17. Show that when a ray of light enters a medium whose refractive index is √2, its greatest deviation is 45°. 18. A small pencil of light is obliquely refracted through a plate of thickness t. The angle of incidence being tanμ, show that the distance between the secondary focus after με - 1 emergence, and the original point of light is 2 με Ꮎ - .t. 19. The angles at the base of a triangular prism are 0-6 and 20, where sin 0 μ sin o; a ray of light falls on the = sin; shorter side of the triangle, the angle of incidence being on the side of the normal next the vertex: show that the ray after reflection from the base and the other side will emerge from the base in a direction parallel to its original direction; and that unless sin is greater than sin, the second reflection will not be total. 20. A ray of light is refracted through a sphere of glass in such a manner that it passes through the extremities of two radii at right angles to each other. If be the angle of incidence, and D the deviation, prove that sin (2-D). sin D μ3 — 1. 21. If n equal and uniform prisms be placed on their ends with their edges outwards, find the angle of each prism that a ray refracted through each of them in a plane perpendicular to their edges may describe a regular polygon. Show that the distance of the point of incidence of such a ray on each prism from the edge of the prism bears to the distance of each edge from the common centre the ratio of π √(μ2 - 2μ cos + 1) to μ +1. 22. Prove that in prisms of the same material, as the refracting angle increases the minimum deviation also in creases. 23. Without knowing the angles of a triangular prism, show that its refractive index can be determined by observing the minimum deviations of rays passing in the neighbourhood of the three angles; and if these deviations be denoted by 27, 28, 2y, then μ is given by μ μ3 — μ3 (cos a + cos ẞ + cos y) + μ {cos (B+ y) + cos (y + a) +cos (a + B)} — cos (a + B + y) = 0. 24. Two prisms whose refracting angles are right angles and refractive indices μ, μ', are placed so that one face of each is in contact: their edges are parallel and their refracting angles opposed. Prove that the minimum deviation of the compound prism is sin ̄1 (μ3 — μ”). CHAPTER VI. ON REFRACTION THROUGH LENSES. 66. A PORTION of a transparent medium bounded by surfaces, two of which are surfaces of revolution with a common axis, is called a lens. In all cases that we shall have to consider, these surfaces of revolution are spheres, and the portion of the medium is symmetrical with respect to the line joining the centres of the spheres, being either entirely bounded by the surfaces of the spheres, or by them and a cylindrical surface, whose axis is the line joining the centres of the spheres. The spherical surfaces are called the faces of the lens. The line joining the centres of the spheres is called the axis of the lens. Lenses of different forms are distinguished by names indicating the nature of their bounding surfaces with respect to the external medium. A lens, of which both spherical boundaries are convex towards the outside, is called a double convex lens. A lens, of which one face is convex, and the other concave to the outside, is called a convexo-concave, or concavoconvex lens, according as the light falls first on the convex or concave face respectively. A lens, of which one face is convex, and the other plane, is called convexo-plane or plano-convex. The terms double concave, concavo-plane, and plano-concave, are intelligible without farther explanation. A lens, which is on the whole thicker at the middle than at the borders, is called generally a convex lens, while one, which is on the whole thinner at the middle than at the borders, is called a concave lens. A concavo-convex or convexo-concave lens, which is thicker in the middle than at the borders, is sometimes called a meniscus. 67. A pencil is incident on a lens of small thickness in such a manner that its axis before refraction coincides with BA the axis of the lens. It is required to find its geomètrical focus after refraction through the lens. It is clear that the axis of the pencil, after refraction into the lens, and again after emergence, will still coincide with the axis of the lens. Let Q be the origin of light, QAB the common axis of the pencil and the lens. Let r, s be the radii of the first and second surfaces of the lens respectively, μ the refractive index from the external medium into the lens. Let q be the geometrical focus of the pencil after the first refraction, Fits geometrical focus after emergence. Let AB, the thickness of the lens, be so small that we may neglect it in comparison with AQ and AF. Then for the first refraction, by Article 29, we have For the second refraction we may consider the pencil to diverge from q, and F to be its geometrical focus, and remembering that the index of refraction from the lens into the If the original pencil consist of parallel rays, u is infinite, and if the corresponding value of v be ƒ, we have This quantity f is called the focal length of the lens, and the geometrical focus of a pencil of parallel rays incident on the lens parallel to the axis, is called the principal focus of the lens. The points Q and F are called conjugate foci. By means of formula (2) the relation (1) can be written If the thickness AB be not neglected, and u be measured from A, and v from B, the second equation becomes |