68. The object is 3 in. from a lens, and its image is 18 in. from the lens on the same side. Is the lens convex or concave, and what is its focal length?

69. The object is 12 ft. from a lens, and the image 1 ft. from the lens on the same side. Find the focal length, and determine whether the lens is convex or

concave.

70. A person who sees best at the distance of 3 ft., employs convex spectacles with a focal length of 1 ft. At what distance should he hold a book, to read it with the aid of these spectacles?

71. A person reads a book at the distance of 1 ft. with the aid of concave spectacles of. 6 in. focal length. At what distance is the image which he sees?

72. A pencil of parallel rays fall upon a sphere of glass of 1 inch radius. Find the principal focus of rays near the axis, the index of refraction of glass being 1.5.

73. What is the focal length of a double-convex lens of diamond, the radius of curvature of each of its faces being 4 millimetres? Index of refraction 2.5.

74. An object 8 centimetres high is placed at 1 metre distance on the axis of an equi-convex lens of crown-glass of index 1.5, the radius of curvature of its faces being 0.4 m. Find the size and position of the image.

75. Two converging lenses, with a common focal length of 0·05 m., are at a distance of 0.05 m. apart, and their axes coincide. What image will this system give of a circle 0·01 m. in diameter, placed at a distance of 1 m. on the prolongation of the common axis?

76. Show that if F denote the focal length of a combination of two lenses in contact, their thicknesses being neglected, we have

fi and f2 denoting the focal lengths of the two lenses.

77. What is the focal length of a lens composed of a convex lens of 2 in. focal length, cemented to a concave lens of 9 in. focal length?

78. Apply the formulæ of § 1015 to find the focal length of a lens, the thickness being neglected.

79. The objective of a telescope has a focal length of 20 ft. What will be the magnifying power when an eye-piece of half-inch focus is used?

80. A sphere of glass of index 1.5 lying upon a horizontal plane receives the sun's rays. What must be the height of the sun above the horizon that the principal focus of the sphere may be in this horizontal plane?

81. A small plane mirror is placed exactly at the principal focus of a telescope, nearly perpendicular to its axis, and the telescope is directed approximately to a distant luminous object. Show that the rays reflected at the mirror will, after repassing the object glass, return in the exact direction from which they came, in spite of the small errors of adjustment of the mirror and telescope.

82. An eye is placed close to the surface of a large sphere of glass (u=3) which is silvered at the back. Show that the image which the eye sees of itself is threefifths of the natural size.

83. The refractive indices for the rays D and F for two specimens of glass are

and an achromatic lens of 20 in. focal length is to be formed by their combination. Show that if the rays D and F are brought to the same focus, the sum of curvatures of the two faces for the crown lens must be double that for the flint, and the focal lengths of the two lenses which are combined will be about 7.9 in. for the crown and 13.1 in. for the flint.

Ex. 9. 547. Ex. 10. 440.

Ex. 11. 825. Ex. 12. 382 sec. Ex. 13. 32 ft., 16 ft., 103 ft.; 343, 68, 103. Ex. 14. 20 ft., 63 ft., 4 ft.; 55, 165, 275. Ex. 15. 55 ft., §1⁄2 ft. Ex. 16. An open tube twice or three times as long will resound, because one of its overtones will coincide with the note of the fork. A stopped tube three times as long will resound, but a stopped tube twice as long will not.

Ex. 17. 4. Ex. 18. 2, 4, 6. Ex. 19. 675-672=3.

Ex. 20. 33843, 38850. Ex. 21. 1110. Ex. 22. The velocity is 1090 at 32° and 1110 at 50°. Ex. 23. 1.018 metre. Ex. 24. 4180 ft. per second. Ex. 25. 33732. Ex. 27. 364° C.

Ex. 26. 1945 vibrations per second.

Ex. 28. 37417. Ex. 29. 567, 1134, 1701. Ex. 30. v=29040, t=v2 m=5481600. Ex. 31. v=43560, t=7589900. Ex. 32. Unison.

times length of platinum.

Ex. 34. 10 ft. Ex. 35. 12800 ft. per second.

Ex. 33. Length of iron=1.68

ANSWERS TO EXAMPLES IN OPTICS.

Ex. 36. 2.95 m. Ex. 37. 1, 5, and 7 ft. behind first mirror; 2, 4, and 8 ft. behind second. Ex. 39. Side of mirror must be of edge of cube.

