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Full text of "Mirrors, prisms and lenses; a text-book of geometrical optics"

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MIRRORS, PRISMS AND LENSES 



?&& & 



THE MACMILLAN COMPANY 

NEW YORK • BOSTON • CHICAGO • DALLAS 
ATLANTA • SAN FRANCISCO 

MACMILLAN & CO., Limited 

LONDON • BOMBAY • CALCUTTA 
MELBOURNE 

THE MACMILLAN CO. OF CANADA, Ltd. 

TORONTO 



MIRRORS, PRISMS AND 
LENSES 

A TEXT-BOOK OF GEOMETRICAL OPTICS 



BY 
JAMES P. C. SOUTHALL 

ASSOCIATE PROFESSOR OF PHYSICS, COLUMBIA UNIVERSITY 

AUTHOR OF " THE PRINCIPLES AND METHODS 

OF GEOMETRICAL OPTICS " 



THE MACMILLAN COMPANY 
1918 

All rights reserved 



Copyright, 1918 

By THE MACMILLAN COMPANY 

Set up and electrotyped. Published December, 1918. 

<2>fc 1 84 

c. 

is 



PREFACE 

In spite of the existence of a number of excellent works 
on geometrical optics, the need of a text-book which will serve 
as an introduction to the theory of modern optical instru- 
ments appears to be generally recognized; and the present 
volume, which is the outgrowth of a course of lectures on 
optics given in Columbia University, has been written in 
the hope that it may answer this purpose. In a certain 
sense it may be considered as an abridgment of my treatise 
on The Principles and Methods of Geometrical Optics, but 
the reader will also find here a considerable mass of more or 
less new and original material which is not contained in the 
larger book. I have endeavored, however, to keep steadily 
in mind the limitations of the class of students for whom 
the work is primarily intended and to employ, therefore, 
only the simplest mathematical processes as far as possible. 
With this object in view I have purposely entered into much 
detail in the earlier and more elementary portions of the 
subject, following in fact the method which has been found 
to be most satisfactory with my own pupils; but I venture 
to hope that the book may be not without interest also to 
readers who already possess a certain knowledge of the 
subject. 

Recent years have witnessed extraordinary progress in 
both ophthalmology and applied optics. Not many persons 
are aware of the rapid rate at which spectacle optics, in par- 
ticular, is developing into a severe scientific pursuit; and 
there are certain portions of this volume which I think will 
be helpful to the modern oculist and optometrist. Thus, 
for example, I have been at some pains to explain the funda- 
mental principles of ophthalmic lenses and prisms. 

In general, however, I have necessarily had to omit much 



vi Preface 

that is essential to a thorough knowledge of the theory of 
optical instruments. In fact, in the space at my disposal 
it has been found quite impossible to describe a single one 
of these instruments in detail. In the latter portion of the 
book the theory of the chromatic and spherical aberrations 
is treated as briefly as possible; and I have given Von SeidePs 
formulae for the five spherical aberrations in the case of a 
system of infinitely thin lenses, chiefly because these formulae 
are exceedingly useful in the preliminary design of an optical 
system. But a complete discussion of these subjects would 
lie far beyond the plan of this volume. 

The problems appended to each chapter were originally 
collected for the use of my pupils and are generally of a very 
elementary description. A few of them have been adapted 
from other text-books, but in such cases I have now lost sight 
of their sources. 

If perchance this book should help to stimulate the study 
of optics in our colleges and universities, the author will feel 
abundantly repaid. Unfortunately, at present geometrical 
optics would seem to be a kind of Cinderella in the curric- 
ulum of physics, regarded perhaps with a certain friendly 
toleration as a mathematical discipline not without value, 
but hardly permitted to take rank on equal terms with her 
sister branches of physics. On the contrary, it might be in- 
ferred that any system of knowledge which had already 
placed at the disposal of scientific investigators such in- 
comparable means of research as are provided by modern 
optical instruments, and which has found so many useful 
applications in the arts of both peace and war, would be de- 
serving of the highest recognition and would be fostered and 
encouraged in all possible ways. According to the maxim, 
fas est et ab hoste doceri, the fact that from the time of Fraun- 
hofer the Germans have not ceased to cultivate this field of 
theoretical and applied science with notable achievements, 
is certainly not without significance for us in this country 
and in England. Indeed, both in England and in France, 



Preface vii 

apparently due to the exigencies of war, schools of applied 
optics have recently been organized. 

Nearly all of the diagrams in this volume were drawn by 
my friends, Professor Joseph Hudnut, Dr. B. A. Wooten and 
Mr. J. G. Sparkes, to whom I am much indebted. I desire 
also to express my grateful acknowledgments to my col- 
league, Professor H. W. Farwell, for numerous valuable 
criticisms from time to time and especially for aid in making 
the photographic illustrations in Chapter II. 

Any suggestions or corrections which may improve and 
extend the usefulness of the book will be appreciated. 

James P. C. Southall. 
Columbia University, 

New York, N. Y., 
April 4, 1918. 



CONTENTS 
CHAPTER I 

Lights and Shadows 

Sections Pages 

1-11. 1-27 

1. Luminous Bodies 1 

2. Transparent and Opaque Bodies 1-3 

3. Rectilinear Propagation of Light 3-5 

4. Shadows, Eclipses, etc 6-9 

5. Wave Theory of Light 9, 10 

6. Huygens's Construction of the Wave-Front 10-13 

7. Rays of Light are Normal to the Wave-Surface .... 13-15 

8. The Direction and Location of a Luminous Point. . . 15-18 

9. Field of View 18, 19 

10. Apparent Size 20-22 

11. The Effective Rays 23-25 

Problems 25-27 



CHAPTER II 

Reflection of Light. Plane Mirrors 

Sections Pages 

12-25. 26-63 

12. Regular and Diffuse Reflection 28-30 

13. Law of Reflection 30-32 

14. Huygens's Construction of the Wave-Front in case 

of Reflection at a Plane Mirror 33-37 

15. Image in a Plane Mirror 37-40 

16. The Field of View of a Plane Mirror 40-43 

17. Successive Reflections from Two Plane Mirrors 43 

18. Images in a System of Two Inclined Mirrors 43-48 

ix 



x Contents 

Sections Pages 

19. Construction of the Path of a Ray Reflected into 

the Eye from a Pair of Inclined Mirrors 48-50 

20. Rectangular Combinations of Plane Mirrors 50, 51 

21. Applications of the Plane Mirror 52, 53 

22. Porte Lumiere and Heliostat 53-55 

23. Measurement of the Angle of a Prism 55 

24. Measure of Angular Deflections by Mirror and 

Scale 56-58 

25. Hadley's Sextant 58-60 

Problems 60-63 



CHAPTER III 

Refraction of Light 

Sections Pages 

26-39. 64-94 

26. Passage of Light from One Medium to Another. ... 64, 65 

27. Law of Refraction 65-67 

28. Experimental Proof of the Law of Refraction 67-69 

29. Reversibility of the Light Path 69 

30. Limiting Values of the Index of Refraction 70 

31. Huygens's Construction of a Plane Wave Refracted 

at a Plane Surface 70-72 

32. Mechanical Illustration of the Refraction of a Plane 

Wave 72,73 

33. Absolute Index of Refraction 74-76 

34. Construction of the Refracted Ray 76-78 

35. Deviation of the Refracted Ray 78 

36. Total Reflection 78-83 

37. Experimental Illustrations of Total Reflection. . . . 83-86 

38. Generalization of the Laws of Reflection and Re- 

fraction. Principle of Least Time (Fermat's 

Law) 86-89 

39. The Optical Length of the Light-Path and the Law 

of Malus 89-91 

Problems 92-94 



Contents xi 

CHAPTER IV 

Refraction at a Plane Surface and also through a Plate 
with Plane Parallel Faces 

Sections Pages 

40-47. 95-112 

40. Trigonometric Calculation of Ray Refracted at a 

Plane Surface 95, 96 

41. Imagery in a Plane Refracting Surface by Rays 

which Meet the Surface Nearly Normally 96-98 

42. Image of a Point Formed by Rays that are Ob- 

liquely Refracted at a Plane Surface 98, 99 

43. The Image-Lines of a Narrow Bundle of Rays Re- 

fracted Obliquely at a Plane 100 

44. Path of a Ray Refracted Through a Slab with Plane 

Parallel Sides 101-103 

45. Segments of a Straight Line 104, 105 

46. Apparent Position of an Object seen through a 

Transparent Slab whose Parallel Sides are Per- 
pendicular to the Line of Sight 105-107 

47. Multiple Images in the two Parallel Faces of a Plate 

Glass Mirror 107-110 

Problems 110-112 

CHAPTER V 

Refraction through a Prism 

Sections Pages 

48-62. 113-148 

48. Definitions etc 113 

49. Construction of Path of a Ray Through a Prism. . 113-116 

50. The Deviation of a Ray by a Prism 116, 117 

51. Grazing Incidence and Grazing Emergence 117, 118 

52. Minimum Deviation 119-122 

53. Deviation away from the Edge of the Prism 122, 123 

54. Refraction of a Plane Wave Through a Prism 123, 124 

55. Trigonometric Calculation of the Path of a Ray in 

a Principal Section of a Prism 124, 125 



xii Contents 

Sections Pages 

56. Total Reflection at the Second Face of the Prism. . 125-128 

57. Perpendicular Emergence at the Second Face of 

the Prism 129 

58. Case when the Ray Traverses the Prism Symmet- 

rically 129 

59. Minimum Deviation 129-133 

60. Deviation of Ray by Thin Prism 133, 134 

61. Power of an Ophthalmic Prism. Centrad and 
Prism-Dioptry 134-138 

62. Position and Power of a Resultant Prism Equiva- 

lent to Two Thin Prisms 138-142 

Problems 142-148 



CHAPTER VI 

Reflection and Refraction of Paraxial Rays at a Spherical 

Surface 

Section^ Pages 

63-86. 149-216 

63. Introduction. Definitions, Notation, etc 149-153 

64. Reflection of Paraxial Rays at a Spherical Mirror ... . 153-156 

65. Definition and Meaning of the Double Ratio 156-159 

66. Perspective Ranges of Points 159-161 

67. The Harmonic Range 161-164 

68. Application to the Case of the Reflection of Par- 

axial Rays at a Spherical Mirror 164-166 

69. Focal Point and Focal Length of a Spherical Mirror 166-168 

70. Graphical Method of Exhibiting the Imagery by 

Paraxial Rays 168-171 

71. Extra-Axial Conjugate Points 171-175 

72. The Lateral Magnification 176 

73. Field of View of a Spherical Mirror 176-179 

74. Refraction of Paraxial Rays at a Spherical Surface . . . 179-182 

75. Reflection Considered as a Special Case of Refrac- 

tion 182, 183 

76. Construction of the Point M' Conjugate to the 

Axial Point M 183-186 



Contents xiii 

Sections Pages 

77. The Focal Points (F, F') of a Spherical Refracting 

Surface 186-190 

78. Abscissa-Equation Referred to the Vertex of the 

Spherical Refracting Surface as Origin 190, 191 

79. The Focal Lengths /, f of a Spherical Refracting 

Surface 191-193 

80. Extra-Axial Conjugate Points; Conjugate Planes 

of a Spherical Refracting Surface 193, 194 

81. Construction of the Point Q' which with respect to 

a Spherical Refracting Surface is Conjugate to 

the Extra- Axial Point Q 194-196 

82. Lateral Magnification for case of Spherical Re- 

fracting Surface 196 

83. The Focal Planes of a Spherical Refracting Sur- 

face 197-199 

84. Construction of Paraxial Ray Refracted at a Spher- 

ical Surface 199, 200 

85. The Image-Equations in the case of Refraction of 

Paraxial Rays at a Spherical Surface 200, 201 

86. The so-called Smith-Helmholtz Formula 201, 202 

Problems 203-216 



CHAPTER VII 

Refraction of Paraxial Rays through an Infinitely Thin 

Lens 

Sections Pages 

87-98. 217-257 

87. Forms of Lenses 217-223 

88. The Optical Center O of a Lens surrounded by the 

same Medium on both sides 223-226 

89. The Abscissa-Formula of a Thin Lens, referred to 

the Axial Point of the Lens as Origin 226-229 

90. The Focal Points of an Infinitely Thin Lens 229-232 

91. Construction of the Point M' Conjugate to the 

Axial Point M with respect to an Infinitely Thin 

Lens 232-234 



xiv Contents 

Sections Pages 

92. Extra-Axial Conjugate Points Q, Q'; Conjugate 

Planes 234-236 

93. Lateral Magnification in case of Infinitely Thin 

Lens 236, 237 

94. Character of the Imagery in a Thin Lens 237-240 

95. The Focal Lengths /, /' of an Infinitely Thin 

Lens 240-242 

96. Central Collineation of Object-Space and Image- 

Space 242-244 

97. Central Collineation (cont'd). Geometrical Con- 

structions 244-247 

98. Field of View of an Infinitely Thin Lens 247-249 

Problems 249-257 



CHAPTER VII 

Change of Curvature of the Wave-front in Reflection and 
Refraction. Dioptry System 

Sections Pages 

99-110. 258-299 

99. Concerning Curvature and its Measure 258-265 

100. Refraction of a Spherical Wave at a Plane Surface. 265-269 

101. Refraction of a Spherical Wave at a Spherical Sur- 

face 269-274 

102. Reflection of a Spherical Wave at a Spherical 

Mirror 274-276 

103. Refraction of a Spherical Wave through an In- 

finitely Thin Lens 276-279 

104. Reduced Distance 279-281 

105. The Refracting Power 281-284 

106. Reduced Abscissa and Reduced "Vergence" 284-286 

107. The Dioptry as Unit of Curvature 286-288 

108. Lens-Gauge 288, 289 

109. Refraction of Paraxial Rays through a Thin Lens- 

System 289-291 

110. Prismatic Power of a Thin Lens 291-295 

Problems 295-299 



Contents xv 

CHAPTER IX 

Astigmatic Lenses 

Sections Pages 

111-116. " 300-328 

111. Curvature and Refracting Power of a Normal Sec- 

tion of a Curved Refracting Surface 300-305 

112. Surfaces of Revolution. Cylindrical and Toric Sur- 

faces 305-310 

113. Refraction of a Narrow Bundle of Rays incident 

Normally on a Cylindrical Refracting Surface. . . 310-314 

114. Thin Cylindrical and Toric Lenses 314-318 

115. Transposing of Cylindrical Lenses 318-320 

116. Obliquely Crossed Cylinders 320-326 

Problems 326-328 

CHAPTER X 

Geometrical Theory of the Symmetrical Optical 
Instrument 

Sections Pages 

117-124. 329-255 

117. Graphical Method of tracing the Path of a Paraxial 

Ray through a Centered System of Spherical Re- 
fracting Surfaces 329-331 

118. Calculation of the Path of a Paraxial Ray through 

a Centered System of Spherical Refracting Sur- 
faces 332-334 

119. The so-called Cardinal Points of an Optical System 334-339 

120. Construction of the Image-point Q' conjugate to an 

Extra-Axial Object-Point Q 339, 340 

121. Construction of the Nodal Points, N, N' 340-342 

122. The Focal Lengths/,/' 342-344 

123. The Image-Equations in the case of a Symmetrical 

Optical System 344-349 

124. The Magnification-Ratios and their Mutual Rela- 

tions 349-351 

Problems 351-355 



xvi Contents 

CHAPTER XI 

Compound Systems. Thick Lenses and Combinations of 
Lenses and Mirrors 

Sections Pages 

125-132. 356-396 

125. Formulae for Combination of Two Optical Systems 356-359 

126. Formulae for Combination of Two Optical Systems 

in terms of the Refracting Power 360-362 

127. Thick Lenses Bounded by Spherical Surfaces 362-365 

128. The so-called "Vertex Refraction" of a ThickLens 365, 366 

129. Combination of Two Lenses 366-370 

130. Optical Constants of Gullstrand's Schematic Eye 370-374 

131. Combination of Three Optical Systems 374-376 

132. "Thick Mirror" 376-384 

Problems 384-396 

CHAPTER XII 

Aperture and Field of Optical System 

Sections Pages 

133-143. 397-424 

133. Limitation of Ray-Bundles by Diaphragms or Stops 397-399 

134. The Aperture-Stop and the Pupils of the System. . . 399-401 

135. Illustrations 401-404 

136. Aperture-Angle. Case of Two or More Entrance-Pupils 404-406 

137. Field of View 406-409 

138. Field of View of System Consisting of a Thin Lens 

and the Eye 409-413 

139. The Chief Rays 413, 414 

140. The so-called " Blur-Circles" (or Circles of Diffu- 

sion) in the Screen-Plane 414-416 

141. The Pupil-Centers as Centers of Perspective of 

Object-Space and Image-Space 416, 417 

142. Proper Distance of Viewing a Photograph 417-419 

143. Perspective Elongation of Image 419 

144. Telecentric Systems 420-423 

Problems 423, 424 



Contents xvii 

CHAPTER XIII 

Optical System of the Eye. Magnifying Power of Optical 
Instruments 

Sections , Pages 

145-159. 425-464 

145. The Human Eye 425-431 

146. Optical Constants of the Eye 431-433 

147. Accommodation of the Eye 433, 434 

148. Far Point and Near Point of the Eye 434, 435 

149. Decrease of the Power of Accommodation with In- 

creasing Age 435, 436 

150. Changes of Refracting Power in Accommodation. . 436, 437 

151. Amplitude of Accommodation 437-^39 

152. Various Expressions for the Refraction of the Eye . . 439 

153. Emmetropia and Ametropia 439-443 

154. Correction Eye-Glasses 443-446 

155. Visual Angle 446-448 

156. Size of Retinal Image 448, 449 

157. Apparent Size of an Object seen Through an Optical 

Instrument 449-452 

158. Magnifying Power of an Optical Instrument Used 

in Conjunction with the Eye 452-455 

159. Magnifying Power of a Telescope 455-460 

Problems 461-464 



CHAPTER XIV 

Dispersion and Achromatism 

Sections Pages 

160-174. 465-507 

160. Dispersion by a Prism 465-471 

161. Dark Lines of the Solar Spectrum 472 

162. Relation between the Color of the Light and the Fre- 

quency of Vibration of the Light- Waves 473-476 

163. Index of Refraction as a Function of the Wave- 

Length 476, 477 



xviii Contents 

Sections Pages 

164. Irrationality of Dispersion 477^L79 

165. Dispersive Power of a Medium 479-481 

166. Optical Glass 481-487 

167. Chromatic Aberration and Achromatism 487-489 

168. "Optical Achromatism" and "Actinic Achroma- 

tism" 489^91 

169. Achromatic Combination of Two Thin Prisms 491-493 

170. Direct Vision Combination of Two Thin Prisms. . . 493-495 

171. Calculation of Amici Prism with Finite Angles 495-497 

172. Kessler Direct Vision Quadrilateral Prism 497-499 

173. Achromatic Combination of Two Thin Lenses 499-502 

174. Achromatic Combination of Two Thin Lenses in 

Contact 502-505 

Problems 505-507 



CHAPTER XV 

Rays of Finite Slope. Spherical Aberration, Astigmatism 
of Oblique Bundles, etc. 

Sections Pages 

175-193. 508-557 

175. Introduction 508, 509 

176. Construction of a Ray Refracted at a Spherical Sur- 

face 509-512 

177. The Aplanatic Points of a Spherical Refracting Sur- 

face 512, 513 

178. Spherical Aberration Along the Axis 513-515 

179. Spherical Zones 515, 516 

180. Trigonometrical Calculation of a Ray Refracted at a 

Spherical Surface 516-519 

181. Path of Ray through a Centered System of Spheri- 

cal Refracting Surfaces. Numerical Calculation 519-522 

182. The Sine-Condition or Condition of Aplanatism . . . 522-525 

183. Caustic Surfaces 525, 526 

184. Meridian and Sagittal Sections of a Narrow Bundle 

of Rays before and after Refraction at a Spherical 
Surface 526-529 



Contents xix 

Sections Pages 

185. Formula for Locating the Position of the Image- 

Point Q' of a Pencil of Sagittal Rays Refracted at 

a Spherical Surface 529, 530 

186. Position of the Image-Point P' of a Pencil of Me- 

ridian Rays Refracted at a Spherical Surface. . . . 530-533 

187. Measure of the Astigmatism of a Narrow Bundle 

of Rays 533, 534 

188. Image-Lines (or Focal Lines) of a Narrow Astig- 

matic Bundle of Rays 534-536 

189. The Astigmatic Image-Surfaces 536-538 

190. Curvature of the Image 538-540 

191. Coma 540-543 

192. Distortion; Condition of Orthoscopy 543-545 

193. Seidel's Theory of the Five Aberrations 545-550 

Problems 551-557 

Index 559-579 



MIRRORS, PRISMS AND LENSES 



MIRRORS, PRISMS AND LENSES 

CHAPTER I 

LIGHTS AND SHADOWS 

1. Luminous Bodies. — The external world is revealed to 
the eye by means of light. With the rising sun night is 
changed into day, and animals, vegetables and minerals in 
all their manifold varieties of form and shade and color, 
which were quite invisible in the dark, are now revealed to 
view. Wherever the eye turns to gaze, there comes to it 
from far or near a messenger of light conveying information 
about the object which is under inspection. In an absolutely 
dark room everything is invisible, because the eye can per- 
ceive objects only when they radiate or reflect light into it. 
In the strict sense a source of light is a self-luminous body 
which shines by its own light, such as the sun or a fixed star 
or a candle-flame; but frequently the term is applied to a 
body which merely reflects or transmits light which has 
fallen upon it from some other body, as, for example, the 
moon and the planets which are illuminated by the light 
from the sun. In this latter sense the blue sky and the 
clouds, which, shining by light derived originally from the 
sun, contribute the greater portion of what is meant by 
daylight, are to be regarded as light-sources. A point- 
source of light or a luminous point is in reality a small ele- 
ment of luminous surface of relatively negligible dimensions 
or else a body like a star at such a vast distance that it ap- 
pears like a point. 

2. Transparent and Opaque Bodies. — In general, when 
light falls on a body, it is partly turned back or reflected at 
or very near the surface of the body, partly absorbed within 

1 



2 Mirrors, Prisms and Lenses [§ 2 

the body, and partly transmitted through it. An absolutely 
black body which absorbs all the light that falls on it does 
not exist; the best example we have is afforded by a body 
whose surface is coated with lamp-black. The color of a 
body as seen by reflected light is explained by the fact that 
part of the incident light is absorbed, whereas only light 
characteristic of the color in question is cast off or reflected 
from the body. Thus, when sunlight falls on a piece of red 
flannel, it is robbed of all its constituent colors except red, 
and thus it happens that the color by which we describe 
the body is in fact due to the light which it rejects. If the 
piece of red flannel were illuminated by pure blue light, it 
would appear black or invisible. 

A substance such as air or water or glass, which is per- 
vious to light, is said to be transparent. None of the light 
that traverses a perfectly transparent body will be absorbed; 
and, on the other hand, a perfectly opaque body is one which 
suffers no light at all to be transmitted through it. No 
substance is either absolutely transparent or absolutely 
opaque. These terms, therefore, as applied to actual bodies 
are merely relative, and so when we say that a body is opaque, 
we mean only that the light transmitted through it is so 
slight as to be practically inappreciable. Naturally, one 
thinks of clear water as transparent and of metallic sub- 
stances generally as opaque; but a sufficiently large mass 
of water will be found to be impervious to light, whereas, 
on the other hand, gold leaf transmits green light. A per- 
fectly transparent body would be quite invisible by trans- 
mitted light, although its presence could be detected by 
observing the distortion in the appearance of bodies viewed 
through it. 

Again there are some substances which, while they are 
not transparent in the ordinary sense, are far from being 
opaque, such, for example, as ground glass, alabaster, por- 
celain, milk, blood, smoke, which contain imbedded or sus- 
pended in them fine particles of matter of a different optical 



§ 3] Rectilinear Propagation of Light 3 

quality from that of the surrounding mass. Light does 
penetrate through materials of this nature in a more or less 
irregular fashion, and accordingly they are described as 
translucent. In the interior of such granular structures or 
"cloudy media" light undergoes a so-called internal diffused 
reflection or scattering; so that while it may be possible to 
discern the presence of a body through an intervening mass 
of such material, the form of the object will be to some ex- 
tent indistinct and unrecognizable. 

An optical medium is any space, whether filled or not 
with ponderable matter, which is pervious to light. In geo- 
metrical optics it is generally assumed that the media are 
not only homogeneous and isotropic (meaning thereby that 
the substance possesses the same properties in all directions), 
as, for example, air, glass, water and vacuum, but perfectly 
transparent as well. 

3. Rectilinear Propagation of Light. — When an opaque 
body is interposed between the observer's eye and a source 
of light, it is well known that all parts of the latter which 
lie on straight lines connecting the pupil of the eye with 
points of the opaque obstacle will be hid from view. We 
cannot see round a corner; we can look through a straight 
tube but not through a crooked one. A child takes note of 
such facts as these among the very earliest of his experiences 
and recognizes without difficulty the truth of the common 
saying that "light travels in straight lines," which in the 
language of science is called the law of the rectilinear propa- 
gation of light. The light that comes to us from a star 
traverses the vast stretches of interstellar space in straight 
fines until it reaches the earth's atmosphere, which is com- 
posed of layers of air of increasing density from the upper 
portions towards the surface of the earth; so that the me- 
dium through which the light passes in this short remainder 
of its downward journey is no longer isotropic, and, hence, 
also this part of the light path will, in general, be no longer 
straight but curved by a gradual and continuous bending 



4 Mirrors, Prisms and Lenses [§ 3 

from the less dense layers of air to the more dense layers 
below. This explains why it is necessary for an observer 
on the earth's surface looking through a long narrow tube 
at a star not directly overhead to point the tube not at the 
star itself but at its apparent place in the sky, which depends 
on the direction which the light has when it enters the eye; 
and, consequently, in accurate determinations of the posi- 
tion of a heavenly body, the astronomer is always careful 
to take account of the apparent displacement due to this 
so-called " atmospheric refraction," and a principal reason 
why astronomical observatories are nearly always located 
on high mountains is to obviate as much as possible the 
disturbing influence of the atmosphere. In aiming a rifle 
or in any of the ordinary processes we call " sighting," which 
are at the basis of some of the most delicate methods of 
measurement known to us, we rely with absolute confidence 
on this proved law of experience concerning the rectilinear 
propagation of light; and, in fact, the most conclusive dem- 
onstration that a line is straight consists in showing that it 
is the path which light pursues. The notion of a "ray of 
light" is derived from this law, and any line along which 

light travels is to be regarded as 

a ray of light. According to this 

f^~^ ^ ^ ^^ ^"^ idea, therefore, the rays of light 

X^-^ "^^^^ in an isotropic medium are 

straight lines. 

A very striking proof of the 

Fig. 1.— Rectilinear Propagation rectilinear propagation of light 

is afforded by placing a lumi- 
nous object (Fig. 1) in front of an opaque screen in which 
there is a very small round aperture. If now a second screen 
or a white wall is placed parallel to the first screen on the 
other side of it, there will be cast on it a so-called inverted 
image of the object, the size of which will be proportional to 
the distance between the two screens. From each point of 
the luminous object rays go out in all directions, and a narrow 




§ 3] Pinhole Camera 5 

cone of these rays will traverse the perforated screen through 
the opening and illuminate a small area on the other screen, 
and thus every part of the object will be depicted in this way 
by little patches of light arranged in a figure which is similar 
in form to the object, but which is completely inverted, since 
not only top and bottom but right and left are reversed in 
consequence of the rectilinear paths of the rays of light. It 
may be remarked that this image is not an optical image in 
the strict sense of the term (see § 11), but the phenomenon 
can be explained only on the supposition that light proceeds 
in straight lines. If another small opening were made in the 
front screen very near the first hole, there would be two 
images formed which would partly overlap each other, so that 
the resultant image would be more or less blurred, and if we 
have a single large aperture, we could no longer see any 
distinct image at all. 

The pinhole camera, invented by Giambattista Della 
Porta (c. 1543-1615), and sometimes called Porta's camera, 
is constructed on the principle of the experiment which has 
just been described. It is very useful in making accurate 
photographic copies of the architectural details of buildings, 
because the image which is obtained is entirely free from 
distortion. 

In the pinhole camera there is a certain relation between 
the size of the pinhole and the distance of the sensitive plate. 
According to Abney, in order to get the best results with 
an apparatus of this kind the diameter of the pinhole ought 
to be directly proportional to the square-root of the distance 
of the plate from the aperture, that is, 

y = k\Zx, 

where x and y denote the distance of the plate and the di- 
ameter of the pinhole, respectively, and k denotes a con- 
stant, the value of which will depend on the unit of length. 
Thus, if x and y are measured in inches, A; = 0.008; in centi- 
meters, k = 0.01275. 



6 



Mirrors, Prisms and Lenses 



[§4 



4. Shadows, Eclipses, etc. — The forms of shadows are also 
easily explained on the hypothesis that light proceeds in 
straight lines, for the outline of the shadow cast by a body 
is precisely similar to that of the object as viewed from the 
place where the source of light is. Thus, for example, the 




Fig. 2. — Shadow (umbra) of opaque globe E illuminated by 
point-source S. 

shadow of a sphere held in front of a point-source of light 
has the form of a circle, and the shadow cast by a circular 
disk will have the outline of an ellipse of greater and greater 
eccentricity as the disk is turned more and more nearly 
edge-on towards the light. Passing a shop- window on 
Sunday when the shade is drawn down, if the sun is shining 




Fig. 3. — Shadow (umbra and penumbra) of opaque globe E 
illuminated by two point-sources Si, S2. 

on the window, one can read the shadow of the sign painted 
on the glass quite as distinctly as the sign itself. The in- 
terposition of an opaque body between a source of light and 
a wall not only darkens a portion of the wall or casts its 
shadow there, but it converts an entire region of space be- 
tween it and the wall into a dark tract either wholly or par- 



§ 4] Shadows 7 

tially screened from the light. Thus, for example, the space 
A (Fig. 2) behind the body E which is comprised within the 
cone of rays proceeding from the point-source S that are 
intercepted by E gets no light from S, and this wholly un- 
illuminated region is called the umbra or true shadow. When 
there are two luminous points Si and S2 (Figs. 3 and 4), the 
region of shadow behind the opaque body E consists of the 




Fig. 4. — Shadow (umbra and penumbra) of opaque globe E 
illuminated by two point-sources Si, S2. 

umbra A which is wholly screened from both sources of light 
and the so-called penumbra or partially illuminated space 
composed of a space Bi which gets light only from Si and 
a similar space B 2 which gets light only from S2. Points lying 
beyond the penumbra will receive light from both sources. 

If the light-source has an appreciable size, light will pro- 
ceed from each of its shining points in all directions. Sup- 
pose, for example, that an opaque globe E (Fig. 5) is placed 
in front of a luminous globe S: then the dark body will 
intercept all rays that fall within the cone which is tangent 
externally to the two spheres, and, consequently, the por- 
tion A of this cone which lies behind E will be completely 



8 Mirrors, Prisms and Lenses [§ 4 

screened from all points of the source S, so that this portion 
constitutes the umbra where no light comes. In this case 
also there are two penumbral regions Bi and B 2 which are 
partially illuminated, but the illumination is not uniform, 




Fig. 5. — Shadow (umbra and penumbra) of opaque globe E 
illuminated by luminous globe S. 

but increases gradually from total darkness at the outer 
borders of the umbra into the complete illumination of the 
region outside the shadow. The shadow cast on a screen 
by an opaque body exposed to an extended source of light 
has no sharp outline but fades by imperceptible gradations 
into the bright space outside. As to the umbra, it terminates 
in a point at a certain distance x behind the opaque body, 
provided the diameter of the latter is less than that of the 
luminous globe in front of it, that is, provided R is greater 
than r, where R, r denote the radii of luminous and opaque 
globes, respectively. If the distance d between the centers 
of the two globes is known, the length x of the umbra may 
be calculated from the proportion : 

R d+x. 
r x 

whence we find : 

d 



x = 



r 



Thus, for example, the diameter of the sun is 109.5 times 
that of the earth, and the distance between the two bodies 



§ 5] Wave Theory of Light 9 

is 93 millions of miles. Accordingly, the umbra of the earth 
is found to extend to a distance of more than 857 000 miles 
behind it. Sometimes the moon whose distance from the 
earth is about 240 000 miles enters inside the shadow, and 
becomes then totally eclipsed. When the moon is only 
partly inside the earth's umbra, there is a partial eclipse of 
the moon. On the other hand, if the earth or any part of it 
comes inside the moon's shadow, there will be an eclipse 
of the sun visible from points on the earth that are in the 
shadow. 

The angular diameter of the sun is 32' 3.3"; whence it is 
easy to calculate that the length of the umbra of an opaque 
globe in sunlight is about 105 times the diameter of the globe. 

On the other hand, if the light-source is smaller than the 
interposed object, the umbra, instead of contracting to a 
point, widens out indefinitely; and thus, whereas the shadow 
cast on the opposite wall by a hand held in front of a broad 
fire is smaller than the object, the shadow made by the same 
hand in front of a small source of light like a candle-flame 
may be prodigious in extent. 

5. Wave Theory of Light. — The term "ray," as we have 
employed it, is a purely geometrical conception, but in or- 
dinary usage a ray of light implies generally an exceedingly 
narrow beam of light such as is supposed to be obtained 
when sunlight is admitted into a dark room through a pin- 
hole opening in a shutter. But when the experiment is 
carefully made to try to isolate a so-called ray of light in 
this fashion, new and unexpected difficulties arise, and, 
contrary to our preconceived notions, we are disconcerted 
by finding that the smaller the opening in the shutter, the 
more difficult it becomes to realize the geometrical concep- 
tion which is conveyed by the word "ray." In fact, in con- 
sequence of this experiment and others of a similar kind, 
we begin to perceive that the statement of the law of the 
rectilinear propagation of light needs to be modified; for 
among other phenomena we discover that when light pro- 



10 Mirrors , Prisms and Lenses [§ 6 

ceeds through a very narrow aperture in a screen, it does 
not pass through it just as though the screen were not pres- 
ent, but it spreads out laterally from the point of perfora- 
tion in all directions beyond the screen, proceeding, in fact, 
very much as it might do if the opening in the screen were 
the seat of a new and independent source of light. 

The truth is, as has been ascertained now for a long time, 
light is propagated not by "rays" at all but by waves; and 
if, in general, it is found that light does proceed in straight 
lines and does not bend around corners as sound-waves do, 
the explanation is because the waves of light are excessively 
short, considerably less than one ten-thousandth of a centi- 
meter. Wave-lengths of light are usually specified in terms 
of a unit called a "tenth-meter" or an "Angstrom unit," 
which is the hundred-millionth part of a centimeter (see 
§ 162) ; that is, 1 Angstrom unit = 10 - 10 meter = 0.000 000 01 
cm. The wave-length of the deepest red light is found to 
be about 7667 of these units and the wave-length of light 
corresponding to the extreme violet end of the spectrum 
is a little more than half the above value or 3970 units. 

According to the wave-theory the phenomena of light 
are dependent on an hypothetical medium called the ether, 
which may be compared to "an impalpable and all-per- 
vading jelly" that not only fills empty space but penetrates 
freely through all material substances, solid, liquid and 
gaseous, and through which particles of ordinary matter 
move easily without apparent resistance, for it is impon- 
derable and exceedingly elastic and subtle, insomuch that 
no one has ever succeeded in obtaining direct evidence of 
its existence. It is this ether which is the vehicle by which 
light-energy is transmitted and through which waves of 
light are incessantly throbbing with prodigious but measur- 
able velocity, which in vacuo is about 300 million meters per 
second or about 186 000 miles per second. 

6. Huygens's Construction of the Wave-Front.— The great 
Dutch philosopher Huygens (1629-1695), who was a contem- 



§6] 



Construction of Wave-Front 



11 



porary of Newton's (1642-1727), and who is usually regarded 
as the founder of the wave-theory of light, encountered his 
greatest difficulty in trying to give a consistent and satis- 
factory explanation of the apparent rectilinear propagation of 
light. His mode of reasoning, as set forth in his " Treatise 
on Light " published in 1690, while by no means free from 
objection, leads to a simple geometrical construction of the 
wave-front which corresponds with the known facts in regard 
to the procedure of light. 

Let O (Fig. 6) designate the position of a point-source of 
light from which as center or origin ether waves proceed in 
an isotropic medium with 
equal speeds in all direc- 
tions. At the end of a 
certain time the disturb- 
ances will have arrived 
at all the points which 
lie on a spherical surface 
Ci described around O as 
center, and at the instant 
in question this surface 
will be the locus of all the 
particles in the medium 
that are in this initial 
phase of excitation, and Fig. 6 
so it represents the wave- 
front at this moment. Now according to Huygens, every 
point in the wave-front becomes immediately a new source 
or center from which so-called secondary waves or wave- 
lets spread out. These innumerable' ripples or wavelets 
starting together from all the points affected by the 
principal wave overlap and interfere with each other, 
and Huygens inferred that their resultant sensible effects 
are produced only at the points of the surface which at any 
given instant touches or, as we say, envelops all the secondary 
wave-fronts, and that accordingly the new principal wave- 




H.uygens's construction of wave- 
front. 



12 



Mirrors, Prisms and Lenses 



[§6 



front will be this enveloping surface; so that the effect is 
the same as though the old wave-front # had expanded into 
the new, the disturbance marching forward along a straight 
line in any given direction. Obviously, in an unobstructed 

isotropic medium, such 
as is here supposed, 
the enveloping surface 
or new wave-front will 
be a sphere concentric 
with the old wave- 
front, and the straight 
lines that radiate out 
from the center will be 
the paths of the dis- 
turbance. 

Now if a plane screen 
MN (Fig. 7) is inter- 
posed in front of the 
advancing waves, and 
if there is an opening 

Fig. 7. — Huygens's construction of spherical AB in the Screen, each 

Tcrlen. PaSSing thr ° Ugh ^^ * & P oint in the opening 

between A, which is 
nearest to the source O, and B, which is farthest from it, 
will become in turn a new center of disturbance whence 
secondary spherical waves will be propagated into the re- 
gion on the other side of the screen. Since the disturbance 
will have arrived at the point A before it has reached a point 
X between A and B, the secondary wave emanating from 
A will at the end of a given time t have been travelling for 
a longer time than the secondary wave coming from X. If 
the radius of the wavelet around X at the time t is denoted 
by r, and if the distance OX is put equal to x, then d = x-\-r 
will denote the distance from O which the disturbance will 
have gone at the end of the time t; and since this distance 
is constant, whereas the distances denoted by x and r are 




Rays Normal to Wave-Surface 



13 




variables depending on the position of the point X, it is 
evident that the farther X is from 0, that is, the greater 
the value of x, the smaller will be the radius r = d — re of 
the secondary wavelet around X. The enveloping surface in 
this case is seen to be that part of the spherical surface de- 
scribed around O as center with radius equal to d which is in- 
tercepted by the cone 
which has O for its vertex 
and the opening AB in 
the screen for a section. 
Within this cone, accord- 
ing to Huygens's view, 
the disturbance is propa- 
gated exactly as though 
the perforated screen had 
not been interposed, 
whereas points on the far 
side of the screen and 
outside this limiting cone 
are not affected at all. 

It is plain that this mode of explanation is equivalent to 
the hypothesis of the rectilinear propagation of light. 

If the luminous point O (Fig. 8) is so far away that the 
dimensions of the opening AB in the screen may be regarded 
as vanishingly small in comparison with the distance of the 
source, the straight lines drawn from to the points A, X, B 
in the opening in the screen may be regarded as parallel, 
and the wave-front in this case will be plane instead of 
spherical, that is, the wave-front is a spherical surface with 
an exceedingly great radius as compared with the dimen- 
sions of the aperture in the screen. 

7. Rays of Light are Normal to the Wave-Surface. — The 
most obvious objection to Huygen's construction is, What 
right has he to assume that the places of sensible effects are 
the points on the surface which is tangent to or envelops 
the secondary waves? And why is the light not propagated 



Fig. 8. — Huygens's construction of plane 
waves passing through opening in a 



14 Mirrors, Prisms and Lenses [§ 7 

backwards from these new centers as well as forwards? 
Moreover, when the opening in the screen is very narrow, 
it is found, as has been already stated (§ 5), that this con- 
struction does not correspond at all with the observed facts. 

It is entirely beyond the scope of this book to attempt to 
answer these questions here or to describe even briefly the 
remarkable and complex phenomena of diffraction (which 
is the name given to these effects due to the bending of the 
light- waves around the edges of opaque obstacles). For 
an adequate discussion of these matters the reader must 
consult a more advanced treatise on physical optics. Suffice 
it to say, that the wave-theory of light and especially the 
principle of interference as developed long after Huygens's 
death (1695) by Young (1773-1829) and Fresnel (1788- 
1827) entirely supports the idea of the rectilinear propaga- 
tion of light as commonly understood; notwithstanding 
the fact that this law, as indeed is the case with nearly all 
so-called natural laws, has to be accepted with certain reser- 
vations; but, fortunately, these latter do not concern us at 
present. 

Accordingly, a luminous point is said to emit light in all 
directions, and the so-called light-rays in an isotropic medium 
are straight lines radiating from the center of the spheri- 
cal wave-surface. These rays may subsequently be bent 
abruptly into new directions in traversing the boundary 
between one isotropic medium and another, and under such 
circumstances the wave-surfaces may cease to be spherical; 
but no matter what may be the form of the wave-surface, 
the direction of the ray at any point is to be considered always as 
normal to the wave-front that passes through that point (see § 39) . 
In an isotropic medium the waves always march at right 
angles to their own front, and the so-called rays of light in 
geometrical optics are, in fact, the shortest optical routes 
along which the disturbances in the ether are propagated 
from place to place. With the aid of the principle of inter- 
ference (alluded to above) and by the use of the higher 



§ 8] Apparent Place of Light-Source 15 

mathematics, it may indeed be shown that the effect pro- 
duced at any point P in the path of a ray of light is due 
almost exclusively to previous disturbances which have 
occurred successively at all the points along the ray which 
lie between the source and the point P in question, and that 
disturbances at other points not lying on the ray which goes 
through P are practically without influence at P, that is, their 
effects there are mutually counteracted. And thus we arrive 
also at the so-called principle of the mutual independence of 
rays of light, which is also one of the fundamental laws of 
geometrical optics. From this point of view a ray of light 
is to be regarded as something more than a mere geomet- 
rical fiction and as having in some real sense a certain physi- 
cal existence, although it is not possible to isolate the ray 
from its companions. 

8. The Direction and Location of a Luminous Point. — 
When a ray of light comes into the eye, the natural infer- 
ence as to its origin is that the source lies in the direction from 
which the ray proceeded. There is no difficulty in pointing out 
correctly the direction of an object which is viewed through 
an isotropic medium; but if the medium were not isotropic, 
the apparent direction of A 

the object might not be, 
and probably would not 
be, its real direction. 
Thus, owing to the ef- 
fects of atmospheric re- 
fraction, to Which allu- FlG - 9- Direction and location of a lumi- 
' nous point. 

sion has been made al- 
ready (§ 3), the sun is seen above the horizon before it has 
actually risen, and so also in the evening the sun is still 
visible for a few moments after sunset. For the same reason 
a star appears to be nearer the zenith than it really is. 

In general, however, when a ray SA (Fig. 9) enters the 
eye at A, it is correctly inferred that the source S lies some- 
where on the straight line AS, but whether it is actually 




16 Mirrors, Prisms and Lenses [§ 8 

situated at S or farther or nearer cannot be determined by- 
means of a single ray. If the eye is transferred from A to 
another point B, the source will appear now to lie in the new 
direction BS. If the spectator views the source with both 
eyes simultaneously, one eye at A and the other at B, or if using 
only one eye he moves it quickly from A to B, the position 
of the source at S will be located at the point of intersection 
of the straight lines AS and BS; and this determination will 
be more accurate in proportion as the distance between the 
two points of observation A and B is greater or the more 
nearly the acute angle ASB approaches a right angle. That 
is the reason why in estimating the distance of a remote 
object one tries to observe it from two stations as widely 
separated as possible, and that explains also why a person 
shifts his head from side to side. If the object is compara- 
tively near at hand, a single movement of the head may be 
sufficient in order to get a fairly good idea of its distance, 
or it may be that it is simply necessary to look at the object 
with both eyes at the same time. It is amusing to watch 
a person with one eye closed attempting to poke a pencil 
through a finger-ring suspended in the middle of a room on a 
level with his eye; by chance he may succeed after repeated 
failures, whereas with both eyes open, the operation is per- 
formed without the slightest difficulty. 

In case the rays come into the eye after having traversed 
two or more isotropic media, it is easy to be deceived about 
the direction of the source where they emanated. In order 
for a bullet to hit a fish under water, the rifle must be 
pointed in a direction below that in which the fish appears 
to be. At the boundary-surface between two isotropic media 
the direction of a ray of light is usually changed abruptly 
by refraction (§ 26) ; so that, in general, the path of a ray 
will be found to consist of a series of line-segments. In 
Fig. 10 the broken line ABCD represents the course taken 
by a ray of light in proceeding through several media such as 
water, air and glass. The line-segments AB, BC and CD 



§8] 



Image of Point-Source 



17 




are portions of different straight lines of indefinite extent. 

For example, the actual route of the ray in air is along the 

straight line between B and C, and if the point P lies on 

this line between B and 

C, we say that the ray 

BC passes " really" 

through P, whereas we 

say that this same ray 

passes "virtually " 

through a point Q or R 

which lies in the prolon- Fig. 10.— Points P, Q and R considered as 

gation of the line-segment ! y ! ng on ray BC *? to be ; e g arded *» 

° . ° . lying in same medium as BC. 

BC in either direction. 

Moreover, thinking of the point Q or R as a point lying on 
the straight line BC which the light pursues in traversing 
the medium between the water and the glass, we must re- 
gard such a point as being optically in the same medium 
as the ray to which it belongs. Thus, the points Q and R 
in the figure considered as points on the ray BC are to be 
regarded as being optically in air, although in a physical 
sense Q is a point in the water and R is a point in the glass 
(see § 104). 
Now let us suppose that two rays emanating originally 

from a point-source S 
(Figs. 11 and 12) are 
bent at A and B into 
new directions AP and 
BQ, respectively, so as 
to enter the two eyes of 
an observer at P and Q. 
In such a case the ob- 




Fig. 11. — S' is said to be a "real' 
image of point-source at S. 



server will infer that the rays originated at the point S' 
where the straight lines AP and BQ intersect. This point 
S', which is called the image of S, may lie in the actual paths 
of the rays AP and BQ that enter the eyes, so that the light 
from S really does go through S', and in this case (Fig. 11) 




18 Mirrors, Prisms and Lenses [§ 9 

the image S' is said to be a real image. On the other hand, 
if the straight lines AP and BQ have to be produced back- 
wards in order to find their point of intersection, the rays 
do not actually pass through S', and in this case the image 

is said to be a virtual 

•^x-- image of the point S 

(Fig. 12). However, it 
must be borne in mind 
in connection with these 
diagrams that in reality 

Fig. 12. — S' is said to be a "virtual" image we do not See objects by 
of point-source at S. r • i 

p means of single rays; 

and, hence, we shall not be in a position to form an ac- 
curate idea of the term optical image until we come to 
consider bundles of rays in § 11. 

9. Field of View. — The open or visible space commanded 
by the eye is called the field of view. Since the eye can turn 
in its socket, the field of view of the mobile eye is very much 
more extensive than that of the stationary eye, and, more- 
over, the field of view of both eyes is greater than that of 
one eye by itself. The spectator may also widen his field 
of vision by turning his head or indeed by turning his entire 
body. For the present, however, we shall employ the term 
field of view to mean that more limited portion of space 
which is accessible to the single eye turning in its socket 
around the so-called center of rotation of the eye. When 
a person gazes through a window, the outside field of view 
is limited partly by the size of the window and partly also 
by the position of the eye with reference to it; so that only 
such exterior objects will be visible as happen to lie within 
the conical region of space determined by drawing straight 
lines from the center of rotation of the eye to all the points 
in the edge of the window. Thus, for example, if the open- 
ing in the window is indicated by the gap GH in the straight 
line GH in Fig. 13, and if the point marked is the position 
of the center of rotation of the eye, a luminous object at P 



§9] 



Field of View 



19 



in front of the window and directly opposite the eye will 
be plainly in view, because some of the rays from P may go 
through the window and enter the eye. But if the object 
is displaced far enough to one side to some position such as 




Fig. 13. — Field of view determined by contour of 
window GH and position of the eye at O. 

that marked R in the diagram, so that the straight line OR 
does not pass through the window, the object will pass out 
of the field of view. The straight line MN drawn parallel 
to GH is supposed to represent a vertical wall opposite the 
window. If this wall is covered with a mural painting, the 
only part of the picture that can be seen through the win- 
dow by the eye at O is the section included between the 
points T and V where the straight lines OG and OH intersect 
the straight line MN. The window acts here as a so-called 
field-stop (§ 137) to limit the extent of the field of view. But 
the limitation of the visible region depends essentially also 
on the position of the eye, becoming more and more con- 
tracted the farther the eye is from the window. The size of 
the window makes very little difference when the eye is 
placed close to it, and a person sitting near an open window 
can command almost as wide a view as if the entire wall of 
the room were removed. If one is looking through a key- 
hole in a door, he must put his eye close to the hole in order 
to see objects that are not directly in front of it. 



