or since in the limit - 4, and '-,' are indefinitely small, replacing sines of small angles by their circular measures, ultimately, μ cos o' Again, cos2 μ cos — cos o = = γ AQAq, AQAO-Aq40; .. uv, sin (4- 4') = ur sin 8 — §v ̧r sin d'; sin . cos - cos sin o'sin o sin o' and substituting for sino its value μsin d' and dividing throughout by sin o', we have 47. We have now discussed to a certain order of approximation the alterations produced in a given small pencil of light by one refraction or one reflection, whether the incidence be oblique or direct. In the three succeeding chapters we have to examine to the same order of approximation the effect produced on such pencils by a number of such reflections or refractions in certain important cases. The student may notice that the results of the last three Articles include the positions of the geometrical foci investigated in Chapter II. These latter may be obtained from the formulæ of this Chapter by giving to 4 and p' the value zero. It may be mentioned here that although, in accordance with the remark of Art. 42, the circle of least confusion is probably the position at which the eye sees an object by a pencil which has been obliquely refracted or reflected, it is sometimes convenient to assume the primary or secondary focus as the position of the image. If the obliquity be small these points will all be close together, and it will not matter much which of them we take. EXAMPLES. CHAPTER III. 1. A small pencil of parallel rays is incident at an angle of 60° on a spherical reflecting surface. Find the position of the focal lines. 2. A small pencil of parallel rays is incident on a spherical refracting surface at an angle of 60°, the refractive index being √3. Find the position of the focal lines. 3. In each of the last two examples find the position of the circle of least confusion on the supposition that the incident pencil is a right circular cylinder. 4. The refractive index of a medium being , find the position of the primary focus of a pencil incident on a sphere formed of that medium, at an angle whose cosine is √7 3√3' 5. A pencil is incident obliquely on a spherical refracting surface at an angle whose tangent is equal to the refractive index of the sphere. Find the position of the focal lines. 6. A small pencil diverges from one extremity of the diameter of a sphere whose interior surface reflects light, and is incident on the sphere, so that its axis after reflection passes through the other end of the same diameter. Find the position of the focal lines, and show that v2 = 3v ̧. 7. Find the position of the point, from which light must diverge so that after refraction at a sphere whose refractive index is μ the primary and secondary foci may coincide. Show that the point must be at a distance ur from the centre of the sphere. 8. A small pencil of parallel rays is incident on a concave spherical reflector at an angle of T. Find the position 4 of the focal lines and the circle of least confusion, assuming (1) the border of the mirror to be circular; (2) to be elliptical with its diameters in the primary and secondary planes in the ratio of /2 to 1. 9. If O be the origin of light, P the point of incidence of the axis, and if the perpendicular to OP through O meet the tangent at P, at the foot of the perpendicular from the secondary focus on the same tangent; prove that the primary focus is at an infinite distance. 10. A pencil of parallel rays is incident obliquely on a convex refracting spherical surface. Find the position of the primary and secondary focal lines. If the angle of incidence π be and the primary focus be on the surface of the sphere, 3 show that the angle of refraction is the complement of the critical angle. 11. A small pencil diverges from a point in the surface of a spherical shell polished internally, and is twice reflected, show that if the normal at the first point of incidence pass through the final primary focus, the angle of incidence was CHAPTER IV. ON REFLECTIONS AT TWO OR MORE PLANE SURFACES. 48. WE have first to prove that if a ray be reflected successively at two plane mirrors so that its course through out lies in a plane at right angles to each of them, its deviation from its original direction after two reflections will be double of the angle between the mirrors. Let a ray be incident on one plane mirror at Q in the direction PQ, and let it be reflected along QR so as to fall on a second mirror at R and be again reflected along RST. Let PQ or PQ produced meet RT in S; then in figure (1) QST, and in figure (2) UST is the angle between the first direction of the ray of light and its last direction, that is, the deviation of the ray, and in either case this deviation is double of the angle ROQ between the mirrors. Let QN, RN be the normals to the mirrors at the points Qand R, meeting, produced if necessary, in N; QN and RN bisect the angles PQR and SRQ respectively. |