easily seen to be true even after becomes negative (if it does become negative before reaches 17). Hence as increases from the deviation continually in creases. If diminishes from 4,, then increases from o, and we have only to consider the reversed ray to see that the same result follows. Hence, when the ray of light passes symmetrically through the prism, the deviation is a unique minimum. The theorem may also be proved by means of the formulæ of the preceding article. = 2 sin 1(+) cos † (4—†)=2μ sin 1 (p' + y′) cos § (p' —¥' ́), that is sin (D+) = μ sin L Suppose that greater than . and are unequal, and that is Then the deviation - is greater -' than the deviation; therefore - is greater than '-', and therefore cos('-') is greater than cos 1 (−). and Similarly cos(' — ') is greater than cos() if be greater than . Hence in all cases in which are unequal sin (D + ) is greater than But when = 4, sin }(D + 1) = μ sin 1ɩ. sin. Hence when = y, D is a unique minimum. EXAMPLES. 1. Rays are incident at a given point of a prism so as to be refracted in a plane perpendicular to its edge. If be the angle of the prism and a the critical angle, show that the ray will pass through the prism if the angle of incidence be such that sin sin a > sin (ɩ — a). 2. If the angle of a prism be 60o, and the refractive index √3, find the limits between which must lie in order that the ray may be able to emerge at the second face. Ans. 300 and 60o. 3. If the angle of a prism be 60°, and the refractive index 2, show that the minimum deviation is 30o. 4. The minimum deviation for a prism is 90°. Show that the least value possible for the refractive index is √2. 5. Prove that in prisms of the same material, as the refracting angle increases, the minimum deviation also increases. 6. The refractive indices of three rays with respect to a given prism are M1, M2, M3; show that if D1, D2, D3 their minimum deviations through it are in Arithmetical Progression, then 7. Two prisms of the same vertical angle but of different refractive indices are placed in contact with their edges parallel and their angles turned opposite ways; prove that the deviation due to the system of a ray which is incident perpendicularly on the first surface of the system increases with the angle of the prisms. 8. If the minimum deviation for rays incident on a prism be a, the refractive index cannot be less than seca, and the angle of the prism cannot be greater than — α. 25. When the refracting angle of the prism is small, then the deviation will be small. By equations (2) and (3), Hence, f' = 1 - &', sin (ɩ + D− p) = μ sin (1 — p'). But, since and D are small, their sines may be very nearly represented by their circular measures, and their cosines do not differ sensibly from unity, and we get If the ray passes nearly perpendicularly through the prism and p' will both be small, so that to the third order of small quantities, the value of the deviation be which, to this approximation, is independent of the angle of incidence. 26. We shall next suppose that the ray does not lie in a principal plane of the prism. Let the same notation as before be applied to the projections of the path of the light on a principal plane. Also let n, n' be the inclinations of the incident and refracted rays to the principal plane at the first refraction, the inclinations of the refracted and incident rays to the same plane, at the second refraction, respectively. Then by § 20 sin = η μ sin n' Also, and n' denote the inclination of the same ray to the same plane, and therefore §' = n' and § = n. This proves that the incident and emergent rays are equally inclined to the principal plane, or to the refracting edge of the prism. and Further, there are the equations of refraction sin & cos n = μ sin d'cos n') These equations contain the whole theory of the refraction of a ray through a prism. MISCELLANEOUS EXAMPLES ON CHAPTER II. 1. A hemispherical bowl whose inner surface is polished, is just filled with water; and a ray is incident on the surface of the water in a vertical plane, which passes through the centre of the sphere. After two internal reflexions it emerges, when it makes the same angle with the normal to the surface of the water as at incidence. Show that the point of incidence must lie on a ring concentric with the rim of the vessel and having its bounding radii respectively √2 and of the radius of the vessel, the index of refraction for water being taken to be 2. The concave side of an equiangular spiral being polished, prove that a ray of light once a tangent to the spiral will always be a tangent to the spiral, however often it may be reflected at the curve. 3. Rays proceeding from the vertex of a parabola are reflected, each one at the diameter where it meets the curve. Prove that the reflected rays all touch a parabola of eight times the dimensions of the given parabola. 4. A ray of light is reflected a number of times between two plane mirrors, not in a principal plane; prove that every segment of the ray reflected from one mirror intersects every segment reflected from the other mirror. 5. A prism, refractive index μ and refracting angle 60°, is enclosed between two others of refractive indices μ and angle 60°, their edges being turned the opposite way to that of the first. Show that if a ray passes through without deviation, its course must be symmetrical, and that 3μ2=μ'2+μ'+1. 6. Two right-angled prisms each of refracting index λ, and a prism whose angle is 60° and refracting index u, are placed so that each of the former touches one face of the latter, and the angle of the middle prism is turned in a direction opposite to that of the angles of the other two. A ray passes through the system in such a direction that its deviation by the middle prism is a minimum, and it emerges parallel to its incident direction; prove that 7. A direct-vision spectroscope is composed of three prisms, two of which are exactly alike and are placed each with a face in contact with the faces of the third and their vertices turned towards its blunt end. Find equations for the angles of the prisms and their refractive indices in order that a ray refracted through the three prisms may be able to emerge parallel to its direction of incidence. If the refractive indices of the two similar prisms and the third be √6 and √3+1, respectively, and the angle of the third prism be 120°, show that the angle of the two like prisms is tan-1(6+3√3). 8. If n equal and uniform prisms be placed on their ends with their edges outwards, symmetrically about a point on the table, find the angle of each prism in order that a ray refracted through each of them in a principal plane 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 9. A ray is refracted at one face of a triangular prism in the principal plane, and after being reflected at each of the other faces emerges through the first face; show that the whole deviation is greater or less than two right angles, according as the vertical angle of the prism is less or greater than a right angle. Show also that if the angle of emergence of the ray be equal to the angle of incidence, the deviation will be a minimum when the vertical angle is less, and a maximum when it is greater, than a right angle. 10. A ray enters a prism of quadrilateral section in a principal plane and after reflexion at three sides in order emerges from the one at which it entered, making the angle of emergence equal to that of incidence but on the opposite side of the normal. Show that the section of the prism by the principal plane can be inscribed in a circle. 11. Sunlight falls on a small isosceles prism standing on a horizontal table and emerges after reflexion at the base, the edge of the prism being inclined at any angle to the sun's rays. Show that the result is the same as if the sunlight had been simply reflected at the table. 12. There are two confocal reflecting ellipses; a ray proceeds from a point P of either of them in a direction passing through one of the foci and is continually reflected between the curves. after 2n − 1 reflexions, it returns to the point P, the length of the path is equal to n times the difference of the major axes. If, 13. A cylindrical pencil of light is incident on a refracting prolate spheroid in a direction parallel to the axis, the excentricity of the spheroid being e, and the refractive index μ; find the positions of the rays which emerge parallel to the axis, supposing μ>1/e2, and show that none of the emergent rays will be parallel to the axis if μ < 1/e2. 14. Three plane mirrors are placed with their planes at right angles to one another. If a ray be reflected by all of them successively, its direction will be parallel to its direction at incidence. 15. A ray is reflected at three plane mirrors successively, so as to be parallel to its original directions after the reflexions, and the three directions which it takes are mutually at right angles to each other. Prove that the mirrors are mutually inclined at angles of 60°. H. O. 3 |