20 



Mirrors, Prisms and Lenses 



[§10 



10. Apparent Size. — The apparent size of an object is 
measured by the visual angle which it subtends at the eye. 
Several objects in the field of view which subtend equal 
angles when viewed from the same standpoint are said to 
have the same apparent size; although their actual sizes will 




Fig. 14. — Apparent size measured by visual angle. 



be different if they are at unequal distances from the eye. 
The objects marked 1, 2 and 3 in Fig. 14 appear to an eye 
at to be all of the same size. Thus an elephant may ap- 
pear no bigger than a man or a boy. Looking through a 
single pane of glass in a window, one may see a large build- 
ing or an entire tree, because the apparent extent of the 
small area of glass is greater than that of the distant object. 
A fly crawling across the window may hide from view a 
large portion of the distant landscape outside. A mountain 
a few miles off may be viewed through a finger-ring. 

The apparent size of an object, being measured by the 
visual angle which it subtends, is expressed in degrees or 
radians. The apparent diameter of the full moon in the sky, 
for example, is not quite half a degree, so that by holding 
a coin a little less than 9 mm. in diameter at a distance of 
one meter from the eye, the entire moon could be hid from 
view. In fact, instead of the angle itself it is customary to 
employ the tangent of the angle, especially in case the vis- 
ual angle is not large. Thus, the apparent size of an object 
of height h at a distance d from the eye (in Fig. 15 AB = 
h, AO = d) is measured by the tangent of the angle BOA, 
that is, 



§ 10] Apparent Size 21 

A . linear dimension of the object h 

Apparent size= ^r— ^ —, — = -• 

distance from the eye a 

Accordingly, in order to determine the actual size (h) of 

the object, it is necessary to know its distance (d) as well 

as its apparent size, because the actual size is equal to the 

product of these two magnitudes. The apparent size of an 




Fig. 15. — Apparent size varies inverse^ as distance d and directly as actual 

size h. 

object at a distance of one foot is an hundred times greater 
than it is at a distance of an hundred feet, or, as we say, the 
appare?it size varies inversely as the distance. As the object 
recedes farther and farther from the eye, its apparent size 
diminishes until at last it looks like a mere speck and the 
details in it have all disappeared. On the other hand, al- 
though the object is quite close to the eye, its actual dimen- 
sions may be so minute that it is not to be distinguished from 
a point. There is, indeed, a limit to the power of the human 
eye to see very small objects, which is reached when the 
object subtends in the field of view an angle that does not 
exceed one minute of arc. Two stars whose angular dis- 
tance apart is less than this limiting value cannot be seen 
as separate and distinct by a normal eye without the aid 
of a telescope. Now tan l' = sTz-g, and consequently the 
eye cannot distinguish details of form in an object which 
is viewed at a distance 3438 times as great as its greatest 



22 Mirrors, Prisms and Lenses [§ 10 

linear dimension. A silver quarter of a dollar is about 
24 mm. in diameter and viewed from a distance of 82.5 
meters (3438 times 24 mm. = 82 512 mm. = 82.5 m.) its ap- 
parent size will be 1' of arc and it will appear therefore 
like a mere point. The apparent width of a long straight 
street diminishes in proportion as the distance increases; 
until, finally, if the street is long enough, the two opposite 
sidewalks seem to run together at the so-called "vanishing 
point." . 

If rays of light coming through a window and entering 
the eye could leave marks in the glass at the points where 
they cross it, and if these marks could be made to emit the 
same kind of light as was sent out from the corresponding 
points of the object, there would be formed on the glass 
a pictorial representation of the object which when held 
before the eye at the proper distance would have almost 
exactly the same appearance as the object itself. This 
principle of perspective is made use of in the art of painting, 
and the artist, with his lights and shades and colors, tries 
to portray on a plane canvas a scene which will produce 
as nearly as possible the same visual impression on a spec- 
tator as would be produced by the natural objects them- 
selves. So far as apparent size is concerned, such a repre- 
sentation may be perfect. In a good drawing the various 
figures are delineated in such dimensions that when viewed 
from the proper standpoint they have the same apparent 
sizes as the realities would have if seen under the aspect 
represented in the picture. No one looking at a photo- 
graph of a Greek temple will notice (unless his attention 
is specially directed to it) that the more distant pillars are 
much shorter in the picture than the nearer ones. Indeed, 
generally we pay little heed to the apparent sizes of things, 
but always try to conceive their real dimensions. When 
two persons meet and shake hands, neither is apt to observe 
that the other appears much taller than he did when they 
were fifty yards apart. 



§ 11] Effective Rays 23 

11. The Effective Rays. — All the rays that enter the eye 
and fall on the retina must pass through the circular window 
in the iris or colored diaphragm of the eye which is called 
the pupil of the eye and which is sometimes spoken of as 
the "black of the eye/' because it appears black against 
the dark background of the posterior chamber of the eye. 
The pupil of the eye is about half a centimeter in diameter, 
although within certain limits its size can be altered to regu- 
late the quantity of light which is admitted to the eye. So 
far as the spectator's vision is concerned, it is only these 
rays that go through the pupil of his eye that are of any 
use, and these are the effective rays. When the pupil dilates, 
more rays can enter, and consequently the source appears 
brighter. The brightness of the source will depend also on 
its distance, because for a given diameter of the pupil, the 
aperture of the cone of rays from a nearer source will be 
wider than that of the cone of rays from a more distant 
source. In general, therefore, the pupil of the eye regu- 
lates the angular apertures of the cones of rays that enter 
the eye from each point of a luminous object and acts as 
the so-called aperture-stop (§ 134). Thus, while the extent 
of the field of view is controlled by the field-stop (§ 9), the 
brightness of the source depends essentially on the size of 
the aperture-stop. 

A series of transparent isotropic media each separated 
from the next by a smooth, polished surface constitutes an 
optical system. An optical instrument may consist of a single 
mirror, prisms or lens, but generally it is composed of a com- 
bination of such elements, which may be in contact with 
each other or separated by air or some other medium. In 
the great majority of actual constructions the instrument 
is symmetrical with respect to a straight line called the 
optical axis. Not all the rays emitted by a luminous object 
will be utilized by the instrument; generally, in fact, only 
a comparatively small portion of such rays will be trans- 
mitted through it, in the first place because its lateral di- 



24 Mirrors, Prisms and Lenses [§11 

mensions are limited, and in the second place because, in 
addition to the lens-fastenings and other opaque obstacles 
(sides of the tube, etc.), nearly all optical instruments are 
provided with perforated screens or diaphragms called 
" stops," specially placed and designed to intercept such rays 
as for one reason or another it is not desirable to let pass 
(§133). The planes of these stops are placed at right angles 
to the optical axis with the centers of the openings on the 
axis. Accordingly, each separate point of the object is to 
be regarded as the vertex of a limited cone or bundle of rays, 
which, with respect to the instrument, are the so-called 
effective rays, because they are the only rays coming from 
the point in question that traverse the instrument from 
one end to the other without being intercepted on the way. 

Moreover, in every bundle of rays there is always a cer- 
tain central or representative ray, coinciding perhaps with 
the axis of the cone or distinguished in some special way, 
called the chief ray of the bundle (§ 139). In a symmetri- 
cal optical instrument the chief ray of a bundle of effective 
rays is generally defined to be that ray which in traversing 
a certain one of the series of media crosses the optical axis 
at a prescribed point, which is usually at the center of that 
one of the stops which is the most effective in intercepting 
the rays and which, therefore, is called the aperture-stop, 
as will be explained more fully hereafter (see Chapter XII). 
According to this definition, the chief rays coming from all 
the various points of the object constitute a bundle of rays 
which in the medium where the aperture-stop is placed 
(sometimes called the "stop medium") all pass through 
the center of the stop. 

We shall employ the term pencil of rays to mean a section 
of a ray-bundle made by a plane containing the chief ray. 

The effective rays in the first medium before entering 
the instrument are called the incident rays or object rays; 
and these same rays in the last medium on issuing from the 
instrument are called the emergent rays or image rays. If we 



Ch. I] Problems 25 

select at random any point X lying on one of the rays of 
the bundle of emergent rays which had its origin at the lu- 
minous object-point P, in general, no other ray of this bundle 
will pass through X, since in a given optical system there 
will usually be one single route by which light starting from 
the point P and traversing the instrument can arrive finally, 
either really or virtually (§ 8), at a selected point X in the 
last medium. However, there may be found a number of 
singular points where two or more rays of the bundle of 
emergent rays intersect; and under certain favorable and 
exceptional circumstances it may indeed happen that there 
is one special point P' where all the emergent rays emanating 
originally from the object-point P meet again; and then we 
shall obtain at P' a perfect or ideal image of P, which is 
described by saying that P' is the image-point conjugate to 
the object-point at P. This image will be real or virtual 
according as the actual paths of the image-rays go 
through P' or merely the backward prolongations of these 
paths (§ 8). 

In order to obtain an image in this ideal sense, the optical 
system must be such as to transform a train of incident 
spherical waves spreading out from the object-point P into 
a train of emergent spherical waves converging to or di- 
verging from a common center P' in the image-space. When 
all the rays of a bundle meet in one point, the bundle of rays 
is said to be homocentric or monocentvic. In general, how- 
ever, a monocentric bundle of rays in the object-space will 
be transformed in the image-space into an astigmatic bundle 
of emergent rays, which no longer meet all in one point; 
and in fact this is a usual characteristic of a bundle of op- 
tical rays. 

PROBLEMS 

1. Why are the shadows much sharper in the case of an 
arc lamp without a surrounding globe than with one? 

2. Draw a diagram to show how a total eclipse of the 



26 Mirrors, Prisms and Lenses [Ch. I 

moon occurs; and another diagram to illustrate a total 
eclipse of the sun. Give clear descriptions of the 
drawings. 

3. An opaque globe, 1 foot in diameter, with its center 
at a point C, is interposed between an arc lamp S and 
a white wall which is perpendicular to the straight line 
SC. If the wall is 12 feet from the lamp, and if the 
distance SC = 3 feet, what is the area of the shadow on 
the wall? Ans. 12.57 sq. ft. 

4. What is the apparent angular elevation of the sun 
when a telegraph pole 15 feet high casts a shadow 20 feet 
long on a horizontal pavement? Ans. 36° 52' 10". 

5. What is the height of a tower which casts a shadow 
160 feet long when a vertical rod 3 feet high casts a shadow 
4 feet long? Ans. 120 feet. 

6. An object 6 inches high is placed in front of a pinhole 
camera at a distance of 6 feet from the aperture. What is 
the size of the inverted image on the ground glass screen if 
the length of the camera-box is 1 foot? Ans. 1 inch. 

7. A small hole is made in the shutter of a dark room, and 
a screen is placed at a distance of 8 feet from the shutter. 
The image on the screen of a tree outside 120 feet away is 
measured and found to be 3 feet long. How high is the tree? 

Ans. 45 feet. 

8. If the sensitive plate of a pinhole camera is 20 cm. 
from the pinhole, what should be the diameter of the pin- 
hole, according to Abney's formula? Ans. 0.57 mm. 

9. What is the apparent size of a man 6 feet tall at a dis- 
tance of 100 yards? How far away must he be not to be 
distinguishable from a point? Ans. 1° 8' 45"; 3.9 miles. 

10. If the moon is 240 000 miles from the earth and its 
apparent diameter is 31' 3", what is its actual diameter? 

Ans. 2168 miles. 

11. A person holding a tube 6 inches long and 1 inch in 
diameter in front of his eye and looking through it at a 
tree moves backwards away from the tree until the entire 



Ch. I] Problems 27 

tree is just visible. What is the apparent height of the 
tree? Ans. 9° 27' 44". 

12. Assuming that the resolving power of the eye is one 
minute of arc, at what distance can a black circle 6 inches 
in diameter be seen on a white background? Ans. 1719 feet. 



CHAPTER II 

REFLECTION OF LIGHT. PLANE MIRRORS 

12. Regular and Diffuse Reflection. — When a beam of 
sunlight, admitted through an opening in a shutter in a 
dark room, falls on a piece of smoothly polished glass, al- 
though the glass itself may be almost or wholly invisible, a 
brilliant patch of light will be reflected from the glass on the 
walls of the room or the ceiling or on some other adjacent ob- 
ject. If a person in the room happens to be looking towards 
the piece of glass along one special direction, he will be al- 
most blinded by the light that is reflected into his eyes. The 
glass acts like a mirror and reflects the sunlight falling on 
it in a definite direction which depends only on the direc- 
tion of the incident rays and on the orientation of the re- 
flecting surface, and in such a case the light is said to be 
regularly reflected. Thus, for example, signals may be com- 
municated to distant and inaccessible stations by reflecting 
thither the rays of the sun by a plane mirror adjusted in a 
suitable position. 

If the surface is not smooth, the light will be reflected in 
many directions at the same time. The long sparkling trail 
of sunlight seen on the surface of a lake or a river on a bright 
day is caused by the reflections of the sun's rays into the 
eyes of the spectator from countless little ripples on the 
surface of the water. 

The bright spot of light on the wall of a dark room at 
the place where a beam of sunlight falls, which shines almost 
as though this portion of the wall were itself a self-luminous 
body, is visible from any part of the room by means of the 
light which is reflected from it; and although the incident 
rays have a perfectly definite direction, the reflected light 

28 



§ 12] Diffuse Reflection 29 

is scattered in all directions. Some of this reflected light 
will fall on other bodies in the room, which will be more or 
less feebly illuminated thereby and rendered dimly visible 
by the light which they reflect in their turn; until at last 
the light after undergoing in this way repeated reflections 
from one body to another becomes too faint to be percep- 
tible. Light which is reflected or scattered in this way is 
said to be diffusely reflected or irregularly reflected, although, 
strictly speaking, there is nothing irregular about it. Ordi- 
narily it is in this way that bodies illuminated by day- 
light or by artificial light are rendered visible to a whole 
group of spectators at the same time. 

The paper on the walls of an apartment which gets very 
little light through the windows should be a dull white in 
order to scatter and diffuse as much as possible the light 
that comes into the room. The walls of a dark chamber 
used for developing photographic plates should be painted 
a dull black in order to absorb the light that falls on them. 
An absolutely black body (§ 2) exposed to the direct rays 
of the sun will be completely invisible, except by contrast 
with its surroundings. If the walls of a dark room and all 
the objects within it were coated with lampblack, and if 
the air inside were entirely free from dust and moisture, 
a beam of sunlight traversing the room could not be seen 
and the only way to detect its presence would be by placing 
the eye squarely in its path. But if a little finely divided 
powder were scattered in the air or if a cloud of smoke were 
blown across the beam of light, the course of the rays would 
immediately become manifest to a spectator in any part of 
the room, because some of the light reflected from the float- 
ing particles of matter in practically every direction would 
enter the eye. But the light itself is quite invisible. 

Any surface that is not too rough, that is, whose scratches 
or ridges are not wider than about a quarter of a wave- 
length of light, will reflect light in a greater or less degree 
depending on the smoothness of the surface. Waves of 



30 Mirrors, Prisms and Lenses [§ 13 

light falling on a sheet of white paper are broken up or 
scattered in all directions, and we can get some idea of the 
quantity of light that is diffusely reflected from such a sur- 
face by letting the light of a lamp shine on the paper when 
it is held near an object that is in shadow. It is almost 
startling to see how under the influence of this indirect 
illumination the details of the obscure body suddenly ap- 
pear as if summoned forth by magic. A highly polished 
metallic surface makes the best mirror, reflecting some- 
times as much as three-fourths of the incident light. Our 
ordinary looking-glasses are really metallic mirrors, because 
they are coated at the back with silver, and the glass merely 
serves as a protection for the reflecting surface. 

13. Law of Reflection. — A ray of light represented in 
Fig. 16 by the straight line AB is incident at B on a smooth 
reflecting surface whose trace in the plane of the diagram 
is the line ZZ. The straight line BN normal to the surface 




Fig. 16. — Law of reflection: 
Z NBA = -Z NBC = Z CBN. 



at B is called the incidence-normal, and the plane ABN which 
contains the incident ray AB and the normal BN is called 
the plane of incidence, which corresponds here with the 
plane of the diagram. The angle of incidence is the angle 
between the incident ray and the incidence-normal; or, to 



§ 13] Law of Reflection 31 

define this angle more exactly, the angle of incidence is the acute 
angle ( a ) through which the incidence-normal has to be turned 
about the point of incidence in order to make it coincide with 
the incident ray; thus, a ~ Z NBA. Counter-clockwise rota- 
tion is to be reckoned as positive and clockwise rotation as 
negative. This rule will be consistently observed in the 
case of all angular measurements. 

The reflected ray corresponding to the incident ray AB is 
represented by the straight line BC; and if in the above 
definition of the angle of incidence we substitute "reflected 
ray" for " incident ray," we shall obtain the definition of 
the angle of reflection (/3); that is, /3 = ZNBC. The sense 
of the rotation is indicated by the order in which the 
letters specifying the angle are named; thus, ZABC is the 
angle described by rotating the straight line AB around the 
point B until it coincides with the straight line BC; whereas 
ZCBA=-ZABC denotes the equal but opposite rotation 
from CB to BA. The student should take note of this 
usage, which will be uniformly employed throughout this 
book. 

The law of the reflection of light, which has been known 
for more than 2200 years, is contained in the following 
statement : 

The inflected ray lies in the plane of incidence, and the in- 
cident and reflected rays make equal angles with the normal 
on opposite sides of it; that is, /S =-a. 

A very accurate experimental proof of this law may be 
obtained by employing a meridian circle to observe the light 
reflected from an artificial mercury-horizon, that is, from 
the horizontal surface of mercury contained in a basin. In 
fact, this is the actual method used by astronomers in meas- 
uring the altitude of a star. The telescope is pointed at the 
star and then at the image of the star in the mercury mirror, 
and it will be found that the axis of the telescope in these 
two observations will be equally inclined to the vertical on 
opposite sides of it. 




32 Mirrors, Prisms and Lenses [§ 13 

A simple lecture-table apparatus for verifying the law 
of reflection of light consists of a circular disk (Fig. 17) made 
of ground glass, about one foot or more in diameter, and 
graduated around the circumference in degrees. This disk 
is mounted so as to be capable of rotation in a vertical plane 

about a horizontal axis 
perpendicular to this 
plane and passing through 
the center of the disk. A 
small piece of a plane 
mirror B with its plane 
perpendicular to that of 
the disk is fastened to 
the disk at its center, and 
the mirror is adjusted so 
that it is perpendicular 
to the radius BN drawn 

Fig. 17. — Optical disk used to verify law on the disk. A beam of 

of reflection. sunlight falling on the 

mirror in the direction NB will be reflected back from the 
mirror in the opposite direction BN, so that in this adjust- 
ment of the disk the paths of the incident and reflected rays 
coincide (/?= -a=0). Now if the disk is turned so that 
the incident ray AB makes with the normal BN an angle 
NBA, the reflected ray will proceed in a direction BC such 
that ZNBC = ZABN=-a. 

If, without changing the direction of the incident ray, the 
disk is turned through an angle 6, the plane of the mirror to- 
gether with the incidence-normal will likewise be turned 
through this same angle, and the angles of incidence and re- 
flection will each be changed in opposite senses by the amount 
6, so that the angle between the incident and reflected rays 
will be changed by 2 6. Accordingly, when a plane mirror 
is turned through a certain angle, the reflected ray will be turned 
through an angle twice as great. 



14] 



Waves Reflected at Plane Mirror 



33 



14. Huygens's Construction of the Wave-Front in Case 
of Reflection at a Plane Mirror. 

1. The case of a plane wave reflected from a plane mirror. 
The rebound of waves from a polished surface affords a very 
simple and instructive 
illustration of Huygens's 
Principle (§ 5) . In Fig. 18 
the straight line AD 
represents the trace in 
the plane of the diagram 
of a plane mirror, and 
the straight line AB rep- 
resents the trace of a 
portion of the front of 
an incident plane wave 
(§ 6) advancing in the 
direction of the wave- 
normal BD. At the first 
instant under considera- 
tion the wave-front is 
supposed to be in the 
position AB when the 
disturbance has just 
reached the point A of 
the reflecting surface, 
and from this time for- 
ward, according to Huy- 
gens's theory, the point 
A is to be regarded as 
itself a center of dis- 
turbance from which 
secondary hemispherical 
waves are reflected back into the medium in front of the 
mirror. Exactly the same state of things will prevail at 
this instant (t = 0) at all points of the plane reflecting sur- 
face lying on a portion of the straight line perpendicular 




Fig. 18. — Huygens's construction of plane 
wave reflected at plane mirror. 



34 Mirrors, Prisms and Lenses [§ 14 

to the plane of the paper at the point A, and the envelop 
of the hemispherical wavelets originating from these points 
will be a semicylindrical surface whose axis is the straight 
line just mentioned. If the speed with which the waves 
travel is denoted by v, then at the end of the time t = YQ/v 
the disturbance that was initially at the point P in the wave- 
front AB will have advanced to a point Q on the reflecting 
plane between A and D; and from this moment a new set 
of hemispherical wavelets having their centers all on a 
straight line perpendicular to the plane of the diagram at 
the point Q will begin to develop, and their envelop will 
also be a semicylinder. And so at successively later and 
later instants the disturbance will arrive in turn at each 
point along AD; until, finally, after the time t = BT)/v the 
farthermost point D will be reached. Meanwhile, around 
all the straight lines perpendicular to the plane of the 
paper at points lying along AD semicylindrical elementary 
wave-surfaces will have been spreading out from the re- 
flecting surface, the radii of these cylinders diminishing 
from A towards D. At the time when the disturbance 
reaches D, the semicylindrical wavelet whose axis passes 
through A will have expanded until its radius is equal to 
BD, and at this same instant the semicylindrical wavelet 
corresponding to a point Q between A and D will have been 
expanding for a time (BD — PQ)/v, and hence its radius will 
be egual to (BD— PQ) = (BD— BK) =KD. 

Now, according to Huygens's Principle, the surface which 
at any instant is tangent to all these elementary semi- 
cylindrical waves will be the required reflected wave-front 
at that instant. We shall show that the reflected wave-front 
is a plane surface which at the moment when the disturb- 
ance reaches the point D contains this point; or, what 
amounts to the same thing, we shall show that if a straight 
line DC in the plane of the diagram is tangent at C to the 
semicircle in which this plane cuts the semicylinder whose 
axis passes through A, it will be a common tangent to all 



§ 14] Waves Reflected at Plane Mirror 35 

such semicircles; for example, it will also be tangent to 
the semicircle in which the plane of the diagram cuts the 
semicylinder belonging to the point Q. From D draw DC 
tangent at C to the semicircle described around A as center 
with radius AC = BD and DR tangent at R to the semi- 
circle described around Q as center with radius QR = KD. 
The right triangles ABD and ACD are congruent, and hence 
ZDAB = ZCDA; and, similarly, in the congruent right tri- 
angles QKD and QRD ZDQK = ZRDQ. But ZDQK = 
ZDAB, and therefore ZRDQ=ZCDA, and hence the two 
tangents DR and DC coincide. Accordingly, the trace of 
the reflected wave-front in the plane of the diagram is the 
straight line CD. This reflected plane wave will be prop- 
agated onwards, parallel with itself, in the direction shown 
by the reflected rays AC, QR, etc. It is evident from the 
construction that the ray incident at A, the normal AN to 
the reflecting surface at the incidence-point A, and the re- 
flected ray AC lie all in the same plane; and the equality of 
the angles of incidence and reflection is an immediate con- 
sequence of the congruence of the triangles ABD and ACD. 
2. The case of a spherical wave reflected at a plane mirror. 
In Fig. 19 the light is represented as originating from a 
point-source L and spreading out from it in the form of 
spherical waves which presently impinge on the plane re- 
flecting surface represented in the diagram by the straight 
line AD. The nearest point of the reflecting plane to the 
source at L is the foot A of the perpendicular let fall from 
L on the straight line AD, and this, therefore, is the first 
point of the mirror to be affected. Obviously, on account 
of symmetry with respect to LA, it will be quite sufficient 
to investigate the procedure of the waves in the plane of 
the figure. The wave-front at the time the disturbance 
reaches A will be represented by the arc of a circle described 
around L as center with radius equal to LA; let P desig- 
nate the position of a point on this arc, and draw the straight 
line LP meeting AD at Q. After a time t = FQ/v the dis- 



36 



Mirrors, Prisms and Lenses 



[§14 



turbance will have advanced from P to Q, and from this 
moment the point Q will begin to send back wavelets from 
the reflecting surface. And so in succession one point of 
the mirror after another will be affected until presently the 
disturbance reaches the farthest point D. Meanwhile, all 

the points along AD on 
one side of AL and 
along AF on the other 
side (AF = DA) will 
have been sending out 
wavelets whose radii will 
be greater and greater 
the nearer these new 
centers are to the point 
A midway between D 
and F. Draw the 
straight line LD meet- 
ing the arc AP in the 
point B: then at the 
moment t = BT>/v when 
the disturbance from L 
has just arrived at D, 
the reflected wavelet 

Fig. 19. — Huygens's construction of spheri- proceeding from A as 
cal wave reflected at plane mirror. < hi u 

center will have ex- 
panded until its radius is equal to BD, and at this same 
instant there will also be a wavelet around Q as center 
of radius (BD— PQ) = (BD— BK)=KD. According to 
Huygens, the problem consists, therefore, in finding the 
surface which is tangent at a given instant to all these 
secondary waves. Produce the straight line LA on the 
other side of the reflecting surface to a point L' such 
that AL' = LA, and draw the straight line L'Q, and mark 
the point R where this straight line produced meets 
the semicircle described around Q as center with radius 
KD = QR. Since LQ + QR = LK + KD = LD, obviously, 




§15] 



Image in Plane Mirror 



37 



L / R = L / D; and therefore a circle described around L' as cen- 
ter with radius equal to L'D will touch at R the semicircle 
described around Q as center with radius equal to QR. 
Moreover, it will also touch at a point C on the straight 
line LA the semicircle described around A as center with 
radius AC = BD. Consequently, this circle will be the. 
envelop of all these semicircles. The reflected wave-front, 
therefore, is obtained by revolving the arc DCF around LL' 
as axis. The straight line QR is the path of the reflected 
ray corresponding to the incident ray PQ; the angle of in- 
cidence at Q is equal to the angle ALQ and the angle of re- 
flection is equal to AI/Q, and these angles are evidently 
equal, in agreement, therefore, with the law of reflection. 
15. Image in a Plane Mirror. — In Fig. 19 the plane mirror 
bisects at right angles the straight line LL', and since the 




Fig. 20. — L' is image of object-point L in plane mirror AD; 

AL = Im- 
position of the point L' is independent of the position of 
the incidence-point Q (Fig. 20), all the rays coming from 
the luminous point L and falling on the plane mirror will 
be reflected along paths which, when prolonged backwards, 



38 Mirrors, Prisms and Lenses [§ 15 

all meet in the point L'. Thus, to a homocentric bundle of 
incident rays reflected at a plane mirror there corresponds also 
a homocentric bundle of reflected rays. This remarkable 
property of converting a homocentric bundle of rays into 
another homocentric bundle is characteristic of a plane 
mirror, because no other optical device is capable of it. ex- 
cept under conditions that are more or less unrealizable in 
practice. Thus, the image 1/ of an object at L is found by 
drawing a straight line from L perpendicular to the plane 
mirror, and producing this line on the other side of the 
mirror to a point L' such that the line-segment LI/ is bi- 
sected by the plane of the mirror; so that an object in front 
of a plane mirror is seen in the mirror at the same distance 
behind it. The image in this case is virtual (§ 8). The late 
Professor Silvanus Thompson in his popular lectures 
published under the title Light Visible and Invisible de- 
scribes the following simple method of showing how the 
rays from a candle flame are reflected at a plane mirror 
(Fig. 21). If a vertical pin mounted on a horizontal base- 
board is illuminated by a lighted candle, the position of 
the shadow is determined by the line joining the top of the 
pin with the source of light. If the pin and the candle are 
both in front of a plane mirror placed at right angles to the 
base-board, a second shadow will be cast by the pin on ac- 
count of the reflected rays from the candle that are inter- 
cepted by it, and this shadow will be precisely such as would 
be produced by a candle flame placed behind the mirror 
at the place where the image of the actual flame is formed, 
as may be proved by removing the mirror and transferring 
the candle to the place where its image was. 

If the bundle of incident rays instead of diverging from 
a point L in front of the plane mirror converged towards 
a point L behind it (as could easily be effected with the aid 
of a convergent lens), a real image (§ 8) will be produced at 
a point L' at the same distance in front of the mirror as the 
virtual object-point L was beyond it. 




Fig. 21. — Shadows cast by an object in front of a plane mirror when object 
is illuminated by point-source (from actual photograph), showing that 
the source and its image are at equal distances from the mirror. 



15] 



Image in Plane Mirror 



39 



The image of an extended object is the figure formed by 
the images of all of its points separately. The diagram 
(Fig. 22) shows, for example, how an eye at E would see 
the image L'M' of an object LM reflected in a plane mirror. 
The series of parallel lines joining corresponding points of 




Fig. 22. — Image L'M' of object LM in plane mirror ZZ. 

object and image will be bisected at right angles by the 
plane of the mirror. 

The dimensions of the image in a plane mirror are ex- 
actly the same as those of the object. Moreover, the top 
and bottom of the image correspond with the top and bot- 
tom of the object, that is, the image is erect. Also, the 
right side of the image corresponds with the right side of 
the object, and the left side of the image with the left side 
of the object (Fig. 23), although it is frequently stated in 
books on optics that when a man stands in front of a mirror 
the right side of the image shows the left side of the person, 
and that if the man extends his right hand, the image will 
extend its left hand. The true explanation of the so-called 
"perversion" of the image in a plane mirror, which is strik- 
ingly seen when a printed page is held in front of the mirror, 
is that it is the rear side of the image that is opposite the front 



40 Mirrors, Prisms and Lenses [§ 16 

side of the object. The image of a printed page in a mirror 
has exactly the same appearance as it would have if the 
page were held in front of a bright light and it was viewed 
from behind through the paper. When a person looks in 
a mirror at his own image, his image appears to be looking 
back at him in the opposite direction, if he faces east, his 
image faces west, and if we call the east side of object or 
image its front side and the west side its rear side, then the 
rear side of the image is turned towards the front side of 
the object; although, because this side of the image cor- 
responds to the front side of the object, it is a natural mis- 
take to regard it as also the front side of the image. The 
explanation of the common impression that, whereas up 
and down remain unchanged in the image of an object in 
a plane mirror, right and left are reversed, is probably be- 
cause a person regarding his own image under such circum- 
stances is unconsciously disposed to transfer himself men- 
tally into coincidence with his image by a rotation of 180°, 
not around a horizontal, but around a vertical axis, thus 
producing a confusion of mind as to right and left but not 
as to top and bottom. The reason why this mental revolu- 
tion is performed around the vertical axis seems to be due 
partly to the circumstance that this movement can be 
readily executed in reality, and partly also perhaps to the 
fact that the human body happens to be very nearly 
symmetrical with respect to a vertical plane. 

16. The Field of View of a Plane Mirror. — In the adjoin- 
ing diagram (Fig. 24) the straight line GH represents the 
trace in the plane of the paper of the surface of a plane mir- 
ror, and the point marked O' shows the position of the center 
of the pupil of the eye of a person who is supposed to be 
looking towards the mirror. Evidently, the straight lines 
HO', GO' drawn to 0' from the points G, H in the edge of 
the mirror will represent the paths of the outermost reflected 
rays that can enter the eye at 0', and therefore the field of 
view (§ 9) is limited by the contour of the mirror just 



Fig. 23. — Image of object in plane mirror (from actual photograph). 



16] 



Field of View of Plane Mirror 



41 



as if the observer were looking into the image-space through 
a hole in the wall that exactly coincided with the place oc- 
cupied by the mirror. Corresponding to the pair of re- 
flected rays HO' and GO' intersecting at O' there would be 
a pair of incident rays directed along the straight lines HO 




: &** 



Fig. 24. — Field of view of plane mirror for given position of eye. 

and GO towards a point on the other side of the mirror, 
and it is evident that 0' will be the real image of a virtual 
object-point at O (§ 15). Any luminous point lying in front 
of the plane mirror within the conical surface formed by 
drawing straight lines such as OG, OH from to all the 
points in the edge of the mirror will be visible by reflected 
light to an eye placed at 0', and hence this cone limits the 
field of view of the object-space. 

Through O' draw a straight line parallel to GH, and take 
on it two points C', B' at equal distances from O' on oppo- 
site sides of it, and let us suppose that B'C represents the 
diameter in the plane of the diagram of the pupil of the eye. 
Construct the image BOC of the eye-pupil B'O'C. Then 
if P designates the position of a luminous point lying any- 



42 



Mirrors, Prisms and Lenses 



[§16 



where within the field of view of the object-space, it is clear 
that the incident rays PO, PC and PB will be reflected at 




Fig. 25. — Deviation of a ray reflected twice in suc- 
cession from a pair of inclined mirrors. 

the. mirror into the pupil of the eye in the directions P'O', 
P'C and P'B', as though they had all come from the point 
P' which is the image of P. This imaginary opening or vir- 




Fiq. 26. — Deviation of a ray reflected twice in suc- 
cession from a pair of inclined mirrors. 

tual stop BOC towards which the incident rays must all be 
directed in order to be reflected into the eye-pupil B'O'C is 



§ 18] Inclined Mirrors 43 

called the entrance-pupil of the optical system consisting 
of the plane mirror and the eye of the observer; and the 
pupil of the eye itself is called here the exit-pupil (see Chap- 
ter XII). Since the entrance-pupil limits the apertures of 
the bundles of rays that ultimately enter the eye, it acts 
as the aperture-stop of the system (§ 11). 

17. Successive Reflections from two Plane Mirrors. — 
Any section made by a plane perpendicular to the line of in- 
tersection of the planes of a pair of inclined mirrors is called 
a principal section of the system. If a ray lying in a prin- 
cipal section is reflected successively at two plane mirrors } it 
will be deviated from its original direction by an angle equal to 
twice the dihedral angle between the mirrors. 

Let the plane of the principal section intersect the planes 
of the mirrors in the straight lines OM, ON (Figs. 25 and 
26) ; and let 7 = Z MON denote the angle between the mir- 
rors. The ray PQ lying in the plane MON is incident on 
the mirror OM at the point Q, whence it is reflected along 
the straight line QR, meeting the mirror ON at the point 
R, where it is again reflected, proceeding in the direction 
RS. Let the point of intersection of the straight lines PQ 
and RS be designated by K. Then Z PKR is the angle be- 
tween the original direction of the ray and its direction after 
undergoing two reflections, and we must show that this 
angle is equal to 2 7. 

Draw the incidence-normals at Q and R, and prolong 
them until they meet at J. Then by the law of reflection 
the straight lines QJ and RJ bisect the angles PQR and 
QRS, respectively. 
In Fig. 25, ZPKR=ZPQR+ZQRS = 2(ZJQR+ZQRJ) 

= 2(180° -ZRJQ) = 27; 
and in Fig. 26, Z PKR = Z PQR - Z SRQ 

= 2(180°-ZRQJ-ZJRQ) 

= 2ZQJR-27. 

18. Images in a System of Two Inclined Mirrors. — When 
a luminous point lies in the dihedral angle between two 



44 



Mirrors, Prisms and Lenses 



[§18 



plane mirrors, some of its rays will fall on one mirror and 
some on the other, and consequently there will be two sets 
of images. In Fig. 27 the plane of the diagram is the prin- 
cipal section which contains the point-source S, and the 
straight lines OM, ON represent the traces of the mirrors 




Fig. 27. — Images of a luminous point S in a pair of inclined 
mirrors OM and ON. 

in this plane. The rays which fall first on the mirror OM 
will be reflected as though they came from the image P x of 
the luminous point S in this mirror. Some of these rays 
falling on the mirror ON will be again reflected and proceed 
thence as though they came from the point P 2 which is 
the image of P x in the mirror ON. Thus, by successive re- 
flections, first at one of the mirrors and then at the other, 
a series of images Pi, P 2 , etc., will be formed by those rays 
which fall first on the mirror OM ; let us call this the P-series 
of images. Similarly, the rays that fall first on the mirror 
ON will produce another series of images Q x , Q 2 , etc., which 
will be called the Q-series. Each of these series will termi- 
nate with an image which lies behind both mirrors in the 



§ 18] Images in Inclined Mirrors 45 

dihedral angle COD opposite the angle MON between the 
mirrors themselves; because rays which, after reflection 
at one of the mirrors, appear to come from a point thus sit- 
uated cannot fall on the other mirror, and so there will be 
no more images after this one. 

Since the straight line OM is the perpendicular bisector 
of the line-segment SP X , the points S and Px are equidistant 
from every point in the straight line OM; and, similarly, 
since P 2 is the image of Pi in the plane mirror ON, these two 
points are likewise equidistant from every point in the 
straight line ON. Accordingly, the three points S, P^ P 2 
are all equidistant from the point where the straight lines 
OM and ON intersect. Applying the same reasoning to 
all the other images, we perceive that the images of both 
series are ranged on the circumference of a circle whose center 
is at and whose radius is OS. 

In the following discussion of the angular distances of 
the images from the luminous point S, the angles will be 
reckoned always in the same sense, either all clockwise or 
all counter-clockwise. Let y = Z AOB denote the angle be- 
tween the two mirrors, the letters A and B referring to the 
points where the circle crosses the planes of the mirrors OM 
and ON, respectively. Also, let a=Z AOS, j8 =ZSOB de- 
note the angular distances of S from A, B, respectively, so 
that a +/3=y. Then 

ZPiOS = 2a; 

ZSOP 2 = ZSOB+ZBOP 2 =0+ZPiOB = 2(a+/3) = 2Y; 

Z P 3 OS = Z P3OA+ a=Z AOP2+ a = Z SOP 2 + 2 a 
= 27+2a; 

Z SOP 4 = Z SOB+ Z BOP4 = P+ Z P3OB 

= 2/3+ZP 3 OS = 2(a+/3+7)=47; 

Z P5OS = Z P5OA+ a = Z AOP4+ a = Z SOP 4 + 2 a 
= 47+2a. 
In general, therefore, 

ZSOP 2k = 2/c7, ZP 2k+1 OS = 2/b7+2a, 

where P 2k , P 2k +i designate the positions of the 2fcth and 



46 Mirrors, Prisms and Lenses [§ 18 

(2fc+l)th images of the P-series, k denoting any integer, 
and where the angles SOP 2k , P2k+iOS are the angles sub- 
tended by the arcs SBP 2k , P 2k+ iAS, respectively. Similarly, 
for the Q-series of images we find : 

ZQ 2k 0S = 2A;7, Z80Q 2k+1 = 2ky+2p, 

where these angles are the angles subtended by the arcs 
Q 2k AS, SBQ 2k+1 , respectively. 

Evidently, the image P 2k+i will fall on the arc CD be- 
hind both mirrors, if arc P 2k+1 AS>arc DAS, that is, if 

2&Y+2 a > 180° -0; 
and, by adding (/3-a) to both sides of this inequation, 
and dividing through by y, this condition may be expressed 
as follows: 

, , ' 180°-a 

In the same way we find that the image P2 k will fall between 
C and D if 

7 180° -a 
2k>—j—' 

Thus, the total number of images of the "P-series t whether it 
be odd or even, will be given by the integer next higher than 
(180°— a)/ y; and, similarly, the total number of images 
of the Q-series will be given by the integer next higher than 
(180°- PHy. 

The only exception to this rule is when the angle y is 
contained in (180° — a) or (180° — /3) an exact whole num- 
ber of times; in the former case the last image of the P-series 
falls at C, and in the latter case the last image of the Q-series 
falls at D; and instead of taking the integers next above 
the quotient (180°— a)/y or (180°— fi)/y, we must take 
the actual integer obtained by the division. An example 
will make the matter clear. Thus, suppose 7 = 27°, a = 8°, 
then /?= 19°, and the integers next higher than (180° — a)jy 
and (180° — f})/y will be 7 and 6, respectively; hence in 
this case there will be 7 images of the P-series and 6 images 
o the Q-series or 13 images in all. But if a = 10° and fi = 17°, 



§ 18] Kaleidoscope 47 

each series will be found to have 7 images, 14 images in 
all. The exceptional case occurs when a = 9° and /3 = 18°, 
for then (180°— /3)/y = 6, and hence there will be 7 P-images 
and 6 Q-images. 

If the angle y between the mirrors is an exact multiple 
of 180°, that is, if 180°/ y = p, where p denotes an integer, 
the integers next higher than (180°- a)/ J and (180°- P)/y 
will both be equal to p, 
no matter what may be 
the special position of 
the object between the 
two mirrors; so that in 
such a case the number 
of images in each series 
will be equal, but the last 
image of one set will co- 
incide with the last of 
the other. In fact, the 
points S, P 2 , P 4 , . . . 

Q47 Q2 an d the points Fig. 28.— Images of a luminous point in a 
p p Q Q ar p pair of plane mirrors inclined to each 

1, 3, • • • v*3> ^1 other at an angle of 60°. 

the vertices 01 two equal 

regular polygons, of p sides each; and if p is odd, the polygon 
P1P3 . . . Q3Q1 will have one of its corners between C and 
D, whereas if p is even, one of the corners of the polygon 
SP 2 P 4 . . . Q4Q2 will fall between C and D; in either case 
this vertex is the position of the last image of both series. 
Thus, for example, if 7 = 60° (Fig. 28), then p = 3, and the 
two polygons are the equilateral triangles SP2Q2 and P1P3Q1 
(orPAQx). 

The toy called a kaleidoscope, devised by Sir David 
Brewster (1781-1868), consists essentially of two long nar- 
row strips of mirror-glass inclined to each other at an angle 
of 60° and inclosed in a cylindrical tube. One end of the 
tube is closed by a circular piece of ground glass whereon 
are loosely disposed a lot of fragments of colored glass or 




48 



Mirrors, Prisms and Lenses 



[§19 



beads, and at the other end of the tube there is a peep-hole. 
When the instrument is held towards the light, an observer 
looking in it will see an exquisitely beautiful and symmetrical 
pattern formed by the colored objects and their images, the 
form of which may be almost endlessly varied by revolving 
the tube around its axis so that the bits of glass assume new 




Fig. 29. — Path of ray reflected into eye from a pair of inclined 
mirrors. 

positions. In fact, this device has been turned to practical 
use in making designs for carpets and wall-papers. 

19. Construction of the Path of a Ray Reflected into the 
Eye from a Pair of Inclined Mirrors. — In order to trace the 
paths of the rays by which a spectator standing in front of a 
pair of inclined mirrors sees the image of a luminous point, 
it is convenient to assume, for the sake of simplicity, that 
the eye at E in Fig. 29 lies in the plane of the paper. The 
first step in the construction of the path of the ray is to draw 
the straight line from the given image-point to the eye, 
because if the eye sees this point the light that enters the 



19] 



Inclined Mirrors 



49 



eye must arrive along this line. If this line does not cross 
the mirror in which the image is produced, this particular 
image will not be visible from the point E. Now join the 
point where this line meets the mirror with the preceding 
image in the same series; the part of this line that lies be- 
tween the two mirrors will evidently show the route of the 




Fig. 30. — Showing how an eye at E sees the images of a lumi- 
nous point S in a rectangular pair of plane mirrors. 

light before its last reflection. Proceeding in this fashion 
from one mirror to the other, we shall trace backwards the 
zigzag path of the ray until we arrive finally at the luminous 
source at S. Consider, for example, the image P 3 formed 
in the mirror OA in Fig. 29. This image is visible to the 
eye at E because the straight line P 3 E cuts at K the mirror 
OA. If J and H designate the points where the straight 
lines KP 2 and JPi meet the mirrors OB and OA, respectively, 
the broken line SHJKE will represent the path of the ray 
from the source at S into the eye at E. Fig. 30 shows how 



50 Mirrors, Prisms and Lenses [§ 20 

an eye at E in front of two perpendicular plane mirrors can 
see the images Pi, P 2 , Qi and Q 2 . 

20. Rectangular Combinations of Plane Mirrors. — In a 

rectangular combination of two plane mirrors (7 = 90°) the 
image formed by two successive reflections will be in- 
verted in the principal section of the system, but in any 
plane at right angles to a principal section the image will 
be erect. For example, if an object is placed in front of two 
vertical plane mirrors at right angles to each other, the 
image produced by two reflections will have the same posi- 
tion and appearance as if the object had been revolved bodily 
through an angle of 180° about a vertical axis coinciding 
with the line of intersection of the planes of the mirrors, as 
represented in Fig. 31. In this case the image remains ver- 
tically erect, whereas it is horizontally inverted. On the 
other hand, if one of the mirrors is vertical and the other 
horizontal, the image by twofold reflection will have the 
same position and appearance as if the object had been 
revolved through 180° around a horizontal axis coinciding 
with the line of intersection of the two mirrors (Fig. 32); 
that is, the image now will be upside down but not inverted 
horizontally. 

Therefore, in order to obtain an image that is completely 
reversed in every respect, two rectangular combinations of 
plane mirrors may be employed with their principal sections 
mutually at right angles, so disposed that the rays coming 
from the object will be reflected in succession from each of 
the four plane surfaces. An auxiliary system of this descrip- 
tion is sometimes used in connection with an optical instru- 
ment for the purpose of rectifying the image which otherwise 
would be seen inverted. A rectifying device depending on this 
principle is the so-called Porro prism-system (1852), utilized 
by Abbe (1840-1905) in the design of the famous prism binocu- 
lar telescope or field-glasses (c. 1883) . A sketch of the arrange- 
ment is shown in Fig. 33. Two rectangular prisms are placed 
in the tube of the instrument, between the objective and the 




Fig. 32. — Image of an object in a rectangular pair of plane mirrors (from 
actual photograph) ; showing how the last image is obtained by rotating 
the object through 180* around the line of intersection of the mirrors. 
One mirror vertical, the other horizontal. 



20] 



Porro Prism System 



51 




ocular, with their principal sections at right angles to each 
other. The axial ray, after traversing the objective, crosses 
normally the hypothenuse-face of the first prism and is 
totally reflected (see § 36), in the plane of a principal section, 
at each of its two per- 
pendicular faces so as to 
emerge from the hypoth- 
enuse-face in a direction 
precisely opposite to that 
which it had when it first 
crossed this surface. This 
ray now undergoes a simi- 
lar cycle of experiences in 
a principal section of the 
second prism, and finally 
emerges from this prism 

in the Same direction as FlG - 33.— Porro prism-system in prism 
• , i j i ., , .i binocular field glasses. 

it had when it met the 

first prism. A ray parallel to the axial ray and lying above 
a horizontal plane containing the axis will be converted by 
virtue of the two reflections in the first prism into a ray 
whose path lies below this plane; and, similarly, a ray par- 
allel to the axis and lying on one side of a vertical plane 
containing the axis will, in consequence of the two reflec- 
tions within the second prism, be converted into a ray whose 
path lies on the opposite side of this vertical plane. Thus, 
the combined effect of the two reflecting prisms together 
will be to reverse completely the position of the ray with 
respect to the horizontal and vertical meridian planes, so 
that the ray will issue from the system on opposite sides of 
both these planes. If the system of prisms were removed, 
the image in the instrument would appear inverted, but by 
interposing the prisms in this fashion the image will be 
rectified and oriented exactly in the same way as the object; 
which in the case of many optical instruments is an essential 
consideration. 



52 



Mirrors, Prisms and Lenses 



[§21 



21. Applications of the Plane Mirror. — It is hardly- 
necessary to say that the plane mirror for various pur- 
poses has been in use among civilized peoples of all ages; 
although the use of mirrors as articles of household fur- 
niture and decoration does not go back farther than the 
early part of the 16th century. By a combination of two 
or more plane mirrors a lady can arrange the back of 




Fig. 34. — Porte lumi&re. 



her dress and in fact see herself as others see her. With 
the aid of a mirror or combination of mirrors many in- 
genious " magical effects" are produced in theaters. The 
plane mirror also constitutes an essential part of numerous 
useful scientific instruments in some of which its only duty 
is to alter the course of a beam of light, whereas in various 
forms of goniometrical instruments and contrivances for de- 
termining an angular magnitude that is not easily measured 



§ 22] Porte Lumiere 53 

directly the angle in question is ascertained indirectly by 
observing the angle turned through by a ray of light which 
is reflected from a plane mirror. 

22. Porte Lumiere and Heliostat. — As good an illustra- 
tion as can be given of the use of a plane mirror for chang- 
ing the direction of a beam of sunlight is afforded by the 




Fig. 35. — Heliostat. 

porte lumiere (Fig. 34), which consists essentially of a plane 
mirror ingeniously mounted so as to be capable of rotation 
about two rectangular axes, whereby it may be readily ad- 
justed in any desired azimuth and reflect a beam of sun- 
light through a suitable opening in the wall of the building 
to any part of the interior of the room. 