Ex. 41. They are the shadows of the object and of its image, cast by the sun's image. The former is due to the intercepting of light after reflection; the latter to the intercepting of light before reflection. Ex. 42. The sun's image throws a shadow of the man's image on the real arch, owing to his intercepting rays on their way to the water. Ex. 43. First let the globe be vertically under the flame, and draw through the flame two equally inclined planes, touching the globe. Their intersections with the table will be parallel lines which will be tangents to the shadow, and will still remain tangents to it as the globe is rolled between the planes to any distance. Ex. 44. 216 radii of earth; 1 radii. Ex. 45. 368 ☛= =1156 metres.

Ex. 46. Focal length 1 ft.; (c) 3 ft. behind mirror; (d) 1 Ex. 48. 69.6 cm. Ex. 49.

(a) 5 ft. in front of mirror; (b) 13 ft. in front; in. behind. Ex. 47. 4, 1, 4, 17.

Distancem., height 3 cm. Ex. 50. 8.73 cm. Ex. 51. Distance 1 m., length dec., new position m. laterally from focus. Ex. 52. The ray in water will emerge, because is greater than 707; the ray

ANSWERS TO EXAMPLES.

1145

is less than 707. Ex. 53. 18.

of focal length of glass lens. All real. Ex. 64. 11 ft., 5 ft.,

Ex. 66. 14. Ex. 70. 9 in.

in glass will be totally reflected, because Ex. 54. 1'48. Ex. 55. 6°. Ex. 56. 60° (by total reflection). Ex. 57. 3 mm. Ex. 58. 1 ft. 4 in. Ex. 59. a. Ex. 61. Focal length of diamond lens is Ex. 63. 1 ft., 2 ft., 3 ft. on other side of lens. ft. on same side of lens. All virtual. Ex. 65. 1, 1, 2, 12, 6, 1. Ex. 67. ft. Ex. 68. 33 in., convex. Ex. 69. 1 ft., concave. Ex. 71. 4 in. Ex. 72. 1·5 in. from centre, or 5 in. from sphere. Ex. 73. 13 mm. Ex. 74. Distance & m. on other side, height 53 cm. Ex. 75. A real image 025 m. beyond second lens; diameter of image 005 m. Ex. 77. 24 in. Ex. 79. 480. Ex. 80. Sine of altitude, altitude = 41° 49′. Ex. 81. Rays from one point of object converge to one point on mirror, and are reflected from this point as a new source. Hence by the principle of conjugate foci they will return to the point whence they came. Ex. 82. The first and second images are at distances of and 3 of radius from centre.

Ex. 83. The dispersive powers are as 32:53. The focal lengths are to be directly as these numbers, and the difference of their reciprocals must be

Newcomb on velocity of light, 960. Pure spectrum, 1062.

Newtonian telescope, 1049.
Newton's rings, 1118.
-spectrum experiment, 1060.
-theory of refraction, 1102.
Nicol's prism, 1123.
Nodal lines on plate, 868.
- points of lens, 1087*.
Nodes and antinodes in air, 891.

in pipes, 920.

Noise and musical sound. 870.

Ohm on elementary tones, 945.
Opera-glass, 1048.
Ophthalmoscope, 991.
Optical centre of lens, 1015.
examination of vibrations, 927-

933.

Optic axes in biaxal crystals, 1131.
axis in uniaxal crystals, 1011,

II22.

Ordinary and extraordinary image,

1012, 1123.
Organ-pipes, 917-925.

effect of temperature on,
925.

overtones of, 919.
Overtones, 912, 914, 919.

Parabolic mirrors, 978.
Parallel mirrors, 972.
Pencil, 979.

Pendulum, acoustic, 892
Penumbra, 954.

Lycopodium on vibrating plate, Pepper's ghost, 975.

Purity numerically measured,

1079.

Pythagorean scale, 901.

Quarter-wave plates, 1134.

Quartz rotates plane of polariza-
tion, 1132.

transparent to ultra-violet rays,
1067.

894.

Recomposition of white light, 1067.
Rectilinear propagation, 948, 1100.
Reed-pipes, 924.
Reflection of light, 967.
– irregular, 969, 1088.
- total, 998.
of sound, 886.
Refraction, 992.

astronomical, 1003, 1108.
at plane surface, 1000.
at spherical surface, 1025.
- atmospheric, 1105.
- double, 1010, 1122.

Newtonian explanation of, 1102.
of sound, 887, 1110.

- table of indices of, 996.

- undulatory explanation of, 1102.
Regnault on velocity of sound,
878.

Resonance, 913.
Resonators, 936.

Resultant tones, 944.

Reversal of bright lines, 1073.
Rhomb, Fresnel's, 1134.

Rings by polarized light, 1127.
-Newton's, 1118.

Rock-salt, its diathermancy, 1064.
Rods, vibrations of, 923.
Rotation of mirror, 976.

of plane of polarization, 1132.