5. A small pencil of rays diverges from a given point within a polished sphere, the axis of the pencil coinciding with a diameter; find the geometrical focus after two reflexions.

6. A luminous point is equidistant from two plane parallel mirrors; find the path of the axis of the small pencil of rays, by which an eye placed in a given position between the mirrors, can see the third image proceeding from either side, and shew that its length is equal to the distance of the image from the eye.

7. BAD, BCE are two plane reflectors inclined at an angle of 15o. A is a given luminous point in one of them. Find at what angle a ray from A must be incident on the other reflector, in order that after 3 reflexions it may be parallel to BA.

8. There are three plane reflectors, two of which are at right angles to each other, and a ray of light is incident upon the third, and reflected successively by each of them; it is required to shew that the angle between the first incident and last reflected rays is equal to twice the angle of incidence upon the first surface.

REFRACTION AT A PLANE SURFACE.

1. Find the thickness of a plane glass mirror, silvered at the back, that the distance of the image from the first surface may be twice as great as in a mirror of inconsiderable thickness.

2. At the bottom of an empty hemispherical basin a crown piece is placed, and an eye is so situated as just to see the edge of the crown piece over the rim of the basin. When the basin is filled with water the whole crown piece becomes visible. Find the radius of the basin.

3.

Is it necessary to aim above or below in order to strike with a bullet a fish swimming in the water?

REFRACTION AT A SPHERICAL SURFACE.

1. A pencil of rays diverging from the centre of a sphere, after refraction at its surface diverge from the opposite extremity of the diameter; required the refractive index.

2. If parallel rays are incident nearly perpendicularly upon a spherical refracting surface, the distance of the geometrical focus of refracted rays from the surface is to its distance from the centre as μ: 1.

3. A small pencil of solar rays incident on the surface of a refracting sphere is brought to a focus upon the opposite surface of the sphere; required the refractive index of the substance of which the sphere is made.

4. A small pencil of rays is incident from a point 3 feet distant from a concave spherical surface of glass (μ = 1.5), the radius of which is 2 feet; find the geometrical focus of refracted rays.

5. When divergent rays are incident from a certain point upon a spherical surface of glass, the refracted rays are found to converge to a focus at exactly the same distance on the opposite side of the surface; is the surface convex or concave? and if the position of the point of incidence be given, determine the radius of the surface.

6. There is a speck in the interior of a glass sphere; determine the apparent distance of the speck from the eye, supposing the line joining them to pass through the centre of the sphere.

REFRACTION THROUGH A PRISM.

1. A speck in the middle of the back of an isosceles prism, will appear double to an eye placed close to its edge. Suppose the angles which the two images so seen subtend at the eye to be a right angle, determine the angle of the prism.

2. If the angle of a prism be 15o 30', and the angle of incidence 14° 18', determine the deviation which the ray suffers in passing through the prism.

3. A ray enters a prism, the refracting angle of which is so large that no ray can pass out of it, it is therefore reflected at the second face of the prism; shew how to determine the angle between the incident and this reflected ray.

4. ABC is an equilateral triangle; PQRSTV the course of a ray refracted at Q and T, and reflected at R and S. The angle of incidence of PQ is 15°; find μ so that the incident and emergent rays may be inclined at an angle of 30o.

5. A ray of light is incident on a prism, in a plane perpendicular to its edge, at an angle of 45o; find the refracting angle of the prism in order that the ray may just emerge parallel to the second surface; the value of μ being √2.

6. If a ray of light be refracted through a right-angled prism in a plane perpendicular to the edge, and if ' be the angles of incidence and refraction, & the deviation, then

7.

If a ray of light QACS be refracted through a prism KIL in a plane perpendicular to its edge, and if the angle of the prism KIL = a, QAK = 0, ACL = p, and the whole deviation = 8, then will

a

α

tan (p-2) tan tan (9+ 3+ 4) tan3 + a.

α 2

=

2

2

8. If be the angle of incidence of a ray passing through a prism in a plane perpendicular to its edge, the angle of emergence, a the angle of the prism, then

9.

A

ray of light is refracted through a prism, the angle

of which is 60° and index of refraction √2, in such a manner

that the angles of incidence and emergence are equal, find the whole deviation. Shew also that no ray can be transmitted through a prism of the same substance when the angle exceeds 90o.

REFRACTION THROUGH A LENS.

1. An object 10 feet below the surface of water, is viewed by an eye 15 feet above the surface. What is the focal length of a lens through which it must be viewed, that its apparent depth may be 10 feet?

2. If an object is placed in the focus of a convex lens, the visual angle is the same whatever is the distance of the eye from the glass.

3. Given the radii of a thin double concave lens, upon which parallel rays are incident, find the radius of a double equiconvex lens, which compounded with the former will refract the rays parallel.

4. The back of a double convex lens is quicksilvered; if a small pencil of rays after entering the lens is reflected, find the focus of the emerging rays.

5. A small pencil of rays diverge from a point in the axis of a double convex lens, the thickness of which equals one of its radii. Required the geometrical focus of the refracted

rays.

6. The concavity of a thin glass meniscus is filled with water; the radii of the surfaces are 5 and 6 inches respectively, and the refractive indices of glass and water are 1.535 and 1.336; find the focal length of the compound lens.

7. Four lenses having a common axis are placed at intervals 1, 5, 10 inches from each other, the focal length of each of the first three being 5 inches, and of the last 1 inch. If a pencil of parallel rays fall on the first, determine the point to which they will converge after passing through the system.

8. The focal lengths of two double convex lenses are to each other as mn, and the radii of their first surfaces are equal; required the radii of their second surfaces.

9.

The radii of the surfaces of a double convex lens are 1 and 6 inches, and the refracting index 1.6; find the focal length.

10. The first surface of a lens is concave and of given radius, determine the form of the second in order that the focal length may be the same as that of a double convex lens of the same material having each of its radii equal to that of the first surface, before mentioned.

11.

Determine the focal length of the lens, which causes rays diverging from a point at a distance of 2 feet to diverge as if from a distance of 3 feet.

12. What single lens is equivalent to a combination of a double convex lens of focal length 2 inches with a double concave lens of focal length 4 inches?

13. Rays diverging from a distance of 3 inches on one side of a lens are made to converge to a point 3 inches on the other side; find the focal length of the lens.

14. A double convex and a concavo-convex lens are placed in contact; the radii of the surfaces of the former are respectively 3 and 4 inches, those of the latter 3 and 5 inches, and the refractive indices of the material of the lenses are respectively 1.52 and 1.6; find the focal length of the combination.

a

IMAGES.

1. A straight line, .15 inches in length, is placed before concave mirror, radius 9 inches, at a distance from the mirror equal to one third of its radius; find the magnitude and position of the image considered as a straight line.

2. A person whose eyes are 5 feet 8 inches from the ground looks into a plane vertical mirror 4 feet high; what portion of his figure will he see?