However, owing to the diurnal movement of the sun, 



54 Mirrors, Prisms and Lenses [§ 22 

a continual adjustment of the mirror is necessary in order 
to keep the spot of light for any length of time at the place 
in the room where it is needed, and sometimes this manipu- 
lation is very inconvenient and annoying, especially in the 
case of a laboratory experiment extending perhaps over 
a considerable part of a day. Thus, for example, in study- 
ing the solar spectrum it is often desirable to illuminate the 
slit in the collimator tube of the spectrometer for hours at 
a time. For such purposes it is better to use a heliostat 
(Fig. 35), which is contrived so that the plane mirror is con- 
tinuously revolved by clockwork around an axis parallel 
to the earth's axis so as to preserve always the same relative 
position with respect to the sun in its apparent diurnal 
motion in the sky. The mirror can also be turned about 

a horizontal axis, and it has first 
to be adjusted about this axis so 
that the rays of the sun are re- 
flected towards the north pole 
of the celestial sphere, that is, 
parallel to the axis of the earth. 
The mirror being adjusted at 
this angle, which will depend on 
the declination of the sun above 
Fig. 36.-Princi P le of heliostat. or below^ the celestial equator, 

and turning at the rate of 15° 
per hour around an axis parallel to the axis of rotation of 
the earth, it is evident that the rays of the sun will continue 
to be reflected constantly in the same direction. Suppose, 
for example, that the mirror is adjusted in the position 
ZZ (Fig. 36) so that the ray SB coming from the sun at S is 
reflected at B in the direction BP parallel to the axis of the 
earth and therefore parallel to the axis of rotation AB of the 
mirror. If the polar distance of the sun is denoted by 
2a = ZPBS, and if the angle between the normal to the 
mirror and the axis of rotation is denoted by r), then, evi- 
dently, rj = a. If the sun's declination on a certain day 




§ 23] Measurement of Angle of Prism 55 

is +10°, then 2 a = 90°- 10° = 80°, and 77 = 40°. If, on the 
other hand, the sun is 10° below the equator, 2a=100° and 
77 = 50°. 

The heliostat is provided also with a fixed mirror which 
reflects the rays from the rotating mirror in a definite di- 
rection, as desired, usually in a horizontal direction into 
the room where the sunlight is to be used. Generally, the 
instrument is mounted on a permanent ledge outside the 
window; sometimes it is placed on the roof of the building 
and the fixed mirror adjusted so as to send the sun's rays 
down a vertical tube at 
the bottom of which there 
is another mirror placed 

at an angle of 45° with S^~ — ' ^^^C^ 

the vertical where the 
rays are once more re- 
flected so that the beam 
of sunlight which enters 
the room will be hori- 
zontal. 

23. Measurement of 
the Angle of a Prism. — 
Another laboratory ap- 
plication of the principle 
of a plane mirror is seen 
in the method of using a 

goniometer to ascertain FlG - 37-Measurement of angle of prism. 

the dihedral angle between two plane faces of a glass prism 
(§ 48) . The angle that is actually measured by the goniom- 
eter is the angular distance between the images of a distant 
object as seen in the two faces of the prism. Parallel rays 
coming from a far-off source at S (Fig. 37) and incident on 
the two faces of the prism that meet in the edge V are re- 
flected as shown in the diagram, and the angle between the 
two directions of the reflected rays is obviously equal to 
twice the dihedral angle /3. 




56 



Mirrors, Prisms and Lenses 



[§24 



24. Measure of Angular Deflections by Mirror and 
Scale. — The angular rotation of a body, for example, the 
deflection of the magnetic needle of a galvanometer, is fre- 
quently measured by attaching a mirror to the rotating 
body from which a beam of light is reflected. This reflected 
light acts as a long weightless pointer whereby the actual 




Fig. 38. — Mirror, telescope and scale for measure- 
ment of angles. 

movement of the body can be magnified to any extent with- 
out in the least affecting the sensitiveness of the apparatus. 

In Fig. 38 the plane mirror which is capable of rotation 
about an axis perpendicular at A to the plane of the paper 
is represented in its initial position by the line-segment 
marked 1. The straight line MN in front of the mirror and 
at a known distance (c? = AB) from it represents a scale 
graduated in equal divisions. An eye at E looking through 
a telescope pointed towards A will see the image in the mirror 
of the scale-division at S, the so-called " zero-reading," be- 
cause the light from S incident at A on the mirror in the 
position 1 (" equilibrium-position") is reflected along AE 
into the eye at E. If now the mirror is turned through an 
angle 6 into the position marked 2, another scale-division 
will come into the field of view of the telescope and coin- 



§ 24j Mirror and Scale 57 

cide with the cross-hair in the eye-piece. If this scale- 
division corresponds to the point marked P, it is the light 
that comes along PA that is now reflected along AE into 
the eye at E; and evidently, according to § 13, Z PAS = 2 0. 
In making a measurement by this method, the three points 
designated by S, B and E are generally adjusted so as to 
be very near together, if not actually coincident. If they 
were coincident, the planes of the mirror and scale would 
be parallel, and the axis of the telescope would coincide 
with the straight line BA perpendicular to the scale at B. 
But in any case the Z B AS = e will be a constant, depending 
partly on the initial position of the mirror and partly on 
the direction of the axis of the telescope; thus, 

tan € = a/d, 
where a = BS. If, therefore, we put x = SP, we have: 

x 

-j = tan ( e+ 2 6)— tan e ; 

whence, since the value of x can be read off on the scale, it 
will be easy to calculate the value of the required angle 6 
through which the mirror has been turned. In many cases 
where this method is employed the angles denoted by 6 and 
€ are both so small that there will be little error in sub- 
stituting the angles themselves in place of their tangents. 
Under these circumstances the above formula will be greatly 
simplified, for the angle e will disappear entirely, and we 
shall obtain : 

x 

e= Td> 

where, however, it must be noted that this expression gives 
the value of the angle 6 in radians. The value of 6 in de- 
grees is found by multiplying the right-hand side of this 
formula by 180/ 7T, so that we obtain: 

u = — j degrees. 
Tr.a 

A lamp and scale is sometimes used instead of a telescope 

and scale, the light of the lamp being reflected from the 



58 



Mirrors, Prisms and Lenses 



[§25 



mirror on to the scale which is usually made of translucent 
glass, so that it is easy to read the position of the spot of 
light. 

25. Hadley's Sextant. — Another instrument which utilizes 
the principle of § 17 is the sextant, which is employed for 




Fig. 39. — Principle of sextant. 

measuring the angular distance between two bodies, for ex- 
ample, the altitude of the sun above the sea-horizon. The 
plan and essential features of this apparatus are shown in 
Fig. 39. At the center A of a graduated circular arc ON 
a small mirror is set up in a plane at right angles to that of 
the arc. This mirror can be turned about an axis perpendic- 
ular to the plane of the paper and passing through A. Rig- 
idly connected to this mirror and turning with it is a long 
solid arm AP whose other end P, provided with a vernier 
scale, moves over the arc ON, whereby the angle through 
which the mirror turns can be accurately measured. A little 
beyond the extremity N of the graduated part of the arc, 
a second mirror B is erected facing the first mirror. The 
plane of this mirror is likewise perpendicular to that of the 
circle, but from the upper half of it the silver has been 
removed, so that this portion of the mirror B is transparent. 
Moreover, this mirror is fixed with respect to the instru- 
ment. An observer looking through a peep-hole or tele- 



§ 25] Mirror Sextant 59 

scope attached to the instrument towards the mirror B may- 
see a distant object through the upper transparent part of 
this mirror, and at the same time he may also see just below 
it the image of a second object reflected in the lower half 
of the glass. When the planes of the two mirrors A and B 
are parallel, the zero-mark of the vernier on the movable 
arm coincides with the zero-mark of the graduated arc. 
Suppose, for example, that when the two mirrors are par- 
allel to each other, the instrument is pointed at a distant 




Fig. 40. — Model of mirror sextant. 

object, say, a star at Si, which will be seen directly through 
the upper half of the fixed mirror B. At the same time the 
observer will see an image of the object Si by rays which 
have been reflected from the mirror A to the mirror B and 
thence into the eye at E; for if the two mirrors are parallel, 
the direction of a ray after two reflections will be the same 
as its initial direction. If now the mirror A is turned until 
the image of another object at S 2 comes into the field of 
vision, the two objects Si and S 2 will be seen simultaneously, 
for with the mirror at A in this new position the incident ray 
S 2 A will be the ray that is reflected from A to B and thence, 
as before, into the eye at E. Moreover, since the angle between 
the original direction S 2 A of this ray and its final direction 



60 Mirrors, Prisms and Lenses [Ch. II 

S X A is equal to double the angle between the planes of the 
two mirrors, that is, is equal to 2 6, where 8 = Z OAP, the 
angular distance between the objects at Si and S 2 must be 
equal to 2 6, that is, Z SiAS 2 = 2 0. In order to save trouble 
in making the readings, half-degrees on the graduated arc 
are reckoned as degrees, so that the value of the angle 2 6 is 
read directly on the scale. As the angular distance between 
the objects will seldom exceed 120°, and since, in fact, the 
method is not very accurate for angles greater than this, 
the actual length of the graduated arc need not be greater 
than about 60° or one-sixth of the circumference; whence 
the name sextant is derived. 

A simple model of a mirror sextant is shown in Fig. 40. 
For accurate measurements the instrument is made of metal 
with a scale etched on a silver strip. Moreover, a telescope 
is used instead of a peep-hole; so that with a fine sextant it 
is comparatively easy to measure the angular distance be- 
tween two points to within one-half minute of arc. One 
great advantage of this instrument is its portability, and 
since it does not have to be mounted on a stand, it is very 
serviceable on shipboard for measurements of altitude and 
determinations of latitude, etc. 

PROBLEMS 

1. The top of a vertical plane mirror 2 feet high is 4 feet 
from the floor. The eye of a person standing in front of the 
mirror is 6 feet from the floor and 3 feet from the mirror. 
What are the distances from the wall on which the mirror 
hangs of the farthest and nearest points on the floor that 
are visible in the mirror? Ans. 6 ft. ; 18 in. 

2. A ray of light is reflected at a plane mirror. Show that 
if the mirror is turned through an angle 6, the reflected ray 
will be turned through an angle 2 0. 

3. Show that the deviation of a ray reflected once at each 
of two plane mirrors is equal to twice the angle between the 
mirrors. 



Ch. II] Problems 61 

4. If a plane mirror is turned through an angle of 5°, 
what is the deflection indicated by the reading on a straight 
scale 100 cm. from the mirror? Ans. About 17.6 cm. 

5. Find the angle turned through by the mirror when the 
deflection on the scale in the preceding example is 10 cm.? 

Ans. 2° 52'. 

6. What must be the length of a vertical plane mirror in 
order that a man standing in front of it may see a full length 
image of himself? Ans. The length of the mirror must be 
equal to half the height of the man. 

7. Show that a plane mirror bisects at right angles the 
line joining an object-point with its image. 

8. A ray of light proceeding from a point A is reflected 
from a plane mirror to a point B. Show that the path pur- 
sued by the light is shorter than any other path from A to 
the mirror and thence to B. 

9. Give Huygens's construction, (1) for the reflection 
of a p'ane wave at a plane mirror, and (2) for the reflection 
of a spherical wave at a plane mirror. 

10. Explain clearly how to determine the limits of the 
field of view in a plane mirror for a given position of the eye 
of the spectator. 

11. A candle is placed between two parallel plane mirrors. 
Show how an observer can see the image of the candle pro- 
duced by rays which have been twice reflected at one mirror 
and three times at the other. Draw accurate diagram show- 
ing the paths of the rays, the positions of the images, etc.; 
and give clear explanation of the figure. 

12. OA and OB are two plane mirrors inclined at an angle 
of 15°, and P is a point in OA. At what angle must a ray 
of light from P be incident on OB in order that after three 
reflections it may be parallel to OA? Ans. 45°. 

13. Show that the image of a luminous object placed 
between two plane mirrors all He on a circle. 

14. Show how by means of two plane mirrors a man 
standing in front of one of them can see the image of the back 



62 Mirrors, Prisms and Lenses [Ch. II 

of his head. Trace the course of the rays from the back of 
his head into his eye and explain clearly. 

15. Show by a diagram, with clear explanations, how 
one sees the image of an arrow in a plane mirror. 

16. Construct the image of an arrow formed by two re- 
flections in a pair of inclined mirrors, (1) when the mirrors 
are at right angles, and (2) when the angle between the 
mirrors is 60°. 

17. Show how a horizontal shadow of a vertical rod can 
be thrown on a vertical screen by a point-source of light with 
the aid of a plane mirror. Draw a diagram. 

18. An object is placed between two plane mirrors in- 
clined at an angle of 45°. Show by a figure how a spectator 
may see the image after four successive reflections. Give 
clear explanation. 

19. Two plane mirrors are inclined at an angle of 50°. 
Show that there will be 7 or 8 images of a luminous point 
placed between them, according as its angular distance from 
the nearer mirror is or is not less than 20°. 

20. Find the number of images formed when a bright point 
is placed between two plane mirrors inclined to each other 
at an angle of 25°. Ans. 14 or 15 images according as the 
angular distance of the luminous point from the nearer mir- 
ror is or is not less than 5°. 

21. A luminous object moves about between two plane 
mirrors, which are inclined at an angle of 27°. Prove that 
at any moment the number of images is 13 or 14 according 
as the angular distance of the luminous point from the nearer 
mirror is or is not less than 9°. 

22. The angle between a pair of inclined mirrors is 80°. 
Find the position of an object which is reproduced by 5 
images. Ans. The object must be less than 20° from the 
nearer mirror. 

23. Describe a sextant with the aid of a diagram, and ex- 
plain its use. 

24. Describe and explain the heliostat. 



Ch. II] Problems 63 

25. Construct the image of the capital letter F as seen in 
a plane mirror. 

26. When a candle-flame is placed in front of a screen 
with a pin-hole opening, an image of the flame is formed on 
a second screen placed parallel to the first. But if the second 
screen is replaced by a plane mirror, the image will be formed 
on the back of the first screen. Explain how this happens. 

27. Explain clearly (with diagram, formula, etc.) the 
method of using a mirror and scale for measurement of 
angles. 

28. Describe how the dihedral angle of a glass prism is 
measured on a goniometer-circle. 



CHAPTER III 



REFRACTION OF LIGHT 




Fig. 41. — Coin at bottom of bowl rendered 
visible by filling bowl with water. 



26. Passage of Light from One Medium to Another. — 

Hardly any one can have failed to observe that the course of 
light in passing obliquely from water to air is abruptly changed 
at the surface of the water. For example, if a coin is placed 
at A in the bottom of a china bowl (Fig. 41), and if the eye 

is adjusted at a point C 
so that the coin is hid 
from view by the side of 
the vessel, then, without 
altering the position of 
the eye, the coin can be 
made visible merely by 
pouring water in the 
bowl up to a certain level. 
The broken line ACB illustrates how a ray proceeding from 
A may be bent at the surface of the water so as to pass over 
the edge of the bowl and enter the eye at C. It is true the 
coin will will not appear to be at A but at a point A' nearer 
the surface of the water and displaced a little sideways to- 
wards the eye, because the rays that come to the eye inter- 
sect at this point A' (§ 42). A clear pool of water seems to 
be shallower than it really is, and this illusion is greater in 
proportion as the line of sight is more oblique, so that bright 
objects at the bottom of the pool appear to be crowded to- 
gether towards the surface. When a stick is partly immersed 
in water, the part under water appears to be bent up to- 
wards the surface (§ 42). 

This bending of the rays which takes place when light 
crosses the boundary between two media is called refraction. 

64 




Fig. 42. — Law of Refraction. 



§ 27] Law of Refraction 65 

The path of a beam of sunlight through water can easily 
be shown by mixing a little milk in the water or by stir- 
ring in it a minute quantity of chalk-dust, while a puff of 
smoke will at once reveal the track of the beam in the air, 
so that the phenomena of 
refraction can readily be 
exhibited to the eye. In 
every case it will be found 
that the ray is bent farther 
from the incidence-normal 
in the rarer or less dense 
medium (see § 30) ; and 
here also, as in the case 
of reflection, there is a 
perfectly definite connec- 
tion between the direction of the incident ray and that of 
the corresponding refracted ray. 

27. Law of Refraction. — In Fig. 42 the straight line AB 
represents the path of a ray incident at the point B on a 
smooth refracting surface separating two media which for 
the present will be designated by the letters a and b. The 
straight line NN' drawn perpendicular to the plane which 
is tangent to the refracting surface at B represents the 
incidence-normal; and the plane of the paper which con- 
tains the incident ray and the incidence-normal is the plane 
of incidence, as already defined (§ 13). The line ZZ repre- 
sents the trace of the refracting surface in this plane. And, 
finally, the path of the refracted ray is shown by the straight 
line BC. The angles of incidence and refraction are defined 
to be the acute angles through which the incidence-normal 
has to be turned in order to bring it into coincidence with 
the incident and refracted rays, respectively. Thus, if 
these angles are denoted by a, a', then 

a = ZNBA, a' = ZN'BC. 

In the figure as drawn the angle a is represented as greater 
than the angle a', so that, according to the statement at 



66 Mirrors, Prisms and Lenses [§ 27 

the end of § 26, the medium a is less dense or "rarer" than 
the medium b. 

Before stating the relation which is found to exist be- 
tween the angles a and a! , it is necessary to allude to 
Newton's great discovery that sunlight and indeed so- 
called " white light" of any kind, as, for example, the light 
of an arc lamp, is composed of light of an innumerable 
variety of colors (see Chapter XIV), as may be shown by 
passing a beam of sunlight through a glass prism, whereby 
it will be seen that white light is a mixture of all the colors 
of the spectrum in all their infinite varieties of hues. On 
the other hand, monochro?natic light, as it is called, is light 
of some one definite color, as, for example, the yellow light 
emitted by a sodium flame which may be obtained by 
burning common salt in the flame of a Bunsen burner. 
In geometrical optics, unless we are specially concerned 
with the investigation of color-phenomena (as in Chapter 
XIV), it is nearly always tacitly assumed that the source 
of the light is monochromatic. 

The law of refraction, as found by experiment, may now 
be stated as follows : 

The refracted ray lies in the plane of incidence on the op- 
posite side of the normal in the second medium from the incident 
ray in the first medium; and the sines of the angles of incidence 
and refraction are to each other in a constant ratio, the value of 
which depends only on the nature of the two media and on the 
color (or wave-length) of the light. 

This constant ratio, denoted by the symbol n ah , is called 
the relative index of refraction from the first medium (a) to 
the second medium (b) for light of the given color; thus, 

sin a 

£^~ %; 

the value of this constant, as a rule, being greatest for violet 
and least for red light, so that the violet rays are the most 
" refrangible" of all. When light is refracted from air (a) to 
water (w) the relative index of refraction is, approximately, 



§ 28] Proof of Law of Refraction 67 

n aw =4/3, and hence under these circumstances sina/ = 
% sin a. Although there are many different varieties of 
optical glass, for rough calculations the value of the rela- 
tive index of refraction from air (a) to glass (g) may be 
taken as n ag = 3/2; which means that the sine of the angle 
which the ray makes with the normal in glass is about two- 
thirds of the sine of the angle which the corresponding ray 
makes with the normal in air. 

Although the law of refraction is quite simple, it some- 
how eluded discovery until early in the seventeenth century 
when the true relation between the angle of refraction and 
the angle of incidence was first ascertained by Willebrord 
Snell (1591-1626) or Snellius, of Leyden, and the law is, 
therefore, often referred to as Snell' s Law of Refraction. 
The law was first published by the French philosopher 
Descartes (1596-1650), who had probably seen Snell's 
papers, although he does not allude to him by name. 

28. Experimental Proof of the Law of Refraction. — The 
relation between the angles of incidence and refraction can 
be very strikingly exhibited with the aid of the optical disk 
that was mentioned in § 13 in connection with a lecture- 
table experiment for verifying the law of reflection of light. 
The vertical ground glass disk is adjusted in the track of 
a narrow beam of sunlight (or parallel rays from a lantern) 
in such a position that the path of the light is shown by a 
band of light crossing the face of the disk along one of its 
diameters. The glass body through which the light is re- 
fracted has the form of a semicylinder, the two plane par- 
allel sides being ground rough so as to be more or less opaque, 
whereas the curved surface and the diametral plane face 
are both highly polished. This half -disk has a radius of 
about 2 inches and is about one-half inch thick or more. It 
can be fastened against the vertical face of the optical disk 
with its axis horizontal and coinciding with the axis of rotation 
of the disk, as represented in the diagram Fig. 43. If this 
adjustment is made, and the disk turned so that the inci- 




68 Mirrors, Prisms and Lenses [§ 28 

dent ray AB meets the polished plane face of the glass body 
at its center B, the refracted ray BC will proceed through 
the glass along a radius of the semicylinder, and therefore 
meeting the curved surface normally, it will emerge again 

into the air without being 
further deviated. The di- 
ameter NN' which is 
marked on the face of the 
optical disk is normal to 
the plane surface of the 
glass body, and if from the 
points A and C where the 
incident and refracted rays 
cross the circumference of 
the disk perpendiculars are 

Fig. 43. — Optical Disk used to verify i , * -m ^_ ,1 «^«^„i 
law of refraction. let fal1 0n the normal 

NN', the lengths of these 
perpendiculars AX and CY will be proportional to the sines 
of the angles of incidence and refraction NBA and N'BC, 
respectievly. Now it will be found that, no matter how we 
turn the disk, the perpendicular AX will always be about 
one-and-a-half times as long as the perpendicular CY. If 
we substitute for the half -disk of solid glass a hollow vessel 
of the same form and size with thin glass walls, and if we fill 
this vessel with water, we shall find now that the length of 
the perpendicular AX will always be about one-and-one- 
third times that of the perpendicular CY, because the relative 
index of refraction from air to water is 4/3, as above stated. 
But the best proof of the law of the refraction of fight 
is to be found in the fact that this law is at the basis of the 
theory and construction of nearly all optical instruments, 
and it has been subjected, therefore, to the most searching 
tests. The law of refraction may also be regarded as com- 
pletely verified by the methods that are employed in the 
determination of the indices of refraction of transparent 
bodies, solid, liquid and gaseous; which are described in 



§ 29] Reversibility of Light Path 69 

treatises on experimental optics usually under the title of 
"refractometry." 

29. Reversibility of the Light Path. — When a ray of light 
AB is reflected at B in the direction BD, a plane mirror 
placed at D at right angles to BD will turn the reflected 
ray back on itself; arriving again at B, the light will ob- 
viously be reflected there so as to return finally to the point 
A where it started. This is a simple instance of a general 
law of optics known as the principle of the reversibility of 
the light path. Experiment shows that the same rule holds 
likewise in the case of the refraction of light, and that if 
ABC is the route pursued by light in going from a point 
A in one medium to a point C in an adjoining medium by 
way of the incidence-point B, and if then the light is re- 
versed by some means so as to be started back along the 
path CB, it will be refracted at B into the first medium 
along the path BA. And, in general, if the final direction 
of the ray is reversed, for example, by falling normally on 
a plane mirror, the light will retrace its entire path, no 
matter how many reflections or refractions it may have 
suffered. Thus, in any optical diagram, in which the di- 
rections of the rays of light are indicated by arrow-heads, 
these pointers may all be reversed, if we wish to ascertain 
how the rays would go through the system if they were to 
enter it from the other end. 

It follows, therefore, since 

sin a sin a' 

sin a' ab ' sin a ba ' 

that we have the relation : 

^ab-^ba = 1 ; 

that is, the relative indices of refraction from (a) to (b) and 
from (b) to (a) are reciprocals of each other. Thus, for ex- 
ample, since n aw = 4/3 is the index from air to water, the 
index from water to air is n wa = 3/4. Similarly, if ?i ag = 3/2, 
the index from glass to air is n ga = 2/3. 



70 Mirrors, Prisms and Lenses [§ 31 

30. Limiting Values of the Index of Refraction. — Accord- 
ingly, we see that the value of the relative index of refrac- 
tion may be greater or less than unity. If n ab >l, the 
second medium (b) is said to be more highly refracting or 
(optically) denser than the first medium (a); and since in 
this case sin a >sina', it follows that a > a' ', which means 
that the refracted ray is bent towards the normal, as happens 
when light is refracted from air to water (w ab = 1.33). On 
the other hand, if n ab <l, the second medium (b) is said to 
be less highly refracting or (optically) rarer than the first 
medium (a), and now the angle of refraction (a') will be 
greater than the angle of incidence (a), so that in this case 
the refracted ray will be bent away from the normal, as, for 
example, when light is refracted from water into air (n wa = 
0.75). Glass is more highly refracting than water, and 
diamond has the greatest light-bending power of all optical 
media, the index of refraction from air to diamond being 
about 2.5. The values of the constant n ab for pairs of 
media a, b that are available for optical purposes are com- 
prised within comparatively narrow limits, say, between 
1/2 and 2. In the exceptional case when n ab =l, the angles 
of incidence and refraction will be equal, and the rays pass 
from a to b without change of direction. This is the reason 
why a glass rod is invisible in oil of cedar. Sometimes ac- 
cidental differences of refrangibility between two adjacent 
layers of the same medium enable us to distinguish one 
part of a transparent medium from another. Similarly, 
also, the presence of air-bubbles in water or glass is made 
manifest by the refractions that take place at the boundaries. 
A fish swimming in water does not see the water around him, 
but the phenomena of refraction may make him aware of the 
existence of a different medium above the surface of the water. 

31. Huygens's Construction of a Plane Wave Refracted 
at a Plane Surface. — The straight lines AB and AD (Fig. 44) 
show the traces in the plane of the diagram of the plane 
wave-front advancing in the first medium (a) in the direc- 



31] 



Waves Refracted at Plane Surface 



71 



tion BD and the plane refracting surface, respectively. The 
disturbance is supposed to have just arrived, at the point A 
of the- refracting plane, which from this moment (£ = 0) 
becomes a new origin 
from which secondary 
hemispherical wavelets 
are propagated into the 
second medium (b) . Now 
light is propagated with 
different velocities in dif- 
ferent media; thus, for 
example, the velocity of 
light in water is only 
about three-fourths of 
what it is in air and the 
velocity in glass is about 
two-thirds of the velocity 
in air. Consequently, 
when waves of light pass 
from air into water or 
glass, the part of the wave-front that is in the denser medium 
advances more slowly than the part that is still in the air, 
so that the direction of the wave-front is changed in passing 
from one medium to another. Let the velocities of light 
in the media a and b be denoted by v a and v^, respectively. 
Then after a time i = BD/y a , when the disturbance which 
was at B has just arrived at D on the boundary between 
the two media, the secondary wavelets which have been 
spreading out from A as center will have been propagated 




Fig. 44. — Huygens's construction of plane 
wave refracted at plane surface. 



in the second medium (6) to a distance AC 



M = --BD; 



and, similarly, at the same instant from any intermediate 
point Q lying on AD between A and D the disturbance will 
have proceeded into the second medium (b) to a distance 

QR=-(BD— PQ)=-KD, 



72 Mirrors, Prisms and Lenses [§ 32 

where K (not shown in the figure) designates the foot of 
the perpendicular let fall from Q on BD. Thus, the radii 
of the elementary cylindrical refracted waves whose axes are 
perpendicular to the plane of the diagram at A and Q are 

SfcD, "- b KD, 

respectively; and, according to Huygens's principle, the 
refracted wave-front at this instant will be the surface which 
is tangent to all these elementary cylindrical surfaces. Ex- 
actly the same method as was used in the similar problem 
of reflection (§ 14) can be applied here; and thus it may be 
shown that at the moment when the disturbance reaches 
the point D of the plane refracting surface, the refracted 
wave-front will be the plane CD containing this point, which 
is perpendicular to the plane of the figure and tangent at C 
to the elementary wave represented by the spherical sur- 
face described about C as center with radius equal to AC. 
In the first medium the wave marches forward in the di- 
rection LA and in the second medium in the direction AC. 

Snell's law of refraction (§ 27) may be deduced from 
the figure by observing that BD = AD.sina, where a = 
Z NAL = Z DAB denotes the angle of incidence, and AC = 
AD. sin a', where a' = Z N'AC = Z ADC denotes the angle of 
refraction. Consequently, 

sin a BD v a 

/ = ttt = — = a constant, 

sin a' AC Vb 

which constant must, therefore, be identical with the relative 

index of refraction n ab . 

The diagram is drawn for the case when the light travels 
faster in the first medium than it does in the second (v a >v h ), 
that is, when the second medium is more retarding or "op- 
tically denser" (§ 30) than the first. 

32. Mechanical Illustration of the Refraction of a Plane 
Wave. — A simple mechanical illustration of the refraction 
of a plane wave at a plane surface may be devised as 
follows : 



32] 



Mechanical Illustration 



73 




Two boxwood wheels each about two inches in diameter 
are connected by an iron axle about 4 inches long passing 
through the centers of the wheels at right angles to their 
planes of rotation (Fig. 45). If this body is placed on a 
smooth rectangular board, about a yard long and about 
18 inches wide, which is 
slightly tilted, and allowed 
to roll diagonally down the 
board, its path will be 
along a straight line. But 
if a piece of felt cloth or 
velveteen cut in the form 
of a rectangle is glued in 
the middle of the board, 
with its long side parallel 
to the edge of the board, 
then when the body de- 
scends the inclined plane 
obliquely, one of the wheels 
will arrive at the edge of FlG 
the cloth before the other, 
so that it will be suddenly slowed up while the other wheel 
continues to move on the bare board under the same condi- 
tions as before. Consequently, the axle will be made to swing 
round until both wheels get on to the cloth piece, the direc- 
tion of motion having been abruptly changed in this process. 
At the opposite edge of the cloth rectangle, a similar change 
of the direction of motion takes place in an opposite sense, 
so that when the roller leaves the retarding surface and 
emerges again on to the bare board, it will be found to be 
going approximately in the same direction as at first. These 
bendings in the course of the roller descending the inclined 
plane at the places where it crosses the parallel sides of the 
cloth rectangle are analogous to the deviations in the line 
of march of a plane wave of light in traversing a glass slab 
surrounded by air. 



45. — Mechanical illustration 
refraction. 



of 



74 Mirrors, Prisms and Lenses [§ 33 

33. Absolute Index of Refraction. — If v & , v h and v c denote 
the velocities of light in the media a, b and c, respectively, 
then, as we have just seen (§ 31), according to the wave- 
theory of light, the relative indices of refraction will be: 



iti naC ~V nbc "~Vc ; 

and, hence, we find : 

n ac. 

n ab=— -; 

Wbc 

so that in case we know the values n ac , n bc of the indices 
of a medium c with respect to each of the two media a and 
b, the value n ab of the index of medium b with respect to 
medium a can be obtained at once by means of the above 
relation. Moreover, since (§ 29) 

1 

Wbc 

the preceding equation may be written as follows : 

Thus, for example, suppose the three media a, b and c are 
water, glass and air, respectively; since n ac = 3/4 and n ch =-- 
3/2, the index of refraction from water to glass is found by 
the above formula to be n ab = 9/8. 

In fact, if there are a number of media a, &, c, . . . , i> j, h 
it is obvious that we shall have the following relation be- 
tween the relative indices of refractions: 

n ab -n bc . . . nij.n jk = n ak , 
which is easily remembered by observing the order in which 
the letters occur in the subscripts. In particular, if the last 
medium k is identical with the first medium a, as is the case 
in an optical instrument surrounded by air, then n ak = n aa = 1, 
and accordingly we obtain : 

^ab-^bc .... nij.n ja = l. 
A special case of this general relation, viz., 

has already been remarked (§ 29). 



§ 33] Absolute Index of Refraction 75 

Since w ac .n ca =7i bc .n cb = l and n ab .n\ iC = n ac , we may write 
also: 

"'ca 

and this formula suggests immediately the idea of employ- 
ing some suitable medium c as a standard optical medium with 
respect to which the indices of refraction of all other media 
may be expressed. The natural medium to choose for this pur- 
pose is the ether itself which light traverses in coming to the 
earth from the sun and stars; and so the index of refraction of 
a medium with respect to empty space or vacuum is called 
its absolute index of refraction or simply its refractive index. 
Thus, the absolute index of refraction of vacuum (c) is equal 
to unity, that is, n c = l. Similarly, the symbols n a , n h will 
be employed to denote the absolute indices of the media 
a, b, respectively; so that here they are really equivalent 
to the magnitudes denoted by n CSL , n ch in the preceding 
formula, which, therefore, may be written : 

n h 

that is, the relative index of refraction of medium b with respect 
to medium a is equal to the ratio of the absolute index of medium 
b to that of medium a. 

The absolute indices of refraction of all known transparent 
substances are greater than unity. The velocity of light 
in ordinary atmospheric air is so nearly equal to its velocity 
in vacuo that for all practical purposes we may generally 
take the absolute index of refraction of air as also 
equal to unity. The actual value for air at 0°C. and 
under a pressure of 76 cm. of mercury, for sodium light, 
is 1.000293. 

With every isotropic medium there is associated, there- 
fore, a certain numerical constant n called its (absolute) 
index of refraction; and, hence, when a ray of light is re- 
fracted from a medium of index n into another of index n f , 



76 



Mirrors, and Prisms Lenses 



34 



the trigonometric formula for the law of refraction may be 
written thus: 

sina n' 



sina' n 
which may also be put in the following symmetric form : 

n'.sina' = ft.sina. 
This latter mode of writing this relation suggests also an- 
other way of stating the fundamental fact in regard to the 




Fig. 46. — Construction of refracted ray (n'>n) 

refraction of light, as follows: Whenever a ray of light is re- 
fracted from one medium to another , the product of the index 
of refraction and the sine of the angle between the ray and the 
normal to the refracting surface has the same value after re- 
fraction (n'.sina/) as before refraction (n.sina). This prod- 
uct K = n.sin a = n'.sina' which does not vary when the 
light crosses a surface separating a pair of isotropic media 
is called the optical invariant of refraction. 

34. Construction of the Refracted Ray. — Let the absolute 
indices of refraction of two media separated from each other 
by a smooth refracting surface be denoted by n, n', and let 
the straight line AB (Figs. 46 and 47) represent the path 
in the first medium (n) of a ray incident on the boundary- 



§34] 



Construction of Refracted Ray 



77 



surface at the point B. The straight line NN' represents 
the normal to the refracting surface at this point, and hence 
the plane of the diagram is the plane of incidence. The 
straight line ZZ shows the trace in this plane of the plane 
tangent to the refracting surface at the incidence-point B; 
in the special case when the refracting surface is itself plane, 
this straight line will be the trace of the surface of separa- 
tion between the two media. With the point B as center 




Fig. 47. — Construction of refracted ray (n'<n) 

and with any radius r describe in the plane of incidence the 
arc of a circle cutting the incident ray AB in a point P lying 
in the first medium; and in the same plane, with radius 
n'jn times as great, that is, with radius n'r/n, describe also 
the arc of a concentric circle intersecting at P' the straight 
line HP drawn through P perpendicular to ZZ at H. If 
the second medium is more highly refracting than the first, 
that is, if n'>n, the radius of the second circle will be greater 
than that of the first, as represented in Fig. 46; whereas 
when n'<n, the second circle is inside the first, as in Fig. 47. 
The path of the refracted ray correspodinng to the given 



78 Mirrors, Prisms and Lenses [§ 36 

incident ray AB will be represented by the prolongation 
BC in the second medium of the straight line P'B. 

The proof of this construction consists simply in showing 
that the ZN'BC between the normal and the straight line 
BC is equal to the angle of refraction a' as given by the 
formula n'.sin a' = n.sin a, where a=ZNBA denotes the 
given angle of incidence. Evidently, from the figure, we have : 

sinZHPB _BP / _n' 

sin/HP / B~BP "»' 
and since ZHPB = ZNBA=a, and Z HP'B = Z N'BC, we 
obtain immediately the relation: n'. sinZN'BC = ft.sina and 
therefore ZN'BC=a'. 

35. Deviation of the Refracted Ray. — The acute angle 
through which the direction of the refracted ray has to be 
turned to bring it into the same direction as that of the in- 
cident ray is called the angle of deviation of the refracted ray 
and is denoted by e; thus, € = ZP / BP (Figs. 46 and 47). 
Obviously, 

e= a — a'. 
The only ray incident at B whose direction will remain un- 
changed after the ray enters the second medium is the one 
that proceeds along the normal NB (a = a'=e = 0). The 
more obliquely the ray AB meets the refracting surface, 
that is, the greater the angle of incidence, the greater also will 
be the deviation-angle. The truth of this statement will be 
apparent from an inspection of the relation between the 
angles a and e as exhibited in Fig. 46 or Fig. 47. The inter- 
cept PP' included between the circumferences of the two 
construction-circles, which remains constantly parallel to the 
incidence-normal, increases in length as the angle of inci- 
dence increases, whereas the other two sides BP, BP' of 
the triangle BPP', being always equal to the radii of the 
circles, remain constant in length; and hence the angle € 
must increase in absolute value as the angle a increases. 

36. Total Reflection. — In ordinary refraction, as we have 
seen, there can only be one refracted ray corresponding to 



§36] 



Total Reflection 



79 



a given incident ray, but the question may be asked: Is it 
possible that, under certain circumstances, there will be 
no refracted ray, so that the incident light will be totally 
reflected at the surface without being refracted at all? Evi- 
dently such will be the case whenever in the foregoing con- 
struction (§ 34) the point P' (Figs. 46 and 47) cannot be 
located, because the path of the refracted ray is determined 
by the straight line P'B. 

Let us examine, first, the case when the second medium 
is more highly refracting than the first, n'>n (Fig. 46). 




Fig. 48. — Limiting refracted ray (n f >ri) 

Suppose that the straight line AB which represents the 
path of the incident ray is initially in the position NB, and 
that it is rotated from this position around the point B as 
a pivot until it has turned through a right angle in the plane 
of the figure. While the point P on AB describes a quadrant 
of the circumference of the circle of radius BP, the point 
P' will trace out an arc of the concentric circle of radius 
BP', which, however, will never be equal to a quadrant of 
this circumference; for when the point P has completed its 
quadrant and arrived at the point D (Fig. 48) on the tan- 
gent plane drawn to the refracting surface at B, the point 



80 Mirrors, Prisms and Lenses [§ 36 

P' will likewise have reached the extremity of its arc where 
the tangent to the inner circle at D meets the circumference 
of the outer circle. The incident ray ZB just grazes the re- 
fracting surface at B or skims along it, and most of the 
light is reflected and does not enter the second medium at 
all, but the portion that is refracted pursues the path BQ 
corresponding to this extreme position of the point P', and 
this will be the outermost of all the refracted rays that 
enter the second medium at the point B. The ZN'BQ=A 
which is the greatest value that the angle of refraction can 
have in the case when n'>n is called the limiting or critical 
angle with respect to the two media. Since 

sinZ N'BQ = sinZ PP'B = BD/BP' = n\n\ 
the magnitude of the angle A may be found from the rela- 
tion: 

sinA = n/n', (n <n f ) ; 
which may likewise be derived by substituting the values 
a = 90°, a/ = A in the refraction-formula. Thus, if the 
first medium is air.(n = l) and the second medium is glass 
(n' = 3/2), sinA = 2/3, so that the critical angle for air-glass 
is found to be A =41° 49'. For air-water sinA = 3/4, A = 
48° 35'; and, consequently, a ray of light whose path lies 
partly in air and partly in water cannot possibly make 
an angle with the normal in the water greater than about 
48° 30'. For example, when a star is just rising or setting, 
the rays coming from it will fall very nearly horizontally 
on the surface of tranquil water and will be refracted into 
the water, therefore, at an angle of approximately 48° 30' 
with the vertical, so that if these rays entered an eye under 
the water, the star would appear to be nearly halfway to 
the zenith. In fact, all the rays coming into an eye placed 
under water from the entire overhanging arch of the sky 
would be comprised in the water within a cone whose axis 
points to the zenith and whose angular aperture is about 
97°. In this connection it is interesting and instructive to 
examine a photograph of an air-scene made with a so-called 



36] 



Total Reflection 



81 



"fish-eye" camera immersed below the level of a clear pool 
of water, which affords some idea of how the world outside 
the pond must look to a fish. Professor Wood, of the Johns 
Hopkins University, has obtained a number of pictures of 
this kind, some of which are reproduced in illustrations in 
his very original book on Physical Optics, where also a brief 
description of the essential features of the ingenious pin- 




Fig. 49. — Limiting incident ray (n'<n) 

hole camera which was used in making these pictures is 
also given. 

Accordingly, when light is refracted from a rarer to a 
denser medium, there will always be a refracted ray cor- 
responding to a given incident ray, because it is always 
possible under these circumstances to locate the position 
of the point P' opposite P, or, to express it in another way, 
because when n<n r there will always be a certain acute 
angle a' that will satisfy the equation sina' = n.sina/tt' for 
values of a comprised between 0° and 90°. But in the op- 
posite case when, the first medium is denser than the second 
(n>n f ), for example, when the light is refracted from water 
to air, the statement just made is no longer true. The es- 



82 



Mirrors, Prisms and Lenses 



[§36 



1 

\ \ 


r 


z 


AM \ \ 




/ 


WATER \ \ \ 


s 


/ / ^i/ // \ 



Fig. 50.- 



-Refraction from water to air; 
total reflection. 



sential difference in the two cases may be seen at once by 
reversing the arrow-heads in the diagram Fig. 48, at the 
same time making corresponding changes in the letters and 

symbols. Fig. 49 is a 
special diagram to illus- 
trate this case. The re- 
fracted ray BQ which 
grazes the surface at the 
point B corresponds to 
the limiting incident ray 
PB which is incident at 
B at the critical angle 
A = ZNBP; and, conse- 
quently, any ray, such 
as RB, which meets the 
surface at an angle of incidence greater than the angle A 
will be totally reflected in the direction BS. Thus, for values 
of a which are greater than the value A of the critical angle 
of incidence, there will be no value of a' that will satisfy 
the equation sina' = 7i.sina/w' 
when n>n'. Only those rays 
incident at B which lie within 
the cone generated by the 
revolution of the limiting in- 
ident ray around the inci- 
dence-normal as axis will be 
refracted into the second 
medium; and all rays falling 
on the refracting surface at 
B and lying outside this cone 
will be totally reflected. 

Fig. 50 shows how rays 
proceed from a radiant point below the horizontal free sur- 
face of still water. 

If a pin is stuck in the under side of a flat circular cork 
floating on water, as represented in Fig. 51, and if the 




Fig. 51. 



-Experiment illustrating 
total reflection. 



37] 



Total Reflection 



83 



diameter of the cork is (say) 6 inches and the head of 
the pin is not more than 2.5 inches below the water-level 
and vertically beneath the center of the cork, an eye placed 
anywhere above the level of the water will be unable to see 
the pin, because all the rays coming from it that meet the 
surface of the water beyond the edge of the cork will be 
totally reflected back into the water. 

In Fig. 49 since sin ZNBP = sinZP'PB = BP'/BP, we find 
in this case when n'<n that sinA = n'/n, which will also be 
obtained by putting a = A, a' = 90° in the refraction- 
formula n.sina = n / .sina'. Comparing this result with 
the formula sin A = n/n' obtained for the case when n'>n, 
and recalling the fact that the sine of an angle is never 
greater than unity, we may formulate the following rule: 

The sine of the so-called 
critical angle (A) with re- 
spect to two media is the 
ratio of the index of refrac- 
tion of the rarer to that of 
the denser medium. Or, 
the sine of the critical angle 
(A) of a substance is the 
reciprocal of the absolute 
index of refraction of the 
substance: thus, 



A-* 




Fig. 52. — Optica Disk used to show total 
reflection. 



sinA = -. 
n 

37. Experimental Il- 
lustrations of Total Re- 
flection. — The phenomenon of total reflection may be ex- 
hibited with the aid of the optical disk and the semicylinder 
of glass described in § 28. If the disk is turned so that the 
beam of incident parallel rays falls first on the curved surface 
of the semicylinder, as shown in Fig. 52, the rays meet this 
surface normally and proceed through the glass to the plane 
face without being deviated. At the plane surface a por- 



84 



Mirrors, Prisms and Lenses 



37 



tion of the beam is reflected and, in general, a portion is re- 
fracted from glass to air. If the disk is turned until the 
angle of incidence at the plane surface is just equal to the 
critical angle (A), the rays emerging into the air will pro- 
ceed along the plane face, and if the disk is turned a little 
farther in the same sense, so that the angle of incidence 
exceeds the critical angle, the light will be totally reflected. 
An ingenious contrivance for exhibiting the procedure of 
light in passing from water to air consists of a compara- 




Fig. 53. — Demonstration of refraction from water to 
air and total reflection. 

tively large glass tank (Fig. 53) filled with water and pro- 
vided with a plane vertical metallic screen the lower half of 
which is under water while the upper half extends into the 
air above. A cylindrical beam of light is directed horizon- 
tally and normally against the lower part of the vertical 
glass wall of the tank, which is behind the screen and par- 
allel to it. The rays entering the water are received first 
on the surface of a solid reflecting cone of aperture-angle 
90° placed in the water under the screen and mostly in front 
of it, the axis of the cone being horizontal and its apex 
turned towards the on-coming light. From the surface of 
this cone the rays are reflected through the water in all di- 
rections in a vertical plane coinciding as nearly as possible 
with the front side of the screen turned towards the spec- 
tators. Surrounding the conical reflector and co-axial with 



§37] 



Total Reflection Prism 



85 



it, there is a cylindrical cavity of diameter very little larger 
than that of the base of the cone. The surface of this cylin- 
der is made of thin sheet-metal blackened on the inside, 
wherein a number of equal horizontal slits are cut at equal 
angular distances apart, and through these slits narrow 
beams of light reflected from the surface of the cone are 
permitted to pass upwards towards the surface of the water, 
their courses being shown by the bright traces on the screen. 
Some of these beams will be refracted out into the air, 
whereas others, meeting the water-surface more obliquely, 
will be totally reflected. 

If rays are incident normally on one of the two perpendic- 
ular faces of a glass prism (§ 48) whose principal section is an 
isosceles right-triangle (Fig. 54), 
they will enter the prism with- 
out deviation, and falling on *~~ 
the hypothenuse-face at an angle z — ^_ 
of 45°, which is greater than the 
critical angle of glass, they will *"~ 
be totally reflected there and 
turned through a right angle, so 
that they will emerge in a direc- 
tion normal to the other of the two 
perpendicular faces of the prism. FlG - 54.— Total reflection prism 
A prism of this kind is frequently employed in optical sys- 
tems. It is used, for example, in connection with a photo- 
graphic lens to rectify the image focused on the sensitive 
plate of the camera, so that the right and left sides of the 
negative will correspond to the right and left sides of the 
object. None of the light is lost by the total reflection in 
the prism, and if the prism is made of good optical glass 
of high transparency there will be comparatively little loss 
of light by absorption in the prism or by reflection on enter- 
ing and leaving it. The same optical effect can be produced 
by a simple plane mirror, but as a rule a polished metallic 
surface absorbs the incident light to a considerable extent. 



86 Mirrors, Prisms and Lenses [§ 38 

However, the loss of light in the case of a mirror silvered 
on glass is very slight; but on the other hand, the fine layer 
of silver may easily be injured mechanically or tarnished 
by exposure to the air. If the glass mirror is silvered on 
the back side, the light will be reflected from both surfaces 
of the glass and there will be confusion. Moreover, a glass 
mirror may easily get broken or become dislocated in an 
optical instrument; whereas a prism made of a solid piece 
of glass is much more substantial and durable. 

Optical prisms consisting of solid pieces of highly trans- 
parent homogeneous glass with three or more polished plane 
faces are very extensively used in the construction of modern 
optical instruments for rectifying images which would other- 
wise be inverted or for bending the rays of light into new 
directions, etc. Usually the light undergoes several interior 
reflections before it issues from the prism, and these reflec- 
tions are often total reflections. If the reflection is not 
total, it is best to silver the surface. 

38. Generalization of the Laws of Reflection and Re- 
fraction. Principle of Least Time (Fermat's Law). — The 
laws of reflection and refraction, which merely describe the 
observed effects when light falls on the common surface of 
separation of two homogeneous media, and which are cap- 
able of simple explanation on the basis of the wave-theory, 
as has been illustrated in certain special cases (§§ 14 and 31), 
may be combined into a general law which was first an- 
nounced about 1665 by the French philosopher Fermat, 
and which may be stated as follows: The actual "path pur- 
sued by light in going from one point to another is the route 
that, under the given conditions, requires the least time. 

In case the reflections and refractions take place only 
at plane surfaces, the truth of the above statement is 
easily proved. Consider, first, the case when the light is re- 
flected from a plane mirror. The straight line ZZ (Fig. 55) 
represents the trace of the plane mirror in the plane of the 
diagram, and A and C designate the positions of a pair of 





Cv 


B 






V 


A 




A' 



§ 38] Principle of Least Time 87 

points lying in this plane in front of the mirror. Now if 
a point X in the plane of the mirror is connected with A 
and C by the straight lines XA, XB, the route AXC will 
be shortest when the normal to the mirror at X lies in the 
plane AXC and bisects the angle 
AXC. The point X must lie, there- 
fore, in the plane of the diagram at 
the point B, so that when AB is the 
direction of an incident ray, BC will 
be the direction of the reflected ray. 
Obviously, if A' is the image of A in 
the mirror, then AB-f-BC=A'B+BC 

-AT and *\nop> thp strfliVht linp FlG * 55 ' — Fermat ' s P rf nci- 

-a^, ana since tne straignt line p i e of least time in case 
A'C is shorter, for example, than of reflection at a plane 
(A'D+DC) = (AD+DC), where D is mirron 
another point on the mirror different from the point B, it 
is evident that the route from A to C by way of B is shorter 
than the route via any other point on the mirror. More- 
over, if the ray is reflected at a number of plane mirrors in 
succession, its entire path will be the shortest possible route 
from the starting point to the terminal point, subject to the 
condition that it must touch at each mirror in turn. The 
principle of least time in the case of reflection of light at a 
plane mirror dates back to the time of Hero of Alexandria 
(150 B. C). 

