas; and give the formula of Pouillet for applying to these experiments the necessary corrections. 6. The effect of the introduction of a liquid capable of giving off vapour into a dry gas is different according as the latter is contained in an unyielding envelope, or in one which expands upon the application of the slightest force. What is the difference in question? 7. If a gaseous mixture having a constant volume v has, at t, the pressure p; and, at t, the pressure p',-how do you determine whether by the change of temperature vapour has been developed or condensed? 8. A dry gas having a volume v at temperature t and pressure p, has its temperature changed to t, and its pressure to p', and at the same time acquires vapour whose elastic force is f. What is the expression for the new volume v', which it acquires? 9. A metallic bar 7 feet long, and whose coefficient of expansion for 1o is 1323, is, by a given rise of temperature, elongated just as much as a bar of another metal whose length is 9 feet. What is the coefficient of expansion of the latter? 10. A hollow glass sphere at 80° has an inner diameter of 2.5 inches What weights of mercury is it capable of holding at 32° and at 212°, the envelope and the liquid being at the same temperature? N.B.-Specific gravity of mercury at 60° is assumed to be 13.6 O 1000' .00014 FROFESSOR GALBRAITH. 1. By what number must we multiply a quantity of heat, expressed in French units, in order to obtain its expression in English units? Express the latent heat of water in French units. 2. To determine the specific heat of oil of vitriol (sp. gr. = 1.85), the following experiment is made with the calorimeter:-One pint of the vitriol is introduced into a glass flask, of which the weight is 12 oz., and the specific heat ; the flask and its contents, having been raised to the temperature of boiling water, are placed in the calorimeter, and in descending to 32° F. melt 19 oz. of ice. Calculate the specific heat of the vitriol. 3. How much ice is melted by the vitriol; how much by the glass ? 4. Water boils in the city of Quito at 194° F. Why is this the case ? and hence infer the height of the city above the sea-level. 5. What is the simplest form of the barometric expression for the height of a mountain? Can you state a formula which does not require the use of logarithmic tables? 6. How do you illustrate the influence of pressure on ebullition by the pulse-glass of Franklin? 7. Describe Thilorier's apparatus for liquefying carbonic acid gas. Why does the liquid solidify when the stop-cock is opened which allows it to escape? 8. By what empirical formulæ are the pressure and density of steam connected together? 9. What is the exact expression for the relative volume of steam in terms of its pressure and temperature? 10. A bath of water, at 56°, which is 6 feet long, 2 feet wide, and 2 feet deep, is to be raised to the temperature of 98° by the condensation in it of steam at 212°; to what height will the level of the water be raised when the required temperature is reached? PROFESSOR HAUGHTON. 1. Supposing the Sun to move in the Equator, and the Earth to be made of uniform material, find the law of mean temperature depending upon latitude. 2. State Newton's laws of cooling, which give the relations between— a. Excess of Temperature and Time. b. Rate of Cooling and Time. 3. How is it proved that Heat obeys the same laws of Reflexion as Light? 4. Why have islands a smaller range of temperature from summer to winter than the interior of continents? 5. State M. Senarmont's laws of conduction of heat in crystals, and how far are they inconsistent with the optical properties of the same bodies. History and Logics. HISTORY. MR. BARLOW. 1. Give some account of the following writers, and their principal works: Ordericus Vitalis; Ingulphus; Roger de Hoveden; Geoffrey of Monmouth; Matthew Paris; John Paston. 2. Name all those sovereigns of England, from Egbert to Henry VIII., who, according to modern notions, would be considered usurpers. State your reasons for each particular case. 3. Discuss fully the claims of Edward III. to the French Crown. 4. Give some account of a. Hereward. 3. Margaret of Anjou. y. Thomas of Woodstock. 5. Write out a short sketch of the constitutional history of England during the reign of Richard II. 6. Discuss the following question Was Perkin Warbeck the Duke of York? 7. Who were the principal persons executed for treason in the reign of Henry VIII.? Comment upon the legality of the sentence in each particular case. 8. Mention all the instances of Regencies in England from the Conquest till the accession of Elizabeth; and discuss those particular cases which are of importance as constitutional precedents. BACON, STEWART, AND WHATELY. MR. M'DOWELL. 1. There is an apparent inconsistency in the accounts given by Stewart of Conception and Memory. In what does it consist; and how is the difficulty to be removed? 2. Describe briefly Stewart's argument to show that the idea of visible figure cannot be derived from Perception alone. |
