1 reconcile this fact with the common theory of images, and A small curvilinear object is placed before a lens in a plane 187. Determine the index of refraction between two media, having given the indices between each of them and a third. Given the refracting and critical angles of a prism, find the greatest inclination at which a ray may enter it so as to emerge. 188. A small pencil of parallel rays passes directly through a plano-convex lens, determine the longitudinal aberration of the extreme rays; shew that it is greater when the rays are incident on the plane than when on the convex surface. 189. A small pencil of diverging rays is incident directly on a plane refracting surface, determine the longitudinal aberration. THEORY OF LIGHT. 1821 1. A ray of light, after being refracted through media possessing equal dispersive powers, will always appear coloured at its emergence, unless the incident and emergent rays are parallel. 2. If the refraction of light be the effect of any causes which act throughout a certain distance from the surface of a medium, and the intensity of which depends solely on the distance from that surface, the ratio of the sines of incidence and refraction must be constant. 1827 • The formulæ for the primary and secondary focal lines are respectively, 22 3V 2f 1 h + M 1831 3. Explain briefly the optical experiments and theories to 1828 which the following terms refer: Fits of easy transmission and reflection; Polarization; Plane of Polarization ; Depolarization ; Depolarizing Axis ; Fringes; Interferences. 4. Mention the facts from which it appears that the phenomena of the extraordinary ray in a double refracting crystal can be accounted for on the supposition of a repulsive force emanating from the axis. 5. Explain the theory of the interferences of light, and 1830 determine the colour, origin, and intensity of a ray resulting from the interference of two similar rays, differing in origin and intensity. 6. What properties of a medium in which the density varies as the pressure, correspond to and serve to explain the following observed properties of light: its rectilinear and uniform transmission; the different intensity of different rays; the difference of intensity of the same ray at different distances from its origin; the difference which the eye distinguishes in rays by colour ; their crossing in all possible ways without mutual disturbance; the interference of two rays, the paths of which nearly coincide in direction and differ in length by a multiple of a certain interval ? 7. Demonstrate the laws of reflection and refraction on the undulatory theory of light. 8. In the undulatory theory of optics, define the terms wave, 1832 length of a wave, front of a wave, phase of a wave. How is the effect of any wave in disturbing a given point found? If the maximum vibration produced at a point by a succession of cλ 276 cos 0 waves be sin it is required to compare the ar cos 0 values of this expression for different values of 0 according as a is much greater or much less than b. 9. Shew how to express algebraically the transmission of an undulation. 10. In the undulatory theory of light, explain the principle 1833 of interferences. By what experiments is the existence of such interference proved? Give the mathematical investigation of some experiment for determining the effect of the interference of rays. 1834 11. A prism is cut out of a biaxal crystal having its edge parallel to one of the axes of elasticity.; describe the nature of the refractions which take place in a plane perpendicular to the edge. 12. State the two theories which have been imagined to account for the phenomena of light. Point out the principal objection which has been made to each, and describe the corresponding experiment. 13. Prove, from the principles of the undulatory theory, that light proceeding from a bright point by a small hole into a room, ought not to spread through the room in the same manner as sound coming in the same direction and through the same hole. 14. A plate of Iceland spar, bounded by planes perpendicular to its axis, is interposed between a polarizing and an analyzing plate, the latter being so placed that no light is reflected without the interposition of the crystal ; investigate the intensity of the light in various parts of the image seen after reflection at the analyzing plate. The difference of retardations of the ordinary and extraordinary rays produced by passing through the crystal for a small angle of incidence, is c2 a? T. sin?i. 2av 15. A luminous point is placed at a given distance from a plate of glass bounded by two parallel planes ; required the form of the emergent wave of light at a given time. 16. Define the axes of elasticity of a crystal ; and supposing the displacement of a molecule to be exceedingly small compared with the distances between the molecules, and proportional to the force resulting from it, prove that there are three axes of elasticity in every crystal at right angles to each other. 17. A ray of light is incident on a plate of Iceland spar, which is bounded by planes perpendicular to the axis of the 1835 sin’d1 =e? crystal; shew that if $ and ø, be the angles of refraction for the ordinary and extraordinary rays, and e the eccentricity of the spheroid of refraction, 1 (1 - e?)? sin & 18. Light diverging from a point falls on one side of a prism which has a very obtuse angle opposite to this side; state the phenomena observed, and explain them on the theory of undulations. 19. Two convex lenses of small curvature are placed in contact; having given the intensities of the reflected (1) and transmitted (2) light, trace fully the phenomena of Newton's rings. By what probable analogy is the circumstance, that the rings formed by the reflected and transmitted light are complementary to each other, illustrated ? 4a?e? sino TV a’ (1 - ea)? (1) (2) (1 – е2)2 + 4e sin? V (1 – ex)2 + 4e’sino V à 20. In a dark room two bright points of light are placed 1836 very near to each other; describe the phenomena exhibited upon a screen placed in any position. 21. A succession of waves of light, whose fronts are parallel to a plane screen which has a small opening in it, is moving towards the screen, and the magnitude of a vibration on any point of a semicircle behind the screen, and of which the сл 276 cos 0 27 centre lies in the opening, sin sin a ~ T (vt – r); investigate the appearance on the semicircle, and shew that it obviates a material objection against the undulatory theory. 22. What is meant by polarized light? Account for the separation of common light into two pencils by a doubly refracting crystal, and for their consequent polarization. If the crystal be uniaxal, shew that an extraordinary wave will diverge from a point in the form of a spheroid of revolution. or cos SECTION XX. QUESTIONS ON ATTRACTIONS, TIDES, FIGURE OF THE EARTH, &c. 1821 1. The attraction of a spherical shell upon a particle placed without it, is the same as if the whole matter in the shell were placed in its centre. 2. In a revolving spheroid of small eccentricity, if polar gravity : equatoreal sensible gravity : radius of equator : semiaxis, gravity is everywhere perpendicular to the spheroidal surface. 3. Shew, by measuring the area ANB in the 12th section, that if a sphere be composed of particles, the attraction of which 1 the attraction of the whole sphere on an external particle varies in the same law. 1822 4. Shew that if M and S represent the height of the tide produced by the moon and sun respectively ; retardation of tide at new and full moon : retardation in quadratures :: M-S : M+S. 5. The density of different parts of a circle varies as the square of the distance from the centre ; find the velocity ac |