When light is refracted at a plane surface, the route pur- 
sued between a point A in one medium to a point C in the 
other is indeed the quickest way but generally not the 
shortest. The following illustration will help to make the 
problem clear in this case. Suppose a level field is divided 
into two parts by a straight line ZZ (Fig. 56), on one side 
of which the ground is bare and smooth while on the other 
side it is plowed and rough; and let us also suppose that 
a man can walk only half as fast over the rough part of the 
field as over the smooth part, and that he desires to march 
as quickly as possible from a point A in the smooth ground 



88 



Mirrors, Prisms and Lenses 



38 




Fig. 56. — Quickest route 
from A to C via 
ABC. 



to a certain other point C in the plowed ground. The 
question is, Where should he cross the dividing line ZZ? 
Of course, his shortest route would be along the straight 
line from A to C which intersects ZZ 
at the point marked E in the figure, 
but unless the straight line AC hap- 
pens to be perpendicular to ZZ this 
will not be his quickest way. In- 
stead of crossing at E, suppose he 
selects a point F on ZZ which is a 
little nearer to his objective at C; 
then although the length FC in the 
plowed ground is shorter than be- 
fore, on the other hand the distance 
path AF over the smooth ground is longer, 
but on the whole we may assume that 
the route AFC will take less time than the shortest route 
AEC. But if the point of crossing ZZ is taken too far from 
E, the advantage of the shorter dis- 
tance in the rough ground will pres- 
ently be more than offset by the 
increasing length of the distance that 
has to be traversed in the smooth 
ground. Accordingly, there is a cer- 
tain point B on ZZ such that the 
time taken along the route ABC 
will be the quickest of all routes. 
Now we shall see that this is also the 
very path that light would take if it 

p j £ » , ^ Fig. 57. — Fermat's princi- 

were refracted from A to C across ple of least time in case 

ZZ, supposing that the ratio of the 

velocities of light on the two sides 

of ZZ were the same as the ratio of the velocities of walking 

in the two parts of the field. 

In the accompanying diagram (Fig. 57) the broken line 
ABC represents the actual path of a ray of light from a 




of refraction at plane 
surface. 



§ 39] Optical Length 89 

point A in the first medium (n) to a point C on the other 
side of the plane refracting surface ZZ in the second medium 
(n f ) ; so that if NBN' is the normal to the surface at B, then 
by the law of refraction : 

sin/NBA _rt_v 
sinZN'BA~n v'' 
where v, v' denote the speeds with which light travels in 
the media n, n', respectively. The time taken to go over 
the route ABC is 

AB BCl 
*~ v + v' * 
and we wish to show that this time t is less than the time 

AD DC 
v V 
along any other route ADC, where D designates the posi- 
tion of any point on ZZ different from the point B. Draw 
DG, DH perpendicular to AB, BC, respectively; then, since 

Z BDG = Z NBA, Z BDH = Z N'BC, 
evidently we have: 

sinZBDG GB v GB HB 



Now 



sinZBDH HB~V ' 0r v V 



AB BC_ AG+GB BC = AG HC. 
v v' v v' v v' ' 



and since AG<AD and HC<DC, therefore 
/AB , BC\ /AD , DC\ 

(v+vj < \ir + y)> 

and hence the time via ABC is less than it would be via any 
other route from A to C. 

It should be remarked, however, that when the boundary- 
surface between two media is curved, the time taken by 
light to go from a point A across the surface to another point 
C is not always a minimum. It may, indeed, be a maximum, 
but it is always one or the other. 

39. The Optical Length of the Light-path, and the Law 
of Malus. — In the time t that light takes to go along the path 




90 Mirrors, Prisms and Lenses [§ 39 

ABC from a point A in one medium (n) to a point C in an 
adjacent medium (n f ) it would traverse in vacuo the distance 

-="( A V B + B T C ). 

where V denotes the velocity of light in vacuo. But by the 
definition of the absolute index of refraction (§ 33), n = V/v, 

n'=V/v'; and hence the 
equivalent distance in 
vacuo is : 

nJUB+n'JBC. 
The optical length of the 
path of a ray in a medium 

Fig. 5S.-Optical|ngth of ray-path ig defined tQ be ^ prod _ 

uct of the actual length (I) 
of the ray-path by the index of the medium (n) that is, n. I. 
Suppose, for example, that light traverses a series of media 
wi, ri2, etc., as represented in Fig. 58; the total optical length 
along a ray will be: 

k =m 

ni.li+rh.k+ +n m .Z m =2Jn k .Z k ; 

k=l 

where ? k denotes the actual length of the ray-path in the 
fcth medium. 

Now the wave-front at any instant due to a disturbance 
emanating from a point-source is the surface which con- 
tains all the farthest points to which the disturbance has 
been propagated at that instant. Thus, the wave-surface 
may be defined as the totality of all those points which are 
reached in a given time by a disturbance originating at a point. 
In a single isotropic medium the wave-surfaces, as we have 
seen, will be concentric spheres described around the point- 
source as center; but if the wave-front arrives at a reflect- 
ing or refracting surface /*, at which the directions of the 
so-called rays of light are changed, the form of the wave- 
surface thereafter will, in general, no longer be spherical; and 
even in those exceptional cases when the reflected or refracted 
wave-front is spherical, the waves will spread out from 




§ 39] Law of Malus 91 

a new center which is seldom identical with the original 
center. The function 2nl has the same value for all ac- 
tual ray-paths between one position of the wave-surface and 
another position of it; so that when the form and position 
of the wave-front and the paths of the rays at any instant 
are known, the wave-front at any subsequent instant may 
be constructed by laying off equal optical lengths along the 
path of each ray. 

A consequence of this definition of the wave-surface is 
that the ray is always normal to the wave-surface (§7), as will 
be evident from the following 
reasoning. Suppose that the 
straight line AB (Fig. 59) repre- 
sents the path of a ray incident 
on the refracting surface ZZ at 
the point B, and that the straight 
line BC represents the path of 
the corresponding refracted ray. 

Moreover, let the wave-SUrface FlG - 59.— Law of Malus: Ray 

. normal to wave-front. 

which passes through the pomt 

C be designated by <r. From the incidence-point B draw 
any other straight line, as BD, meeting the wave-surface a 
in the point D. Then by the principle of least time, the 
route ABC is quicker, that is, optically shorter, than the 
route ABD, because the natural or actual route between the 
points A and D would not be by way of the incidence-point 
B. Hence, the straight line BC must be shorter than BD, 
and therefore BC is the shortest line that can be drawn from 
the incidence-point B to the wave-surface a. 

The same reasoning is applicable to all cases of reflection 
and refraction, and hence we may make the following gen- 
eral statement: 

Rays of light meet the wave-surface normally; and, con- 
versely, The system of surfaces which intersect at right angles 
rays emanating originally from a point-source is a system of 
wave-surfaces. 

This law was published by Malus in 1808. 



92 Mirrors, Prisms and Lenses [Ch. Ill 

PROBLEMS 

1. (a) A ray is refracted from vacuum into a medium 
whose index of refraction is\/2, the angle of incidence being 
45°: find the angle of refraction. 

(b) Find the angle of incidence of a ray which is re- 
fracted at an angle of 30° from vacuum into a medium of 
index equal to \/3. 

(c) Find the relative index of refraction when the 
angles of incidence and refraction are 30° and 60°, respec- 
tively. Ans. (a) 30°; (b) 60°; (c) y/%: 3. 

2. Assuming that the indices of refraction of air, water, 
glass and diamond have the values 1, -|, f and -§, respec- 
tively, calculate the angle of refraction in each of the 
following cases: 

(a) Refraction from air to glass, angle of incidence 40°; 
(b) from air to water, angle of incidence 60°; (c) from air 
to diamond, angle of incidence 75°; (d) from glass to water, 
angle of incidence 30°; (e) from diamond to glass, angle of 
incidence 36° 52' 11.6". Ans. (a) 25° 22' 26"; (6) 40° 30' 19"; 
(c) 22° 43' 44"; (d) 34° 13' 44"; (e) 90°. 

3. The height of a cylindrical cup is 4 inches and its di- 
ameter is 3 inches. A person looking over the rim can just 
see a point on the opposite side 2.25 inches below the rim. 
But when the cup is filled with water, looking in the same 
direction as before, he can just see the point of the base 
farthest from him. Find the index of refraction of water. 

Ans. 4:3. 

4. The index of a refracting sphere is\/3; it is surrounded 
by air. A ray of light, entering the sphere at an angle of 
incidence of 60° and passing over to the other side, is 
there partly reflected and partly refracted. Show that the 
reflected ray and the emergent ray are at right angles to 
each other. 

5. In the preceding problem, show that the reflected ray 
will cross the sphere again and be refracted back into the 



Ch. Ill] Problems 93 

air in a direction exactly opposite to that which the ray had 
before it entered the sphere. 

6. A straight line drawn through the center C of a spher- 
ical refracting surface meets the surface in a point desig- 
nated by A. If J, J' designate the points where an inci- 
dent ray and the corresponding refracted ray intersect the 

TL 71 

straight line AC, and if CJ = — .AC, show that CJ'=— .AC, 

n n' 

where n, n' denote the indices of refraction of the first 

and second media, respectively. 

7. Construct the path of a ray refracted at a plane sur- 
face. Draw diagrams for the cases when n' is greater and 
less than n. Construct the critical angle in each figure. 

8. The velocity of light in air is approximately 186000 
miles per second. How fast does it travel in alcohol of 
index 1.363? Ans.. Approximately, 136 460 miles per sec. 

9. A fish is 8 feet below the surface of a pool of clear water. 
A man shooting at the place where the fish appears to be 
points his gun at an angle of 45°. Where will the bullet 
cross the vertical line that passes through the fish? (Take 
index of water as 1.33, and neglect any deflection of the 
bullet caused by impact with the water.) 

Ans. 3 feet above the fish. 

10. Assuming that the velocity of light in air is 
30 000 000 000 cm. per sec, calculate its velocity in water 
and in glass. 

11. Prove that n ab = n cb : n ac . 

12. Show that the sine of the critical angle of an optical 
medium is equal to the reciprocal of the absolute index of 
refraction. 

13. Assuming same values of the indices of refraction as 
in problem No. 2, calculate the values of the critical angle 
for each of the following pairs of media: (a) air and glass, 
(b) air and water, (c) air and diamond. 

Ans. (a) 41° 48' 40"; (b) 48° 35' 25"; (c) 23° 34' 41". 

14. A 45° prism is used to turn a beam of light by total 
internal reflection through a right angle. What must be 



94 Mirrors, Prisms and Lenses [Ch. Ill 

the least possible value of the index of refraction of the 
glass? Ans. \/2. 

15. Show that when a ray of light passes from air into 
a medium whose index of refraction is equal to\/2> the de- 
viation cannot be greater than 45°. 

16. The absolute index of refraction of a certain trans- 
parent substance is -§. Show that a luminous point at the 
center of a cube of this material cannot be seen by an 
eye in the air outside, if at the center of each face of the 
cube a circular piece of opaque paper is pasted whose radius 
is equal to three-eighths of the edge of the cube. 

17. What will be the greatest apparent zenith distance of 
a star to an eye under water? 

18. Explain why it is that it is not possible for a person 
by merely opening his eyes under water to see distinctly 
objects in the water around him or in the air above the 
water; whereas, if he is provided with a diver's helmet with 
a plate glass window in it, he will experience no difficulty 
in distinguishing such objects clearly. 

19. Rays of light are emitted upwards in all directions 
from a luminous point at the bottom of a trough contain- 
ing a layer of a transparent liquid 3 inches in depth and of 
refractive index 1.25. Show that all rays which meet the 
surface outside a certain circle whose center is vertically 
above the point will be totally reflected; and find the radius 
of this circle. Ans. 4 inches. 

20. A pin with a white head is stuck perpendicularly in 
the center of one side of a flat circular cork, and the cork 
is floated on water with the pin downwards. Assuming 
that the head of the pin is 2 inches below the surface of the 
water, find the smallest diameter the cork can have so that 
a person looking down through the water (index -|) from 
the air above (index unity) could not see the head of the pin. 

Ans. 4.535 inches. 

21. Plot a curve showing the deviation e as a function 
of the angle of incidence a for the case when the refraction 
is from water (n = 4/3) to air (n f = 1) . 



CHAPTER IV 

REFRACTION AT A PLANE SURFACE, AND ALSO THROUGH A 
PLATE WITH PLANE PARALLEL FACES 




40. Trigonometric Calculation of Ray Refracted at a 
Plane Surface. — A geometrical construction of the path of 
the refracted ray was 
given in § 34. The path 
of a ray refracted at a 
plane surface may also 
be easily determined by 
trigonometric calculation. 
The straight line yy in 
Figs. 60 and 61 represents 
the plane refracting sur- 
face Separating the two FlG - 60.-Refractio D .of ray at plane sur- 
1 , . & face: a = AL, v =AL (n >n). 

media of indices n, n', 

and the straight line LB shows the path of a ray which is in- 
cident on yy at the point marked B. The straight line LA 

perpendicular to yy at A 
is the axis of the refract- 
ing plane with respect 
to the position of the 
point L. The magni- 
tudes ?; = AL a = 
Z ALB which determine 
completely the position 

Fig. 61. — Refraction of ray at plane surface: q£ ^\q incident rav are 

v = AL, v =AL' (n'<n). J 

sometimes called the 
ray-coordinates. Let L' designate the point where the re- 
fracted ray L'B intersects the axis xx, and let z/ = AL', 
a' = Z AL'B denote the coordinates of the refracted ray. The 

95 





B 


V 


x ^\ a 


-"vV 


X 


L 


n 


n' 
y 



96 



Mirrors, Prisms and Lenses 



41 



problem is: Given the incident ray (v, a), determine the re- 
fracted ray (*/, a'). 

From either diagram we obtain immediately the relation: 

z/_tan a 

v "tana" 

and since n.sma = n'.sma', we obtain finally the following 
formulae for calculating the refracted ray: 



v V? 



n 



cos a 



?? 2 .sm 2 a . , n . 

-, sin a =— .sin a. 

n' 



Now if the point L is a luminous point, rays will emanate 
from it in all directions, and, whereas the magnitude v will 

remain the same for all these 
rays, the angle a will vary from 
ray to ray. But for different 
values of a, in general we shall 
obtain different values of the 
magnitude v', and, consequently, 
the position of the* point 1/ on 
the axis will be different for dif- 
ferent incident rays coming from 
L. Accordingly, the bundle of 
Fig. 62.— Refraction of paraxial re f r acted rays corresponding to 

rays at plane surface: u= , . " 

a homocentnc bundle of mcident 




AM, w' = AM' 

(n'>n). 



u : n =u:n, 



rays will not be homocentric. 
41. Imagery in a Plane Refracting Surface by Rays 
which Meet the Surface Nearly Normally. — The more 
or less blurred and distorted appearance of objects seen 
under water is familiar to everybody. When the rays 
that enter the eye meet the surface of the water very 
obliquely, the distortion is almost grotesque. If the pupil 
of the eye were not comparatively small, it would indeed 
be practically almost impossible to recognize an object under 
water, even if the eye were placed in the most favorable 
position vertically over the object. It is only because the 
apertures of the bundles of effective rays that enter the eye 



41] 



Plane Refracting Surface 



97 



are quite narrow, that there is any true image-effect at all 
in the case of refraction at a plane surface. 

When the eye looks directly along the normal to the 
plane refracting surface at an object-point M on the other 
side of the surface (Figs. 62 and 
63), the effective rays coming 
from M will meet the surface 
very nearly perpendicularly, 
and the incidence-points will 
all be so close to the point A 
that there will be practically no 
difference between the lengths 
of the straight lines MA and 
MB, and accordingly under 
these circumstances we may 
write sin a in place of tan a. 
Similarly, also, with respect to 
the refracted ray, sin a' can be substituted here for tan a/. 
And if in this case we put KM = u, AM , = ii', where M, M' 
designate the points where a ray which is very nearly nor- 
mal to the refracting plane crosses the normal before and 
after refraction, we have therefore, 




Fig. 63. — Refraction of paraxial 
rays at plane surface: m = AM, 
u' = AM', u' : n' = u:n , (n'<n). 



tan a sin a 



u tan a' sin a' 
and, hence, by the law of refraction : 

n' n . n' 

— — - , or u = —.u. 
u' u n 

The angle a has disappeared entirely from this formula, and 
the value of v! may be found as soon as the value of u is given. 
This means that corresponding to a given position of the 
object-point M there is a perfectly definite image-point M', 
and the points M, M' are said to be a pair of conjugate points. 
Accordingly, when a narrow bundle of homocentric rays is 
incident nearly normally on a refracting plane, the correspond- 
ing bundle of refracted rays will be homocentric also. And if 



98 



Mirrors, Prisms and Lenses 



[§42 



the aperture of the bundle is infinitely narrow, the imagery 
will be ideal. 

For example, a pebble at the bottom of a pool of water 
12 inches deep will be seen distinctly from a point in the air 
vertically above it, but it will appear to be only 9 inches 
below the surface of the water, since n'/w = 3/4. On the other 
hand, an object 9 inches above the surface will seem to be 

12 inches above it to an eye in 
the water vertically beneath the 
object, beca-use in this case 
n'/n= 4/3. 

42. Image of a Point Formed 
by Rays that are Obliquely Re- 
fracted at a Plane Surface. — 
But if the bundle of rays com- 
ing from the luminous point S 
(Fig. 64) is a wide-angle bundle 
of considerable aperture, no dis- 

Fig. 64,-Caustic by refraction at tinct ima S e wil1 be formed b ^ 

plane surface from water to these rays after refraction at a 
air ' plane, but the points of inter- 

section of the refracted rays will be spread over a so- 
called caustic surface, which in this case is a surface of 
revolution around the normal SA drawn from S to the re- 
fracting plane. The figure shows a meridian section of this 
surface for the case when the rays are refracted from a 
denser to a rarer medium (n'< n), the curve in this case being 
the evolute of an ellipse. Each refracted ray produced back- 
wards touches the caustic surface. The cusp of the meridian 
curve is on the normal SA at the point M' where the image of 
S is formed by rays that meet the refracting plane nearly per- 
pendicularly, as explained in the preceding section. Wherever 
the eye is placed in the second medium, only a narrow 
bundle of rays coming from S can enter it through the pupil 
of the eye. The nearest approach to an image of the source 
at S as seen by rays that are refracted more or less obliquely 




§ 42] Caustic Surface 99 

will be the little element of the caustic surface which is the 
assemblage of the points where the effective rays that enter 
the eye touch this surface. Thus, rays entering the eye at E 
appear to come from the point S' where the tangent from E 
touches the caustic. It is evident now why an object S under 





WATER 




Fig. 65. — Rod partly immersed in water appears to be bent 
upwards. 

water appears to be raised towards the surface and at the 
same time also to be shifted towards the spectator more and 
more as the eye at E is brought nearer to the surface of the 
water, until finally when the eye is on a level with the surface 
of the water, the image of S appears now to be at V on the 
refracting plane. Rays from S that meet the surface beyond 
this limiting point V where the caustic curve is tangent to 
the straight line ZZ will be totally reflected. The image of S 
seen by the eye at E is blurred and distorted, because 
the image-point S' is the point of intersection of a very 
limited portion of the bundle of refracted rays that enter 
the eye. 

The above explanation makes it clear why a straight line 
ABC (Fig. 65) which is partly in air and partly in water will 
appear to an eye at E to be bent at B into the broken line 
ABC'. The image BC of the part BC under water can be 
plotted point by point for any position of the eye. 



100 



Mirrors, Prisms and Lenses 



[§43 




43. The Image-lines of a Narrow Bundle of Rays Re- 
fracted Obliquely at a Plane. — The diagram (Fig. 66) shows 
the paths of two rays SBD and SCE which originating at S 
and falling on the refracting plane ZZ at the points B and C 
are refracted in the directions CE and BD into the eye of an 
observer. The refracted rays produced backwards intersect 

at S' and cross the normal 
SA at the points marked 
WandV. Evidently, all 
the rays from S that fall 
on the refracting plane at 
points between B and C 
will, after refraction, in- 
tersect SA at points be- 
tween V and W. Sup- 
pose that the figure is 

Fig. 66. — Oblique refraction at plane sur- revolved around SA as 
face (n'<n). ^ then each my ^yj 

generate a conical surface, and the vertices of these cones will 
be at the points S, V, and W for the rays that are actually 
drawn in the diagram. The bundle of rays that enter the eye 
at DE will be a small portion of the refracted rays that are 
contained between the conical surfaces whose vertices are 
at V and W. These conical surfaces intersect each other in 
the circle which is described by the point S' when the figure 
is rotated around the axis SA, and it is a little element of 
arc of this circle perpendicular to the plane of the diagram 
at S' that contains the points of intersection of the rays that 
enter the eye. This is called the primary image-line (§188) of 
the narrow bundle of refracted rays. There is another 
image line at V called the secondary image-line, which lies in 
the plane of the paper, and which is generally taken as per- 
pendicular to the axis of the bundle of refracted rays, though 
sometimes it is considered as the segment VW of the axis 
of revolution. But these are intricate matters that can be 
only alluded to in this place. (See Chapter XV.) 



§44] 



Path of Ray through Plate 



101 



44. Path of a Ray Refracted Through a Slab with Plane 
Parallel Sides. — When a ray of light traverses several media 
in succession, then 

ui . sin ai = rbi . sin a/, n^ . sin a 2 = n 3 . sin a 2 ' ', etc. , 
where n h n^, n 3 , etc., denote the indices of refraction of the 
media, and a h oi'; a 2 , 02'; etc., denote the angles of incidence 
and refraction at the various surfaces of separation. In the 




Fig. 67. — Path of ray refracted through plate with plane parallel sides. 

special case when these refracting surfaces are a series of 
parallel planes, the angle of incidence at one plane will be 
equal to the angle of refraction at the preceding plane 
( a k + i = a/, where the integer k denotes the number of the 
plane). 

The simplest case of this kind occurs when there are only 
two parallel refracting planes, and when the last medium is 



102 Mirrors, Prisms and Lenses [§ 44 

the same as the first, as, for example, in the case of a slab 
of glass bounded by plane parallel sides and surrounded by 
air, as represented in Fig. 67. Then 

n<i = ni = n, ri2 = n', 
and ai=a 2 =a'. 

Accordingly, we have the following pair of equations: 

n . sin ai = n' . sin a f , n' . sin a' = n . sin a 2 f ; 
and, therefore: 

0.2 — CLi = a; 
which means that the ray emerges from the slab in the same 
direction as it entered it. Thus, when a ray of light traverses 
a slab with plane parallel sides which is bounded by the same 
medium on both sides, the emergent ray will be parallel to the 
incident ray. Obviously, this statement may be amplified 
as follows: When a ray of light traverses a series of media each 
separated from the next by one of a series of parallel refracting 
planes, the final and original directions of the ray will be 
parallel, provided the first and last media have the same index 
of refraction. 

The only effect of the interposition of the glass plate 
(Fig. 67) in the path of the ray is to shift the path to one 
side without altering the direction of the ray. It might be 
inferred, therefore, that the apparent position of an object 
as seen through such a plate of glass would not be altered, 
but this is not true in general, as we shall proceed to explain. 
Every ray that traverses the plate will be found to be dis- 
placed at right angles to its original position through a dis- 
tance 

sin(a-aO 
cos a' ' 
where d denotes the thickness of the plate. Since 

\/n' 2 -n 2 .sin 2 a 

cos a = , 

n' 

the formula above may be put also in the following form : 
"R t>_ sma (~\/^ /2 -^ 2 sin 2 a-n.cosa ) , 
\/V 2 -ft 2 .sin 2 a 



§ 44] Plate with Plane Parallel Faces 103 

Accordingly, the shift B 2 D varies with the slope of the in- 
cident ray. If the object is very far away, the rays that 
enter the eye will be parallel, so that the apparent position 
of a distant object will not be altered in the slightest by view- 
ing it through a plate of glass with plane parallel sides, no 
matter what may be the angle of incidence of the rays, and 
consequently the plate may be turned to the rays at dif- 



0&^ 

^k* 




Fig. 68. — Apparent position of object seen through plate with 
plane parallel sides 

ferent angles without producing any change in the appear- 
ance of the object as seen through it. But if the object- 
point S (Fig. 68) is near at hand, an eye at E will see it in 
the direction ES, but when the glass is interposed, it will 
appear to lie in the direction ES' which is sensibly different 
from ES, and this difference can be increased or diminished 
by rotating the plate around an axis perpendicular to the 
plane of the figure. This principle is utilized very ingeniously 
in the original form of ophthalmometer designed by Helm- 
holtz (1821-1894) for measuring the curvatures of the re- 
fracting surfaces of the eye. It is employed also in an instru- 
ment for measuring the diameter of a microscopic object, 
which Professor Poynting has called the " parallel plate mi- 
crometer" (see Proc. Opt. Convention, London, 1905, p. 79). 



104 Mirrors, Prisms and Lenses [§ 45 

45. Segments of a Straight Line. — The finite portion of 
a straight line included between two points is called a segment 
of the line, while each of the other two parts of the line is to 
be regarded as a prolongation of the segment. Considered 
as generated by the motion of a point along a straight line 
from a starting-point or origin A to an end-point or terminus 

A& B, the segment AB is 

_ g frequently spoken of also 

< bZ m as the step from A to B 

Fig. 60.— Segments of a straight line: Or the Step AB. The 

AB = -ba. order of naming the two 

capital letters placed at the ends of a segment describes 
the sense of the motion or the direction of the segment. Thus, 
with respect to direction the step BA (Fig. 69) is exactly the 
reverse of the step AB. 

Two steps AB and CD are said to be congruent, that is, 
AB = CD, 
provided these steps are not only equal in length but ex- 
ecuted in the same sense. 

If A, B, C are three points ranged along a straight line in 
any order, that is, if AB and CD are two steps along the same 
straight line such that the end of one step is the starting 
point of the other, then the step AC is said to be equal to 
the sum of the steps AB and BC; thus, 

AB+BC = AC; 
and hence also : 

AB = AC-BC, BC = AC-AB. 
Moreover, if we suppose that the point C is identical with 
the point A, it follows that 

AB+BA = 0orAB= -BA. 
Thus, if one of the two directions along a straight line is 
regarded as the positive direction, the opposite direction is 
to be reckoned as negative. For example, if the distance 
between A and B is equal to 12 linear units, and if we put 
AB= +12, thenBA= -12. 



46] 



Apparent Position 



105 



Similarly, also, we may write: 

AB+BC+CA = 0; 
or if X designates the position of any fourth point on the 
straight line, then 

AB+BC+CX = AX. 

These ideas will be found to be of great service in treating 
a certain class of problems in geometrical optics; and an 
application of this method of adding line-segments occurs 
in the following section. 

46. Apparent Position of an object seen through a 
transparent Slab whose Parallel Sides are perpendicular 
to the Line of Sight. — In 
Fig. 70 the line of sight 
joining the object-point 
Mi with the spectator's 
eye at E is perpendicular 
at Ai and A 2 to the paral- 
lel faces of the transpar- 
ent slab, and all the rays 
that enter the eye will 
pass through the slab 
close to this axial line. 
Inside the slab they will proceed as if they had originated at 
a point Mi' on the line of sight, but being again refracted, 
they will emerge into the surrounding medium as if they had 
come from a point M 2 ', which is the apparent position of the 
object-point as seen by rays that are very nearly perpen- 
dicular to the faces of the slab. If n, n' denote the indices 
of refraction of the two media, then, according to §§ 41 and 
45, we may write the following equations: 



n 


- '.*' "' 


n 


k\ M, M^ A, 


y-. .j a * ' E 



Fig. 70. — Displacement of object viewed 
perpendicularly through plate with 
plane parallel sides. 



AiMi AiMi' 



A 2 Mi , = A 2 A 1 +A 1 M 1 / , 



A 2 Mi' A 2 M 2 ' 



106 Mirrors, Prisms and Lenses [§ 46 

Hence, the apparent displacement of the object is: 
MiM 2 ' = MiAi+AiA 2 +A 2 M 2 ' 

77 

= MiA 1 +A 1 A 2 +-A 2 M 1 / 

lb 

77 

= M 1 A 1 +A 1 A 2 +- (A 2 A 1 +A 1 M 1 ') 

lb 

=M 1 A 1 +A 1 A 2 (l-^)+A 1 M 1 =^A I A 2 ; 

lb lb 

accordingly, if the thickness of the plate is denoted by 
d = AiA 2 , 

Ml M 2 ' = ^d. 

Thus, we see that the apparent displacement in the line of 
sight depends only on the thickness of the plate and on the 
relative index of refraction (V: n), and is entirely independent 
of the distance of the object-point from the slab. Hence, 
also, the size of the image of a small object viewed directly 
through a glass plate is the same as that of the object, but 
its apparent size will be different, because since the image 
and object are at different distances from the eye, the angles 
which they subtend will be different. 

An object viewed perpendicularly through a glass plate 
surrounded by air (n r : n = 3 : 2) will appear to be one-third 
the thickness of the plate nearer the eye than it really is. 

If the displacement of the object is denoted by x, that is, 
if we put MiM' 2 = x, then 

n'_ d 
n d-x' 
This relation has been utilized in a method of determining 
the relative index of refraction {n'\ n). A microscope S 
pointed vertically downwards is focused on a fine scratch 
or object-point O. A plate of the material whose index 
is to be determined is then inserted horizontally between 
the object and the objective of the microscope. The inter- 
position of the plate necessitates a re-focusing of the micro- 
scope in order to see the object distinctly, which will 



§47] 



Multiple Images 



107 



now appear to be at a point 0' nearer the microscope by 
the distance # = 00'. This distance x is easily ascertained 
in terms of the distance through which the objective of the 
microscope has to be raised in order to obtain a distinct image 
of the object. The thickness of the plate is easily measured, 
and, consequently, we 
have all the data for de- 
termining the value of 
n'jn. This method is 
especially convenient for 
obtaining the index of 
refraction of a liquid 
(Fig. 71). 

47. Multiple Images 
in th e two Parallel Faces 
of a plate glass Mir- 
ror. — An object is repro- 
duced in a metallic mir- 




Fig. 71.- 



-Measurement of index of refrac- 
tion of a liquid. 



ror by a single image, but in a glass mirror which is silvered 
on the back side there will be a series of images of an object 
in front of the glass, which may be readily seen by looking a 
little obliquely at the reflection of a candle-flame in an or- 
dinary looking glass. The first image will be comparatively 
faint, the second one the brightest and most distinct of all, 
and behind these two principal images other images more 
or less shadowy may also be discerned whose intensities 
diminish rapidly until they fade from view entirely. These 
multiple images by reflection may also be seen in a trans- 
parent block of glass with plane parallel sides. 

The light falling on the first surface is partly reflected and 
partly refracted. It is this reflected portion that gives rise 
to the first image of the series. The rays that are refracted 
across the plate will be partly reflected at the second face, 
and, returning to the first face, a portion of this light will be 
refracted back into the air and give rise to the second image 
of the series; while the other portion of the light will be re- 



108 



Mirrors, Prisms and Lenses 



[§47 



fleeted back into the glass to be again reflected at the back 
face, and so on. In the diagram (Fig. 72) the source of the 
light is supposed to be at the point marked S, and the straight 




Fig. 72. — Multiple images by reflection from 
the two parallel faces of a plate of glass. 

line drawn from S perpendicular to the parallel faces of the 
glass slab meets these faces in the points marked Ai and A 2 . 
The path of one of the rays coming from S is indicated in 



§47] Multiple Images 109 

the figure, and it can be seen how it zigzags back and forth 
between the two sides of the slab, becoming feebler and 
feebler in intensity at each reflection. We consider here only 
such rays from S as meet the surface very nearly normally. 
The series of images of S will be formed at S', S", S"', etc., 
all lying on the prolongation of the normal SAiA 2 , and it is 
because these images are all ranged in a row one behind the 
other, that ordinarily when we look in a mirror we do not see 
the images separated. 

The reflected ray 1 proceeds as if it had come from S', the 
position of this point being determined by the relation A]S' = 
SAi. But the refracted ray crosses the slab as if it had come 
from the point T, the position of which is determined by the 
relation TAi = n.SAi, where n denotes the index of refraction 
of the glass (the other medium being assumed to be air of 
index unity). Arriving at the second face, this ray will be 
reflected as if it had come from a point U such that A 2 U = 
TA 2 . Returning to the first surface, it will be partly re- 
fracted out into the air as the ray marked 2 proceeding as 
if it came from the second image-point S", the position of 
which is determined by the relation AiS" = AiU/n; and also 
partly reflected as if it had come from a point V such that 
VAi = AiU. The ray is reflected a second time at the second 
face, as if it came from the point W, where A 2 W = VA 2 ; and 
being once more refracted at the first face, emerges into the 
air as the ray marked 3, appearing now to come from the 
image-point marked S'" determined by the relation AiS'" = 
AiW/n. 

What is the interval between one image and the next? 
For example, let us try to obtain an expression for the inter- 
val S"S'". This may be done as follows: 
S"S'" = S"Ai+AiS'"; 

AiS"' = AiW/n= (AiA 2 +A 2 W)/n= (AiA 2 +VA 2 )/n 
= (A!A 2 +VAi+AiA 2 )/tt= (AiU+2AiA 2 )/n 
= AiS"+2d/n; 



110 Mirrors, Prisms and Lenses [Ch. IV 

where d = AiA 2 denotes the thickness of the glass plate. 
Hence, we find : 

n ' 

It appears, therefore, that the distance between one image 

2 
and the next is constant and equal to - times the thickness 

of the plate. Thus, for a glass plate for which n = 3/2 the 
distance from one image to the next is equal to 4/3 the thick- 
ness of the plate. 

PROBLEMS 

1. A ray of light traverses in succession a series of isotropic 
media bounded by parallel planes, and emerges finally into 
a medium with the same index of refraction as that of the 
first medium. Show that the final path of the ray is parallel 
to its original direction. 

2. Construct accurately the paths of six rays proceeding 
from a point below the horizontal surface of water and re- 
fracted into air; and show where the object-point will appear 
to be as seen by an eye above the surface of the water, for 
three different positions of the eye. 

3. Why does the part of a stick obliquely immersed in 
water appear to be bent up towards the surface of the water? 
Explain clearly. 

4. Derive the formula —, = - for the refraction of paraxial 

u u 

rays (§63) at a plane surface. 

5. A ray of light incident on a plane refracting surface at 
an angle a crosses a straight line drawn perpendicular to the 
surface at a distance v from this surface. How far from the 
surface does the refracted ray cross this line? 

6. If a bird is 36 feet above the surface of a pond, how high 
does it look to a diver who is under the water? What is the 
apparent depth of a pool of water 8 feet deep? 

Ans. 48 feet above the surface; 6 feet. 



Ch. IV] Problems 111 

7. What will be the effect on the apparent distance of an 
object if a slab of transparent material with plane parallel 
sides is interposed at right angles to the line of vision? 

Ans. It will appear to be nearer the eye by the amount 
(n— I) jd, where d denotes the thickness of the slab and n de- 
notes the index of refraction of the material. 

8. A cube of glass of index of refraction 1.6 is placed on a 
fiat, horizontal picture; where does the picture appear to be 
to an eye looking perpendicularly down on it? 

Ans. It will appear to be raised three-eighths of the thick- 
ness of the cube. 

9. A microscope is placed vertically above a small vessel 
and focused on a mark on the base of the vessel. A layer of 
transparent liquid of depth d is poured in the vessel, and then 
it is found that the image of the mark has been displaced 
through a distance x which is determined by re-focusing the 
microscope. Show that the index of refraction of the liquid 
is equal to d/(d — x). 

10. In an actual experiment made by the above method to 
determine the index of refraction of alcohol, the depth of the 
liquid was 4 cm., and the displacement of the image was 
found to be 1.06 cm. What value was found for the index of 
alcohol? Ans. 1.36. 

11. A candle is observed through a tank of water with 
vertical plane glass walls. The line of sight is perpendicular 
to the sides of the tank, the candle being 15 cm. from one 
side and 39 cm. from the opposite side. What is the apparent 
position of the candle? (Neglect the effect of the thin glass 
walls.) Ans. It appears to be 9 cm. from the near side. 

12. If an object viewed normally through a plate of glass 
with plane parallel faces seems to be five-sixths of an inch 
nearer than it really is, how thick is the glass? 

Ans. 2.5 inches. 

13. A layer of ether 2 cm. deep floats on a layer of water 
3 cm. deep. What is the apparent distance of the bottom of 



112 Mirrors, Prisms and Lenses [Ch. IV 

the vessel below the free surface of the ether? (Take index of 
refraction of water =1.33 and of ether = 1.36.) 

Ans. 3.73 cm. 

14. A person looks perpendicularly into a mirror made 
of plate glass of thickness one-half inch silvered on the back. 
If his eye is at a distance of 15 inches from the front face, 
where will his image appear to be? 

Ans. 15 2 /3 inches from the front face. 

15. When a stick is partly immersed in a transparent 
liquid of index n at an angle 6 with the free horizontal sur- 
face, what is the angle 6 ' which the part of the stick below 
the surface appears to make with the horizon as seen by an 
eye looking vertically down on it from the air above the 
liquid? 

tan0 
Ans. tancr = . 



CHAPTER V 

REFRACTION THROUGH A PRISM 

48. Definitions, etc. — An optical prism is a limited portion 
of a highly transparent substance with polished plane faces 
where the light is reflected or refracted. Prisms in a 
great variety of geometrical forms and combinations are 
employed in many types of modern optical instruments (cf. 
§§ 20, 37) ; but in this chapter the term prism will be re- 
stricted to mean a portion of a transparent, isotropic sub- 
stance included between two polished plane faces that are 
not parallel. The straight line in which the planes of the 
two faces meet is called the edge of the prism, and the di- 
hedral angle between these planes is called the refracting angle. 
This angle, which will be denoted by the symbol /3, may be 
more precisely defined as the convex angle through which the 
first face of the prism has to be turned around the edge of the 
prism as axis in order to bring this face into coincidence with 
the second face. The first face of the prism is that side where 
the rays enter and the second face is the side from which the 
rays emerge. Every section made by a plane perpendicular 
to the edge of the prism is a principal section, and we shall 
consider only such rays as traverse the prism in a principal 
section, not only because the problem of oblique refraction 
through a prism presents some difficulties which are beyond 
the scope of this volume, but especially because in actual 
practice the principal rays are usually confined to a principal 
section of the prism. It will also be assumed, for simplicity, 
that the prism is surrounded by the same medium on both 
sides. 

I. Geometrical Investigation 

49. Construction of Path of a Ray Through a Prism. — 
The plane of the diagram (Fig. 73) represents the principal 

113 



114 Mirrors, Prisms and Lenses [§ 49 

section of a prism whose edge meets this plane perpendicu- 
larly at the point marked V. The traces of the two plane 
faces are shown by the straight lines ZiV, Z 2 V intersecting at 
V. The straight line ABi represents the path of the given 
incident ray lying in the plane of the principal section and 




Fig. 73. — Construction of path of ray through principal section 
of prism (n'>n). 

falling on the first face of the prism at the incidence-point Bi. 
The problem of constructing the path of the ray both within 
the prism and after emergence from it is solved by a method 
essentially the same as that employed in § 34. 

Let n denote the index of refraction of the medium sur- 
rounding the prism and n' the index of refraction of the prism- 
medium itself. With the point V as center, and with radii 

equal to r and — .r, where the radius r may have any con- 
venient length, describe the arcs of two concentric circles both 
lying within the angle Z 2 VE, where E designates a point on 
the prolongation of the straight line ZiV beyond V. Through 



§49] Construction of Ray through Prism 115 

V draw a straight line VG parallel to ABi meeting the arc 
of radius nr/n' in the point designated by G; and through 
the point G draw a straight line GE perpendicular at E to 
the first face of the prism (produced if necessary), and let H 
designate the point where the straight line GE (likewise 
produced if necessary) meets the circumference of the other 
of the two circular arcs. Then the straight line BiB 2 drawn 
parallel to the straight line VH will represent the path of 
the ray within the prism. For if the straight line NiN/ is the 
incidence-normal to the first face of the prism at the point 
Bi, and if the angles of incidence and refraction at this face 
are denoted by ai = ZNiB i A,ai , = ZNi / BiB 2 , then by the 
law of refraction : 

n.sinai = n'.sinai'.- 
But by the construction : 

sinZEGV VH n' 

sinZEHVVG-n' 

and since Z EGV= Z NiBiA = ai, it follows that Z EHV = a/; 
and hence the path of the ray within the prism must be 
parallel to VH. 

Again, from the point H let fall a perpendicular HF on 
the second face of the prism, where F designates the foot of 
this perpendicular; and let J designate the point where HF 
intersects the arc of radius nr/n'. Then the straight line 
B 2 C drawn from the incidence-point B 2 parallel to the straight 
line VJ will represent the path of the emergent ray. For if 
we draw N 2 N 2 ' perpendicular to the second face of the prism 
at B 2 , and if the angles of incidence and refraction at this 
face are denoted by a 2 = ZN 2 B 2 Bi, a 2 ' = ZN 2 'B 2 C, re- 
spectively, then n' . sin a 2 = n . sin a 2 '. But 

sinZFJV VH n' 

sinZFHV" VJ~n' 

and since by construction ZFHV= a 2 , it follows that 
ZFJV= a 2 ', and hence the path of the emergent ray will 
be parallel to VJ. 



116 Mirrors, Prisms and Lenses [§ 50 

The diagram (Fig. 73) is drawn for the case when n f >n, as 
in the ordinary case of a glass prism surrounded by air. The 
student should draw also a diagram for the other case when 
n'<n, showing the procedure of a ray through a prism of less 
highly refracting substance than that of the surrounding 
medium, for example, an air prism surrounded by glass, such 
as is formed by the air-space between two separated glass 
prisms. 

50. The Deviation of a Ray by a Prism. — The total de- 
viation of a ray refracted through a prism, which is equal to 
the algebraic sum of the deviations produced by the two 
refractions (§35), may be defined as the angle e= ei~\r e 2 
through which the direction of the emergent ray must be 
turned in order to bring it into the direction of the incident 
ray; thus, in Fig. 73, e = ZJVG; and if the angle e is meas- 
ured in radians, the arc JG = e . J V. In order to specify 
completely an angular displacement, it is necessary to give 
not only the magnitude of the angle and the sense of rotation 
of the radius vector, but also the plane in which the displace- 
ment occurs. This plane may be specified by giving the 
direction of a line perpendicular to it, which in the case of 
the angle here under consideration may be the edge of the 
prism or any line parallel to it; because any such line will 
be perpendicular to the principal section of the prism in 
which the ray lies. In fact, the angle e may be completely 
represented in a diagram by a straight line drawn parallel 
to the edge of the prism, which by its length indicates the 
magnitude of the angle and by its direction shows the sense 
of rotation. Thus, for example, the line may be drawn along 
the edge of the prism itself from a point V in the plane of the 
principal section and always in such a direction that on 
looking along the line towards that plane Z JVG = e will 
be seen to be a counter-clockwise rotation. A deviation of 
20° in a principal section coinciding, say, with the plane of 
the paper would be represented, therefore, by a straight line 
perpendicular to this plane of length 20 cm., if each degree 



§ 51] Ray "Grazes" one Face of Prism 117 

were to be represented by one centimeter. If e= +20°, this 
line would point out from the paper towards the reader, and 
if € = -20°, it would point away from him. Thus, if the 
prism, originally "base down," is turned "base up" (as the 
opticians say), everything else remaining the same, the sign 
of the angle e will be reversed, and so also will be the direc- 
tion of the vector which represents this angle. 

51. Grazing Incidence and Grazing Emergence. — The 
angle GHJ between the normals to the two faces of the prism 
is equal to the refracting angle (3; and hence for a given prism 
this angle will remain always constant. No matter how the 
direction of the incident ray ABi (or VG) may be varied, 
the vertex H of this angle will lie always on a certain portion 
of the circumference of the construction-circle of radius r, 
and the sides HG, HJ will remain always in the same fixed 
directions perpendicular to the faces of the prism. Obviously, 
there will be two extreme or limiting positions of the point H 
marking the ends of the arc on which it is confined, namely, 
the positions winch H has when one of the sides of the angle 
GHJ is tangent to the circle of radius nrjn'; which can occur 
only for the case when n'>n, because otherwise the point H 
will lie inside the circumference of this circle and therefore 
it will be impossible for either HG or HJ to be tangent to it. 

If the side HG is tangent to the inner circle at G, as shown 
in Fig. 74, the point G will lie in the plane of the first face 
of the prism, and accordingly the corresponding ray incident 
on the first face of the prism at the point Bi, which must 
have the direction VG, will be the ray ZiBi which, entering 
the prism at "grazing" incidence (ai = 90°), traverses the 
prism as shown in the figure. 

On the other hand, when the side HJ of the angle GHJ is 
tangent at J to the construction-circle of radius nr\n' (Fig. 75), 
the point J will lie in the second face of the prism, and the 
straight line VJ will coincide with the straight line VZ 2 . 
Under these circumstances the ray emerges from the prism at 
B 2 along the second face in the direction B 2 Z 2 (a 2 / '= -90°). 



118 



Mirrors, Prisms and Lenses 



51 



The straight line KB X shows the path of the ray incident on 
the first face of the prism at Bi which " grazes" the second 
face on emerging from the prism. Any ray incident at Bi and 
lying in the principal section of the prism within the angle 
KB1Z1 will succeed in getting through the prism and emerging 




Fig. 74. — Case when ray "grazes" first face of prism. 



into the surrounding medium again; whereas if the ray in- 
cident at Bi lies anywhere within the angle VBiK, it will 
be totally reflected at the second face of the prism. The 
ray KBi is called the limiting incident ray and ZNiBiK = t 
is the limiting angle of incidence. These relations will be 
discussed more fully in the analytical investigation of the 
path of a ray through a prism (§§ 55, foil.) ; but it may be 
remarked that ZGHV= a/ in Fig. 74 and ZJHV= a 2 in 
Fig. 75 are both equal to the critical angle A (§ 36) with 
respect to the two media n, n' (sinA=n/n'). 



52] 



Minimum Deviation 



119 



52. Minimum Deviation. — Between the two extreme or 
terminal positions of the vertex H of ZGHJ shown in 
Figs. 74 and 75, there is also an intermediate place which is 
of special interest and importance and to which, therefore, 
attention must be called. In general, the sides HG, HJ inter- 




Fig. 75. — Case when ray "grazes" second face of prism. 

cepted between the two construction-circles (Fig. 73) will 
be unequal in length, but if HG = H J, as in Fig. 76, the angles 
GVJ, GHJ and EVF will evidently all be bisected by the 
diagonal VH of the quadrangle VGHJ. When this happens, 
the path BiB 2 of the ray inside the prism, which is parallel 
to VH, crosses the prism symmetrically, that is, the triangle 
VBiB 2 is isosceles. In fact, the points designated in the dia- 
gram by the letters V, D and O will be the summits of isos- 
celes triangles having the common base BiB 2 , and they will 
all lie therefore on the bisector of the refracting angle /3 = 



120 Mirrors, Prisms and Lenses [§ 52 

ZZiVZ 2 , which is perpendicular to VH. The angle of in- 
cidence at the first face and the angle of emergence at the 
second face are equal in magnitude, although they are de- 
scribed in opposite senses, so that 012' = — ai. The same is 
true also in regard to the angles which the ray makes inside 




Fig. 76. — Ray traverses prism symmetrically (VBi = VB2) ; case 
of minimum deviation. 

the prism with the normals to the two faces, that is, ci2 = 
-a/. 

Now when the ray traverses the prism symmetrically, as 
represented in Fig. 76, the deviation e has its least value 
€ Q . In order to show that this is true, it will be convenient 
to reproduce the symmetrical quadrangle VGHJ in Fig. 76 
in a separate diagram, as in Fig. 77. Suppose that H' desig- 
nates the position of a point infinitely near to H lying likewise 
on the arc of the circle of radius r, and draw H'G', H'J' 
parallel to HG, HJ and meeting the arc of the other circle 
in the points G', J', respectively. In the figure the point H' 
is taken below the point H, and in this case it is plain that 




Fig. 77. — Case of minimum deviation. 



