6. Explain the construction and use of Nicholson's hydrometer, and give the method of determining the specific gravity of a body lighter than the fluid in which it is weighed. 7. If a body float on a fluid, the weight of the fluid displaced is equal to the weight of the body, and the centres of gravity of the body and fluid displaced are in the same vertical line. 8. If a vessel be constantly filled with fluid, and a very small orifice be made in the side or bottom of the vessel, the velocity of the issuing fluid is nearly that due to the depth of the orifice. 9. Find the time in which a given cone will empty itself by a small orifice at the vertex, the slant side being placed parallel to the horizon. 10. Find the resistance on the surface of a solid of revolution moving in a fluid in the direction of its axis. How is the hypothesis incorrect on which the laws of this resistance are founded? 11. Compare the resistances on a cone and paraboloid of equal bases, moving in a fluid with equal velocities in the direction of their axes. 12. Explain the action of the common pump, and find the height to which the water rises after (n) strokes of the piston. 13. Show how the heights of mountains may be determined by means of the barometer and thermometer, and explain how to correct for the variations in the latitude, and in the temperature at the top and bottom of the mountain, OPTICS. TRINITY COLLEGE, 1822. 1. If a plane mirror be made to move in a conic section, so as always to touch it, find the path described by the image of an object placed in the focus. 2. Parallel rays may be made to converge much more nearly to the same point by means of a reflector generated by the revolution of a small arc of a catenary round its axis, than by a spherical reflector of the same dimensions. 3. If F be the focal length of a spherical reflector, or refractor, and D the distance from the centre of a straight line perpendicular to its axis, then the polar equation of the image is r = F F D Prove this, and find the axes of the conic section to which it belongs. 4. If a ray of light be refracted through any number of mediums contained by parallel plane surfaces, it will be as much bent from its original course as if it passed immediately out of the first medium into the last. 5. When a ray of homogeneal light is incident obliquely upon a spherical refracting surface, determine the intersection of the refracted ray with the axis of the pencil to which it belongs. 6. If the focus of rays incident on a convex lens of inconsiderable thickness be near its axis, then the focus of refracted rays may be de1 1 1 termined from the formula+ f F" where d, f, F denote the distances from the centre of the lens of the foci of incident, and re fracted rays, and of the principal focus respectively. Prove this, and shew how the formula may be applied to all the other lenses. 7. Supposing the sun's rays, after being refracted by the earth's atmosphere, to pass through it in right lines, and no reflection to take place; find the dimensions of the illumined part of the atmosphere which is opposite to the sun. 8. The density of rays in the sun's image formed by a reflector area of the aperture reflecting power (focal length of the reflector)? 9. If rays be incident parallel to its axis on the plane surface of a plano-convex lens, whose thickness = t, and radius =r; and emerge after two refractions at the plane surface, and one reflection from the spherical; prove that the distance of the geometrical focus of the reflecto-refracted rays from the plane surface = r 2t ; where 2n 10. Construct Galileo's telescope, find its magnifying power, its greatest field of view, and shew how it must be adjusted to the eye of a short sighted-person. Describe the experimentum crucis, by which Newton shewed that the primary colors cannot be separated into others by refraction. 12. If n = sin. incidence from air into glass for the rays of mean refrangibility, and n±♪n denote the same for the greatest and least refrangible rays; also if angle at which the sun's rays are incident on an isosceles prism whose angle = 2 a, then the angle contained between the rays of greatest and least refrangibility 4 sin.a.Jn cos. 13. In a convex lens with surfaces of equal radii, the spherical aberration will exceed the chromatic, if the semi-aperture of the lens be greater than of its radius. 14. If the dispersive powers of two prisms placed one against the other in opposite directions, be inversely as their refracting angles; a ray of light incident nearly perpendicularly on either prism, aud refracted through both will emerge free from color, 15. Explain fully the formation of the primary and secondary rainbows, and find their altitudes and breadths. 16. (1). If a tangent be drawn to the generating circle of a cycloid at its generating point, and be taken equal to the ordinate of the circle at that point, prove that the extremity of the tangent will trace out the caustic formed by rays incident on the cycloid in a direction parallel to the base. (2). Find the length of this caustic, its highest point, its point of regression, and the area contained between the cycloid, the caustic, and the reflected ray. 17. The caustic by refraction of a plane surface, when rays diverge from a point, is the evolute of an ellipse, or an hyperbola, according as the rays pass from a denser into a rarer, or from a rarer into a denser medium. 18. If the quantity of light emitted by any particle of a luminous spherical superficies towards a point placed within it, be supposed to vary as the sine of the angle, which the emitted rays make with the surface then the point will receive the same quantity of light, whatever be its position, and whatever be the magnitude of the superficies. TRINITY COLLEGE, 1824. 1. WHAT are the two principal theories which have been formed on the nature and propagation of light? Would the mathematical explanation of the common phenomena of reflected and refracted light be the same upon both hypotheses? 2. If A and A' be the distances of the foci of incident and reflected rays from the surface of a spherical reflector, whose radius is r, then the direction of incidence being nearly perpendicular to the surface. 8. The reflecting curve is a circle, and the radiating point is the extremity of the diameter: to describe the caustic, 4. What is the angle at which two plane reflectors must be placed with respect to each other, so that the images of an object placed between them may be found in the angles of an equilateral pentagon ? 5. If rays are reflected at the back of a plane looking-glass, whose thickness is given, given the focus of incident rays, to find the focus of emergent rays, the direction of incidence being nearly perpendicular to the surface. 6. Find the deviation of a ray passing through a prism, whose refracting angle is inconsiderable. 7. The conjugate foci in a spherical refractor move in the same direction upon the axis and coincide at its surface and centre. 8. Find the focal length of a double convex lens. 9. The power of a compound lens is equal to the sum of the powers of the component lenses, the power of a lens being defined to be the reciprocal of its focal length. 10. Find the focal length of a double concave lens by experiment. 11. Newton, in describing the experiment with the prism for determining the unequal refrangibility of light, says, that the image of the sun, whilst the prism was turned, first descended, and then ascended, and for one position was stationary; in that position, the refraction of the light at the two sides of the refracting angle of the prism was the same: prove this, and shew the use of this circumstance in the experiment in question. 12. In a double achromatic object-glass, the powers of the lenses must be inversely as the dispersive powers of the glass of which they are respectively formed. 13. A short-sighted person can see distinctly at the distance of six inches: what must be the power of a lens to enable him to see distinctly at the distance of 18 inches? 14. In the common Astronomical telescope, given the focal lengths and diameters of the object-glass and eye-glass, find expressions for its magnifying power, field of view, and for the brightness of the image compared with that of the object. 15. Explain the construction of the Gregorian telescope, and find its magnifying power. |