M. A. Degree. 7. Shew how to find the points of inflection of a curve y=ƒ (x): and apply the method to the curve (ay x2) 3 = 8. Find the expression for the radius of curvature in terms of p and r. When the angle between the perpendicular and radius vector is a maximum or minimum then the radius of 7.2 12. The radii of the ends of a frustum of a sphere are πη "", and its height is h: prove that its volume = 6 2 3r, 2+ h2). OPTICS AND ASTRONOMY. Examiner.-K. S. MACDONALD, M. A. 1. Prove that rays diverging from a point and incident nearly at right angles on a concave spherical surface, converge after reflection nearly to another point such that the sum of the reciprocals of the distances of the two points from the reflector is double of the reciprocal of the radius. 2. Light diverging from a point 10 inches in front of a mirror, after reflection appears to diverge from a point situated Examination Returns, &c. 3 inches behind the mirror: find the radius of the latter and whether it is couvex or concave. 3. Give a diagram illustrative of the vision by a convex mirror. 4. When is light totally reflected? find the angle of total reflection for a substance whose index of refraction is 2. Mention an experiment bearing upon this subject. 5. Describe the Galilean Telescope, and prove the relation between its magnifying power and the focal length of its lenses. 6. A person can read with the naked eye small print at the distance of 11 inches; by using a pair of spectacles he finds he can read the same print with ease at a distance of 7 ft. 7 inches find the focal length of the spectacles used. : 7. Three persons whose distances of distinct vision are 10, 20 and 30 inches use the same telescope: find the magnifying power relative to each observer when the focal length of the eye-glass is one inch. 8. Demonstrate the rotation of the earth on its axis from the vibrations of a pendulum on its surface. Suppose the pendulum to be made to oscillate in the plane of the meridian at a place 30o N Latitude, what angle would the pendulum make with the meridian after one hour's vibrations? 9. Define parallax, stating how the position of a heavenly body is affected by it. In what positions of a star are its right ascension and declination respectively unaffected by it, and state how the right ascension and declination of a star may be determined by means of the transit instrument and mural circle. 10. Explain the effect produced by Atmospheric refraction on the apparent position of the heavenly bodies, where is the effect least, and where greatest? 11. From Kepler's third law and assuming the orbits of M. A. Degree. the planets to be circular, show that their linear velocities are inversely proportional to the square roots of their distances. from the sun, and their angular velocities proportional to the cubes of their linear velocities. 12. Give a brief general view of the solar system, remarking on the number and distinguishing characteristics of planets, comets, satellites, asteroids, and aerolites. HYDROSTATICS AND HYDRODYNAMICS. 1. State and illustrate the property which is assumed as the basis of all reasonings upon fluid action. 2. Describe the Hydrostatic bellows. If the tube leading into the bellows be inch diameter and the area of the bellows be one square yard, what weight can be supported by a pressure of 1 lb on the water in the tube? 3. A cubical vessel is filled with two liquids of given densities, the volume of each being the same; it is required to find the pressure on the base and on any side of the vessel. 4. A side of the base of a square pyramid is 10 inches, the altitude is 22 inches; if the pyramid be filled with water, compare the pressure on the base with the pressure on each side and with the weight of the water. 5. A syphon is filled with mercury and held with its legs pointing downwards and the ends closed; what will be the effects of opening the ends 1st when they are, and 2ndly when they are not, in the same horizontal plane? State your reasons. 6. A body weighs 250 grains in vacuum, 40 grains in water, and 50 grains in a spirit; find the specific gravities of the body and of the spirit. A solid (a) whose weight is 1000 grains, loses 400 grains in water and 750 in a liquid (c), required the density of a and c and the volume of a. Examination Returns, &c. 7. At great altitudes it is sometimes found that a sensation of discomfort is felt, the lips crack and the skin of the hands is roughened; how do you account for these facts? 8. Obtain formulæ for the determination of the centre of pressure. A quadrant of a circle is just immersed in a heavy homogeneous fluid with one end in the surface; find its centre of pressure. 9. Find the time in which a given quantity of fluid will flow through a small orifice. A right cone is filled with fluid and placed with a generating line horizontal and uppermost, and a small orifice is made at the lowest point: find the time in which it will be emptied. 10. Assuming the height of the homogeneous atmosphere to be 27,690 feet, find the velocity with which air rushes through a small aperture into a vacuum. 11. Describe the action of the single exhausting syringe and prove the theoretical formula for determining the degree of exhaustion produced by a given number of strokes. If the contents of the receiver and the syringe are as 9 to 1, how many strokes will reduce the density of the air to ? N. B. log 3 = 0.48. 12. Describe the differences between the atmospheric Steam Engine, and Watt's double-acting Engine. STATICS AND DYNAMICS. Examiner.-K. S. MACDONALD, M. A. 1. Any number of forces act at the same point, their directions all lying in the same plane: find the direction and magnitude of their resultant. 2. Four forces represented by 1, 2, 3 & 4 act on a point. The directions of the first and third are at right angles to each other; and so are the directions of the second and fourth; and M. A. Degree. the second is inclined at an angle of 60° to the first. Find the magnitude and direction of the resultant. 3. Prove that two equal and opposite couples whose planes. are parallel and arms equal and parallel, are in equilibrium. 4. A uniform rod 8 ft. long and 16 oz. weight rests hori zontally on 2 fixed spheres each 10 ft. in diameter, and whose centres are in the same horizontal direction at a distance of 14 feet. What is the pressure exerted by it on the spheres ? State and prove the connexion between the height of the centre of gravity and the stability of its equilibrium. 5. 6. If a common balance have unequal arms, show that the real weight of a substance is a geometrical mean between its apparent weights when put successively in the two scales. Show also that these apparent weights are to each other as the squares of the arms inversely. 7. Prove that in uniformly accelerated motion from rest, the spaces described in equal successive periods are as the odd numbers. If the space described in the 30th sccond is 11.8 feet, find the acceleration, the velocity at the end of the 30th second, and the whole space from rest. 8. Upon a steeple 150 ft. high is a spire of 40 ft.; at the same instant that a stone was let fall from the top of the steeple another was projected vertically upwards from the bottom of it with a velocity sufficient to carry it to the top of the spire only; at what point will these stones meet? 9. Show that the times down any inclined planes are proportional to the lengths of the planes, when the height is the same. 10. Two bodies start from the top of an inclined plane, one falling down the length of the plane, and the other down its height; it is observed that the former is 3 times as long as the latter in reaching the base. Required the inclination of the plane. |