§ 52] Minimum Deviation 121 

the two parallels HJ, H'J' will meet the circumference of 

the inner circle more obliquely than the other pair of parallel 

lines HG, H'G', and, consequently, the infinitely small arc 

J' J intercepted between 

the first pair will be 

greater than the arc G'G 

intercepted between the 

second pair. Hence, the 

small angle J'VJ will be 

greater than Z G'VG, and 

therefore 

ZJ'VG'>ZJVG. 
The angle JVG here is the 
angle of deviation ( e ) of 
the ray that goes sym- 
metrically through the prism; whereas Z J' VG /= € is the angle 
of deviation of a ray which traverses the prism along a very 
slightly different path. And according to the above reason- 
ing (for we shall arrive at the same result if we take the 
point H' also above H), we find: 

e> e . 
Accordingly, we see that the ray which traverses the prism sym- 
metrically in the plane of a principal section is also the ray 
which is least deviated. 

It is easy to verify this statement experimentally. Thus, 
for example, if a bundle of parallel rays is allowed to fall on 
an isosceles triangular prism, so that while some of the rays 
are incident on one of the equal faces and are transmitted 
through the prism, the other rays of the bundle are reflected 
from the base of the prism, as represented in (1) in Fig. 78; 
and if then the prism is gradually turned around an axis 
parallel to its edge, first, into position (2), which is the posi- 
tion of minimum deviation, and then past this position into 
a third position (3), it will be observed that when the prism 
is in the position of minimum deviation the rays reflected 
from the base will be parallel to the rays which emerge at 




122 Mirrors, Prisms and Lenses [§ 53 

the second face of the prism; which can only be the case 

when the rays cross the prism symmetrically. 

In spectroscopic work and in many other scientific uses of 

the prism, the position of mini- 
mum deviation, which is easily 
found, is frequently the most 
convenient and advantageous ad- 
justment of the prism for purposes 
of observation. 

53. Deviation away from the 
Edge of the Prism. — When a ray 
of light passes through a prism of 
more highly refracting material 
than that of the surrounding me- 
dium (n'>n), the deviation is al- 
ways away from the edge towards 
the thicker part of Ihe prism. 

If the angles of the triangle 
VBiB 2 (Fig. 79) at B x and B 2 are 
both acute, the incident and 
emergent rays lie on the sides of 
the normals at Bi and B 2 away 

Fig. 78. — Experimental proof e r* • j ,1 , , 

that ray which traverses fr0m the P^sm-edge, SO that at 

prism symmetrically is ray both refractions the ray will be 

of minimum deviation. u , £ ,-, •, T£ 

bent away trom the edge. It 
one of the angles, say, the angle at B 2 , is a right angle, 
the ray will not be deviated at all by the refraction at 
this point, but at the other incidence-point it will be bent 
away from the edge. And, finally, if one of the angles at Bi 
or B 2 is obtuse, for example, the angle at Bi (Fig. 80), the 
deviation on entering the prism will, it is true, be towards 
the edge of the prism, but this deviation will not be so great 
as the subsequent deviation away from the edge which is 
produced at the second refraction when the ray issues from 
the prism, as may be easily seen from the diagram. Thus, 




§ 54] Plane Wave Refracted Through Prism 123 

in every case when n f >n, the total deviation will be away 
from the prism-edge. 

If n'<n, all these effects will be reversed. 




Fig. 79. — Deviation away from edge of prism. 

54. Refraction of a Plane Wave Through a Prism. — 

The diagram (Fig. 81) shows a principal section of the prism, 
and the straight line BiD represents the trace of a plane 
wave (supposed to be perpendicular to the plane of the 
paper and parallel therefore to the 
edge of the prism) advancing to- 
wards the first face of the prism in 
the direction DV at right angles to 
BiD. If around the point Bi, which 
lies in the first face of the prism, the 
arc of a circle is described with ra- 

71 

dius BE = -,DV, then, according to 

HUYGENS'S principle (§ 5), the Fig. 80 — Deviation away 

straight line VE tangent to this circle from edge of prism ' 

at E will represent the trace of the wave-front inside the prism. 

Let the straight line BiE meet the second face of the prism 

n' 
at B 2 . Around V as center with radius VF= - EB 2 describe 




the arc of a circle; then the straight line B 2 F tangent to this 



124 



Mirrors, Prisms and Lenses 



55 



circle at F will represent the trace of the emergent wave- 
front. 

The disturbance at any point C will have emanated from 
some point on ABi, and the time 
taken by the light to go from Bi to 
B 2 inside the prism will be the 
same as that required to go from 
D to F in the surrounding me- 
dium (§ 39) ; that is, the optical 
lengths along these two routes 
are equal. For, as appears from 

Fig. 81.— Refraction of plane ^he Construction, 

wave through prism. n (DV+ VF) = n'.BiB 2 . 

An excellent and most instructive mechanical illustration 
of the refraction of a plane wave through a prism can be ob- 
tained by using the roller and tilted board described in § 32 
with a triangular piece of plush cloth glued in the middle 
of the board to represent the prism (see Fig. 45). 




II. Analytical Investigation 

55. Trigonometric Calculation of the Path of a Ray in a 
Principal Section of a Prism. — The angles of incidence and 
refraction at the first and second faces of the prism, denoted 
by ai, ai' and a 2 , a 2 ', are, by definition (§ 27), the acute angles 
through which the normals to the refracting surfaces at the 
incidence-points have to be turned in order to bring them into 
coincidence with the incident and refracted rays at the two 
faces of the prism; thus, in Fig. 73, ZNiBiA= rii, ZNi'BiB 2 
= ai', ZN 2 B 2 Bi= a 2 , ZN 2 / B 2 C= a 2 '. 

Assuming that the prism is surrounded by the same me- 
dium on both sides, and being careful to note the sense of 
rotation of each of the angles, we obtain by the law of re- 
fraction, taken in conjunction with the obvious geometrical 
relations as shown in the figure, the following system of 



§ 56] Total Reflection in Prism 125 

equations for calculating the path of a ray through a prin- 
cipal section of a prism : 

n'.sinai' = 7i.sinai, a 2 = a/— fi, n . sin a 2 ' = n' . sin a 2 . 
Combining these formulse so as to ehminate ai' and a 2 , we 
may derive the following convenient expression for deter- 
mining the angle of emergence (a 2 ') at the second face of the 
prism : 

, o nVn' 2 -n 2 . sin 2 a\ 

sin a 2 = sm ai . cos p - sin p . 

n 

Thus, if we know the value of the relative index of refraction 

(n'/ri) and the refracting angle of the prism (/3=ZZiVZ 2 ), 

we can calculate the angle of emergence ( a 2 ') corresponding 

to any given direction (ai) of the ray incident on the first 

face of the prism. 

The total deviation ( e ) of a ray refracted through a prism 

is measured, as defined above (§50), by ZJVG, and since 

this angle is equal to the external angle at D in the triangle 

DBiB 2 , we have: 

€=ZB 2 BiD+ZDB 2 Bi 

= Z Ni'BiD - Z NiBiB 2 -f Z DB 2 N 2 - Z BiB 2 N 2 

= ai— a/— a 2 '+ a 2 ; 
and since a\ — a 2 =/3, we obtain finally the following ex- 
pression for the angle of deviation: 

e= ai- a 2 '-/3. 
These formulae contain the whole theory of the refraction of 
a ray through a prism in a principal section. It will be in- 
teresting to discuss analytically some of the special cases 
which we have already studied in the preceding sections of 
this chapter. 

56. Total Reflection at the Second Face of the Prism. — 
If the angle of emergence at the second face of" the prism is 
a right angle, that is, if a 2 ' = -90°, the emergent ray B 2 C 
will issue from the prism along the second face in the direc- 

Tl Tl 

tion B 2 Zi (Fig. 75) . Hence, sin <x 2 = — . sin a 2 ' = -„ and there- 

Tl Tl 

fore a 2 = - A, where A denotes the critical angle (§ 36) of the 



126 Mirrors, Prisms and Lenses [§ 56 



n 



media n, n', denned by the relation sinA = — . If the absolute 

value of the angle a 2 is greater than A, the ray will be totally 
reflected at the second face of the prism, and there will be no 
emergent ray. This case may be discussed in some detail. 

For a prism of given refracting angle (/J), there is a certain 
limiting value (t) of the angle of incidence ( cti) at the first 
face of the prism (§ 51) for which we shall have at the second 



/ 
/ 
/ 
1 
1 




\ 

\ 

X 


1 


\ 


^V. 1 


t 


&r 


Xl 


t 


n t 


— 7& 




1 
/ 


/ 1 


'•tk 


/ y 


r 1 




i jS 


1 






1 
1 



, t*4 = 



^=A. 



-K 



Fig. 82. — Prism with refracting angle /3 = 2A. 

face the values a 2 = -A, a 2 ' = -90°; so that a ray which is 

incident on the first face of the prism at an angle less than the 

limiting angle i will not pass through the prism but will be 

totally reflected at the second face. Putting a 2 = — A, we find 

a\ = /3 - A, and therefore, since ai = i, 

n' 
. sin i=— sin 'jfl~A), 

which is the trigonometric formula for computing the value 
of the limiting angle of incidence for a given prism. It will 



§ 56] Limiting Incident Ray 127 

be worth while to examine this formula for certain particular 
values of the refracting angle /3. 

(1) If /3>2A, then, since sinA = -?, the formula shows that 

sin l will be greater than unity, so that for a prism of this 
form there is no angle corresponding to the limiting angle t. 
No ray can be transmitted through a prism whose refracting 
angle is more than twice as great as the critical angle of the two 
media in question. A prism of this size is called a totally 




Fig. 83. — Prism with refracting angle /3 = A. 

reflecting prism; if it is made of glass of index 1 . 5 and sur- 
rounded by air, the refracting angle should be about 84° at 
least. 

(2) If jft = 2A, we find that t = 90°; which is the case repre- 
sented in Fig. 82. The only ray that can get through this 
prism is the ray that traverses it symmetrically, entering the 
prism along one face and leaving it along the other. 

(3) If /3> A but <2A (that is, if 2A> (3> A), the value of 
the angle i as determined by the formula above will be com- 
prised between 90° and 0°. This is the case which was shown 
in Fig. 73. The direction of the limiting incident ray is be- 



128 



Mirrors, Prisms and Lenses 



56 




Fig. 84. 



tween ZiBi and NiBi; that is, ZViBK will be an obtuse 
angle. 

(4) If j8 = A, we find i = 0°, and then the limiting incident 

ray will proceed along the 
normal NiBi, as shown in 
Fig. 83, and ZVBiK (or 
ZVBiA) will be a right 
angle. 

(5) FinaUy, if /5<A,the 
limiting angle of inci- 
dence (i) will be negative 
in sign; and therefore in a 
more or less thin prism of 
this description the limit- 
ing incident ray KBi will 
fall on the side of the 
normal NiBi towards the 
apex V of the prism, so that the angle VBiK will be an 
acute angle (Fig. 84). 

Any ray incident on the first face of the prism at Bi and 
lying within the angle KBiZi will be transmitted through 
the prism; whereas if the ray falls within the supple- 
mentary angle VBiK, it will be totally reflected at the 
second face. 

In Kohlrausch's method of measuring the relative index 

n' 

of refraction (— ), the prism is adjusted so that the incident 

lb 

ray " grazes" the first face, and then if the refracting angle 
of the prism (/3) is known, and if the angle of emergence 
(a 2 r ) is measured, the value of n'\ n may be calculated by 
means of the formula : 

cos/3-sina2' 



-Prism with a refracting angle 
/5<A. 



v/ 



;n 



& "I 



■, (ai = 90°). 



n ■ sin P 

The principle of total reflection is also employed in the 
prism refractometers of Abbe and Pulfrich for measure- 
ment of the index of refraction. 



§ 59] Minimum Deviation 129 

57. Perpendicular Emergence at the Second Face of the 
Prism. — For this case we have d 2 = d 2 ' = 0°, and therefore 
di'= fi, cii= £-€, and hence: 

n'_ sin(ft- e) . 

n sin/3 
which is also a convenient formula for the experimental de- 
termination of the value of the relative index of refraction. 
A description of the apparatus and the method of procedure 
may be found in the standard treatises on physics. 

58. Case when the Ray Traverses the Prism Symmet- 
rically. — As has been pointed out already (§ 52), a special 
case of great interest occurs when the ray traverses the prism 
symmetrically. Under these circumstances, the general 
prism-equations given in § 55 take the following forms: 

di= - d2 = — 2 — ' a i = ~" a 2= 2> 

. /3+€o 
sin — - — 



sin- 



where e Q denotes the angle of deviation of this symmetric 
ray. The last of these formulae is the basis of the Fraun- 
hofer method of determining the relative index of refrac- 
tion, the angles fi and e being both capable of easy measure- 
ment. 

This last formula may also be transformed into the fol- 
lowing form: 

n.sin| 
tan-^ = 



2 , e ' 

n — n.cos-^ 



whereby the refracting angle f3 can be calculated in terms 
of n, n' and e . 

59. Minimum Deviation. — The prism itself is defined by 
its refracting angle (/3) and the relative index of refraction 
(n'/ri). The total deviation (e) of a ray refracted through 



130 Mirrors, Prisms and Lenses [§ 59 

a given prism depends only on the angle of incidence (ai), 
according to the formula: 

€= ai- a 2 '- P; 
for the angle a 2 ' may be expressed in terms of a h /3 and n'[n, 
as we have seen (§ 55). Hence, for a given value of these 
three magnitudes the angle e will be uniquely determined. 
On the other hand, for a given value of the angle e there 
will always be two corresponding values of the angle of in- 
cidence ai; for it is obvious from the principle of the reversi- 
bility of the light-path (§ 29) that a second ray incident on 
the first face of the prism at an angle equal to the angle of 
emergence of the first ray will emerge at the second face at 
an angle equal to the angle of incidence of the first ray at 
the first face, and these two rays will be equally deviated in 
passing through the prism. For example, suppose that the 
values of the angles of incidence and emergence in the case 
of the first ray are ai= 7, a 2 ' = 7': a second ray incident on 
the first face of the prism at the angle ai= — y' will emerge 
at the second face at an angle a 2 ' = — y, and each of these rays 
will suffer precisely the same deviation, viz., e= 7 — 7' — /3. 
Thus, corresponding to any given value of the angle €, 
within certain limits, there will always be a pair of rays which 
are deviated by this same amount. One pair of such rays 
consists of the two identical rays determined by the relation 

di=7=- a 2 '. 
In fact, this is the ray which traverses the prism symmet- 
rically, and a little reflection will show that the deviation of 
this ray must be either a maximum or a minimum. 

But while the best way of demonstrating that the ray 
which goes symmetrically through the prism is the ray of mini- 
mum deviation (§ 52) involves the employment of the methods 
of the differential calculus, the following analytical proof 
demands of the student a knowledge of only elementary 
mathematics. 

The deviation at the first face of the prism is ei = Hi— a/, 



§ 59] Minimum Deviation 131 

and that at the second face is e 2 =a 2 -a 2 ' (§35), and 
hence the total deviation is 

e= €i-f-e 2 = (ai- a/) + (a 2 - a 2 '), 
or, since a/ - a 2 = 13, e = ai — a 2 ' - /3, as has been already 
remarked, for example, in § 55. Assume now that n'>n, and, 
consequently, that the angle e is positive, as is always the 
case when the ray is bent away from the edge of the prism 
(§ 53) ; then it is evident that the angle e will have its least 
value (e ) in the case of that ray for which the function 
(ai— a 2 ') is least. Now since 

n . sin di = n' . sin a/, n . sin a 2 ' = n' . sin a 2 , 
we obtain by subtraction : 

n(sin ai — sin a 2 ') = 7i'(sin a/ — sin a 2 ), 
and hence by an obvious trigonometric transformation : 
. d\- a 2 ' ai+tt/ , . ai'— a 2 ai'+a 2 
n. sin — ^ — - cos — o — =w — 9 — ,cos — 2 — ' 

which may be written as follows: 

. ai-a 2 n . p 2 

sm — ^ — =„ .sm-^.- 



According as ai= — a 2 r , the deviation €i at the first face of 

the prism will (see § 35) be greater than, equal to, or less than, 
the deviation e 2 at the second face; that is, according as 

ai=— a 2 ', we shall have ( ai— di')=( a 2 — a 2 '), and hence also 

ai+a2 / > a 2 +ai / 
2 < 2 ' 

If we suppose, first, that ai> — a 2 r , then a/> — a 2 and 
(a 2 +a/) > 0; and since the cosine of a positive angle de- 
creases as the angle increases, it follows that here we must 
have: 

a/+a 2 ai-f-a 2 ' 

cos x >cos — ^ — • 

On the other hand, if we suppose, second, that ai< — a 2 ', then 



132 Mirrors, Prisms and Lenses [§ 60 

ai'<-a 2 and (a 2 +ai')<0; but in this case (a 2 +ai')> 

( ai+ a 2 '), so that although ( a 2 + a/) and ( cti+ a 2 ') are both 

negative, the absolute value of the former is greater than that 

of the latter, and hence here also we find exactly the same 

result as before. 

Thus, whether ai is greater or less than — a 2 ', the ratio 

ai'+ft2 

cos — ^ — 

aH-a 2 ' ' 
cos- 



2 

and only in the case when ai = — a 2 ' will this ratio equal to 

unity. Hence, sin — s has its least value when a\ = — a/, 

and then also the deviation ( e) is a minimum and equal to 
€ = 2ai-/3. 
The same process of reasoning applied to the case when 
n'<n leads to the conclusion that the angle € will be a maxi- 
mum for the ray which traverses such a prism symmetrically, 
for example, an air-prism surrounded by glass; but in this case 
the angle € will be negative in sign, and since a maximum value 
of a negative magnitude corresponds to a minimum absolute 
value, the actual deviation of the ray is least in this case also. 
60. Deviation of Ray by Thin Prism. — If the refracting 
angle of the prism (/3) is small, as represented, for example, 
in Fig. 85, the deviation (e) will likewise be a small angle 
of the same order of smallness; for if /5 = a/ - a 2 is small, then 
( ai — a 2 ') will be small also, and the angle e is the difference 
between these two small magnitudes. In fact, the deviation 
€ produced by a thin prism will not only always be small, 
but it will never be very different from its minimum value 
e . Accordingly, in the case of a thin prism, we may put 
e = € without much error; and therefore very approxi- 
mately (see § 58) : 

n 2 



n . & 
sin ^ 



60] 



Deviation in Thin Prism 



133 



Consequently, the deviation e, as calculated by this formula, 
will depend only on the prism-constants ( /3, n' : n) and not 
on the angle of incidence ( ai). The smaller the angle /5, the 
more nearly correct this formula will be; and if the angle /3 

is so small that we may substitute ~ and — ^ — in place of 
sin 2 and sin — ^—, respectively, we obtain the exceedingly 




Fig. 85. — Prism with comparatively small refracting 
angle. 



useful and convenient practical relation for the angle of 
deviation of a ray refracted through a thin prism, viz.: 

which, however, is more frequently written: 

«-(»-Dft 

where n is employed now to denote the relative index of 
refraction. Accordingly, in a thin prism the deviation is di- 



134 Mirrors, Prisms and Lenses [§ 61 

redly proportional to the refracting angle. For example, the 
deviation in the case of a thin glass prism surrounded by 
air for which n = 1 . 5 is one-half the refracting angle. 

61. Power of an Ophthalmic Prism. Centrad and Prism- 
Dioptry. — An ophthalmic prism is a thin glass prism, whose 
index of refraction is usually about 1.52, which is used to 
correct faulty tendencies and weaknesses of the ocular 
muscles which turn the eye in its socket about the center of 
rotation of the eye-ball. In an ordinary laboratory prism 
the two faces are usually cut in the form of rectangles having 
the edge of the prism as a common side; but the contour of 
an ophthalmic prism which has to be worn in front of the eye 
in a spectacle-frame is circular or elliptical like that of any 
other eye-glass, and its edge is the line drawn tangent to this 
curve at the thinnest part of the glass. The line drawn 
perpendicular to this tangent at the point of contact and 
lying in the plane of one of the faces of the prism is the so- 
called "base-apex" line, which is a term frequently employed 
by writers on spectacle-optics. 

The formula 

e=(n-l)/3 
obtained in § 69 is peculiarly applicable to the weak prisms 
used in spectacles. As long as the refracting angle of the 
prism does not exceed, say, 10°, the error in the value of e 
as calculated by this approximate formula will be less than 
5 per cent. 

Formerly it was customary to give the strength or power 
of an ophthalmic prism in terms of its refracting angle ft 
expressed in degrees; but the proper measure of this power 
is the deviation produced by the prism. However, instead 
of measuring this angle in degrees, Dennett has suggested 
that the deviation of an ophthalmic prism shall be measured 
in terms of a unit angle called a centrad, which is the one- 
hundredth part of a radian and equal therefore to the angle 
subtended at the center of a circle of radius one meter by an 
arc of length one centimeter. Since 7r radians = 180°, the 




§ 61] Centrad and Prism-Dioptry 135 

relation between the centrad and the degree is given as 
follows : 

1°= ,ft centrads, 

or 

1° = 1 . 745 ctrd., 1 ctrd. = . 573°. 
Prior to this suggestion, Mr. C. F. Prentice, of New 
York, had proposed in 
1888 to measure the de- 
viation of an ophthalmic 
prism in terms of the 
linear or tangential dis- 
placement in centime- 

i j Fig. 86. — Deviation of prism: 

ters on a screen placed tan € =ab:OA. 

at a distance of one 

meter from the prism. If the straight lines OA, OB (Fig. 86) 
represent the directions of the incident and emergent 
rays, respectively, then ZAOB will be the angle of devi- 
ation of the prism; and if a plane screen placed at right angles 

AB 

to OA at A is intersected by OB at B, then tanZ AOB=^-. 

Now if the distance OA = 100 cm. and if AB = z cm., 
then, according to Prentice's method, the ZAOB would 
be an angle of x units and the power of the prism would be 
denoted by x. Dr. S. M. Burnett suggested that the name 
prism-diopter or prism-dioptry be given to this unit. (The term 
"prismoptrie" was proposed by Professor S. P. Thompson.) 
The prism-dioptry is the angle corresponding to a deviation of 
one centimeter on a tangent line at a distance of one meter; 
and, accordingly, when the angle of deviation is equal to the 
angle whose trigonometric tangent is x/100, the power of the 
prism is said to be x prism-dioptries or z A, where the symbol 
A stands for prism-dioptry. The chief objection to be urged 
against this unit of angular measurement is that the angle 
subtended at a given point O (Fig. 87) by equal line-segments 
on a line Ay perpendicular to Ox at A diminishes as the 



136 Mirrors, Prisms and Lenses [§ 61 

segment on Ay is taken farther and farther from A. In 
other words, since tan — * z/100 is less than x . tan - x 1/100, x 
prism-dioptries is less than x times one prism-dioptry. Or- 
dinarily, the variability in the magnitude of a unit would 
constitute an insuperable objection to it; but so long as the 




Fig. 87. — Unequal angles subtended at O by equal intervals on straight 
line Ay drawn perpendicular to OA. 



angles to be measured are always small, as is the case with 
ophthalmic prisms, the prism-dioptry may be regarded as in- 
variably equal to the tan - 1 1/100 or about 34' 22. 6" without 
sensible error; and hence we may say, for example, that 
2A+3A = 5A, although this statement is not quite accu- 
rate. At any rate, whatever may be the theoretical objec- 
tions, this unit of measurement of the strength of a thin 
prism is so convenient and satisfactory that it has been gen- 
erally adopted in ophthalmic practice. 

In point of fact, with the small angular magnitudes which 
are here pre-supposed (the power of an ophthalmic prism 
seldom exceeds 6 ctrd.), there is practically no distinction to 
be made between the angle itself and the tangent of the angle, 



§ 61] Centrad and Prism- Dioptry 137 

so that we may regard the centrad and the prism-dioptry as 
identical in most cases; that is, 

1A = 1 ctrd. =0.573°. 
Accordingly, we obtain the following relation between the 
power (p) of an ophthalmic prism expressed in prism- 
diop tries or centrads and the refracting angle (/3) given in 
degrees: 

P=^f (n-l)/3=l. 745(n-l)ft 

where n denotes the relative index of refraction. If n= 1.5, 
then the power of a prism of refracting angle /3 degrees is 
0.873 prism-dioptries. 

However, in order to exhibit the actual relations still 
more clearly, the following table gives the values in degrees, 
minutes and seconds of all integral numbers of prism-dioptries 
and centrads from 1 to 20; and incidentally it will be seen 
that whereas an angle of k centrads contains k times as many 
degrees, minutes and seconds as an angle of 1 centrad, where 
k denotes any integer from 1 to 20, the same statement is 
not strictly true of the prism-dioptry. 



138 



Mirrors, Prisms and Lenses 



[§62 



Prism- 
Dioptries 


Equivalent in degrees, 
minutes and seconds 


Centrads 


Equivalent in degrees, 
minutes and seconds 


1 


0° 34' 22.6" 


1 


0° 34' 22.7" 


2 


1° 8' 44. 8" 


2 


1° 8' 45.3" 


3 


1°43' 6.1" 


3 


1°43' 8.0" 


4 


2° 17' 26.2" 


4 


2° 17' 30.6" 


5 


2° 51' 44.7" 


5 


2° 51' 53.3" 


6 


3° 26' 1.1" 


6 


3° 26' 15.9" 


7 


4° 0'15.0" 


7 


4° 0'38.6" 


8 


4° 34' 26.1" 


8 


4° 35' 1.2" 


9 


5° 8' 33.9" 


9 


5° 9' 23.9" 


10 


5° 42' 38.1" 


10 


5° 43' 46.5" 


11 


6° 16' 38.3" 


11 


6° 18' 9.2" 


12 


6° 50' 34.0" 


12 


6° 52' 31.8" 


13 


7° 24' 24.9" 


13 


7° 26' 54.5" 


14 


7° 58' 10.6" 


14 


8° 1'17.1" 


15 


8° 31' 50.8" 


15 


8° 35' 39.8" 


16 


9° 5' 25.0" 


16 


9° 10' 2.4" 


17 


9° 38' 53.0" 


17 


9° 44' 25.1" 


18 


10° 12' 14.3" 


18 


10° 18' 47.7" 


19 


10° 45' 28.7" 


19 


10° 53' 10.4" 


20 


11° 18' 35.8" 


20 


11° 27' 33.0" 



62. Position and Power of a Resultant Prism Equivalent 
to Two Thin Prisms. — In ascertaining the prismatic cor- 
rection of the eye of a patient, the oculist or optometrist 
sometimes finds it convenient and advantageous to employ 
a combination of two thin prisms placed one in front of the 



62] 



Combination of Two Thin Prisms 



139 



other with their edges inclined to each other at an angle 7 

which can be measured; and having obtained the necessary 

correction in this way, he has to prescribe a single prism which 

will produce precisely the 

same resultant effect as 

the two superposed 

prisms of the trial-case. 

In general, it would be 

exceedingly laborious and 

difficult to calculate the 

power of this resultant 

prism, but, fortunately, 

the problem in this case 

is enormously simplified 




Fig. 88, a. — Parallelogram law for find- 
ing single prism equivalent to a com- 
bination of two thin prisms. 



by the fact that the refracting angles are so small that it is 
quite simple to obtain an approximate solution which is 
sufficiently accurate and reliable for ordinary practical 
purposes. 

Let the deviation-angles or powers of the two prisms, de- 
noted by pi and p 2 , be represented, according to the method 
•r/ explained in § 50, by 

the vectors OA, OB, 
respectively (Fig. 88), 
which are drawn parallel 
to the edges of the prism, 
so that Z AOB = 7. Com- 
plete the parallelogram 
OACB and draw the di- 
agonal OC. The vector 
OC will represent on 
the same scale the deviation-angle or power p of the resultant 
prism, as we shall proceed to show. 

If a point P is taken anywhere in the plane of the parallelo- 
gram OACB, it may easily be proved that the area of the 
triangle POC is equal to the sum or difference of the areas of 
the triangles POA and POB according as the point P lies 




Fig. 88, b. — Parallelogram law for finding 
single prism equivalent to a combina- 
tion of two thin prisms. 



140 Mirrors, Prisms and Lenses [§ 62 

outside the Z AOB, as in Fig. 88 (a), or inside this angle, as in 

Fig. 88 (b) , respectively. And, therefore, if PQ, PR and PS are 

drawn perpendicular to OA, OB and OC, respectively, then 

SP.OC = QP.OA=*=RP.OB. 

For simplicity, let us assume that the deviations p\, p 2 
produced by the two component prisms are indefinitely 
small. Now suppose that the point P is turned, first, about 
OA as axis through a very small angle pi and then about OB 
as axis through the small angle p 2 . In consequence of the 
first rotation it will move perpendicularly out from the plane 
of the paper towards the reader through a tiny distance 
corresponding to the arc of a circle described around Q as 
center with radius QP, the length of this arc being equal 
to the product of the radius by the angle, that is, equal to 
QP . OA, since the length of OA is made equal to the magni- 
tude of the angle p±. If now in this slightly altered position 
the point P is again rotated, this time, however, around OB 
as axis, through another small angular displacement pi = OB, 
either it will move a little farther out from the plane AOB, 
as in the case shown in Fig. 88 (a), or it will move back 
away from the reader, as in the case shown in Fig. 88 (b), 
by an additional amount equal to RP.OB. And as this 
latter displacement will also be very nearly at right angles to 
the plane of the paper, the resultant angular displacement 
of the point P may be regarded as equal to the algebraic 
sum of its two successive displacements and numerically 
equal, therefore, to 

QP.OA± RP.OB, 
where the upper sign is to be taken in case the point P lies 
outside the angle AOB and the lower sign in case it lies inside 
this angle. In either case, therefore, the resultant displace- 
ment of P will be equal to SP . OC. But this product is equal 
to the linear displacement which the point P would have if 
it experienced an angular displacement represented by the 
vector OC. 

Hence, if the straight lines OA, OB drawn parallel to the 



§ 62] Combination of Two Thin Prisms 141 

edges of the two thin prisms represent the components of the 
total deviation of a ray which traverses both prisms, the 
diagonal OC of the parallelogram OABC will represent the 
resultant or total deviation, and this effect will be produced 
by a single prism of power p = OC placed with its edge in- 
clined to the edge of the prism of power pi ( = OA) at an angle 
6 = Z. OAC. If the powers p h p 2 of the two component prisms 
are given in prism-dioptries (or in terms of any other suit- 
able unit, for example, degree, centrad, etc.), and if also the 
angle y between the edges of the prisms is given in degrees, 
the power p of the resultant prism may, therefore, be com- 
puted by the formula: 

P=Vpi 2 +P2 2 +2pi. Pi- cost , 
and the angle 6 which shows how the resultant prism is to 
be placed may be calculated by the formula: 

tanfl= ^ Sin7 ■ 
Pi+P2.cosy 

In particular, if 7 = 90°, then p= -\/pi 2 +P2 2 > tan 6 =— . 

As an illustration of the use of these formulae, suppose 

that the deviations produced by the two prisms separately 

are 3° and 5°, and that the edges of the prisms are inclined to 

each other at an angle of 60°. Then pi = 3°, p 2 = 5°, y = 60°, 

and hence the deviation produced by the two prisms together 

5V3 

will be p = \/9+ 25+ 15 = 7°; and since tan#= — — , the 

resultant prism in this case is found to be a prism of power 
7° placed with its edge at an angle of nearly 38° 13' with that 
of the weaker of the two component prisms. 

A " rotary prism" used for finding the necessary prismatic 
correction of a patient's eye is an instrument, circular in form, 
which consists of two ophthalmic prisms of equal power 
(pi = 7?2) conveniently mounted so that the prisms can be 
rotated about an axis perpendicular to the plane of the in- 
strument, one in front of the other, the angle between the 
prism-edges being shown by the positions of two marks which 



142 Mirrors, Prisms and Lenses [Ch. V 

move as the prisms are turned over a circular arc graduated 
in degrees. In the initial position when the two marks are 
at opposite ends of a diameter of the circular scale the base 
of one prism corresponds with the edge of the other, so that 
in this position the two prisms are equivalent to a glass 
plate with plane parallel faces (7 = 180°, p = pi— P2 = 0). 
The maximum effect is obtained when the edges of the prism 
correspond (7 = 0°, p = pi~{-p2 = 2pi). With a device of this 
kind, we can obtain, therefore, any prismatic power from 
p = to p = 2pi. 

On the other hand, we can resolve the effect of a given 
prism of power p into a component p . cos 6 in one direction 
and a component p . sin 6 in a direction perpendicular to the 
first. Thus, a prism of power 5 centrads with its edge at an 
angle of 30° to the horizontal is equivalent to a combination 

of two prisms of powers -—- and ~ centrads, with their 

edges horizontal and vertical, respectively. 

PROBLEMS 

1. Show how to construct the path of a ray refracted 
through a prism in a principal section; and prove the con- 
struction. Discuss the following special cases, and draw 
separate diagrams for each of them : (a) Incident ray normal 
to first face of prism, (b) Emergent ray " grazes " second 
face; (c) Ray traverses prism symmetrically; (d) Ray is in- 
cident on first face on side of normal towards the edge of 
the prism. ^ 

2. Show that the total deviation of a ray in a principal 
section of a prism of more highly refracting material than 
the surrounding medium is always away from the prism- 
edge. Discuss each of the three possible cases, viz., When 
the point where the two incidence-normals intersect falls 
(a) inside the prism, (b) outside the prism, and (c) on one of 
the two faces of the prism. Draw diagram for each case. 

3. Obtain a formula for calculating the magnitude of the 



Ch. V] Problems 143 

angle of incidence at the first face of the prism of the ray 
which emerges from the prism along the second face; and dis- 
cuss this formula for the cases when the refracting angle of 
the prism is (a) greater than 2 A, (b) equal to 2 A, (c) less 
than 2A but greater than A, (d) equal to A, and (e) less than 
A; where A denotes the so-called critical angle of the two 
media concerned. Draw diagram for each case. 

4. Show that the deviation of a ray which goes symmet- 
rically through a prism in a principal section is less than 
that of any other ray. 

5. Show that the point of intersection of the incidence- 
normals to the two faces of a prism is equidistant from the 
incident ray and its corresponding emergent ray. 

6. Construct the path of a ray refracted through a prism 
of small refracting angle; and show that the angle of deviation 
will also be a small angle of the same order of smallness, no 
matter how the ray falls on the prism. 

7. What is the smallest angle that a glass prism (n = 1 . 5) 
can have so that no ray can be transmitted through it? 
What is the magnitude of this angle for a water prism 
(n = 1.33)? (Assume in each case that the prism is sur- 
rounded by air of index unity.) 

Ans. 83° 37' 14"; 97° 10' 52". 

8. What must be the refracting angle of a prism whose 
index of refraction is equal to \/2 in order that rays that 
are incident on one of its faces at angles less than 45° will 
be totalfy reflected at the other face? Ans. 75°. 

9. The refracting angle of a prism is 60° and the index of 
refraction is equal to \/2. Show that the angle of minimum 
deviation is 30°, and draw accurate diagram showing the 
construction of the path of this ray through the prism. 

10. The refracting angle of a glass prism (n = 1.5) is 60°, 
and the angle of incidence is 45°. Find the angle of deviation. 
What is the angle of minimum deviation for this prism? 

Ans. 37° 22' 52.5"; 37° 10' 50". 

11. If the angle of minimum deviation of a ray traversing 



144 Mirrors, Prisms and Lenses [Ch. V 

a principal section of a prism is 90°, show that the index of 
refraction cannot be less than s/2. 

12. Find the angle of minimum deviation in the case of a 
glass prism (n = 1 . 54) of refracting angle 60°. 

Ans. 40° 42' 28". 

13. The minimum deviation for a prism of refracting angle 
40° is found to be 32° 40'. Find the value of the index of 
refraction. Ans. 1.7323. 

14. A glass prism of refracting angle 60° is adjusted so 
that the ray "grazes" the first face, and in this position the 
angle of emergence is found to be 29° 25' 49". Determine 
the index of refraction. Ans. 1 . 52. 

15. A prism is made of glass of index 1.6, and the angle 
of minimum deviation is found to be 28° 31' 20". Calculate 
the refracting angle. Ans. 42° 39' 44". 

16. The efracting angle of a water prism (n = -|) is 30°. 
How must a ray be sent into this prism so that it will emerge 
along the second face? 

Ans. Ray must He on the side of the normal towards the 
edge of the prism, and make with the normal an angle of 
25° 9' 15". 

17. The angle of incidence for minimum deviation in the 
case of a prism of refracting angle 60° is 60°. Find the 
index of refraction. Ans. v3. 

18. Find the index of refraction of a glass prism for sodium 
light for the following measurements: Refracting angle of 
prism = 45° 4'; angle of minimum deviation = 26° 40'. 

Ans. 1.53. 
19 The refracting angle of a prism is 30° and its index of 
refraction is 1.6. Find the angles of emergence and deviation 
for each of the following rays: (a) Ray meets first face nor- 
mally; (b) Angle of incidence at first face is equal to 24° 28'; 

(c) Angle of incidence at first face is equal to 53° 8'; and 

(d) Ray " grazes" first face. 

Ans. (a) 53° 8'; 23° 8'; (6) 24° 28'; 18° 56'; (c) 0°; 23° 8'; 
(d) 13° 59'; 46° 1'. 



Ch. V] Problems 145 

20. Find the refracting angle of a glass prism (n = 1.52) 
for which the minimum deviation is 15°. Ans. 27° 24' 15". 

21. The refracting angle of a flint glass prism is measured 
and found to be 59° 56' 22.4"; and the angles of minimum 
deviation for rays of light corresponding to the Fraunhofer 
lines D, F and H are also measured and found to have the 
following values: 46° 31' 4.15"; 47° 35' 59.2"; and 49° 30' 
5.7", respectively. Calculate the values of the indices of 
refraction n B , n F , and n H - 

Ans. rc D = 1 . 603528; n F = 1 . 614771 ; n H = 1 . 634183. 

22. The refracting angle of a crown glass prism is measured 
and found to be 60° 2' 10.8"; and the angles of minimum 
deviation for rays of light corresponding to the Fraunhofer 
lines D, F and H are also measured and found to have the 
following values: 38° 38' 14.3"; 39° 10' 51.8"; and 40° 3' 
49.4", respectively. Calculate the values of the indices of 
refraction n D , n F , and n K . 

Ans. n D = 1 . 516274 ; n F = 1 . 522437 ; n u = 1 . 532370. 

23. A prism is to be made of crown glass of index 1.526, 
and it is required to produce a minimum deviation of 17° 20'. 
To what angle must it be ground? Ans. 31° 20'. 

24. A ray of light falls on one face of a prism in a direction 
perpendicular to the opposite face. Assuming that the re- 
fracting angle of the prism (/3) is an acute angle, show that 
the ray will emerge along the opposite face if 

cot/3 = cotA— 1, 
where A denotes the critical angle of the prism-medium. 

25. A ray "grazes" the first face of a prism and emerges 
at the second face in a direction perpendicular to the first 
face: show that the refracting angle (/3) is such that 

cot/3=Vw2-l-l, 
where n denotes the index of refraction of the prism-medium. 

26. The refracting angle of a prism is 60° and the index of 
refraction is s/7/3. What is the limiting angle of incidence 
of a ray that will be transmitted through the prism? 

Ans. 30°. 



146 Mirrors, Prisms and Lenses [Ch. V 

27. Show that if € denotes the angle of minimum devia- 
tion of a prism of refracting angle /3, the angle fi cannot be 
greater than (7r— e ) and the index of refraction cannot be 

less than sec-^-- 

28. Show that the minimum deviation of a prism of given 
index of refraction increases with increase of the refracting 
angle of the prism. 

29. Derive the formula for the angle of deviation of a thin 
prism, and show that the deviation is approximately con- 
stant for all angles of incidence. 

30. Show that when a thin glass prism of index f is im- 
mersed in water of index |- the deviation of a ray will be 
only one-fourth of what it would be if the prism were sur- 
rounded by air. 

31. The refracting angle of a prism of rock salt is 1° 30'. 
How much will a ray be deviated in passing through it? 
And what should be the refracting angle of a rock salt prism 
which is to produce a deviation of 48'? (Index of refraction 
of rock salt = 1 . 54.) Ans. 48' 36" ; 1° 29\ 

32. What must be the refracting angle of a water prism of 
index |- to produce the same deviation as is obtained with 
a glass prism of index f whose refracting angle is equal to 
2°? Ans. 3°. 

33. A glass prism of index 1.5 has a refracting angle of 
2°. What is the power of the prism in prism-dioptries? 

Ans. 1 . 745 prism-dioptries. 

34. The power of a prism is 2 prism-dioptries and n= 1 .5. 
Find the refracting angle. Ans. 2 . 29°. 

35. A prism of refracting angle 1° 25' bends a beam of 
light through an angle of 1° 15'. Calculate the index of 
refraction and the power of the prism in prism-dioptries. 

Ans. n = 1 . 882; 2 . 18 prism-dioptries. 

36. Two thin prisms are crossed with their edges at an an- 
gle of 30°. The first prism produces a deviation of 6° and 
the second a deviation of 8°. Find the deviation produced 



Ch. V] Problems 147 

• 
by the single prism which is equivalent to this combination 

and the angle which the edge of the resultant prism must 
make with the edge of the first prism. 

Ans. Deviation of resultant prism = 13.53°; angle be- 
tween its edge and that of the 6°-prism=17° 11'. 

37. Two prisms, each of power 5 prism-dioptries, are 
combined base down with their base-apex lines inclined to 
the horizontal at angles of 45° and 135°. Find the equivalent 
single prism. 

Ans. A prism of power a little more than 7 prism-dioptries, 
base down, vertical meridian (edge horizontal). 

38. What will be the horizontal effect of a prism of power 
10 placed with its base-apex line at an angle of 20° with the 
horizontal? 

Ans. It will be the same as the effect of a prism of power 
nearly 9 . 4 in horizontal meridian (edge vertical) . 

39. The base-apex line of a prism of power 4 centrads makes 
an angle of 120° with the horizontal. Show that it is equiva- 
lent to a combination of two prisms, one of power 2 centrads in 
the vertical meridian (edge horizontal) and the other of power 
3.46 centrads in the horizontal meridian (edge vertical). 

40. Find the single prism equivalent to a combination of 
two prisms superposed with their base-apex lines at right 
angles to each other, the power of one being 3 and that of 
the other 4. 

Ans. A prism of power 5 with its base-apex line inclined to 
that of the weaker prism at an angle of nearly 53° 8'. 

41. Two equal prisms, each of power 3, are superposed 
in meridians inclined to each other at an angle of 120°. 
Find the equivalent single prism. 

Ans. A prism of power 3 in a meridian halfway between 
the meridians of the two components. 

42. The angle between the base-apex lines of a combina- 
tion of two unit prisms is 82° 50', and the bisector of this 
angle is horizontal. What is the horizontal effect of the 
combination? Ans. 1 . 5 units. 



148 Mirrors, Prisms and Lenses [Ch. V 

• 

43. ABCDE is the principal section of a pentagonal prism. 
AB = BC, AE = CD, ZABC = 90°, ZEAB = Z BCD = 112.5°. 
A ray of light RS lying in the principal section is incident on 
the face BC at the point S. The ray enters the prism at this 
face, and is reflected, first, from the face AE, and then from 
the face DC, and emerges finally at a point P in the face AB 
in the direction PQ. Show that PQ makes a right angle 
with RS. 

44. ABC is a principal section of a triangular prism, 
Z B = 2Z A. A ray of light lying in the plane ABC is refracted 
into the prism at the side BC, and after undergoing two 
internal reflections, first, from the side AB and then from 
the side CA, emerges into the surrounding medium at the 
side AB. Show that the total deviation of the ray will be 
equal to the angle at B. 



CHAPTER VI 

EEFLECTION AND REFRACTION OF PARAXIAL RAYS AT A 
SPHERICAL SURFACE 




63. Introduction. Definitions, Notation, etc. — The center 
of the spherical refracting or reflecting surface ZZ (Fig. 89) 
will be designated by C. The axis of the surface with respect 
to a given point M is the 
straight line joining M 
with C, and the point A 
where the straight line 
MC (produced if neces- 
sary) meets ZZ is called the 
pole or vertex of the surface 
with respect to the point 
M. Evidently, the spheri- 
cal surface will be sym- 
metrical around MC as 
axis, and the plane of the 
diagram which contains the axis is a meridian section of the 
surface. 

It will be convenient to take the vertex A as the origin 
of a system of plane rectangular coordinates; the axis of 
the surface being chosen as the z-axis and the tangent to the 
surface at its vertex, in the meridian plane of the diagram, 
being taken as the ?/-axis. The positive direction of the x-axis 
is the direction of the incident ray which coincides with this 
line, and since the diagrams are all drawn on the supposition 
that the incident light goes from left to right, a point lying on 
the z-axis to the right of A will be on the positive half of 
the axis. The positive direction of the y-axis is the direction 
found by rotating the positive half of the x-axis through a 

149 



Fig. 89, a. — Ray incident on convex 
spherical surface crosses axis at 
point M in front of surface. 



150 



Mirrors, Prisms and Lenses 



l§ 63 



right angle in a sense opposite to that of the motion of the 
hands of a clock in the meridian plane of the diagram. Ac- 
cordingly, if the positive direction of the x-axis is along a 




Fig. 



89, b. — Ray incident on convex spherical surface crosses axis at point 
M on the other side of the surface. 



horizontal line from left to right, the positive direction of 
the i/-axis will be vertically upwards. 

According as the center C lies on the same side of the 
spherical surface as that from which the incident light comes 
or on the opposite side, it is said to be concave (Fig. 89, c 
and d) or convex (Fig. 89, a and 6), respectively. The radius 
r of the spherical surface is the abscissa of the center C, that 
is, r = AC. It is the step from A to C, and this is always a 
positive step for a convex surface (Fig. 89, a and b) and a 
negative step for a concave surface (Fig. 89, c and d). The 
radius of a convex surface whose center is 60 cm. from its 
vertex is r = +60 cm., and the radius of a concave surface of 
the same size is r— — 60 cm. 



§63] 



Ray Incident on Spherical Surface 



151 




Fig 



c. — Ray incident on concave 
spherical surface crosses axis at point 
M in front of the surface. 



It will be assumed in this chapter that any ray with which 
we are concerned lies in a meridian plane of the spherical 
surface; so that any straight line such as RB which repre- 
sents the path of an inci- y 
dent ray will intersect the 
axis either " really" (Fig. 
89, a and c) or " virtually " 
(Fig. 89, b and d) at some 
point designated here by 
M (see § 8). The point 
designated by R is any 
point on the incident ray 
RB at which the light 
arrives before it gets to 
either M or the incidence- 
point B. The straight line BC which joins the point of 
incidence with the center of the surface will be the incidence- 
normal, and if N designates a point on this normal lying in 
front of the spherical surface, then Z NBR = a will be the 

angle of incidence (§§ 13 & 
27). The plane of this 
angle is the plane of inci- 
dence, which is the merid- 
ian plane of the diagram. 

From the incidence-point 
B draw BD perpendicular 
to the x-axis at D ; the or- 
dinate h = DB is called the 

Fig. 89, d.-Ray incident on concave incidence-height of the ray. 

spherical surface crosses axis at point The slope of the Tdy is the 
M on the other side of the surface. acute angle thrQugh wMch 

the rr-axis has to be turned around the point M in order 
that it may coincide in position (but not necessarily in 
direction) with the rectilinear path of the ray. If this angle 
is denoted by 6, then ZAMB= 6. Here, as always in the 
case of angular magnitudes (§ 13), counter-clockwise rotation 




152 Mirrors, Prisms and Lenses [§ 63 

is to be reckoned as positive. And, finally, the acute angle 
at the center C of the spherical surface subtended by the 
arc BA will be denoted by (f>. This angle, sometimes called 
the "central angle," is denned as the angle through which 
the radius CB must be turned around C in order to bring B 
into coincidence with the vertex A; thus, = ZBCA. The 
angles A, 6 and <£, defined as above, are given by the fol- 
lowing relations: 

h h 

tan# = — t^Tt> sm<f> = ~, a= 6-\-<j). 

These formulae should be verified for each of the diagrams 
Fig. 89, (a), (&), (c), {d). 

Moreover, since BM = -, and since (see § 45) 

COS0' v * / 

DM = DC+CA+AM = r.cos</>-r+AM, 
we find: 

EM = r(cos<ft-l)+AM 

COS0 

Now in the special case when the incidence-point B is very 
close to the vertex A of the spherical surface, the angle of in- 
cidence a will be exceedingly small as will be also the angles 
denoted by 6 and </> ; and if these angles expressed in radians 
are all such small fractions that we may neglect their second 
and higher powers, so that in place of the sines (or tangents) 
we can write the angles themselves and put cos 6 = cos <f> = 
cos a = 1. Obviously, in such a case we shall have BM = AM. 
Under these circumstances the ray RB is called a paraxial 
ray, sometimes also a "central" or "zero" ray, a= d = 4> = 0, 
approximately. 

A paraxial ray is one whose path lies very near the axis of 
the spherical surface and which therefore meets this surface at 
a point close to the vertex and at nearly normal incidence: the 
angles denoted by a, 6 and <j> being all so small that their second 
powers may be neglected. 

In this chapter and for several subsequent chapters we 
shall be concerned entirely with the procedure of paraxial 



64] 



Paraxial Rays: Spherical Mirror 



153 




Fig. 



90, O. — Reflection of ray at con- 
cave mirror. 



rays; that is, we shall consider only such rays as are com- 
prised within a very narrow cylindrical region immediately 
surrounding the axis of the spherical surface which is like- 
wise the axis of the cylinder. Accordingly, the only portion 
of the spherical surface that will be utilized for reflection or 
refraction will be a small zone whose summit is at A; so that, 
so far as paraxial rays are 
concerned, the rest of the 
spherical surface may be 
regarded as if it had no 
optical existence or at any 
rate as if it were opaque 
and non-reflecting. Thus, 
for example, the surface 
might be painted over 
with lampblack leaving 
bare and exposed only 
the small effective zone 
in the* immediate vicinity of the vertex; or a screen might 
be set up at right angles to the axis close to the vertex with 
a small circular opening in it. Even then a source of light 

lying at a considerable dis- 
tance off the axis would 
send rays which notwith- 
standing that they were 
incident near the vertex 
would not be paraxial rays. 
64. Reflection of Par- 
axial Rays at a Spherical 
Mirror. — In the accom- 
panying diagrams (Fig. 90, 
a and 6) the straight line 
RB represents the path of 
an incident ray crossing the axis of a spherical mirror ZZ at the 
point M and incident on the mirror at the point B, and the 
straight line BS shows the path of the corresponding re- 




FlG. 



90, 6. — Reflection of ray at con- 
vex mirror. 



154 Mirrors, Prisms and Lenses [§ 64 

fleeted ray crossing the axis, "really" (Fig. 90, a) or "virtu- 
ally" (Fig. 90, b), at the point marked M'. By the law of 
reflection Z NBR = Z SBN where BN is the incidence-normal 
and N designates a point on it which lies in front of the 
mirror. Since the normal bisects the interior or exterior 
angle at B of the triangle MBM', the following proportion 
may be written : 

CM_M / C 

BM BM' # 
Now if the ray RB is a paraxial ray, the letter A may be sub- 
stituted in the above equation in place of B, and thus * we 
obtain : 

CMM'C 

■ AM AM'' 

Denoting the abscissae, with respect to the vertex A, of 

the axial points M, M' by u, u', respectively, that is, putting 

AM = u, AM' = u', and also, as stated in § 63, putting AC = r, 

we may write: 

CM = CA+AM= -r+u = u-r, 
M'C = M'A+AC= -u'+r=-(u'-r); 
so that, introducing these symbols in the equation above, 
we obtain: 

u — r_ u' — r 

u u' 

which may be put in the form (see § 67) : 

u u r 

If, therefore, the form and dimensions of the mirror are 
known (that is, if the value of r is assigned as to both mag- 
nitude and sign), and if also the position of the point M 

* In writing this proportion, care must be taken to see that the two 
members of it shall have the same sign. For example, in each of the 
diagrams in Fig. 90, as they are drawn, the segments CM and AM 
have the same direction along the axis, so that for each of these figures 
the ratio CM : AM is positive. Now if the ratio M'C : AM' is to be 
put equal to this ratio, it must be positive also, that is, the segments 
M'C and AM' in each diagram must have the same direction. 



§64] Paraxial Rays: Spherical Mirror 155 

where the incident paraxial ray crosses the axis of the 

spherical mirror is given, the abscissa u' of the point M' 

where the corresponding reflected ray crosses the axis may 

be calculated by means of the expression: 

, r.u 

u = . 

2u-r 

But the most noteworthy conclusion to be drawn from this 
formula is the fact that, provided the rays are paraxial, their 
actual slopes do not matter, for none of the angular magni- 
tudes a, 6, or cf> appears in the formula; which means that 
all paraxial rays which cross the axis at the point M before 
reflection will cross the axis after reflection in the spherical 
mirror at one and the same point M'. Thus, a homocentric 
bundle of paraxial rays incident on a spherical mirror remains 
homocentric after reflection. If, therefore, M designates the 
position of a luminous point in front of the mirror, and if 
the mirror is screened so that only such rays as proceed close 
to the axis are incident on it, the bundle of reflected rays 
will form at a point M' on the straight line MC an ideal 
image of the luminous point M. According as the image- 
point M' lies in front of the mirror (Fig. 90, a) or beyond it 
(Fig. 90, 6), the image will be real or virtual, respectively. 
Thus, for a real image in a spherical mirror, the value of u' 
as found by the formula above will be negative, whereas 
for a virtual image it will be positive. 

It may be noted also that the formula is symmetrical with 
respect to u and n f , so that the equation will not be altered 
by interchanging the symbols u and u' ; and hence it follows 
that if M' is the image of M, then likewise M may be regarded 
as the image of M'. This is indeed merely an illustration of 
the general law known in optics as the "principle of the 
reversibility of the light-path" (§29). But the symmetry 
of the equation implies more than is involved in this prin- 
ciple; for it indicates that in the case of reflection object- 
space and image-space coincide completely, the actual paths 
of the incident and reflected rays both lying in the space in 



156 Mirrors, Prisms and Lenses [§ 65 

front of the mirror. Accordingly, an incident ray and its 
corresponding reflected ray are always so related that when 
either is regarded as object-ray the other will be an image-ray. 

The Double Ratio of Four Points on a Straight Line 

65. Definition and Meaning of the Double Ratio. — It 
will be convenient and profitable at this place to turn aside 
from the special problem which is here under investigation 
in order to devote a few paragraphs to a brief explanation 
of the simpler metrical processes of modern projective 
geometry, which are of great utility in geometrical optics, 
especially when we are concerned with imagery by means 
of the so-called paraxial rays. 

A B 

• , 1 , 

C D 

(ft/) 



■ 1 1 

Fig. 91. — Line-segment AB divided (a) internally at 
C and externally at D, and (b) internally at C 
and D. 

If L designates the position of a point on a straight line 
determined by the two points A, B, the line-segment AB is 
said to be divided at L in the ratio AL : BL. If the point L 
lies between A and B, the steps (see § 45) AL and BL are 
in opposite senses along the line, and the ratio AL : BL will 
be negative, and in this case we say that the segment AB is 
" divided internally" at L. On the other hand, if the point L 
does not lie between A and B, the ratio AL : BL will be 
positive, and we say that the segment AB is " divided ex- 
ternally" at L. 

Accordingly, if A, B, C, D (Fig. 91, a and 6) designate a 



§ 65] Double Ratio 157 

series of four points all ranged along a straight line in any 
order of sequence, the segment AB will be divided at C and 
D in the ratios AC : BC and AD : BD, respectively; and 
the quotient of these two ratios is called the double ratio (or 
" cross ratio") of the four points A, B, C, D. This double 
ratio is denoted symbolically by inclosing the four letters 
ABCD in parentheses; thus, according to the above def- 
inition, 

where the first two letters in the parentheses mark the end- 
points of the segment and the last two letters designate the 
points of division. The line-segment CD is divided in the 
same way by the points A and B ; for 

According as the two ratios AC : BC and AD : BD have 
the same sign or opposite signs, the value of the double ratio 
(ABCD) will be positive or negative, respectively. Suppose, 
for example, that the segment AB is divided internally at C, 
as represented in both a and b of Fig. 91. Then the ratio 
AC : BC will be negative. Now if AB is divided also in- 
ternally at D, as in Fig. 91, a, the ratio AD : DB will likewise 
be negative. Accordingly, if C and D are both points of in- 
ternal division (or both points of external division), the 
double ratio (ABCD) will be positive. But if one of these 
points divides AB internally while the other divides it ex- 
ternally (Fig. 91, b), the double ratio (ABCD) will be nega- 
tive. 

In order to form a clear idea of the values which (ABCD) 
may assume, let us suppose that the points designated by 
A, B and C in Fig. 92 represent three stationary points on a 
straight line x, and that designates another fixed point not 
on this line. The straight line x and the point O together 
determine a plane which is the plane of the diagram. Now 
let y designate a second straight line lying in this plane and 




158 Mirrors, Prisms and Lenses [§ 65 

passing through O, and let the point of intersection of the 
straight lines x and y be designated by Y. And if the 
straight line y is supposed to turn around as a pivot in a 

sense, say, opposite to 
that of the motion of the 
hands of a clock, the 
point Y will be a variable 
point moving along the 
straight line x constantly 
in the same sense, namely, 
in Fig. 92 from left to 

Fig. 92.— Central projection from Oof the right- Assume, for ex- 
point-range ABCDE lying on the ample, that the three 
straight line x. ... - • k r~\ -n 

stationary points A, C, B 
are ranged along the straight line x from left to right in the 
order named, as shown in the figure; and suppose that the 
variable point Y starts originally at B, so that the revolving 
line OY or y coincides initially with the "ray" marked b in 
the figure and BY = BB = 0, and, consequently, the ratio 
AY : BY = oo , Hence, under these circumstances the initial 
value of the double ratio of the four points A, B, C, Y will be: 

/ ABcY >=i : S=°- 

When the revolving ray has turned through ZBOD, where 
D designates a point lying on the straight line x to the 
right beyond B, the point Y will be at D outside the segment 
AB and the double ratio (ABCY) will be negative, as ex- 
plained above. As y continues to revolve around O, the point 

Y will move farther and farther to the right along the straight 
line x, until when y is parallel to x, and in the position of the 
ray marked e in the figure, the point Y will then coincide 
with the infinitely distant point E of the straight line x. Now 
AE = BE = oo , and hence AE : BE = 1 ; and therefore when 

Y is at E, 



§ 66] Perspective Ranges of Points 159 

When the revolving ray y has turned beyond the position 
represented by the straight line e, the point Y which had 
just vanished at one end E of the straight line x now re- 
appears from the other end E, proceeding along it still in 
the same sense from left to right. Thus, before the ray y 
has executed a complete revolution, the point Y will pass 
through A, and at this moment, AY = AA = 0, and 
aBmn AC AY AC AA 
(ABCY)= BC : BY = BC : BA = -°° ; 
and thus we see that as the point Y has traversed the straight 
line x from B via the infinitely distant point E to A, the double 
ratio (ABCY) has assumed all negative values from to — oo . 
Finally, as the ray y completes its revolution by turning from 
the position a to its initial position b, the point Y moves from 
A via C to B. When Y is at C, AY = AC, BY = BC, and 

(ABCY) = |§:g = + l; 

so that in passing along x from A to C, (ABCY) assumes all 
positive values comprised between + oo and +1. Between 
C and B, it has all positive values less than unity. Thus, 
as the point Y traverses the straight line x continually in 
the same sense until it has returned to its starting point, 
the double ratio (ABCY) will assume all possible values 
both positive and negative. 
In general, since 

(ABUD) BC : BD AD - AC DA'DB CB ' CA ' 
we may write: 

(ABCD) = (BADC) = (CDAB) = (DCBA). 
66. Perspective Ranges of Points. — If A, B, C, etc., desig- 
nate the positions of the points of a point-range x (Fig. 92) 
these points are said to be " projected" from a point out- 
side of x by the straight lines or "rays" OA, OB, OC, etc.; 
and if these rays intersect another straight line x' (Fig. 93) 
in the points A', B', C, etc., the two point-ranges x, x' are 
said to be in perspective with respect to the point as center 



160 



Mirrors, Prisms and Lenses 



[§66 



of perspective. The points A, A'; B, B'; C, C; etc., are 
called pairs of corresponding points of the two perspective 
point-ranges x, x' . 



b f 




Fig. 93.— The point-ranges ABCD and A'B'C'D' 

are in perspective relation with respect to the 
point O as centre of perspective. 

If A, B, C, D designate the positions of any four points of 
x, and if A', B', C, D' designate the corresponding points 
on x', then 

(A'B'C'D') = (ABCD), 

as we shall proceed to show. 




Fig. 94. — Straight lines x, x' are bases of two point-ranges in 
perspective, so that (ABCD) = (A'B'C'D'). 

Through the points A, B, A r and B' (Fig. 94) draw four 
parallel lines AA C , BB C , A'A C ' and B'B/ meeting the ray OG 



§ 67] Harmonic Range 161 

or c in the points A c , B c , A c ' and B c ', respectively; and 
through these same points draw four other parallel lines 
AA d , BB d , A' Ad' and B'B d ' meeting the ray OD or d in the 
points Ad, B d , Ad' and B d ', respectively. Then, evidently, 

AC = AA C AD _ AA d 

BC BB C ' BD BB d ' 
A'C A'A C ' A'D' A'A d ' 



hence, 



B'C B'B C ' 


' B'D' B'B d ' ' 


AP 

(ABCD) = — 

BC 


AD_AA C AA d 
' BD BB C ' BB d ' 


AT/ 

3'C'D') =— , 

B'C 


A'D'A'Ae 7 A'A d ' 
' B'D' B'B C ' " B'B d ' 


A A c A A d 


B B c B B d 



Now 



A'A C ' A'A/ B'B c ' B'B d ' ' 
and, consequently, 

(A , B'C'D') = (ABCD), 
as was to be proved. 

67. The Harmonic Range. — The special case when the 
points C and D divide the line-segment AB internally and 
externally in the same numerical ratio, so that 

AC = _AD 

BC BD' 
demands attention, particularly because it is a case that we 
shall meet again in the theory of the reflection of paraxial rays 
at a curved mirror. Under these circumstances, the value 
of the double ratio is 

(ABCD)=-1; 
and then we say that the segment AB is divided harmonically 
at C and D, or also the segment CD is divided harmonically 
at A and B. For example, the perpendicular bisectors of 
the exterior and interior angles of a triangle divide the op- 
posite side of the triangle harmonically in the ratio of the 
other two sides. 



162 



Mirrors, Prisms and Lenses 



67 




Four harmonic points may be denned not merely by the 
metrical relation that their double ratio is equal to - 1, but 
also by a geometrical relation, as we shall now show. 
Let P, Q, R, S (Figs. 95 and 96) designate the positions of 
\z four points lying all in 

one plane, no three of 
which are in the same 
straight line. These 
four points will deter- 
mine six straight lines, 
viz., PQ, PR, PS, QR, 

Fig. 95.— Complete quadrilateral PQRS; QS, and RS, which are 
(ABCD)=-1. called the sides Q f the 

complete quadrilateral whose four vertices are at the points 
P, Q, R, and S. Any two of these lines which together con- 
tain all the vertices form a pair of opposite sides of the 
quadrilateral. Accordingly, there are three pairs of opposite 
sides, viz., PQ and RS . 

which meet in a point 
designated by A, PS and 
QR which meet in a 
point designated by B, 
and QS and PR which 
meet in a point desig- 
nated by O. The three 

points A, B and are FlG - 96.— Complete quadrilateral PQRS; 

sometimes called the 

secondary vertices of the quadrilateral. We shall explain now 

what connection this figure has with a harmonic range of 

points. 

The secondary vertices A and B are determined by the 
two pairs of opposite sides PQ, RS and PS, QR; and the 
points C and D where the third pair of opposite sides QS 
and PR meet the straight line AB divide the segment AB 
harmonically. For, since A, B, C, D and P, R, O, D are in 




§ 67] Harmonic Range 163 

perspective relation with respect to the point Q as center of 
perspective (§ 66), therefore 

(ABCD) = (PROD). 
But P, R, 0, D and B, A, C, D are also in perspective to 
each other with respect to the point S as center of perspective; 
consequently, 

(PROD) = (BACD). 
It follows therefore that 

(ABCD) = (BACD). 
But by the definition of the double ratio 

(BACD) = . 

(ABCD) 

Accordingly, here we must have: 

(ABCD)= , 

(ABCD) 

or 

(ABCD) 2 =1. 

According to this equation, therefore, the double ratio 

(ABCD) must be equal to +1 or —1. But we saw above 

(§ 65) that the double ratio of four points A, B, C, D in a 

straight line can be equal to +1 only in case one of the 

points A, B is coincident with one of the pair C, D; which 

cannot happen in case of the four points A, B, C, D of the 

quadrilateral PQRS. Therefore, we must have here: 

(ABCD)=-1; 

and hence, by definition, the points A, B are harmonically 

separated by the points C, D. Similarly, also, the points 

P, R are harmonically separated by the points O, D. 

If A, B, C, D is a harmonic range of points, then 



that is, 



BC = DB BA+AC _ DA+AB 
AC AD' ° r AC AD 



AC-ABAB-AD 
AC AD 



164 Mirrors, Prisms and Lenses [§ 68 

which may finally be written in the form: 
11 2 



AC AD A B' 

an equation that is characteristic of a harmonic range of 
four points A, B, C, D (c/. § 64). 

68. Application to the Case of the Reflection of Paraxial 
Rays at a Spherical Mirror. — When paraxial rays are re- 
flected at a spherical mirror whose center is at C, we saw 
(§ 64) that CM : AM = M'C : AM', where M, M' designate 
the positions of a pair of conjugate points lying on a central 
ray which crosses the mirror at the point marked A (Fig. 90, 
a and b) ; and therefore 

Consequently, the four points C, A, M, M' are a harmonic 
range of points lying on the central ray AC, and we may say 
that the pair of conjugate points M, M' is harmonically 
separated by the center of the mirror C and the point A 
where the central ray meets the mirror. Thus, if we know 
the positions of three of these points, we can construct the 
position of the fourth point by the aid of the properties of 
the complete quadrilateral (§ 67). For example, the image- 
point M' conjugate to a given point M with respect to a 
spherical mirror may be constructed as follows: 

Draw a straight line x (Fig. 97, a and b) to represent the 
-axis of the mirror, and mark on it the positions of the three 
given points, A, C and M, which may be ranged along this 
line in any sequence whatever depending on the form of 
the mirror and on whether the object-point M is real or 
virtual. Through M draw another straight line in any con- 
venient direction, and mark on it two points which we shall 
call Q and S, and draw the straight lines AQ and CS meeting 
in a point R and the straight lines AS and CQ meeting in a 
point P. Then the straight line PR will intersect the straight 
line x in the point M' which is conjugate to M with respect 
to a spherical mirror whose vertex is at A and whose center 



§ 68] Spherical Mirror: Conjugate Axial Points 165 

is at C. It will be remarked that in performing this con- 
struction the only drawing instrument that is needed is a 
straight-edge. 




Fig. 97, a. — Concave Mirror: Construction of point 
M' conjugate to axial point M in front of the mirror. 

If the mirror is concave, the possible sequences of these 
four points on the axis are M, C, M', A; M', C, M, A; and 
C, M, A, M', when the object-point M is real, and C, M', 




Fig. 97, b. — Convex Mirror: Construction of point M' 
conjugate to virtual object-point M on axis of mirror. 

A, M, when the object-point M is virtual. In the case of a 
convex mirror the points may occur in any one of the follow- 
ing arrangements: M, A, M', C, when the object-point M 



166 



Mirrors, Prisms and Lenses 



69 




FlG. 



a. — Focal point 
mirror (AF = FC) 



of 



is real, and M', A, M, C; A, M, C, M' and A, M', C, M, 
when the object-point M is virtual. The student should 
satisfy himself as to the accuracy of these statements by 
drawing a diagram for each of these eight sequences accord- 
ing to the directions for the construction as given above. 
Fig. 97, a shows the case of a concave mirror with the points 
in the order M, C, M', A; whereas Fig. 97, b represents 

a convex mirror with a 
virtual object-point at M, 
the order in this case be- 
ing A, M, C, M'. 

69. Focal Point and 
Focal Length of a Spheri- 
cal Mirror. — I n the 
special case when the ob- 
ject-point M coincides 
with the infinitely distant 
point E of the x-axis, the conjugate point M' will lie at a 
point F' (Fig. 98, a and b) determined by the relation: 

(CAEF')=-1, 
and since here CE = AE= oo , we must have: 

AF'=F'C. 
This means that a cyl- 
indrical bundle of inci- 
dent paraxial rays parallel 
to the axis of a spherical 
mirror will be transformed 
into a conical bundle of 
reflected rays with its 
vertex at a point F' which 
is midway between the 
vertex A and the center C. 

If, on the other hand, the image-point M' coincides with 
the infinitely distant point E, the conjugate object-point M 
will He on the axis at a point F determined by the relation: 
(CAFE') = - 1, 



To E a*«° 




\s 


TeE lltOO 




■* -* 


TbE «*« / 




\ a 


ToE at CO 






^° 


r/r 


aYs 





Fig. 98, 



b. — Focal point of 
mirror (AF = FC) 



concave 



§ 69] Focal Length of Mirror 167 

and therefore we obtain here in the same way as above: 

AF = FC. 
Accordingly, a conical bundle of incident rays with its 
vertex at a point F midway between the vertex of the mirror 
and its center will be transformed into a cylindrical bundle 
of reflected rays parallel to the axis of the mirror. The 
letters F and F' will be used to designate the positions of 
the so-called focal points of an optical system which is sym- 
metric around an axis. They are not a pair of conjugate 
points, as might naturally be inferred from the fact that 
they are designated by the same letter. In the case of a 
spherical mirror these two points, as we have seen, are coin- 
cident with each other, which is a consequence of the identity 
of object-space and image-space to which reference was made 
at the conclusion of § 64. The focal point of a concave mirror 
lies in front of the mirror, as shown in Fig. 98, b, so that 
paraxial rays parallel to the axis will be reflected at a con- 
cave mirror to a real focus at F; whereas in the case of a 
convex mirror the focal point F lies behind the mirror (vir- 
tual focus), as shown in Fig. 98, a. 

The focal length f of a spherical mirror may be defined as 
the abscissa of the vertex A with respect to the focal point 
F as origin; that is, /=FA. Hence, according as the mirror 
is concave or convex, the focal length will be positive or negative, 
respectively. It may be remarked that the signs of / and 
r are always opposite, the relation between these magnitudes 
being given by the following formula: 

/=-^orr=-2/. 

Hence, also, the abscissa-relation obtained in § 64 may be 
written in terms of / instead of r as follows : 

u v! f 
where, however, it must be borne in mind that, whereas the 
abscissae u, u' are measured from the vertex A as origin, 
the focal length / is measured from the focal point F. 



168 



Mirrors, Prisms and Lenses 



[§70 



If the abscissae, with respect to the focal point F, of the 
pair of conjugate axial points M, M' are denoted by x, x' t 
that is, if FM = x, FM' = x', then, since 

AM = AF-f-FM, AM' = AF+FM', 
the connection between the w's and the x's is given by the 
following equations: 

u=x-f, u ' = x'-f; 
and substituting these values in the formula above and 
clearing of fractions, we derive the so-called Newtonian 
formula, viz.: 

x.x'=f; 
which is an exceedingly simple and convenient form of the 
abscissa-relation between a pair of conjugate axial points. 
The right-hand side of this equation is essentially positive, 
and hence the abscissae x, x r must always have like signs. 
Consequently, in a spherical mirror the conjugate axial points 
M, M' lie always both on the same side of the focal point F. 

70. Graphical Method of exhibiting the Imagery by 
Paraxial Rays. — The points M, M' in Fig. 99, a and b desig- 




Fig. 99, a. — For paraxial rays the reflecting (or 
refracting) surface must be represented in diagram 
by the straight line Ay, not by the curved line AZ. 



nate the positions on the axis of a spherical mirror of a pair 
of conjugate points constructed according to the method 
explained in § 68. On the reflecting sphere ZZ take a point 
D, and draw the straight lines MD, M'D meeting the tan- 



70] 



Diagrams for Paraxial Rays 



169 



gent Ay in the plane of these lines in the points B, G, re- 
spectively. Also, draw the straight line M'B. Now if the 
point D were very close to the vertex A of the mirror, then 
the straight line MD would represent the path of an incident 




Fig. 99, b. — For paraxial rays the reflecting (or 
refracting) surface must be represented in diagram 
by the straight line Ay, not by the curved line AZ. 

paraxial ray crossing the axis at M, and the path of the 
corresponding reflected ray would be along the straight line 
DM'. But under these circumstances, the three points 
designated here by the letters D, B, G would all be so near 
together that even when we cannot regard D as absolutely 
coincident with A, we may consider D, B and G as all coin- 
cident with one another. Therefore, when the ray is paraxial, 
we may, and, in fact, in the diagram we must, regard the 
straight line BM' as showing the path of the reflected ray. 
It is quite essential that this point which is seldom clearly 
explained should be rightly apprehended by the student. In 
diagrams showing the imagery by means of paraxial ra3 r s 
the duty of the straight lines that are drawn is not primarily 
to represent the actual paths of the rays themselves but to 
locate by their intersections the correct positions of the pairs 
of corresponding points in the object-space and image-space. 
In the construction of such diagrams, a practical difficulty 



170 Mirrors, Prisms and Lenses [§ 70 

is encountered due to the fact that, whereas in reality par- 
axial rays are comprised within the very narrow cylindrical 
region immediately surrounding the axis of the spherical 
surface (§ 63), it is obviously quite impossible to show them 
this way in the figure, because it would be necessary to take 
the dimensions of the drawing at right angles to the axis 
so small that magnitudes of the second order of smallness 
would no longer be perceptible at all; thus, for example, the 
points B, D, G in Fig. 99 would have to be shown as one 
point. On the other hand, if the lines in the diagram are 
not all drawn close to the axis, the relations which have been 
found above will cease to be applicable, so that, for instance, 
the rays shown in such a drawing would not intersect in the 
places demanded by the formulae. 

Accordingly, in order to overcome this difficulty, a method 
of constructing these figures has been very generally adopted, 
which, although it is confessedly in the nature of a com- 
promise, has been found to be on the whole quite satisfactory, 
and wherein at any rate the geometrical relations are in 
agreement with the algebraic conditions, which is the essen- 
tial requirement. In this plan, while the dimensions parallel 
to the axis remain absolutely unaltered, the dimensions at 
right angles to the axis are all prodigiously magnified in the 
same proportion. Thus, for example, if the incidence-height 
h = ~DB (Fig. 89) is a small magnitude of the order, say, of 
one-thousandth of the unit of length, it will be shown in 
the figure magnified a thousand times; whereas another or- 
dinate whose height was only one one-millionth of the unit 
of length and which, therefore, would be of the second order 
of smallness as compared with h, would appear even in the 
magnified diagram as a magnitude of the first order of small- 
ness. And if the ordinate denoted by h, although in reality 
infinitely small, is represented in the drawing by a line of 
finite length, an ordinate of the second order of smallness 
as compared with h will be entirely unapparent in the 
magnified diagram. 



§ 71] Extra-Axial Conjugate Points 171 

Of course, as already intimated, one effect of this lateral 
enlargement will be to misrepresent to some extent the rela- 
tions of the lines and angles in the figure. For instance, the 
circle in which the spherical mirror (or refracting surface) 
is cut by the plane of a meridian section will thereby be 
transformed into an infinitely elongated ellipse with its 
major axis perpendicular to the axis of the spherical surface, 
and this ellipse will appear in the diagram as a straight line 
Ay tangent to the circle at A. The minor axis of the ellipse 
remains unchanged and equal to the diameter 2r of the circle, 
and moreover the center of the ellipse remains at the center C 
of the circle. But the most apparent change will be in the 
angular magnitudes which will be completely altered and 
distorted. For example, every straight line drawn through 
the center C really meets the circle ZZ (Fig. 89) normally, 
but in the distorted figure the axis of symmetry will be the 
only one of such lines which will be perpendicular to the 
straight line Ay which takes the place of the circular arc ZZ. 
Angles which in reality are equal will appear unequal, and 
vice versa. However — and after all this is the really essential 
matter — the absolute dimensions of the abscissa? and the rela- 
tive dimensions of the ordinates will not be changed at all; and 
therefore lines which are really straight will appear as 
straight lines in the figure, and straight lines which are 
parallel will be shown as such. The abscissa of the point of 
intersection of a pair of straight lines in the drawing will be 
the true abscissa of this point. 

In such a diagram, therefore, any ray, no matter what 
slope it may have nor how far it may be from the axis, is to 
be considered as a paraxial ray. The meridian section of 
the spherical reflecting or refracting surface must be repre- 
sented in the figure by the straight fine Ay (?/-axis), and the 
position of the center C with respect to the vertex A will 
show whether the surface is convex or concave. 

71. Extra-Axial Conjugate Points. — If we suppose that the 
axis of the spherical mirror is rotated about the center C 



172 



Mirrors, Prisms and Lenses 



[§71 



through a small angle ACU, so that the vertex A moves 
along the mirror to a neighboring point U, the conjugate 
axial points M, M' will describe also small arcs MQ, M'Q' 
of concentric circles; and, evidently, the points Q, Q' will be 




A x 



Fig. 100. — Concave mirror: Object is a small line MQ perpendicular to 
axis; its image M'Q' is real and inverted. 



harmonically separated (§§67, 68) by the points C, U, so that 
(CUQQO = (CAMM') = - 1. Thus, we see how the point Q' 
is the image-point conjugate to the extra-axial object-point Q. 
In the diagram (Fig. 100) the circular arcs AU, MQ and 
M'Q' will appear as straight lines perpendicular to the axis, as 
explained in § 70. We derive, therefore, without difficulty 
the following conclusions: 

(1) Th image, in a spherical mirror, of a plane object per- 
pendicular to the axis is likewise a plane perpendicular to the 
axis; (2) A straight line passing through the center of the 
spherical mirror intersects a pair of such conjugate planes in a 
pair of conjugate points; and (3) To a homocentric bundle of 
incident paraxial rays proceeding from a point Q in a plane 
perpendicular to the axis of a spherical mirror there corre- 



§ 71] Spherical Mirror: Construction of Image 173 

sponds a homocentric bundle of reflected rays with its vertex Q' 
lying in the conjugate image-plane. 

In order to construct the image-point Q' of the extra-axial 
object-point Q, we have merely to find the point of inter- 
section after reflection at the spherical mirror of any two 




Fig. 101, a. — Lateral magnification and construction of image in 
concave mirror. 



rays emanating originally from Q. The diagrams (Fig. 101, 
a and b) , which are drawn according to the method explained 
in § 70, exhibit this construction for the cases when the mirror 
is concave and convex. Of the incident rays proceeding 
from Q, it is convenient to select for this purpose two of the 
following three, namely: the ray QC which proceeding to- 
wards the center C meets the spherical mirror normally at 
U, whence it is reflected back along the same path; the ray 
QV which proceeding parallel to the axis and meeting the 
mirror in the point designated by V is reflected at V along 
the straight line joining V with the focal point F; and the 
ray QW which being directed towards the focal point F is 
reflected at W in a direction parallel to the axis. The point 
where these reflected rays intersect will be the image- 



174 



Mirrors, Prisms and Lenses 



[§71 



point Q'. Moreover, having located the position of Q', we 
can draw QM, Q'M' perpendicular to the axis at M, M', re- 
spectively; and then M'Q' will be the image of the small 
object-line MQ. In Fig. 101, a the image M'Q' is real and 
inverted, whereas in Fig. 101, b it is virtual and erect. 




;\\f 



^ c 



ir 



Fig. 101, b. — Lateral magnification and construction of image in convex 

mirror. 



Whether the image is real or virtual and erect or inverted 
will depend both on the position of the object and on the 
form of the mirror. 

If the object-point Q is supposed to move, say, from left 
to right along the straight line QV drawn parallel to the 
axis of the mirror, the corresponding image-point Q' will 
traverse the straight line VF continuously in the. same 
direction. Thus, in the diagrams (Fig. 102, a & b) the 
numerals 1, 2, 3, etc., ranged in order from left to right along 
a straight line parallel to the axis of the mirror, show a 
number of successive positions of the object-point, while the 
primed numbers 1', 2', 3', etc., lying along the straight line 
VF, show the corresponding positions of the image-point. 
The straight lines 11', 22', 33', etc., all meet at the center 
C of the mirror. 



§71] 



Imagery in Spherical Mirror 175 



TO 4 AT 00 



^ ' 



Sf 



TO E 
AT0O ^-^V OBJE , CT f f / J ) 



RAY 




/8 7 



TO E AT CO 



CONCAVE MIRROR 

(a) 



TO 5 AT 00 



IMAGE^ray 



TO E AT OO 



OBJECT RAY 



TO E ATCO 

3' 4- 5 6 7 8 

^WTr — r- 



<V 



\ c 

K 
X 



CONVEX MIRROR 

lb) 



K 



% 

TO S AT 00 



Pig. 102, a and b. — Imagery in (a) concave mirror, (6) convex 

mirror. 



176 Mirrors, Prisms and Lenses [§ 73 

72. The Lateral Magnification. — If the ordinates of the 
pair of extra-axial conjugate points Q, Q' are denoted by 
y, y', respectively, that is, if in Fig. 101, a and b, MQ = y, 
M'Q' = y', the ratio y'jy is called the lateral magnification at 
the axial point M. This ratio will be denoted by y; thus, 
y = y'jy. The sign of this function y indicates whether the 
image is erect or inverted. The lateral magnification may 
have any value positive or negative depending only on the 
position of the object. 

In the similar triangles MCQ, M'CQ' 

M'Q' :MQ=M'C:MC; 
and since 

M'C = r-u', MC = r-u, 
where u=AM, u' = AM', r = AC; and since according to the 
abscissa-formula (§ 64) 

r-u' _ v! 
r—u u 
we derive the following formula for the lateral magnification 
in the case of a spherical mirror: 

V u 
Also, from the figure we see that 

M'Q' _ AW _ FA _ M'Q' _ FM' , 
~MQ~MQ"FM AV FA' 
and since FM = z, FM' = z r , and FA=/, we derive also an- 
other formula for the lateral magnification, as follows: 

y y x f 

This expression shows that the lateral magnification is in- 
versely proportional to the distance of the object from the 
focal plane. 

73. Field of View of a Spherical Mirror.— When the 
image of a luminous object is viewed in a spherical mirror, 
the axis of the mirror is determined by the straight line O'C 



§73] 



Field of View of Spherical Mirror 



177 



(Fig. 103, a and b) joining the center O' of the pupil of the 
observer's eye with the center C of the mirror; and, on the 
assumption that the image is formed by the reflection of 
paraxial rays, the actual portion of the mirror that is utilized 




Fig. 103, a. — Field of view for eye in front of convex mirror. 

consists of a small circular zone immediately surrounding 
the vertex A where the axis meets the reflecting surface. Ac- 
cording to the method of drawing these diagrams which was 
described in § 70, the line-segment GH which is perpendicu- 
lar to the axis at A and which is bisected at A will represent 
a meridian section of this zone in the plane of the figure, so 
that the points designated by G, H are opposite extremities 
of a diameter of the effective portion of the mirror. 

All the reflected rays that enter the eye at O' must neces- 
sarily lie within the conical region determined by revolving 
the isosceles triangle O'GH around the axis of the mirror. 



178 



Mirrors, Prisms and Lenses 



[§73 



The outermost rays that can possibly be reflected into the eye 
at O' will be the rays that are reflected along the straight 
lines HO' and GO'. In order to see a real image in a concave 
mirror (Fig. 103, 6), the eye must be placed in front of the 




Fig. 103, b. — Field of view for eye in front of concave mirror. 



mirror at a distance greater than the length of the radius. 
The incident rays corresponding to the extreme reflected 
rays will intersect in a point O which is conjugate to 0'; 
and hence the field of view (§ 9) within which all object-points 
must lie in order that their images in the mirror may be 
visible to an eye at O' will be limited by the surface of a 
right circular cone generated by the revolution of the isosceles 
triangle OHG around the axis of the mirror. Thus, exactly 
as in the case of the corresponding problem in connection 
with the field of view of a plane mirror (§ 16), the contour of 
the effective portion of the spherical mirror acts also as a 
field-stop for the imagery produced by paraxial rays. 

Through O r draw B'J' at right angles to the axis of the 
mirror, and mark the points B', J' at equal distances from 
0' on opposite sides of the axis. Then B'J' may be supposed 
to represent the diameter in the plane of the diagram of the 
iris opening of the pupil of the observer's eye. Construct 
by the method described in § 71, the object-line BJ whose 
image in the mirror is B'J'. Evidently, any ray which after 



§ 74] Spherical Refracting Surface 179 

reflection enters the pupil of the eye between B' and J' must 
before reflection have passed, really or virtually, through 
the conjugate point on the straight line between B and J. 
In fact, the circle described around as center in the trans- 
versal plane perpendicular to the axis at with radius OB 
will act like a material stop to limit the apertures of the 
bundles of incident rays. It is the so-called entrance-pupil 
of the system, while the pupil of the eye plays the part of 
the exit-pupil (see § 16). Thus, for example, if S designates 
the position of a luminous point lying anywhere within the 
field of view, the eye at O' will see the image of S at S' by 
means of a bundle of rays which are drawn from S to all points 
of the entrance-pupil and which after reflection at the 
mirror are comprised within the cone which has its vertex 
at S' and the exit-pupil as base. The entrance-pupil BJ is 
the aperture-stop of the system (§ 11). 

74. Refraction of Paraxial Rays at a Spherical Surface. — 
In the accompanying diagrams Fig. 104, a and b, the straight 
line RB represents an incident ray meeting the spherical 
refracting surface ZZ at B, while the straight line BS shows 




Fig. 104, a. — Convex spherical refracting surface (n'>n). 

the path of the corresponding refracted ray. If the position 
of the point M where the incident ray crosses the axis is 
given, the problem is to determine the position of the point 
M' where the refracted ray meets the axis. The angles of 



180 Mirrors, Prisms and Lenses [§ 74 

incidence and refraction are ZNBR=a, ZN'BS= a', and 
by the law of refraction : 

n'.sina' = n.sina, 
where n, n f denote the indices of refraction of the first and 
second media, respectively. In the triangles MBC, M'BC, 
we have: 

CM : BM = sina : sin0, CM' : BM' = sina' : sin0, 




Fig. 104, b. — Concave spherical refracting surface {nf > n) . 

where $ = ZBCA. Dividing one of these equations by the 
other, we obtain: 

CM .BM _n' 

CM' : BM' n ' 
Now if the ray RB is a paraxial ray, the incidence-point B 
will be so near the vertex A of the spherical refracting surface 
that A may be written in place of B, according to the def- 
inition of a paraxial ray as given in § 63. Therefore, in the 
case of the refraction of paraxial rays at a spherical surface 
the four points C, A, M, M' on the axis are connected by 
the following relation : 

CM .AM _n' 

CM ,: AM' n 



§ 74] Spherical Refracting Surface 181 

which may be written (§ 65) : 

(CAMM')=-; 

n 

that is, the double ratio of the four axial points C, A, M, M' 
is constant and equal to the relative index of refraction from 
the first medium to the second. 

Thus, for a given spherical surface (that is, for known 
positions of the points A and C) , separating a pair of media 
of known relative index of refraction (n'/ri), the point M' 
on the axis corresponding to a given position of the axial 
point M has a perfectly definite position, entirely independent 
of the actual slope of the incident paraxial ray RB; whence 
it may be inferred that M' is the image of M, so that to a 
homocentric bundle of incident paraxial rays with its vertex 
lying on the axis of the spherical refracting surface there corre- 
sponds also a homocentric bundle of refracted rays with its 
vertex on the axis. 

In Fig. 104, a the image at M r is real, whereas in Fig. 104, 6 
it is virtual. Since the relative index of refraction is never 
less than zero, the value of the double ratio (CAMM') in 
the case of refraction at a spherical surface is necessarily 
positive; consequently, the pair of conjugate points M, M' 
is not "separated" (§ 65) by the pair of points A, C, as was 
found to be the case in reflection at a spherical mirror (§ 68) . 
Thus, if M, M' designate the positions of a pair of conjugate 
axial points with respect to a spherical refracting surface, it 
is always possible to pass from M to M' along the axis one 
way or the other without going through either of the points 
A or C, although in order to do this it may sometimes be 
necessary to pass through the infinitely distant point of the 
axis (see § 65) . Accordingly, depending only on the form 
of the surface and on whether n is greater or less than n', 
there will be found to be sixteen possible orders of arrange- 
ment of these four points, viz.: 

A, C, M, M'; A, C, M', M; A, M, M', C; A, M', M, C; 
M, A, C, M'; M', A, C, M; M, M', A, C; M', M, A, C; 



182 Mirrors, Prisms and Lenses [§75 

together with the eight other arrangements obtained by re- 
versing the order of the letters in each of these combinations; 
in other words, exactly the series of combinations that are 
not possible in the case of a spherical mirror where the pair 
of conjugate axial points M, M' is harmonically separated 
by the pair of points A, C, so that (CAMM') = -1 (§ 68). 
The student should draw a diagram similar to Fig. 104 for 
each of the possible arrangements of the four points above 
mentioned. Fig. 104, a shows the case M, A, C, M' and 
Fig. 104, b shows the case M, M', C, A. 

Moreover, if (CAMM')=w'/n, then also (CAM'M) = 
ft/ft', as follows from the definition of the double ratio (§ 65). 
Consequently, if a paraxial ray is refracted at a point B of 
a spherical surface from medium n to medium n' along the 
broken line RBS, a ray directed from S to B will be refracted 
from medium n f to medium n in the direction BR; which is 
in accordance with the general principle of the reversibility 
of the light-path (§ 29). If therefore M' is the image of M 
when the light is refracted across the spherical surface in a 
given sense, then also M will be the image of M' when the 
refraction takes place in the reverse sense. 

75. Reflection Considered as a Special Case of Refrac- 
tion. — It was implied above that if it were possible for the 
ratio n'/n to have not only positive values but also the unique 
negative value —1, the single formula (CAMM.')=n'ln 
would express the relation between a pair of conjugate axial 
points M, M' both for a spherical refracting surface and 
for a spherical mirror. The question naturally arises, there- 
fore, Is there a general rule of this kind applicable also to 
other problems in optics that are not necessarily concerned 
with paraxial rays or particular conditions? Returning to 
fundamental principles and recalling the laws of reflection 
and refraction, we observe that while the angles of incidence 
and refraction always have like signs, the angles of incidence 
and reflection, on the contrary, have opposite signs. In 
order, therefore, that the refraction-formula n f . sina' = n . sina 



§ 761 Construction of Conjugate Axial Points 183 

may include also the law of reflection as well, the values 
of n and n' in the latter case must be such that a'= — a 
is a solution of the equation in question; and obviously this 
solution can be obtained only by putting 

n = —n, or — = — 1. 
n 

Accordingly, the rule discovered above to be true in a special 
case is found to be entirely general, so that, at least from a 
purely mathematical point of view, the reflection of light 
may be regarded as a particular case of refraction back again 
into the medium of the incident light, provided we assign to 
this medium two equal and opposite values of the absolute 
index of refraction. The convenience of this artifice is ap- 
parent, since it makes it quite unnecessary to investigate sep- 
arately and independently each special problem of reflection 
and refraction; for when in any given case the relation be- 
tween an incident ray and the corresponding refracted ray 
has been ascertained, it will be necessary merely to impose 
the condition n'= —n in order to derive immediately the 
analogous relation between the incident ray and the corre- 
sponding reflected ray. Thus, for example, any formula 
hereafter to be derived concerning the refraction of paraxial 
rays at a spherical surface may be converted into the corre- 
sponding formula for the case of a spherical mirror by 
putting n'= —n. 

76. Construction of the Point M' Conjugate to the Axial 
Point M. — In order to construct the point M' conjugate to 
the axial point M with respect to a spherical refracting sur- 
face, we may proceed as follows : 

Through the vertex A (Fig. 105, a, b, c and d) and the center 
C draw a pair of parallel straight lines (preferably but not 
necessarily) at right angles to the axis; and on the line going 
through C take two points and O' such that 

CO:CO' = n':n. 
Join the given axial point M by a straight line with the point 
O, and let B designate the point where this straight line, 



184 



Mirrors, Prisms and Lenses 



76 



R M 




Fig. 105. — Spherical refracting surface: Construction of image-point M' 
conjugate to axial object-point M; construction of focal points F, F\ 
(a) Convex surface, n'>n; order MACM'. 
(6) Concave surface, n'>n; order MM'CA. 

(c) Convex surface, n' <n; order MM'AC. 

(d) Concave surface, n'<n; order MCAM'. 



§ 76] Construction of Conjugate Axial Points 185 

produced if necessary, meets the line drawn through A 
parallel to CO; then the required point M' will be at the 
place where the straight line BO', produced if necessary, 
intersects the axis. 

The straight line Ay drawn perpendicular to the axis at A 
will be tangent to the spherical surface at its vertex ; and this 
line will represent the spherical surface in the diagram, since 
we are concerned here only with paraxial rays (§ 70) . Thus, 
to the incident ray RB crossing the axis at M and incident 
on the surface at B, there will correspond the refracted ray 
BS crossing the axis at M'. 

The proof of the construction consists in showing that 

n' 
the double ratio (CAMM') is equal to — , in accordance with 

n 

the relation which, as we saw above (§ 74), connects the two 
conjugate points M, M'. 
In the pair of similar triangles CMO, AMB, 
CM:AM = CO:AB; 
and in the pair of similar triangles CM'O', AM'B, 

AM':CM' = AB:CO'. 
Multiplying these two proportions, we obtain: 
CM AM' CO 



or 



and hence 



CM' * AM CO' ' 
CM AM n' 



CM' AM' n 



(CAMM') = -. 

n 

The diagrams illustrate four cases, viz., the cases when 
the points A, C, M, M' are ranged along the axis from left 
to right in the orders MACM', MM'CA, MM'AC and 
MCAM'. In the diagrams Fig. 105, a and 6, the second 
medium is represented as more highly refracting than the 



186 Mirrors, Prisms and Lenses [§ 77 

first (n'>n), whereas in the two other diagrams Fig. 105, 
c and d, the opposite case is shown (n'<ri); in a and c the 
surface is convex, and in b and d it is concave. 

77. The Focal Points (F, F') of a Spherical Refracting 
Surface. — The object-point F which is conjugate to the in- 
finitely distant image-point E and the image-point F' which 
is conjugate to the infinitely distant object-point E of the 
axis are the so-called focal points of the spherical refracting 
surface. A conical bundle of incident paraxial rays with its 
vertex at the primary focal point F will be converted into a 
cylindrical bundle of refracted rays all parallel to the axis 
and meeting therefore in the infinitely distant point E of 
the axis; and, similarly, a cylindrical bundle of paraxial rays 
proceeding from the infinitely distant point E of the axis 
will be transformed into a conical bundle of refracted rays 
with its vertex at the secondary focal point F'. 

According to the method explained in § 76, the focal point 
F may be constructed by drawing the straight line O'H 
(Fig. 105, a, b, c and d) through 0' parallel to the axis meeting 
the straight line AB in the point designated by H; and then 
the straight line OH will intersect the axis in the primary focal 
point F. Similarly, if the straight line OK is drawn through 
O parallel to the axis meeting AB in a point K, the point of 
intersection of the straight line KO' with the axis will de- 
termine the position of the secondary focal point F'. In 
brief, the diagonals of the parallelogram OO'HK meet the 
axis in the focal points F, F'. The spherical refracting surface 
is said to be convergent or divergent according as the focal 
point F' is real or virtual, respectively. Thus, in the dia- 
grams Fig. 105, a and d, incident rays parallel to the axis are 
brought to a real focus at F', so that the surface is convergent 
for each of these cases; whereas in the diagrams Fig. 105, 
b and c, incident rays parallel to the axis are refracted as if 
they proceeded from a virtual focus at F'. 

Moreover, certain characteristic metric relations may be 
derived immediately from the diagrams Fig. 105, a, 6, c, and d. 



§ 77] Spherical Refracting Surface: Focal Points 187 

For example, in the two pairs of similar triangles FAH, HO'O 
and F'CO', O'HK, we obtain the proportions: 

FA : HO' = AH : O'O, CF' : HO 7 = CO' : HK, 

and since CO' = AH, HK = 0'0, we find: 

FA = CF'; 
and hence also : 

F'A = CF. 

Accordingly, concerning the positions of the focal points of 
a spherical refracting surface we have the following rule: 

The focal points of a spherical refracting surface lie on the 
axis at such places that the step from one of them to the center 
is identical with the step from the vertex to the other focal point. 
- This statement should be verified for each of the diagrams. 
Not only will the center C be seen to be at the same distance 
from the primary focal point as the secondary focal point is 
from the vertex A, but the direction from F to C will always 
be the same as that from A to F'. 

This relation may also be expressed in a different way; 
for, since 

FA = CF' = CA+AF', 
we have the following equation : 

FA-f-F'A = CA; or AC = AF+AF'; 
which may be put in words by saying that the step from the 
vertex to the center of a spherical refracting surface is equal to 
the sum of the steps from the vertex to the two focal points. 

And, finally, since in the pair of similar triangles FAH, 
FCO, we have: 

FC:FA=CO:AH=CO:CO'=n':w, 
and since FC= — CF= — F'A, we obtain also another useful 
and important relation, viz. : 

F'A__n' 
FA n ' 
and, consequently: The two focal points F, F' of a spherical 
refracting surface lie on opposite sides of the vertex A, and at 
distances from it which are in the ratio of n to n f . If, there- 
fore, we are given the positions of one of the two focal points, 



188 



Mirrors, Prisms and Lenses 



[§77 



AIR / GLASS 




w 




Fig. 106, a, b, c and d. — Focal points of spherical refracting surface sep- 
arating air, of index 1, and glass, of index 1.5. 
(a) Refraction from air to glass at convex surface. 
(6) " " " " " concave " 

(c) " glass to air " convex " 

(d) " " " " " concave " 



§ 77] Spherical Refracting Surface: Focal Points 189 

F or F', as well as the positions of the points A, C which de- 
termine the size and form of the spherical surface, we have 
all the data necessary to enable us to locate the point M' 
conjugate to a given axial object-point M. For we can 
locate the position of the other focal point and thus determine 
the value of the ratio n' : n. 

Whether the secondary focal point will lie on one side or 
the other of the spherical refracting surface, that is, whether 
the surface will be convergent or divergent, will depend on 
each of two things, viz.: (1) Whether the surface is convex 
or concave, and (2) Whether n' is greater or less than n. For 
example, if the rays are refracted from air to glass (n'/n = 
3/2), according to the above relations we find that AF= 2 CA, 
AF' = 3 AC ; so that starting at the vertex A and taking the 
step CA twice we can locate the primary focal point F; and 
returning to the vertex A and taking the step AC three 
times, we arrive at the secondary focal point F\ The dia- 
grams Fig. 106, a and b, show the positions of the focal points 
for refraction from air to glass for a convex surface and for 
a concave surface. In this case the convex surface is con- 
vergent and the concave surface is divergent. On the other 
hand, when the light is refracted from glass to air (n'/n= 
2/3), we find AF = 3 AC, AF' = 2 CA (Fig. 106, c and d), and 
in this case the concave surface is convergent and the convex 
surface is divergent. 

In conclusion, it may be added that the constructions and 
rules which have been given above for the case of a spherical 
refracting surface are entirely applicable also to a spherical 
mirror. In fact, here we have an excellent illustration of 
the method of treating reflection as a special case of refrac- 
tion, which was explained in § 75. For if we take n'= -n, 
the two points O, 0' (Fig. 107, a and b) will lie on a straight 
line passing through the center C of the mirror at equal dis- 
tances from C in opposite directions. The point M' con- 
jugate to the axial object-point M and the focal points F, 
F' will be found precisely according to the directions for 



190 



Mirrors, Prisms and Lenses 



l§78 






S \ 


O 








\\^) 


O 


-"-"M 




A 


/ d» \ 




o' 









Fig. 107, a and b. — Reflection at spherical mirror: Con- 
struction of image-point M' conjugate to axial object- 
point M ; construction of focal point. 

(a) concave mirror, (b) convex mirror. 

drawing 'the diagrams of Fig. 105. Obviously, the focal 
points of a spherical mirror will coincide with each other 
at a point midway between the vertex and center (§ 69). 

78. Abscissa-Equation referred to the Vertex of the 
Spherical Refracting Surface as Origin. — If the vertex A of 
the spherical refracting surface is taken as the origin (§ 63) 
from which distances or steps along the axis are reckoned, 
and if the symbols r, u and v! are employed as in the case 
of a spherical mirror (§ 64) to denote the abscissa? of the 



§ 79] Spherical Refracting Surface 191 

center C and the pair of conjugate axial points M, M', that 
is, if AC = r, AM = u, AM' = u', then 

CM = CA+ AM = u-r, CM' = CA+ AM' = u'- r; 

n' 
and since the formula (CAMM / ) = — may evidently be 

written as follows: 

, CM'_ CM 
n AM' n AM' 



we obtain 



u —r_ u—r 



u u 

Dividing both sides by r, we derive the so-called invariant 
relation in the case of refraction of paraxial rays at a spherical 
surface, in the following form: 

.f.-iy-.p-!). 

\r ul \r ul 
Usually, however, this equation is written as follows: 



-, = -+ 
u u r 

which is to be regarded as one of the fundamental formulae of 
geometrical optics. If the two constants r and n'/n are known, 
the abscissa u' corresponding to any given value of u may 
easily be determined. Putting n' = — n (§ 75), we obtain the 
abscissa-formula for reflection of paraxial rays at a spherical 
mirror (§ 64) ; and if we put r = oo , we derive the formula 

— , = - for the refraction of paraxial rays at a plane surface 

(§41). It is because this linear equation connecting the 
abscissae of a pair of conjugate axial points includes these 
other cases also that some writers have proposed that the 
formula above should be called the characteristic equation of 
paraxial imagery. 

79. The Focal Lengths f, f of a Spherical Refracting 
Surface. — The steps from the focal points F and F' to the vertex 



192 Mirrors, Prisms and Lenses [§ 79 

A are called the focal lengths of the spherical refracting surface; 
the primary focal le?igth, denoted by f, is the abscissa of A with 
respect to F (/=FA), and the secondary focal length, denoted 
by f, is the abscissa of A with respect to F'(/' = F'A). 

Since FA+F'A = CA (§ 77), and since CA= -r, the focal 
lengths and the radius of the surface are connected by the 
following relation: 

f+f'+r=0, 

and hence if two of these magnitudes are known, the value 
of the third may always be determined from the fact that their 
algebraic sum is equal to zero. For example, starting at any 
point on the axis and taking in succession in any order the 
three steps denoted by /, /' and r, one will find himself at 
the end of the last step back again at the starting point. 

Moreover, the focal lengths are connected with the indices 
of refraction by the following relation (§ 77) : 

V n' 

J -=--orn.f'+n'.f=0; 
f n 

and, hence, the focal lengths of a spherical refracting surface 
are opposite in sign and in the same numerical ratio as that of 
the indices of refraction. This formula, as we shall see (§ 122), 
represents a general law of fundamental importance in geo- 
metrical optics. 

Expressions for the focal lengths in terms of the radius 
r and the relative index of refraction {n! : n) may be derived 
immediately from the pair of simultaneous equations above 
by solving them for / and /'. The same expressions may 
likewise be easily obtained by substituting in succession in 
the abscissa-formula (§ 78) the two pairs of corresponding 
values, viz., u= — /, u'= go and w=oo, u'=—f. And, 
finally, they may also be obtained geometrically from one of 
the diagrams of Fig. 107 by observing that, since by con- 
struction CO : CO' — n'\n, it follows that 

CO': 0'0 = n: {n'-n), CO: 0'0 = n': (n'-n). 



§ 80] Spherical Refracting Surface 193 

Now from the two pairs of similar triangles FAH, HO'O and 
F'AK, O'HK we obtain the two proportions: 

FA: HO' = AH: O'O, F'A: 0'H = AK: HK; 
and since 

FA=/, HO' = AC = r, AH = CO', F'A=/', AK = CO, and 
HK = 0'0, 

we have, finally: 

,_ n „ n' 

J n'-n Tl J n'-n' T ' 

which are exceedingly useful forms of the expressions for the 
focal lengths. 
Since 

n n' — n n f 



f r f" 

the abscissa-relation connecting u and v! may be expressed 
in terms of one of the focal lengths instead of in terms of the 
radius r, for example, in terms of the focal length/, as follows: 

n' _n.n 
u u f 

80. Extra-Axial Conjugate Points ; Conjugate Planes of a 
Spherical Refracting Surface. — If the axis AC of a spherical 
refracting surface is revolved in a meridian plane through 
a very small angle about an axis perpendicular to this plane 
at the center C, so that the vertex of the surface is displaced 
a little to one side of its former position A to a point U on 
the surface, the pair of conjugate points M, M' will likewise 
undergo slight displacements into the new positions Q, Q'; 
and, evidently, the same relation will connect the four points 
C, U, Q, Q' on the central line UC as exists between the four 
points C, A, M, M' on the axis AC, and accordingly (§ 76) 
we may write: 

(CUQQ')=£'; 



194 Mirrors, Prisms and Lenses [§ 81 

and hence it is obvious that the points Q, Q' are a pair of 
extra-axial conjugate points with respect to the spherical 
refracting surface. Thus, if the points belonging to an ob- 
ject are all congregated in the immediate vicinity of the axis 
on an element of a spherical surface which is concentric with 
the refracting sphere, the corresponding image-points will 
all be assembled on an element of another concentric spherical 
surface, and any straight line going through C will determine 
by its intersections with this pair of concentric surfaces two 
conjugate points Q, Q'. In order that the rays concerned 
may all be incident near the vertex A, it is necessary 
to assume that ZUCA is very small, which means that 
the little elements of the surfaces described around C may 
in fact be regarded as plane surfaces perpendicular to the 
axis AC. Accordingly, the imagery produced by the re- 
fraction of paraxial rays at a spherical surface may be de- 
scribed by the following statements: 

(1) The image of a plane object perpendicular to the axis 
of a spherical refracting surface is similar to the object, and 
will lie likewise in a plane perpendicular to the axis; (2) A 
straight line drawn through the center C will intersect a pair of 
conjugate planes in a pair of conjugate points Q, Q'; and (3) 
Incident rays which interesct in Q will be transformed into 
refracted rays which intersect in Q'. 

Diagrams showing the refraction of paraxial rays at a 
spherical surface should be drawn therefore according to the 
plan explained in § 70, as has been already stated. The 
spherical refracting surface must be represented in the figure 
by the plane tangent to the surface at its vertex A, whose 
trace in the meridian plane of the drawing is the straight 
line Ay which is taken as the y-axis of the system of rect- 
angular coordinates whose origin is at A (§ 63). 

81. Construction of the Point Q' which with Respect to a 
Spherical Refracting Surface is Conjugate to the Extra- 
axial Point Q. — The point Q' conjugate to the extra-axial 
point Q is easily constructed. Having first located the focal 



81] 



Spherical Refracting Surface 



195 



points F, F' (§ 77), we draw through Q (Figs. 108 and 109) a 
straight line parallel to the z-axis meeting the y-axis in the 
point designated by V; then the point of intersection of the 




Fig. 108. — Spherical refracting surface: Lateral magnification and 
construction of image. Convex surface, n' > n. 

straight lines VF' and QC will be the required point Q'. 
A third line may also be drawn through Q, viz., the straight 
line QF meeting the y-axis in the point marked W; and if a 




Fig. 109. — Spherical refracting surface: Lateral magnification and 
construction of image. Concave surface, n'>n. 

straight line is drawn through W parallel to the x-axis, it 
will likewise pass through Q'. 

If M, M' designate the feet of the perpendiculars let fall 



196 Mirrors, Prisms and Lenses [§ 82 

from Q, Q' respectively, on the z-axis, then M'Q' will be 
the image of the small object-line MQ. In Fig. 108, which 
represents the case of a convex refracting surface, the image 
is real and inverted, whereas in Fig. 109 the surface is 
concave and the image is virtual and erect. Both diagrams 
are drawn for the case when n'>n. 

If the object-point Q coincides with the point marked V, 
the image-point Q' will also be at V, and image and object 
will be congruent. The pair of conjugate planes of an optical 
system for which this is the case are called the principal 
planes (see § 119); and hence the principal planes of a spher- 
ical infracting surface coincide with each other and are identical 
with the tangent-plane at the vertex. 

82. Lateral Magnification for case of Spherical Refract- 
ing Surface.— The ratio M'Q': MQ (Figs. 108 and 109) is the 
so-called lateral magnification of the spherical refracting sur- 
face with respect to the pair of conjugate axial points M, M'. 
Since 





M'Q':MQ = CM':CM 


and since (§ 74) 






CM' n' AM' 




CM n ' AM ' 


we find: 






M'Q'n AM' 



MQ n' AM 

If y, y r denote the heights of object and image, that is, if 
2/ = MQ, y' = M'Q', and if we put the lateral magnification 
equal to y, as in § 72, then, evidently: 



_y _n u 

y n' " u } 

where u = AM, v! = AM'. The lateral magnification depends, 
therefore, on the position of the object, and the image is 
erect or inverted according as this ratio is positive or nega- 
tive. 



§ 83] Spherical Refracting Surface: Focal Planes 197 

83. The Focal Planes of a Spherical Refracting Surface. 
— The focal planes are the pair of planes which are perpendic- 
ular to the axis at the focal points F, F'. "The infinitely 
distant plane of space/' which, according to the notions of 
the modern geometry, is to be regarded as the locus of the 
infinitely distant points (§ 65) of space, is the image-plane 
conjugate to the primary focal plane, which is the plane 
perpendicular to the axis at F. On the other hand, re- 
garded as belonging to the object-space, the infinitely dis- 
tant plane is imaged by the secondary focal plane perpendicu- 
lar to the axis at F'. 

The rays proceeding from an infinitely distant object-point 
I (Fig. 110) constitute a cylindrical bundle of parallel in- 




Toj'at <*> 

Fig. 110. — Focal planes and focal lengths of spherical refracting surface. 



cident rays. Since I lies in the infinitely distant plane of 
space, its image V will be formed in the secondary focal 
plane, and the position of V in this plane may be located by 
drawing through the center C of the spherical refracting 
surface a straight line parallel to the system of parallel 
rays which meet in the infinitely distant point I. Thus, for 
example, the image of a star which may be regarded as a 
point infinitely far away will be formed in the secondary 
focal plane ; and if the apparent place of the star in the firma- 
ment is in the direction CI, the star's image will be at the 



198 Mirrors, Prisms and Lenses [§ 83 

point I' where the straight line CI meets the secondary focal 
plane. 

Similarly, if J designates the position of an object-point 
lying in the primary focal plane, its image J' will be the in- 
finitely distant point of the straight line JC. Thus, to a 
homocentric bundle of incident paraxial rays with its vertex in 
the primary focal plane, there corresponds a cylindrical bundle 
of refracted rays; and to a cylindrical bundle of incident paraxial 
rays there corresponds a homocentric bundle of refracted rays 
with its vertex in the secondary focal plane. 

The directions of the infinitely distant points I and J' are 
given by assigning the values of the slope-angles 

= ZFCI = ZF'CP, 0' = ZFCJ = ZF'CJ'; 
and the points I' and J conjugate to them will lie in the sec- 
ondary and primary focal planes on straight lines passing 
through the center C and inclined to the axis at the angles 
6 and 0', respectively. The angle 0, which is the measure 
of the angular distance from the axis of the infinitely distant 
object-point I, determines the apparent size of an object 
in the infinitely distant plane of the object-space; and, sim- 
ilarly, the angle 0' is the measure of the apparent size of the 
infinitely distant image of the object FJ. 

Draw the straight lines JG and FK paralled to the optical 
axis and meeting the ?/-axis in the points designated by G 
and K, respectively; then the straight lines FK and CP will 
be parallel to each other, and the same will be true with 
respect to the straight lines GF' and JC. Hence, 
ZAFK= 0, ZAF'G = 0'; and since AK=FT and AG = FJ, 
we find: 

|l=tan0,!^ = tan0'. 

Putting FA=/ and F'A=/' (§79), we obtain the following 
expressions for the focal lengths: 

FT FJ 

/•=__, /' = . 



tan tan 6' 



§ 84] Spherical Refracting Surface 199 

and since the tangents of the small angles 8, 8' are indis- 
tinguishable from the angles themselves (see § 63) , we obtain 
new definitions of the focal lengths, as follows: 

The primary focal length is the ratio of the height of the image, 
in the secondary focal plane, of an infinitely distant object to 
the apparent size of the object; and the secondary focal length 
is the ratio of the height of an object in the primary focal plane 
to the apparent size of the infinitely distant image. 

The ratio of the apparent size of the infinitely distant 
image to the height of an object in the primary focal plane 
is a measure of the magnifying power of the optical system 
(see § 158), and in this sense we may say that the magnifying 
power of a spherical refracting surface is equal to the reciprocal 
of the secondary focal length. 

84. Construction of Paraxial Ray Refracted at a Spherical 
Surface. — The refracted ray corresponding to a paraxial ray 
IB (Fig. 110) incident on a spherical refracting surface at 
the point B may easily be constructed, for example, in one 
of the following ways: 

(a) Through the primary focal point draw the straight 
line FK parallel to IB meeting the y-axis in the point K ; and 
through K draw a straight line parallel to the z-axis meeting 
the secondary focal plane in the point I'; the path of the 
refracted ray will lie along the straight line BI'. 

(b) Through the center C draw a straight line CI' parallel 
to the given incident ray meeting the secondary focal plane 
in the point I'; the path of the corresponding refracted ray 
will be along the straight line BI'. 

(c) Let J designate the point where the given incident 
ray crosses the primary focal plane, and draw the straight 
line JG parallel to the z-axis meeting the y-axis in the 
point designated by G; then the path of the required 
refracted ray will lie along the straight line BI' drawn 
through the incidence-point B parallel to the straight line 
GF', where F' designates the position of the secondary focal 
point. 



200 Mirrors, Prisms and Lenses [§ 85 

(d) Finally, the required refracted ray will be along the 
straight line BP drawn parallel to the straight line JC. 

85. The Image-Equations in the case of Refraction of 
Paraxial Rays at a Spherical Surface. — The rectangular co- 
ordinates of the image-point Q' may easily be expressed in 
terms of the coordinates of the object-point Q. But the 
forms of these expressions will depend partly on the particu- 
lar pair of constants (n'/n, r and /, /') which define the sur- 
face and partly on the system of axes to which the coordinates 
are referred. The axis of the spherical surface will always 
represent the axis of abscissae (x-axis), and the ?/-axis will 
be at right angles to it; but the origin may be taken at any 
place along the x-axis. If the vertex A is taken as the origin 
(§ 63), the coordinates of Q, Q' will be (u, y) and (u', y'); 
that is, u = AM, u' = AM', £/ = MQ, 2/'=M'Q'; and since 
(§§ 78 and 82) 

n' _ n.n' — n y r _nu' 
v! u r ' y n'u ' 
we obtain by solving for u' and y'; 

,_ n'ru ,_ nry 

(n' '— n) u+nr' (n'-n) u+nr' 

In terms of the same coordinates, but with a different 
pair of constants, viz., /, /', instead of n f :n, r, the image- 
equations may be put also in other forms, as follows: 

It will be recalled that in § 79 the abscissa-formula was 
written : 

n' _n . n 
u u f 
and since (§ 79) n'/n = —f'/f, n and n' may be eliminated 
and the image-equations will become: 

/+ /'. 1=0 yf - i J'+ u '- /< 

tt -r tt ,i-i u, y f+u ff f , u , 

which are also frequently employed. These formulae may 
also be easily derived from the geometrical relations in 
Figs. 108 and 109, since we have the proportions: 
FM: AM = VA: VW = AF': AM'. 



86] 



Smith-Helmholtz Equation 



201 



Instead of a single system of rectangular coordinates, we 
may have two systems, one for the object-space and the 
other for the image-space. For example, if the focal points 
F, F' are selected as the origins of two such systems, and 
if the abscissae of the pair of conjugate axial points M, M' 
are denoted by x, x', that is, if # = FM, z^F'M', then, since 
u = AM = AF+ FM = x -/, u' = AM' = AF'+ F'M' = x' -/', 
the abscissae, u, v! may be eliminated from the equations 
above, and the image-equations will be obtained finally in 
their simplest forms, as follows: 

y * /'" 

These relations may be derived directly from the two pairs 
of similar triangles FMQ, FAW and F'M'Q', F'AV in 
Figs. 108 and 109. The abscissa-relation 

x.x'=f.f 
is the so-called Newtonian formula (see § 69) . If the x's are 
plotted as abscissae and the x"s as ordinates, this equation 
will represent a rectangular hyperbola. 

86. The^o-called Smith-Helmholtz Formula. — In Fig. Ill 
if M'Q' = 2/' represents the image in a spherical refracting 




Fig. 111. — Spherical refracting surface: Smith-Helmholtz law. 

surface Ay of a small object-line MQ, = y perpendicular to 
the axis at M, and if B designates the incidence-point of a 



202 Mirrors, Prisms and Lenses [§ 86 

paraxial ray which crosses the axis before and after refrac- 
tion at M and M', respectively, then in the triangle MBM' 

sin0:sin0' = BM':BM, 
where = ZAMB, 0' = ZAM'B denote the slopes of the 
incident ray MB and the corresponding refracted ray BM'. 
Since the ray is paraxial, we may put = sin 0, 0' = sin 0' 
and also BM = AM = u, BM' = AM' = v! (§ 63) . Hence, 



^ = -, or u'. 0' = u. 0. 
6' u' 



But (§ 82) 



ri .y' _n.y m 



and, therefore, by multiplying these two equations so as to 
eliminate u and u f , we obtain the important invariant- 
relation in the case of refraction of paraxial rays at a 
spherical surface, viz.: 

ri.y'. 6' = n.y. 0. 
This formula states that the function obtained by the con- 
tinued product of the three factors n, y, has the same value 
after refraction at a spherical surface as it had before re- 
fraction. It is a special case of a general law which is found 
to apply to a centered system of spherical refracting sur- 
faces (§ 118) and which is usually known as Lagrange's law; 
but undoubtedly Robert Smith who announced the law for 
the case of a system of thin lenses as early as 1738 is entitled 
to the credit of it. The importance of the relation was 
recognized by Helmholtz( 182 1-1894), and the form in which 
it is written above is due to him. On the whole it seems 
proper to adopt the suggestion of P. Culmann and to refer 
to this equation as the Smith-Helmholtz formula. 



Ch. VI] Problems 203 

PROBLEMS 

1. If A designates the vertex and C the center of a spher- 
ical mirror, and if M, M' designate the points where a 
paraxial ray crosses the straight line AC before and after 
reflection, respectively, show that 

1+1=2 

u u r 
where r = AC, w = AM, u' = AM'. 

2. The radius of a concave mirror is 30 cm. Paraxial rays 
proceed from a point 60 cm. in front of it; find where they 
are focused after reflection. 

Ans. At a point 20 cm. in front of the mirror. 

3. The radius of a concave mirror is 60 cm. A luminous 
point is placed in front of the mirror at a distance of (a) 120 
cm., (b) 60 cm., (c) 30 cm., and (d) 20 cm. Find the position 
of the image-point for each of these positions of the object. 

Ans. (a) 40 cm. in front of mirror; (b) 60 cm. in front of 
mirror; (c) at infinity; and (d) 60 cm. behind mirror. 

4. A candle is placed in front of a concave spherical mir- 
ror, whose radius is 1 foot, at a distance of 5 inches from 
the mirror. Where will the image be formed? 

Ans. 30 inches behind the mirror. 

5. An object is 24 inches in front of a concave mirror of 
radius 1 foot; where will its image be formed? If the object 
is displaced through a small distance z, through what dis- 
tance will the image move? 

Ans. Image is 8 inches in front of mirror; distance through 
which image moves will be 2z/(z— 18). 

6. An object is placed 1 foot from a concave mirror of 
radius 4 feet. If the object is moved 1 inch nearer the mirror, 
what will be the corresponding displacement of the image? 

Ans. The image moves 3.7 inches nearer the mirror. 

7. An object-point is 10 cm. in front of a convex mirror of 
radius 60 cm. Find the position of the image-point. 

Ans. 7.5 cm. behind the mirror. 



204 Mirrors, Prisms and Lenses [Ch. VI 

8. Given the positions on the axis of a spherical mirror 
of the vertex A, the center C and an object-point M; show- 
how to construct the position of the image-point M'. There 
are eight possible arrangements of these four points; draw 
a diagram for each one of them. 

9. If x, x' denote the abscissae, with respect to the focal 
point F as origin, of a pair of conjugate points on the axis 
of a spherical mirror, show that 

x.x'=}\ 
where / denotes the focal length of the mirror. How are 
object and image situated with respect to the focal plane? 

10. An object is placed at a distance of 60 cm. in front of 
a spherical mirror, and the image is found to be on the same 
side of the mirror at a distance of 20 cm. What is the focal 
length of the mirror, and is it concave or convex? 

Ans. Concave mirror of focal length 15 cm. 

11. How far from a concave mirror of focal length 18 
inches must an object be placed in order that the image 
shall be magnified three times? 

Ans. 1 ft. or 2 ft. from the mirror, according as image is 
erect or inverted. 

12. A candle-flame one inch high is 18 inches in front of 
a concave mirror of focal length 15 inches. Find the position 
and size of the image. 

Ans. The image will be real and inverted, 90 inches from 
the mirror, and 5 inches long. 

13. A small object is placed at right angles to the axis of 
a spherical mirror; show how to construct the image, and 
derive the magnification-formula: 

y u' 

14. A luminous point moves from left to right along a 
straight line parallel to the axis of a spherical mirror. Show 
by diagrams for both concave and convex mirrors how the 
conjugate image-point moves. 

15. The center of a spherical mirror is at C, and the 



Ch. VI] Problems 205 

straight line QQ' joining a pair of conjugate points meets 
the mirror in a point U. If P designates the position of a 
point which is not on the straight line QQ', and if a straight 
line is drawn cutting the straight lines PU, PQ, PC and PQ' 
in the points V, R, Z and R', respectively; show that R, R' 
are a pair of conjugate points with respect to another spher- 
ical mirror whose center is at Z and whose radius is equal to 
VZ. 

16. Show by geometrical construction that the focal point 
of a spherical mirror lies midway between the center and the 
vertex. 

17. An object is placed 5 inches from a spherical mirror of 
focal length 6 inches. Assuming that the object is real, 
where will the image be formed, and what will be the mag- 
nification? Draw diagrams for both convex and concave 
mirrors. 

Ans. For concave mirror, image is 30 in. behind the mirror, 
magnification =+6; for convex mirror, image is 2 T 8 T inches 
behind the mirror, magnification = + tV 

18. How far from a concave mirror must a real object be 
placed in order that the image shall be (a) real and four 
times the size of the object, (b) virtual and four times the 
size of the object, and (c) real and one-fourth the size of the 
object? Draw diagrams showing the construction for each 
of these three cases. 

Ans. Distance of mirror from the object is equal to (a) 
5//4, (b) 3//4, and (c) 5/, where / denotes the focal length. 

19. What kind of image is produced in a concave mirror 
by a virtual object? Illustrate and explain by means of a 
diagram. 

Ans. Image is real and erect and smaller than object. 

20. Determine the position and magnification of the image 
of a virtual object lying midway between the vertex and 
focal point of a convex mirror. Draw diagram showing 
construction. 

Ans. The vertex of the mirror will be midway between the 



206 Mirrors, Prisms and Lenses [Ch. VI 

axial point of the image and the focal point of the mirror, and 
the image will be real and erect and twice as large as object. 

21. Show that when an object is placed midway between 
the focal point and the vertex of a concave mirror the image 
will be virtual and erect and twice as large as the object. 

22. An object 3 inches high is placed 10 inches in front of 
a convex mirror of 30 inches focal length. Find the position 
and size of the image. 

Ans. Virtual image 7.5 inches from the mirror and 234 
inches high. 

23. An object is placed in front of a concave mirror at a 
distance of one foot. If the image is real and three times as 
large as the object, what is the focal length of the mirror? 

Ans. 9 inches. 

24. The radius of a concave mirror is 23 cm. An object, 
2 cm. high, is placed in front of the mirror at a distance of 
one meter. Find the position and size of the image. 

Ans. A real image, 0.26 cm. high, 13 cm. from the mirror. 

25. Find the position and size of the image of a disk 3 
inches in diameter placed at right angles to the axis of a 
spherical mirror of radius 6 feet, when the distance from the 
object to the mirror is (a) 1 ft., (6) 3 ft., and (c) 9 ft. 

Ans. For a concave mirror: (a) Virtual image, 4.5 inches 
in diameter, 18 inches from mirror; (6) Image at infinity; 
(c) Real inverted image, 1.5 inches in diameter, 4.5 feet from 
the mirror. 

26. Assuming that the apparent diameter of the sun is 
30', calculate the approximate diameter of the sun's image 
in a concave mirror of focal length 1 foot. 

Ans. A little more than one-tenth of an inch. 

27. A gas-flame is 8 ft. from a wall, and it is required to 
throw on the wall a real image of the flame which shall be mag- 
nified three times. Determine the position and focal length 
of a concave mirror which would give the required image. 

Ans. The mirror must have a focal length of 3 ft. and must 
be placed at a distance of 4 ft. from the object. 



Ch. VI] Problems 207 

28. It is desired to throw on a wall an image of an object 
magnified 12 times, the distance of the object from, the 
wall being 11 feet. Find the focal length of a concave 
mirror which will do this, and state where it must be 
placed. 

Ans. The focal length of the mirror must be f| ft., and 
it must be placed 1 ft. from the object. 

29. Assuming that the eye is placed on the axis of a spher- 
ical mirror, and that the rays are paraxial, explain how the 
field of view is determined. Draw accurate diagrams for 
concave and convex mirrors. 

30. A man holds, halfway between his eye and a convex 
mirror 3 feet from his eye, two fine parallel wires, so that 
they are seen directly and also by reflection in the mirror. 
Show that if the apparent distance between the wires as 
seen directly is 5 times that as seen by reflection, the radius 
of the mirror is 3 feet. 

31. A scale etched on a thin sheet of transparent glass is 
placed between the eye of an observer and a convex mirror 
of focal length one foot. When the distance between the 
eye and the scale is three feet, one of the scale divisions 
appears to cover three divisions of the image in the mirror. 
Find the position of the mirror. 

Ans. The mirror is one foot from the scale. 

32. A scale etched on a thin sheet of transparent glass is 
interposed between the eye of an observer and a convex 
mirror of focal length /. When the distance of the scale from 
the eye is b feet, one division of the scale appears to cover 
m divisions of its image in the mirror. If now the scale is 
displaced through a distance c in the direction of the axis 
of the mirror, it is found that one division of the scale ap- 
pears to cover k divisions in the mirror. Find an expression 
for / in terms of m, k, b and c. 

Ans. 

(k—m) (b—c) be 



f= 



{b{k-m)-(k-l)c\ {6(fc-m)-(fc+l)c 



208 Mirrors, Prisms and Lenses [Ch. VI 

33. A concave and a convex mirror, each of radius 20 cm., 
are placed opposite to each other and 40 cm. apart on the 
same axis. An object 3 cm. high is placed midway between 
them. Find the position and size of the image formed by 
reflection, first, at the convex, and then at the concave mirror. 
Draw accurate diagram, and trace the path of a ray from 
a point in the object to the corresponding point in the image. 

Ans. The image is 12i 8 r cm. from the concave mirror, 
real and inverted, and i\ cm. high. 

34. Same problem as No. 33, except that in this case the 
image is formed by rays which have been reflected first from 
the concave mirror and then from the convex mirror. 

Ans. The image is 6f cm. behind the convex mirror, 
virtual and inverted, and 1 cm. high. 

35. Two concave mirrors, of focal lengths 20 and 40 cm., 
are turned towards each other, the distance between their 
vertices being one meter. An object 1 cm. high is placed 
between the mirrors at a distance of 10 cm. from the mirror 
whose focal length is 20 cm. Find the position and size of 
the image produced by rays which are reflected first from 
the nearer mirror and then from the farther mirror. 

Ans. A real inverted image, 1 cm. long, at a distance of 
60 cm. from the mirror that is farther from the object. 

36. The distance between the vertices Ai and A 2 of two 
spherical mirrors which face each other is denoted by d, 
that is, <2 = A 2 Ai. The focal points of the mirrors are at Fi 
and F 2 , and the focal lengths are /i = FiAi and / 2 = F 2 A 2 . 
An object is placed between the mirrors at a distance u\ 
from Ai. Rays proceeding from the object are reflected, 
first, from the mirror Ai and then from the mirror A 2 ; show 
that the distance of the final image from the mirror A 2 is 

</i.m~(/i+m) Ah . 

and that the magnification is 

fi.f* 
(/i+«o (/ 2 +d)-/i.«r 



Ch. VI] Problems 209 

37. If the rays fall first on the mirror A 2 and then on Ai, 
these letters having exactly the same meanings as in No. 36, 
then the distance of the image from mirror Ai will be 

fi\(f2+d) (ttrH)+/ 2 rfl 
B (/i 7 d)/2+(t*i+d) Ui-h-dY 
and the magnification will be 

Uh 

(A-d) f2+(ui+d) Ui~h-d) m 

38. If the mirror A x in Nos. 36 and 37 is a plane mirror, 
show that when the light is reflected froni the plane mirror 
first the distance of the image from the curved mirror is 

(ui—d)f 2 



and that the magnification is 



h+d—ui 



fr\-d-ui 

and that when the light is reflected from the curved mirror 
first, the distance of the image from the plane mirror is 

ih+d) (U!+d)+f 2 d 

\ h+ui+d 
and that the magnification is 

h 

f2+Ui+d ' 

If both the mirrors are plane, the magnification will be 
unity, and the image after two reflections, first at Ai and 
then at A 2 , will be formed at a distance of (u\—d) from 
A 2 ; whereas if the light falls first on mirror A 2 , the distance 
of the image from the other mirror will be (uy\-2d). 

39. If M, M' are a pair of conjugate points on the axis 
of a spherical refracting surface which divides two media 
of indices n and n f , show that 

(CAMM')=-, 

n 

where A and C designate the vertex and the center of the 
spherical surface. 

40. Show how to construct the position of the point M' 



210 Mirrors, Prisms and Lenses [Ch. VI 

conjugate to a given point M on the axis of a spherical re- 
fracting surface; and draw diagrams for all the possible ar- 
rangements of the four points A, C, M, M'. Prove the con- 
struction, and derive the formula n'/u' = n/u-\-(n'—ri)/r, 
where n, n' denote the indices of refraction, and w = AM, 
u' = AM',r = AC. 

41. Show how the formula in No. 40 includes as special 
cases the case of refraction of paraxial rays at a plane sur- 
face and the case of reflection at a spherical mirror. 

42. From the formula in No. 40 derive expressions for 
the focal lengths /, /' of a spherical refracting surface, and 
show that 

f+f+r = 0, ra./'+rc'./=0. 

43. Does the construction found in No. 40 apply to the 
case of a spherical mirror? Explain with diagrams. 

44. Apply the construction employed in No. 40 to de- 
termine the positions of the focal points F, F' of a spherical 
refracting surface, and show that 

FA = CF', F'A = CF, F'A: FA= -»': n. 

45. Where are the focal points of a plane refracting sur- 
face? Explain clearly. 

46. Explain how the results of No. 44 are applicable to 
a spherical mirror. 

47. Air and glass are separated by a spherical refracting 
surface of radius 7' = AC. Find the positions of the focal 
points F, F' for the cases when the refraction is from air to 
glass and from glass to air and when the surface is convex 
and concave; illustrating your answers by four accurately 
drawn diagrams. (Take indices of refraction of air and 
glass equal to 1 and 1.5, respectively.) 

48. From the figures used in No. 44 for constructing the 
positions of the focal points F, F', derive the formulae for 
the focal lengths which were obtained in No. 42. 

49. Light falling on a concave surface separating water 
(n=1.33) from glass (n' = 1.55) is convergent towards a 
point 10 cm. beyond the vertex. The radius of the surface 



Ch. VI] Problems 211 

is 20 cm. Find the point where the refracted rays cross the 
axis. 

Ans. 13.19 cm. beyond the vertex of the sphere in the 
glass medium. 

50. Light is refracted from air to glass (n r : n = 3: 2) at a 
spherical surface. If the vertex of the bundle of incident 
rays is in the glass and 20 cm. from the vertex of the re- 
fracting surface, and if the refracted rays are converged to 
a point in the glass and 5 cm. from the vertex, determine 
the form and size of the surface. 

Ans. Convex surface of radius 2 cm. 

51. A small air-bubble in a glass sphere, 4 inches in di- 
ameter, viewed so that the speck and the center of the sphere 
are in line with the eye, appears to be one inch from the 
point of the surface nearest the eye. What is its actual dis- 
tance, assuming that the index of refraction of glass is 1.5? 

Ans. 1.2 inches. 

52. The radius of a concave refracting surface is 20 cm. 
A virtual image of a real object is formed at a distance of 
40 cm. from the vertex, and the distance from the object 
to the image is 60 cm. The first medium is air (n = 1). Find 
the index of refraction of the second medium. 

Ans. n' = 1.6. 

53. Light diverging from a point M in air is converged 
by a spherical refracting surface to a point M' in glass of 
index 1.5. The distance MM' =18 cm., and the point M 
is twice as far from the surface as the point M'. Find the 
radius of the surface. Ans. 1.5 cm. 

54. Find the positions of the focal points F, F' of a con- 
cave spherical refracting surface separating air from a me- 
dium of index 1.6, having found that the image of a luminous 
point 30 cm. in front of the surface is midway between the 
luminous point and the surface. 

Ans. AF = + 13.63 cm.; AF'= -21.81 cm. 

55. A convergent bundle of rays is incident on a spherical 
refracting surface of radius 10 cm. The relative index of 



212 Mirrors, Prisms and Lenses [Ch. VI 

refraction from the first medium to the second medium is 
equal to 2 (n r : n = 2:l). If the incident rays- cross the axis 
at M and the refracted rays at M', and if M'M = +60 cm., 
determine the positions of the points M, M'. 

Ans. If the surface is convex, AM = +77.72 cm., AM' 
= + 17.72 cm. If the surface is concave, then either AM = 
+30 cm., AM' = —30 cm. or AM = +20 cm., AM' = -40 cm. 

56. A beam of parallel rays passing through water (n = 
1.3) is refracted at a concave surface into glass (n' = 1.5). 
If the radius of the surface is 20 cm., where will the light be 
focused? Ans. Virtual focus, 150 cm. from the surface. 

57. A small air-bubble is imbedded in a glass sphere at 
a distance of 5.98 cm. from the nearest point of the surface. 
What will be the apparent depth of the bubble, viewed from 
this side of the sphere, if the radius of the sphere is 7.03 cm., 
and the index of refraction from air to glass is 1.42? 

Ans. 5.63 cm. 

58. Assuming that the cornea of the eye is a spherical 
refracting surface of radius 8 mm. separating the outside air 
from the aqueous humor (of index f), find the distance 
of the pupil of the eye from the vertex of the cornea, if its 
apparent distance is found to be 3.04 mm. Also, if the ap- 
parent diameter of the pupil is 4.5 mm., what is its real 
diameter? Ans. 3.6 mm.; 4 mm. 

59. Construct the image M'Q' of a small object MQ per- 
pendicular at M to the axis of a spherical refracting surface, 
and derive the magnification-formula in terms of the dis- 
tances of M and M' from the vertex of the surface. Draw 
two diagrams, one for convex, and one for concave surface. 

60. Derive the image-equations of a spherical refracting 
surface referred to the focal points as origins. 

61. Derive the image equations of a spherical refracting 
surface in the forms 

f/u+f'/u'+ 1 = 0, y'/y =f/(f+u) = (f'+u')/f. 

62. Show that there are two positions on the axis of a 
spherical refracting surface where image and object coincide. 



Ch. VI] Problems 213 

63. Locate the two pairs of conjugate planes of a spheri- 
cal refracting surface for which image and object have the 
same size. 

64. A real object, 1 cm. high, is placed 12 cm. from a con- 
vex spherical refracting surface, of radius 30 cm., which 
separates air (n = l) from glass (n' = 1.5). Find the position 
and size of the image. 

Ans. Image is virtual and erect, 1.25 cm. high, 22.5 cm. 
from vertex. 

65. In the preceding example, suppose that the object 
is a virtual object at the same distance from the spherical 
refracting surface. Find the position and size of the image 
in this case. 

Ans. Image is real and erect, I cm. high, and 15 cm. 
from vertex. 

66. Solve Nos. 64 and 65 for the case when the surface 
is concave; and draw diagrams showing construction of the 
image in all four cases. 

67. Solve No. 64 on the supposition that the first medium 
is glass and the second medium air. 

Ans. Image will be virtual and erect, if cm. high, and 
f ? cm. from vertex. 

68. (a) The human eye from which the crystalline lens 
has been removed (so-called "aphakic eye") may be re- 
garded as consisting of a single spherical refracting surface, 
namely, the anterior surface of the cornea. If the radius 
of this surface is taken as 8 mm., and if the index of refrac- 
tion of the eye-medium (both the aqueous and vitreous 
humors) is put equal to |, what will be the focal lengths 
of the aphakic eye? (b) Assuming that the length of the 
eye-ball of an aphakic eye is 22 mm., where will an object 
have to be placed to be imaged distinctly on the retina at 
the back of the eye? 

Ans. (a)/=+24mm.,/'=-32mm.; (b) ^ = +52.8 mm., 
which means that the object must be virtual and lie behind 
the eye. 



214 Mirrors, Prisms and Lenses [Ch. VI 

69. Listing's " reduced eye" is composed of -a single 
convex spherical refracting surface of radius 5.2 mm. sep- 
arating air (n = l) from the vitreous humor (n' = 1.332). 
Calculate the focal lengths. 

Ans. /= + 15.68 mm., /'= -20.90 mm. 

70. In Donder's "reduced eye" the focal lengths are 
assumed to be +15 and —20 mm. Calculate the radius of 
the equivalent spherical refracting surface and the index of 
refraction of the vitreous humor for these values of the focal 
lengths. Ans. r = +5mm.; n' = i. 

71. The angular distance of a star from the axis of a 
spherical refracting surface which separates air (n = l) from 
glass (n' = 1.5) is 10°. The surface is convex and of radius 
10 cm. Find the position of the star's image. 

Ans. A real image will be formed in the secondary focal 
plane about 3.5 cm. from the axis. 

72. What is the size of the image on the retina of List- 
ing's " reduced eye" (No. 69) if the apparent size of the 
distant object is 5°? Ans. 1.36 mm. 

73. A hemispherical lens, the curved surface of which has 
a radius of 3 inches, is made of glass of index 1.5. Show 
that rays of light proceeding from a point on its axis 4 inches 
in front of its plane surface will emerge parallel to the axis. 

74. A paraxial ray parallel to the axis of a solid refracting 
sphere of index n' is refracted into the sphere at first towards 
a point X on the axis, and after the second refraction crosses 
the axis at a point F'. If the first and last media are the 
same and of index n, show that the point F' lies midway be- 
tween the second vertex of the sphere and the point X. 

75. A small object of height y is placed at the center of 
a spherical refracting surface in a plane at right angles to 
the axis. Determine the position and size of the image. 
Show how the Smith-Helmholtz formula (§ 86) is appli- 
cable to a part of this problem. 

Ans. Image is in same plane as object, erect, and of size 
y' = n.y/n'. 



Ch. VI] Problems 215 

76. A plane object is placed parallel to a plane refracting 
surface. Show that its image formed by paraxial rays is 
erect and of same size as object. Is the Smith-Helmholtz 
formula (§ 86) applicable to a plane refracting surface? Is 
it applicable to a spherical mirror? Explain clearly. 

77. In a convex spherical refracting surface of radius 
0.75, which separates air (n = l) from water (n' = -|), the 
image is real, inverted and one-third the size of the object. 
Find the positions of object and image. If a ray pro- 
ceeding from the axial point of the object is inclined to the 
axis at an angle of 3°, what will be the slope of the correspond- 
ing refracted ray? 

Ans. Object is in air and image is in water, their distances 
from the surface being 9 and 4, respectively; slope of re- 
fracted ray is —4.5°. 

78. In a spherical refracting surface 

a=6+<p, a'=d'+<p, 
where a, a/ denote the angles of incidence and refraction, 
6, 6' denote the inclinations of the ray to the axis before 
and after refraction, and <p denotes the so-called central 
angle (ZBCA). For a paraxial ray the law of refraction 
may be written 

n'.a' = n.a. 
From these formulae deduce the abscissa-relation in the form 

n' _n , n' — n 

u f u r 

79. The curved surface of a glass hemisphere is silvered. 
Rays coming from a luminous point at a distance u from 
the plane surface are refracted into the glass, reflected from 
the concave spherical surface, and refracted at the plane 
surface back into the air. If r denotes the radius of the 
spherical surface and n the index of refraction of the glass, 
show that 

u u r 



216 Mirrors, Prisms and Lenses [Ch. VI 

where v! denotes the distance of the image from the plane 
surface. 

80. A plane object of height one inch is placed at right 
angles to the axis of a spherical mirror. The slope of the re- 
flected ray corresponding to an incident paraxial ray which 
emanates from the axial point of the object at a slope of +5° 
is +10°. Is the image erect or inverted, and what is its size? 
Ans. Inverted image, one-half inch high. 



CHAPTER VII 

REFRACTION OF PARAXIAL RAYS THROUGH AN INFINITELY 
THIN LENS 

87. Forms of Lenses. — In optics the word lens is used 
to denote a portion of a transparent substance, usually 
isotropic, comprised between two smooth polished surfaces, 
one of which may be plane. These surfaces are called the 
faces of the lens. The curved faces are generally spherical, 
and this may always be considered as implied unless the 
contrary is expressly stated. Lenses with spherical faces 
are sometimes called " spherical lenses" to distinguish them 
from cylindrical, sphero-cylindrical and other forms of 
lenses which are also quite common, especially in modern 
spectacle glasses. A plane face may be regarded as a spher- 
ical or cylindrical surface of infinite radius. 

The axis of a lens is the straight line which is normal to 
both faces, and, consequently, a ray whose path lies along 
the axis (the so-called axial ray) will pass through the lens 
without being deflected from this line. The axis of a spher- 
ical lens is the straight line joining the centers d, C 2 of 
the two spherical faces, and since a lens of this kind is sym- 
metric around the axis, it may be represented in a plane 
figure by a meridian section showing the arcs of the two 
great circles in which this plane intersects the spherical 
faces. Depending on the lengths of the radii in comparison 
with the length of the line-segment CiC 2 , these arcs inter- 
sect in two points equidistant from the axis or else they do 
not intersect each other at all. 

(a) If they intersect, then CiC 2 is less than the arith- 
metical sum but greater than the arithmetical difference 
of the radii, and the lens may be a double convex lens 

217 



218 Mirrors, Prisms and Lenses [§ 87 




Fig. 112, a. — Double convex lens. 




V 



>TO n AT 00 

2 1 



Fig. 112, &. — Plano-convex lens. 



§ 87] Forms of Lenses 219 




Fig. 112, c. — Convex meniscus. 

(Fig. 112, a) or a convex meniscus (Fig. 112, c). A particular 
case of a double convex lens is a plano-convex lens (Fig. 112, b). 

(b) If they do not intersect, then either one circle lies 
wholly outside the other, the distance between the centers 
being, therefore, greater than the arithmetical sum of the 
radii, so that the lens is a double concave lens (Fig. 113, a), 
or, in case one of the surfaces is plane, a plano-concave lens 
(Fig. 113, b); or else one circle lies wholly inside the other, 
so that the distance between the centers is less than the 
arithmetical difference of the radii, and then the lens has 
the form of a concave meniscus (Fig. 113, c). 

The first face of a lens is the side turned towards the in- 
cident light. The points where the axis meets the two faces 
are called the vertices, and the distance from the vertex Ai of 
the first face to the vertex A 2 of the second face, which is 
denoted by d, is called the thickness of the lens; thus, d — 
AiA 2 . Since the direction which the light takes in going 
across the lens from Ai to A 2 is the positive direction along 
the axis (see § 63), the thickness d is essentially a positive 
magnitude. 



220 Mirrors, Prisms and Lenses [§ 87 



s 



\ 



A I IA„ 

1/ 12. 



\ 

\ 



\ 



/ 



Fig. 113, a. — Double concave lens. 



TO C, AT oo 

* fc 



N 



Fig. 113, b. — Plano-concave lens. 




\ 



\ \ 



A \ \ A z C z C i ' 



/ 



Fig. 113, c. — Concave meniscus. 



87] 



Forms of Lenses 



221 



The radii of the surfaces, denoted by r h r 2 , are the ab- 
scissa? of the centers Ci, C 2 with respect to the vertices 
Ai, A 2 , respectively; thus, ri = AiCi, r 2 = A 2 C 2 . 

Certain special forms of spherical lenses may be mentioned 
here, viz. : 

(a) Symmetric Lenses, which are double convex or double 
concave lenses whose surfaces have equal but opposite 
curvatures (ri-f-r 2 = 0). A particular case of double convex 
symmetric lens is one whose two faces are portions of the 
same spherical surface; a lens of this kind being sometimes 
called a solid sphere (d = ri — r 2 = 2r 1 ) . 

(b) Concentric Lenses, whose two faces have the same 
center of curvature (CiC 2 = 0). A concentric lens may be 




Fig. 114. — Concentric concave meniscus. 



a double convex lens characterized by the relation d = r\— r 2 , 
of which a "solid sphere" is a special case; or it may have 
the form of a concave meniscus for which either ri>r 2 >0 
and d = ri — r 2 (Fig. 114) or ri<r 2 <0 and d = r 2 - r\. 

(c) Lenses of Zero Curvature, in which the axial thickness 



222 Mirrors, Prisms and Lenses [§ 87 




Fig. 115. — Lens of zero curvature (ri = r2). 

of the lens is equal to the distance between the centers (d = 
AiA 2 = CiC 2 ). This lens is a convex meniscus characterized 
by the condition that r\— r 2 = (Fig. 115). 

Lenses may also be conveniently classified in two main 
groups, viz., convex lenses and concave lenses, depending on 
the relative thickness of the lens along the axis as compared 
with its thickness at the edges. The thickness of a convex 
lens is greater along the axis than it is out towards the edge, 
whereas a concave lens is thinnest in the middle. Each of 
these two main divisions includes three special forms which 
have already been mentioned. Thus, the three types of con- 
vex lenses are the double convex, the plano-convex and the 
convex or " crescent-shaped " meniscus, as shown in Fig. 112; 
and, similarly, the types of concave lenses are the double 
concave, plano-concave and the concave or " canoe-shaped" 
meniscus (Fig. 113). 

A convex glass lens of moderate thickness held in air with 
its axis towards the sun has the property of a burning glass 
and converges the rays to a real focus on the other side of 



§ 88] Optical Center of Lens 223 

the lens. A convex lens is called therefore also a convergent 
lens or a positive lens. On the other hand, under the same 
circumstances, a concave lens will render a beam of sun- 
light divergent, and, accordingly, a concave lens is called 
also a divergent lens or a negative lens. The explanation of 
the terms " positive" and " negative" as applied to lenses 
will be apparent when we come to speak of the positions of 
the focal points of a lens (§ 90). 

Finally, if the curvatures of the two faces of the lens are 
opposite in sign, the lens is double convex or double con- 
cave; if the curvatures have the same sign, the lens is a 
meniscus; and if the curvature of one face is zero, the lens 
is plano-convex or plano-concave. 

88. The Optical Center of a Lens surrounded by the 
same medium on both sides. — When a ray of light emerges 
at the second face of a lens into the surrounding medium 
in the same direction as it had when it met the first face, 
the path of the ray inside the lens lies along a straight line 
which crosses the axis at a remarkable point O called the 
optical center of the lens, which is indeed the (internal or 
external) " center of similitude" of the two circles whose 
arcs are the traces of the spherical faces of the lens in the 
meridian plane which contains the ray. 

In order to prove this, draw a pair of parallel radii CiBi and 
C 2 B 2 (Fig. 116), and suppose that a ray enters the- lens at 
Bi and leaves it at B 2 , so that the straight line BiB 2 repre- 
sents the path of the ray through the lens. If the straight 
line RBi represents the path of the incident ray, a straight 
line B 2 S drawn through B 2 parallel to RBi will represent 
the path of the emergent ray; because, since the tangents 
to the circular arcs at Bi, B 2 are parallel to each other, the 
lens behaves towards this ray which enters it at Bi exactly 
like a slab of the same material with plane parallel sides 
(§44). Consequently, the position of the point O where 
the straight line BiB 2 , produced if necessary, crosses the 
axis of the lens is seen to be entirely dependent on the 



224 



Mirrors, Prisms and Lenses 



[§88 




Fig. 116. — Optical center of lens. 

geometrical form of the lens. In particular, the position 
of this point will not depend on the direction of the inci- 
dent ray, as will be shown by the following investigation. 
From the similar triangles OCiB! and OC 2 B 2 , we derive 
the proportion : 

OCi: OC 2 = BiCi: B 2 C 2 =AiCi: A 2 C 2 . 
Accordingly, we may write: 

OAi+AiC 1 = AiCi 
OA 2 +A 2 C 2 A 2 CV 
and, consequently: 

AiO_n 

A 2 r 2 ' 

Now A 2 = A 2 Ai+AiO = AiO— d; so that we obtain finally: 



AiO 



d. 



ri-r 2 

The function on the right-hand side of this equation depends 
only on the form of the lens, so that the position of the 



88] 



Optical Center of Lens 



225 



point with respect to the vertex of the first face of the 
lens may be found immediately as soon as we know the 
magnitudes denoted by r h r 2 and d. 



TO C A AT oo * 




Fig. 117. — Optical center of lens with one plane face is at the vertex 
of curved face. 

If the lens is double convex or double concave, the optical 
center O will lie inside the lens between the vertices Ai and 




Fig. 118. — Optical center of meniscus lies outside lens. 

A 2 ; if one face of the lens is plane (Fig. 117), the optical 
center will coincide with the vertex. of the curved face; and, 



226 Mirrors, Prisms and Lenses [§ 89 

finally, if the lens is a meniscus (Fig. 118), the optical center 
will lie outside the lens entirely. 

In general, the positions of the points designated in the 
diagrams by the letters N, N' will vary for different ray- 
paths Bi B 2 within the lens; but if the rays are paraxial, 
the positions of N, N' are fixed. In fact, if the ray RBi B 2 S 
is a paraxial ray, the points N, N' are the so-called nodal 
points of the lens (see § 119). 

89. The Abscissa-Formula of a Thin Lens, referred to 
the axial point of the lens as origin. — Ordinarily, the axial 
thickness of a lens is much smaller than either of the radii 
of curvature, so that in many lens-problems this dimension 
is negligible in comparison with the other linear dimensions 
that are involved. Moreover, the lens-formulae are greatly 
simplified by ignoring the thickness of the lens. However, 
in using these formulae one must be duly cautious about 
taking too literally results that are strictly applicable only 
to an infinitely thin lens, whose vertices are regarded as 
coincident, that is, Ai A 2 = d = 0. The approximate formulae 
that are obtained for lenses of zero-thickness are often of 
very great practical utility, especially in the preliminary 
design of an optical instrument composed, it may be, of 
several lenses whose thicknesses are by no means negligible. 

The optical center O of an infinitely thin lens coincides 
with the two vertices Ai, A 2 , and hereafter these three co- 
incident points in which the axis meets an infinitely thin lens 
will be designated by the simple letter A. An infinitely thin 
lens is represented in a diagram by the segment of a straight 
fine which is bisected at right angles by the axis of the lens; 
the actual form of the lens being indicated by assigning the 
positions of the centers Ci, C 2 of the two faces. In order to 
tell at a glance the character of a lens, the form of it at the 
edges may be indicated, as shown in Fig. 119. Fig. 119,a 
is a conventional representation of an infinitely thin con- 
vex lens, and Fig. 119, b is a similar diagram for an infinitely 
thin concave lens. 



89] 



Abscissa-Formula of Thin Lens 



227 



Let us assume that the lens is surrounded by the same 
medium on both sides; and let n denote the index of refrac- 




Fig. 119, a. — Infinitely thin convex lens; M, M' conjugate points on axis. 

tion of this medium, while n' denotes the index of refraction 
of the lens-substance itself. 

The broken line RBS (Fig. 119) represents the path of 
a paraxial ray which enters and leaves the infinitely thin 




Fig. 119, fr. — Infinitely thin concave lens; M, M' conjugate points 
on axis. 

lens at the point marked B. The points where the ray 
crosses the axis before and after passing through the lens 
will be designated by M, M', respectively. The straight 



228 Mirrors, Prisms and Lenses [§ 89 

line BM/ which intersects the axis at the point marked Mi' 
shows the path the ray takes after being refracted at the first 
face of the lens. Obviously, the points M, M/ are a pair of 
conjugate axial points with respect to the first surface of the 
lens, and, similarly, the points MY, M' are a pair of con- 
jugate axial points with respect to the second surface of 
the lens, and, therefore, M, M' are a pair of conjugate axial 
points with respect to the lens as a whole, so that M' will 
be the image in the lens of an axial object-point M. The 
abscissae of these points with respect to the axial point A 
as origin will be denoted by u, u'; thus, w = AM, w' = AM'. 
Also, put wi / = AMi / . The radii of curvature of the two 
faces are ri = ACi, r 2 = AC2- 

Accordingly, in order to obtain the formulae connecting 
u and u', we have merely to apply the fundamental equa- 
tion (§ 78) for the refraction of paraxial rays at a spherical 
surface to each face of the lens in succession, bearing in 
mind that the first refraction is from medium n to medium 
n', while the second refraction is from medium n' to me- 
dium n. Thus, we obtain : 

n' n n'—n n n' n'—n 



U\ u ri u' U\ r<t 

Eliminating U\ by adding these equations, and dividing 
through by n, we derive the abscissa-formula for the refrac- 
tion of paraxial rays through an infinitely thin lens, in the 
following form : 



1 l n'-n /l 1\ 

u' u n \ri rj' 



The expression on the right-hand side of this equation, in- 
volving only the lens-constants r± f r 2 and n'/n, has for a 
given lens a perfectly definite value, which may be com- 
puted once for all. And so if we put 

l_ n'^n /l 1\ 

f n \ri r 2 /' 
where the magnitude denoted by / is a constant of the lens 



90] 



Focal Points of Thin Lens 



229 



(which we shall afterwards see is the focal length of the 
lens), the formula above may be written: 

1 _1 = 1 
u' u /' 
which is the form of the lens-formula that is perhaps most 
common. For a given value of u we find u' =f.u/(f-\-u). 

Incidentally, it may be observed that the equation above 
is symmetrical with respect to u and -vl ; that is, the equa- 
tion will remain unaltered if —it is written in place of u' and 
— u' in place of u. Accordingly, if the positions of a pair of 
conjugate points on the axis are designated by M, M' 




Fig. 120.— Infinitely thin lens: AP = M'A = BM, AP' = MA. 

(Fig. 120), the pair of axial points designated by P, P' will 
likewise be conjugate, provided AP = M'A and AP' = MA; so 
that the thin lens at A bisects the two segments PM' and 
P'M. Another and more striking way of exhibiting this 
characteristic property of an infinitely thin lens consists 
in saying, that if M' is the image of an axial object-point 
at M, and if then the lens is shifted from its first position 
at A to a point B such that MB = AM', the object-point M 
will again be imaged at M'. 

90. The Focal Points of an Infinitely Thin Lens.— If 
the object-point M is at the infinitely distant point on the 
axis of the lens, its image will be formed at a point F' whose 
position on the axis may be found by putting u= oo ,u' = AF' 



230 



Mirrors, Prisms and Lenses 



90 



in the formula l/u'—l/u=lff; thus, we find AF'=/. Sim- 
ilarly, the object-point F conjugate to the infinitely dis- 
tant point of the axis is found by substituting in the same 
equation the pair of values u = AF, u r — oo J whence we ob- 
tain AF = — /. These points F, F' are the primary and sec- 
ondary focal points, respectively, and, accordingly, it is 
evident that the focal points of an infinitely thin lens are equi- 
distant from the lens and on opposite sides of it. 

The character of the imagery in the case of an infinitely 
thin lens is completely determined as soon as we know the 
positions of the two focal points F, F'; and since the point A 
where the axis meets the lens lies midway between F and 
F', it is obvious that the natural division of lenses is into 
two classes depending on the order in which the three points 
above mentioned are ranged along the axis. 

(1) 7/ the primary focal point is in front of the lens (Fig. 
121, a), that is, if the order of the points named in the se- 




Fig. 121, a. — Focal points (F, F'),of infinitely thin lens 
(FA = AF'=/). In a positive (or convex or conver- 
gent) lens the first focal point (F) lies on same side 
of lens as incident light (real focus) . 

quence in which they are reached by light traversing the 
axis of the lens is F, A, F', then incident rays parallel to 



90] 



Focal Points of Thin Lens 



231 



the axis will be converged to a real focus at F' on the other 
side of the lens, and the lens is a convergent lens (§ 87). It 
is also called a positive lens, because the lens-constant (or 
primary focal length) /=FA = AF' is measured along the 
axis in the positive sense. If it is assumed that n'>n (as, 
for example, in the case of a glass lens in air), the sign of 
this constant /, according to the formula above which de- 
fines 1//, will be the same as that of the term (l/ri— l/r 2 ), 
which is the algebraic expression of the difference of curva- 
tures (§ 99) between the two faces of the lens. If the lens 
is double convex, plano-convex or a crescent-shaped me- 
niscus — that is, in all forms of lenses that are thicker in 
the middle than out towards the edges — the difference of 
curvatures (l/ri — l/r 2 ) will be found to be positive. And 
hence, as already stated (§ 87), thin lenses of this descrip- 
tion are convergent if n'>n. 

(2) // the secondary focal point is in front of the lens (Fig. 
121, b), that is, if the points F', A, F are ranged along the 



W 



F' 




Fig. 121, b— Focal points (F,F') of infinitely thin 
lens (FA = AF'=/). In a negative (or concave or 
divergent) lens the first focal point (F) lies on 
the other side of the lens from the incident light 
(virtual focus) . 



232 Mirrors, Prisms and Lenses [§91 

axis in the order named, incident parallel rays will be made 
to diverge from a virtual focus at F', and in this case the 
lens is said to be a divergent or negative lens, since now the 
lens-constant /=FA=AF' is measured along the axis in 
the negative sense. For lenses which are thinner in the 
middle than at the edges, that is, for double concave, plano- 
concave and canoe-shaped meniscus lenses the difference of 
curvatures (1/ri— l/r 2 ) will be found to be negative; and 
hence for such lenses the constant / will be negative if n r >n. 

A case of rather more theoretical than practical interest is 
afforded by an infinitely thin concentric lens (§ 87) for which 
r 2 = r h and which is therefore uniformly thick in a direc- 
tion parallel to the axis, so that according to the above 
classification it should be neither convergent nor divergent. 
In fact, the value of the lens-constant / for this lens is in- 
finity, and hence u\=u, so that object-point M and image- 
point M' are coincident always. A bundle of parallel rays 
traversing an infinitely thin concentric lens will emerge 
from the lens just as though the lens had not been inter- 
posed in the path of the rays. 

91. Construction of the Point M' Conjugate to the Axial 
Point M with respect to an Infinitely Thin Lens. — The 
planes which are perpendicular to the axis of the lens at the 
focal points F, F' are called the primary and secondary focal 
planes, respectively. 

The point M' conjugate to a point M on the axis of an 
infinitely thin lens surrounded by the same medium on both 
sides may be constructed as follows : 

Through the given point M (Fig. 122, a and b) draw a 
straight line MB meeting the lens at B, and through the 
axial point (A) of the lens draw a straight line AI' parallel 
to MB and meeting the secondary focal plane in the point 
V; then the point where the straight line BF, produced if 
necessary, crosses the axis will be the required point M' 
conjugate to M. 

The point M' may also be constructed in another way, 



91] 



Thin Lens: Conjugate Axial Points 



233 



as follows: Let J designate the point where the straight 
line MB crosses the primary focal plane, and through B 
draw a straight line parallel to the straight line JA, which 





Fig. 122, a and 6. — Infinitely thin lens: Construction of point 
M' conjugate to axial object-point M. (a) Convex, 
(6) Concave lens. 

will intersect the axis of the lens in the required point M'. 
Fig. 122, a shows the construction in the case of a convex 
lens and Fig. 122, b shows it for a concave lens. 

The proof is obvious. From the two pairs of similar 
triangles MAB, AFT and MM'B, AMT, we obtain the 
proportions: 

MA = MB = MM'. 
AF' AI' AM' ' 



234 



Mirrors, Prisms and Lenses 



92 



and if we introduce the symbols w = AM, w' = AM', /=AF', 
we get: 

— u _u'— u 

which is the same as the abscissa-relation found in § 89. 
92. Extra-Axial Conjugate Points Q, Q'; Conjugate 
Planes. — Since the axial point A of an infinitely thin lens 
is also the optical center of the lens (§ 89), a straight line 
drawn through A will represent the path of a ray both be- 
fore and after passing through the lens at this point. If 
the axis of the lens is rotated in a meridian plane through 




Fig. 123. — Infinitely thin lens: Image-point Q' conjugate to extra-axial 
object-point Q. 



a very small angle FAJ (Fig. 123) around the point A as 
vertex, the focal points F, F' will describe the small arcs 
FJ, FT and the straight line JI' will represent the path of 
a paraxial ray traversing the lens at A. The points Q, Q' 
at the ends of the arcs MQ, M'Q' traced out in this angular 
movement of the axis by a pair of conjugate axial points 
M, M' will evidently occupy the same relation to each 
other on the straight line JI' as M, M' have to each other 



92J 



Image in Infinitely Thin Lens 



235 



on the straight line FF', and therefore Q, Q' are a pair of 
extra-axial conjugate points. 

Accordingly, if the points of an object lie in the vicinity 
of the axis on an element of a spherical surface described 




Fig. 124, a and b. — Infinitely thin lens: Lateral magnification 
and construction of image M'Q' conjugate to short object- 
line MQ perpendicular to axis, (a) Convex, (6) Concave lens. 

around the vertex A of the infinitely thin lens as center, 
the corresponding points of the image will be assembled 
on a concentric spherical surface; and since, within the 
region of paraxial rays, these spherical elements may be 
regarded as plane, it follows that a small plane object at 



236 Mirrors, Prisms and Lenses [§ 93 

right angles to the axis will be reproduced by a similar 
plane image also at right angles to the axis. 

Conjugate planes are pairs of parallel planes perpendicular 
to the axis of the lens; and any straight line drawn through 
the center of an infinitely thin lens will pierce a pair of conju- 
gate planes in a pair of conjugate points. 

In particular, the planes conjugate to the focal planes 
are the infinitely distant planes of the image-space and 
object-space, according as the infinitely distant plane is 
regarded as belonging to one or the other of these 
regions. 

The construction of the point Q' conjugate to an extra- 
axial object-point Q (Fig. 124, a and b), with respect to 
an infinitely thin lens, is made by a method precisely sim- 
ilar to that employed in the corresponding problem in the 
cases both of a spherical mirror (§71) and of a spherical 
refracting surface (§ 81) ; the only difference in this case 
being that the center of the lens takes the place of the center 
of the spherical surface and that the focal points of the 
lens are at equal distances on opposite sides of it. 

93. Lateral Magnification in case of Infinitely Thin Lens. 
— The lateral magnification in the case of an infinitely thin 
lens, defined, as in §§ 72 and 82, as the ratio of the height 
of the image (y' = M'Q') to the height of the object (y = MQ), 
may be obtained from the diagram (Fig. 124) and is evi- 
dently given by the following formula: 

y' W 

y=-=-; 

y u 

so that the linear dimensions of object and image are in the 
same ratio as their distances from the thin lens. Moreover, 
it appears that the image is erect or inverted according as 
object and image lie on the same side or on opposite sides of 
the lens. 

Another expression for the lateral magnification may 
be derived by considering the two pairs of similar right 



§ 94] Imagery in Thin Lens 237 

triangles FMQ, FAW and F'M'Q', F'AV, from which we 
obtain the proportions: 

AW_ FA M'Q'_ F'M' . 

MQ FM' AV F'A ; 
and since 

AW = M'Q' = ?/, AV = MQ = y, FA = AF'=/, 
we find: 

y' f x' 
y * f 

where z = FM, x' = F'M' denote the abscissae of M, M' 
with respect to the focal points F, F', respectively, as ori- 
gins. Accordingly, the lateral magnification varies inversely 
as the distance of the object from the primary focal plane, and 
directly as the distance of the image from the secondary focal 
plane. 

94. Character of the Imagery in a Thin Lens. — The 
Newtonian form of the abscissa-relation (c/. § 85) for an 
infinitely thin lens surrounded by air is : 

x.x'=~f, 
which shows that object and image lie on opposite sides of 
the focal planes; so that if M is a point on the axis to the 
right of the primary focal point F, the conjugate point M' 
will be found on the axis at the left of the secondary focal 
point F', and vice versa. 

The character of the imagery produced by the refraction 
of paraxial rays through an infinitely thin lens is exhibited 
in the diagrams Fig. 125, a and b. The numerals 1, 2, 3, 
etc., mark the successive positions of an object-point which 
is supposed to traverse a straight line parallel to the axis 
(so-called " object-ray") from an infinite distance in front 
of the lens to an infinite distance on the other side of it. 
Until it reaches the lens at the point marked V the object 
is real, thereafter it is virtual. The corresponding numerals 
with primes, viz., 1', 2', 3', etc., ranged along the straight 
line VF' (called the " image-ray") mark the successive 
positions of the image-point, which, starting, from the 



238 



Mirrors, Prisms and Lenses 



[§94 



secondary focal point F', moves along this line always in 
the same direction out to infinity and back again to its 
starting point. The straight lines 11', 22', 33', etc., con- 



T0 5ATW 



N«' 



1 


2 J 4 


V * g> 6 7 


OBJECT 


RAY ^ 


\ \ 






1 


? 


A 
\ 


"\IMAGE 

/„i \ray 



TO 3 ATOO 



^ 




' ( / / TO 5' AT 00 

Fig. 125, a and b. — Character of imagery in infinitely thin lens, 
(a) Convex, (b) Concave lens. 

necting corresponding positions of object-point and image- 
point form a pencil of rays all passing through the optical 
center A of the lens, which is the center of perspective of 



[§ 94] Imagery in Thin Lens 239 

object-space and image-space. At the point V object and 
image coincide with each other in the lens itself, and here 
object and image are congruent. The so-called principal 
planes (§ 119) of an infinitely thin lens coincide with each 
other in the plane perpendicular to the axis of the lens at Us 
optical center A. The fact that object-point and image- 
point coincide with each other at V is expressed geometri- 
cally by saying the y-axis is the base of a range of self conju- 
gate points. 

In a convex lens (Fig. 125, a) the image of a real object 
is seen to be real and inverted as long as the object lies in 
front of the lens beyond the primary focal plane; whereas 
the image is virtual and erect if the object is placed between 
the primary focal plane and the lens. The image of a vir- 
tual object in a convex lens is formed between the lens and 
the secondary focal plane and is real and erect. 

In a concave lens (Fig. 125, b) the image of a real object 
lies between the lens and the secondary focal plane, and it 
is virtual and erect. If the object is virtual, its image in a 
concave lens will be real and erect if the object lies between 
the lens and the primary focal plane, but it will be virtual 
and inverted if the object lies beyond the primary focal 
plane. 

If 2 = MM' denotes the distance between a pair of con- 
jugate axial points M, M', then u' = u-\-z, where u — AM, 
u' = AM'. Substituting this value of u in the formula 
l/u'—l/u = l/f, we obtain a quadratic in u, which implies, 
therefore, that for a given value of the interval z between 
object and image, there are always two positions of the 
object-point M with respect to the lens (§ 89). But under 
some circumstances the assigned value of the interval z 
may be such that the roots of the quadratic prove to be 
imaginary, and then it will be quite impossible with the 
given lens to produce an image at the given distance z from 
the object. For example, if the object lies in front of a 
convex lens (/> 0) at a distance greater than the focal length, 



240 



Mirrors, Prisms and Lenses 



[§95 



then u<0 and z>0. Put a = MA = -w, so that the magni- 
tudes denoted by /, z and a are all positive. Eliminating 
v! from the abscissa-formula, we obtain a quadratic in a 
whose roots are given by the following expression: 

s±Vz(s~4/). 



which will be imaginary if (2— 4/)< 0. Hence, the distance 
(z) between a real object and its real image in a convex lens 
cannot be less than four times the focal length f. 

95. The Focal Lengths f, V of an Infinitely Thin Lens. — 
The focal lengths of a thin lens are defined exactly in the 
same way as the focal lengths of a spherical refracting sur- 





I 


^B 








K ^"^\ 


l' 






// 


H 




TO E ATOO y 


4 


^^-v^^ TO B AT00 


/M/ 


F 


A 


F~*^-^^ 


^^*"^^^^ TO j'ATOO 


//TO I 










AT 00 


1 


' 







Fig. 126. 



-Focal planes and focal lengths of infinitely thin lens 
(/=FA = — f' = KF'). 



face (§ 83). Thus, the primary focal length of a lens is the 
ratio of the height of the image, in the secondary focal plane, 
to the apparent size of the infinitely distant object. In Fig. 126 
FT' is the height of the image of the infinitely distant ob- 
ject EI which is seen under the angle 6 = Z EFI = Z AFK, 
and the primary focal length is, therefore, FT/tan 6 = 
AK/tan 6 = FA =/; and hence, as already observed, the 
primary focal length is identical with the lens-constant 
denoted by/, which, as we have seen (§ 90), is the abscissa 
of the axial point A of a thin lens with respect to its primary 
focal point F. Similarly, the secondary focal length (/') is 



§ 95] Infinitely Thin Lens: Focal Lengths 241 

the ratio of the height of an object in the primary focal plane of 
the lens to the apparent size of its infinitely distant image. 
For example, in the diagram the image of the object FJ 
lying in the primary focal plane is EM', which lying in the 
infinitely distant plane of the image-space, subtends the 
angle 0' = ZEF'J' = Z AF'H; and hence /' = FJ/tan 6' = 
AH/tan 0' = F'A; so that the secondary focal length may 
also be defined as the abscissa of the axial point A of an in- 
finitely thin lens with respect to the secondary focal point F'. 
And since F'A = -AF' = -FA, evidently: 

Accordingly, the focal lengths (/, /') of a lens surrounded by 
the same medium on both sides are equal in magnitude and 
opposite in sign. 

If the lens is reversed by turning it through 180° about 
an axis perpendicular to the axis of the lens, that is, if the 
light is made to traverse the lens in a sense exactly opposite 
to that which it had at first, the focal lengths /, /' will not 
be altered. This is evident from the fact that the expres- 
sion for the focal length/, viz., 

J J (n'-n) (r 2 ~n)\ 

remains the same when -r h -r 2 are substituted in place of 

7*1, r 2 , respectively. Thus, the character of the lens (§ 90) 

and its action are not changed by presenting the opposite 

face to the incident rays. 

The focal length of an infinitely thin symmetric lens 

n r 
(§87), for which ri=~r 2 = r (say) is f=^—f — r* and if 

71 = 1, w'=1.5, we find f=r. Accordingly, the focal length 
of an infinitely thin symmetric glass lens surrounded by air 
(n=l, n' =1.5) is equal to the radius of the first face. Spec- 
tacle glasses were at first symmetric lenses, and in the old 
inch system of designation a No. 10 spectacle glass, for ex- 
ample, was a lens whose radius of curvature on each surface 
was 10 inches and whose focal length was 10 inches. 



242 Mirrors, Prisms and Lenses [§ 96 

If one face of the lens is plane, for example, if r\= go , 

71 V 71 T 

r 2 = r, we find/= ——^— ; or if i\ = r, r 2 = oo , then/= 



n—n n—n 

where in each case r denotes the radius of the curved sur- 
face. Comparing this with the value of / obtained in the 
preceding case, we see that if one of the faces of a symmetric 
lens be ground off plane, the focal length of the lens will thereby 
be doubled. 

96. Central Collineation of Object-Space and Image- 
Space. — Comparing the methods and results of this chap- 
ter with those obtained in the preceding chapter, the serious 
student cannot have failed to remark a striking parallelism 
that exists between the imagery by paraxial rays in a spher- 
ical refracting surface and the imagery under the same con- 
ditions in an infinitely thin lens. In some instances the 
formulae are actually identical, and a closer examination 
will show that this similarity extends even to comparative 
details. For, example, the focal points lie on opposite sides 
of a lens just as they were found to do in the case of a spher- 
ical refracting surface, and the resemblance goes still far- 
ther. For in a spherical refracting surface the connection 
between the focal lengths (/, /') and the indices of refraction 
(n, n') is expressed by the formula n'.f-\-n.f' = (§79); 
and if in this formula we put 7i' = n, we obtain the relation 
/-f/ / = 0, which is the algebraic statement of the fact that 
the focal lengths of a lens surrounded by the same medium 
on both sides are equal and opposite (§ 95) . 

It has already been pointed out that the imagery in a 
spherical mirror may be regarded as a special case of refrac- 
tion at a spherical surface (§§ 75, 77 and 78) ; and now it 
is proposed to advance a step farther in this generalization 
process and to show that all these types of imagery which 
have been investigated separately and independently are 
in reality embraced in a concept of geometry known as 
collinear correspondence between one space and another 
(called in the theory of optical imagery " object-space" 



§ 96] Central Collineation 243 

and "image-space")- Moreover, these types of imagery 
belong to a particularly simple kind of collinear correspond- 
ence to which the name central collineation has been given. 

A lens or an optical instrument is said to divide the sur- 
rounding space into two parts, viz., the object-space and 
the image-space; but these are not to be thought of as sep- 
arate and distinct regions but as interpenetrating and in- 
cluding each other; so that a point or ray may be regarded 
at one time as belonging to the object-space and at another 
time as belonging to the image-space, depending merely 
on the point of view. Thus, for example, the infinitely 
distant plane of space may be viewed as the image of the 
primary focal plane of a lens, and then it is a part of the 
image-space; but if the secondary focal plane is regarded 
as the image of the infinitely distant plane, the latter is a 
part of the object-space. 

Now the distinguishing characteristics of the optical 
imagery which is produced by the refraction of paraxial 
rays at a single spherical surface or through an infinitely 
thin lens may be summarized in the two following state- 
ments: 

(a) All straight lines joining pairs of conjugate points in- 
tersect in one point, viz., the center (C) of the spherical re- 
fracting surface or the optical center (A) of the thin lens. 
This point which is the center of perspective of object- 
space and image-space is called the center of collineation, 
and will be referred to here as the point C. 

(b) Any pair of corresponding incident and refracted rays 
lying in a meridian plane meet in a straight line Ay called the 
axis of collineation (or the y-eixis) which is perpendicular at 
A to the optical axis (or the rc-axis) . 

Any straight line going through the center of collinea- 
tion is called a central ray. Every central ray is a self -cor- 
responding ray; that is, image-ray and corresponding object- 
ray lie along one and the same straight line. Moreover, 
any point lying on the axis of collineation is a self-conjugate 



244 



Mirrors, Prisms and Lenses 



[§97 



point; that is, along this line object-point and image-point 
are coincident with each other. The center of collineation 
is also a self-conjugate point, and hence, in general, there are 
two self-conjugate points on a central ray, viz., the center 
of collineation and the point where the ray meets the axis 
of collineation. Only in case the center of collineation lies 
on the axis of collineation will there be only one self-conju- 
gate or so-called double point on a central ray. 

97. Central Collineation (cont'd). Geometrical Con- 
structions. — Starting from these simple propositions, we 
can easily develop a complete theory of optical imagery 
for the simple cases mentioned above. Thus, for example, 




Fig. 127. — Central collineation: Construction of pairs of conjugate points 
M, M'; P, P'; Q, Q'; R, R'; S, S'; T, T'; and U, U'. Axis of collineation 
Ay; center of collineation C. 

being given the axis of collineation (Ay) and the center of col- 
lineation (C), together with the positions of a pair of conjugate 
points P, P', we can construct the position of a point Q' con- 
jugate to a given point Q, as follows: 

(a) In general, the straight line PQ (Fig. 127) will not 
pass through the center of collineation. Let the self- 
conjugate point in which the straight line PQ meets the 
axis of collineation be designated by T; the image-ray cor- 



97] 



Central Collineation 



245 



responding to the object-ray PT will lie along the straight 
line TP', and since this ray must pass likewise through the 
point Q' conjugate to Q, the required point will be at the 
intersection of the straight lines TP', QC. 

(b) But in the special case when the straight line PQ is 
a central ray (Fig. 128) the construction which has just 
been given fails, and we must resort to a different procedure, 





V 


*a 




j 


T 


>^WL^L 


^ X 


F 




A / Cl ^A 


^ 



Fig. 128. — Central collineation: Straight line PQ passes through center 
of collineation (C) . Diagram shows case when C does not lie on axis 
of collineation Ay; as in spherical refracting surface (c> 1). 

as follows: Through the points P and C draw a pair of 
straight lines PO, CO meeting in a point O, and let the 
point where the straight line PO meets the axis of collinea- 
tion be designated by T. Also, let O' designate the point 
of intersection of the straight lines TP' and CO. Then if 
the point where the straight line QO meets the axis of col- 
lineation is designated by U, the required point Q' will be 
the point of intersection of the straight lines UO' and QC. 
The image-point I' conjugate to the infinitely distant 
object-point I of the pencil of parallel rays whose central 
ray is PP' may be constructed exactly as described above 
in (b), provided we have the same data. The straight line 



246 



Mirrors, Prisms and Lenses 



[§97 



OG is drawn parallel to PP' meeting the axis of collineation 
in G, and the required point I' is the point of intersection 
of the straight lines GO' and PP'. 

Similarly, the position of the object-point J conjugate 
to the infinitely distant image-point J' of the central ray 
PP' is found by drawing the straight line O'H parallel to 
PP' meeting the axis of collineation in H; then the point 
of intersection of the straight lines OH, PP' will be the re- 
quired point J. 

u 




Fig. 129. — Central collineation: Straight line PQ passes through center 
of collineation (C). Diagram shows case when C lies on axis of col- 
lineation Ay, as in infinitely thin lens (c— 1). 

The focal points F, F' on the optical axis are constructed 
in precisely the same way as the two points J, I' on the 
central ray PP'. 

The special case when the center of collineation (C) lies on 
the axis of collineation, that is, when the two points A and C 
are coincident, is shown in Fig. 129, which evidently cor- 
responds to the case of an infinitely thin lens surrounded 
by the same medium on both sides. 



§ 98] Field of View of Thin Lens 247 

It would be easy to show by the methods of projective 
geometry that the straight lines FJ, FT are parallel to 
the axis of collineation and that we have the following re- 
lations between the points J, I' and the two self -con jugate 
points B, C on the central ray JF: 

JB = CI', I'B = CJ, ^? = c, 

where c denotes a constant called the invariant of central 
collineation, which has the value n'\ n for a spherical re- 
fracting surface and the value +1 for a thin lens surrounded 
by the same medium on both sides. For a spherical mirror, 
c = — 1. For the axial ray the above relations may be written : 

FA = CF', F'A = CF, ^ = c. 
AF 

The reader who wishes to pursue this subject will find a 
complete discussion at the end of Chapter V of the author's 
Principles and Methods of Geometrical Optics published by 
The Macmillan Company of New York. 

98. Field of View of an Infinitely Thin Lens. — If it is 
assumed that there are no artificial stops present except 
in the plane of the lens, and that the imagery is produced 
by means of paraxial rays only, the field of view in the case 
of an observer looking through the lens along its axis is 
easily determined by drawing the straight lines O'G, O'H 
(Fig. 130, a and b) in a meridian plane of the lens from the 
center 0' of the eye-pupil to the ends G, H of the diameter 
of the lens-opening. For the lens-opening acts here just 
like a round window or port-hole in an opaque wall to limit 
the field of view in the image-space of the lens. If O desig- 
nates the position of the axial object-point which is repro- 
duced by the image-point O', then the straight lines OG, OH 
determine the limits in the meridian plane of the diagram of 
the field of view of the object-space. Let the straight line 
B'C bisected at right angles at O' by the axis of the lens 
represent the diameter of the pupil in the meridian plane of 
the lens; and construct the line BOC whose image in the lens 



248 



Mirrors, Prisms and Lenses 



[§98 



is the diameter B'O'C of the pupil of the eye. Then the 
image S' of the luminous point S lying within the object-side 
field of view may be constructed by drawing through S the 
straight lines SB, SC to meet the lens in two points which 




ENTRANCE 
PUPIL 



PUPIL OF EYE 



S ^% 


^ST; 








A 


/* 


B><^ 


l B ' 



ENTRANCE 

PUPIL EXIT 

PUPIL 

Fig. 130, a and b. — Field of view of infinitely thin lens for given position of 
eye on axis of lens, (a) Convex, (6) Concave lens. 

must be joined with B', C, respectively; and the point of 
intersection of these latter fines will be the required point 
S' conjugate to S. In brief, the circular opening whose di- 
ameter is BC is the common base of all the cones of effective 
rays in the object-space of the lens, just as the pupil of the 
eye itself is the common base of the cones of effective rays 



Ch. VII] Problems 249 

in the image-space. Assuming that the lens-opening is 
large enough to permit the entire pupil of the eye to be filled 
with rays emanating from an axial object-point, the lens- 
opening GH acts as field-stop and the pupil of the eye as 
aperture-stop (Chapter XII). 

PROBLEMS 

1. Show how to construct the optical center of a lens. 
Draw diagrams for the various forms of convex and con- 
cave lenses; and prove that the distance of the optical 
center from the vertex of the first face is equal to rid/(ri — r 2 ), 
where r h r 2 denote the radii of the two surfaces and d de- 
notes the axial thickness of the lens. 

2. In each of the following lenses the axial thickness is 
2 cm. Find the position of the optical center, and draw 
a diagram for each lens showing the position of this point. 

(a) Double convex lens of radii 10 and 16 cm.; (6) Double 
concave lens of radii 10 and 16 cm.; (c) Plano-convex lens; 
(d) Positive meniscus of radii 10 and 16 cm.; (e) Negative 
meniscus of radii 20 and 16 cm.; (/) Lens of zero curvature. 

3. Rays of light diverging from a point one foot in front 
of a thin lens are brought to a focus 4 inches beyond it. 
Find the focal length. Ans. /=-f-3 inches. 

4. An object is placed one foot in front of a thin convex 
lens of focal length 9 inches. Where is the image formed? 

Ans. 3 feet from the lens on the other side. 

5. Rays coming from a point 6 inches in front of a thin 
lens are converged to a point 18 inches on the other side of 
the lens. Find the focal length. Ans. /= +4.5 inches. 

6. An object is placed in front of a thin lens at a distance 
of 30 cm. from it. The image is virtual and 10 cm. from 
the lens. Find the focal length. Ans. /=— 15 cm. 

7. The radius of the first face of a thin double convex 
lens made of glass of index 1.5 is 20 cm. If the focal length 
of the lens is 30 cm., what must be the radius of the second 
face? Ans. 60 cm. 



250 Mirrors, Prisms and Lenses [Ch. VII 

8. A thin convex lens made of glass of index 1.5 has a 
focal length of 12.5 cm. If the radius of the second face is 
+ 17.5 cm., what is the radius of the first face? And if the 
lens is concave, and the radius of the first face is +17.5 cm., 
what is the radius of the second face? 

Ans. In both cases the radius is +4.6 cm. 

9. The focal length of a double convex lens was found 
to be 30.6 cm., and its radii 30.4 and 34.5 cm. Find the 
index of refraction of the glass. Ans. 1.528. 

10. The focal length of a glass lens in air is 5 inches. 
What will be the focal length of the lens in water, assuming 
that the indices of refraction of air, glass and water are 1, 
| and I-, respectively? Ans. 20 inches. 

11. Show that any thin lens which is thicker in the middle 
than out towards the edges is convergent, provided the 
lens-medium is more highly refracting than the surrounding 
medium. 

12. Show that the focal length of a thin plano-convex 
lens is twice that of a double convex lens, if the curvatures 
of the curved surfaces are all equal in magnitude. 

13. Find the focal length of a thin double convex diamond 
lens, of index 2.4875, the radius of each surface being 4 cm. 

Ans. 13.4 mm. 

14. The curved surface of a thin plano-convex lens of glass 
of index 1.5 has a radius of 12 inches. Find its focal length. 
What must be the radii of a symmetric double convex lens 
of same material which has same focal length? 

Ans. /=24 inches; r = 24 inches. 

15. The radii of a thin double convex lens are 9 cm. and 
12 cm. The lens is made of glass of index 1.5. If light di- 
verges from a point 18 cm. in front of the lens, where will 
it be focused? Ans. Real image, 24 cm. from lens. 

16. A thin lens is made of glass of index n. If the focal 
length of the lens in air is a, and if its focal length in a liquid 
is 6, show that the index of refraction of the liquid is 

bn 
6+a(n~l)' 



Ch. VII] Problems 251 

17. Draw figures, approximately to scale, showing the 
paths of the rays of light, and the positions of the images 
formed when a luminous object is placed at a distance of 
(a) 1 inch, (6) 6 inches from a convex lens of focal length 
2 inches. 

18. An object is placed 8 inches from a thin convex lens, 
and its image is formed 24 inches on the other side of the lens. 
If the object were moved nearer the lens until its distance 
was 4 inches, where would the image be? 

Ans. Virtual image, 1 foot from lens. 

19. A virtual image of an object 30 cm. from a thin lens is 
formed on the same side of the lens at a distance of 10 cm. 
from it. Find the focal length of the lens. 

Ans. /= ~ 15 cm. 

20. Light converging towards a point M on the axis of 
a lens is intercepted and focused at a point M' on the same 
side of the lens as M. The distances of M and M' from the 
lens are 5 cm. and 10 cm., respectively. Find the focal 
length of the lens. Ans. /= — 10 cm. 

21. A far-sighted person can see distinctly only at a dis- 
tance of 40 cm. or more. How much will his range of dis- 
tinct vision be increased by using spectacles of focal length 
+32 cm.? 

Ans. The spectacles will enable him to see distinctly 
objects as near to his eye as 17.78 cm., so that his range of 
distinct vision will be increased by 22.22 cm. 

22. The projection lens of a lantern has a focal length of 
one foot. If the screen is 1024 feet away, how far back of 
the lens must the glass slide be placed? Ans. 1024/1023 ft. 

23. An engraver uses a magnifying glass of focal length 
+4 inches, holding it close to the eye. At what distance 
must the lens be from the work so that the magnification 
may be fourfold? Ans. 3 inches. 

24. Assuming that the optical system of the eye is equiva- 
lent to a thin convex lens of focal length 15 mm., what will 



252 Mirrors, Prisms and Lenses [Ch. VII 

be the size of the retinal image of a child 1 meter high at a 
distance of 15 meters from the eye? Ans. 1 mm. 

25. A millimeter scale is placed at a distance of 84 cm. 
in front of a convex lens, and it was found that 10 mm. of 
the scale corresponded to 29 mm. of its real inverted image. 
Find the focal length of the lens. Ans. /= +62.5 cm. 

26. If X, X' and Y, Y' are two pairs of conjugate points 
on the axis of an infinitely thin lens, and if the lens is mid- 
way between X and Y', show that it is also midway be- 
tween X' and Y. 

27. M and M' are a pair of conjugate axial points with 
respect to an infinitely thin lens whose optical center is at 
a point designated by A. Show that when the lens is shifted 
from A to a point B such that MB=AM', the points M 
and M' will be conjugate to each other with respect to the 
lens in this new position. 

28. Given the positions of the focal points F, F' of an 
infinitely thin lens, show how to construct the image-point 
M' conjugate to an axial object-point M. Draw diagrams 
for convex and concave lenses. 

29. At the optical center (A) of a thin lens erect a per- 
pendicular to the axis of the lens, and take a point L on 
this perpendicular such that AL=/, where / denotes the 
primary focal length. Through A draw a line AP in such 
a direction that ZF'AP = 45°, where F' designates the sec- 
ondary focal point of the lens. Take a point M on the axis 
of the lens, and draw the straight line ML meeting the 
straight line AP in a point S. If M' designates the foot of 
the perpendicular let fall from S to the axis of the lens, 
show that M, M' are a pair of conjugate axial points. Draw 
two diagrams, one for a convex and the other for a concave 
lens. 

30. Derive the image-equations in the case of an infinitely 
thin lens in the form : l/u' = \ju-\- 1//, y'/y = u'\u. 

31. Show that the focal points of an infinitely thin lens 
are at equal distances on opposite sides of the lens. 



Ch. VII] Problems 253 

32. A candle is placed at a distance of 2 meters from a 
wall, and when a lens is placed between the candle and the 
wall at a distance of 50 cm. from the candle, a distinct image 
of the latter is cast upon the wall. Find the focal length of 
the lens and the magnification of the image. 

Ans. /=37.5 cm.; image is 3 times as large as object. 

33. The distance between a real object and its real image 
in an infinitely thin lens is 32 inches. If the image is 3 times 
as large as the object, find the position and focal length of 
the lens. 

Ans. The lens is a convex lens of focal length 6 inches 
placed between object and image at a distance of 8 inches 
from the object. 

34. When an object is held at a distance of 6 cm. from 
one face of a thin lens, the image of the object formed by 
reflection in this face is found to lie in the same plane as the 
object. If the object is placed at a distance of 20 cm. from 
the lens, the image produced by the lens is inverted and of 
the same size as the object. The lens is made of glass of 
index 1.5. Find the radii of the two surfaces. 

Ans. The lens is a convex meniscus of radii 6 and |4 cm. 

35. In a magic lantern the image of the slide is thrown 
upon a screen by means of a thin convex lens. Show that 
the adjustment for focusing is always possible provided 
that the distance from the slide to the screen is not less 
than 4 times the focal length of the lens, and provided that 
the lens can move in its tube to a distance from the slide 
equal to twice the focal length. 

36. A person holds a lens in front of his eye and ob- 
serves that by reflection at the nearer surface an object 
which is 6 feet from the lens appears upright and diminished 
to one-twentieth of its height. Looking through the lens 
at an object on the other side 6 feet from the lens, its image 
is inverted and diminished in height to one-tenth. The 
lens is a glass lens of index 1.5. Find the radii of its sur- 
faces. Ans. A double convex lens of radii || and 44 ft. 



254 Mirrors, Prisms and Lenses [Ch. VII 

37. How far from a lens must an object be placed so 
that its image will be erect and half as high as the object? 

Ans. The object must be in the second focal plane of the 
lens. (Draw diagram showing construction of image for 
convex lens and also a diagram for concave lens.) 

38. How far from a thin lens must an object be placed 
so that its image will be inverted and half as high as the 
object? Draw two diagrams, showing construction of image 
for convex lens and for concave lens. 

Ans. If the optical center of the lens and the primary 
focal point are designated by A and F, respectively, and if 
the axial point of the object is designated by M, then 
AM = 3AF. 

39. An object is to be placed in front of a convex lens of 
focal length 18 inches in such a position that its image is 
magnified 3 times. Find the two possible positions, and 
draw diagram for each position showing the construction 
of the image. 

Ans. If image is inverted, object must be 2 ft. from lens; 
if it is erect, object must be 1 ft. from lens. 

40. In the preceding example if the lens were concave, 
where would the object have to be? 

Ans. The object would be virtual, at a distance of 1 ft. 
from the lens for an erect image, and at a distance of 2 ft. 
for an inverted image. 

41. A person can see distinctly at a distance of 1 foot, 
and he finds that when he holds a certain lens close to his 
eye small objects are seen distinctly and magnified 6 times. 
Find the focal length of the lens. Ans. /= +2.4 inches. 

42. Derive the Newtonian formula x.x' = — f 2 for a lens. 

43. A convex lens is used to produce an image of a fixed 
object on a fixed screen. Show that, in general, there will 
be two possible positions of the lens, and prove that the 
height of the object is the geometrical mean between the 
heights of the two images. 

44. A copper cent is 19 mm. in diameter and a silver 



Ch. VII Problems 255 

half dollar is 30.4 mm. in diameter. How far from a con- 
vex lens of focal length 10 cm. must the smaller coin be 
placed so that its image in the lens will be just the size of 
the larger one? 

Ans. It must be placed in front of the lens at a distance 
of either 16.25 cm. or 3.75 cm. 

45. What must be the radius of the curved surface of a 
thin plano-convex lens made of glass of index 1.5 which 
will give a real image of an object placed 2 cm. in front of 
the lens and magnified 3 times? Ans. 9 mm. 

46. Find the magnification of a convex lens of focal 
length 0.2 inch for an eye whose distance of most distinct 
vision is 14 inches. Ans. 71 times. 

47. An object is placed in front of a convex lens at a dis- 
tance from it equal to 1.5 times the focal length. Find the 
linear magnification. If the object is removed to twice this 
distance, what will be the magnification? Ans. - 2; — |. 

48. An object 5 cm. high is placed 12 cm. in front of a 
thin lens of focal length 8 cm. Find the position, size and 
nature of the image (a) for a convex lens, and (6) for a 
concave lens; and draw accurate diagram for each case. 

Ans. (a) Real, inverted image, 10 cm. high, 24 cm. from 
lens; (6) Virtual, erect image, 2 cm. high, 4.8 cm. from 
lens. 

49. When an object is placed at a point R on the axis of 
a thin lens of focal length /, the image is erect, and when 
the object is moved to a point S the image is the same size 
as before but inverted; show that 

m 
where m is a positive number denoting the value of the 
ratio of the size of the image to that of the object. 

50. A screen, placed at right angles to the axis of a thin 
lens of focal length /, receives the image of a small object. 
If the image is 20 times as large as the object, show that 
the distance of the screen from the lens is equal to 21/. 



256 Mirrors, Prisms and Lenses [Ch. VII 

51. Given a convex lens, a concave lens, a concave mirror 
and a convex mirror, each of focal length 20 cm. An object 
is placed in front of each in turn at distances of 40, 20 and 
10 cm. Draw diagrams showing the construction of the 
image for each lens and each mirror and for each of the 
three given positions of the object; and find the position 
and character of the image in each case. 

52. A plane mirror is placed anywhere behind a convex 
lens with its plane at right angles to the axis of the lens. 
A needle is set up perpendicular to the axis in the primary 
focal plane of the lens. Show that the image of the needle 
produced by rays that have passed twice through the lens 
will lie also in the primary focal plane and will be of the 
same size as the object but inverted. 

53. An object is placed in front of a thin convex lens at 
a distance a from it not greater than twice its focal length /; 
and a plane mirror is adjusted in the secondary focal plane 
of the lens. Show that a real image formed by rays which 
have passed twice through the lens will be formed at a dis- 
tance b in front of the lens; and that f=(a-\-b)/2. Show 
also that the image is of the same size as the object but in- 
verted. Draw a diagram showing the construction of the 
image. 

54. A convex lens of focal length 10 cm. is placed at a 
distance of 2 cm. in front of a plane mirror which is per- 
pendicular to the axis of the lens. Where must an eye be 
placed in front of the lens so that it may see its own image 
by means of rays which, after having traversed the lens 
twice, return into the eye as bundles of parallel rays? 

Ans. 3.75 cm. from the lens. 

55. A thin convex lens of focal length 10 inches is placed 
in front of a concave mirror of focal length 5 inches. The 
distance between the lens and the mirror is 10 inches. An 
object is placed in front of the lens at any distance from it. 
Show that its image formed by rays which have passed 
twice through the lens will lie at an equal distance from the 



Ch. VII] Problems 257 

lens on the other side of it, and that it will be of the same 
size as the object but inverted. 

56. A thin convex lens of focal length 12 inches is placed 
12 inches in front of a concave mirror of focal length 8 inches. 
An object is placed 3 inches in front of the lens. Show that 
its image formed by rays which have passed twice through 
the lens is in the same plane as the object and of the same 
size, but inverted. 

57. The focal length of a thin symmetric double concave 
lens made of glass of index 1.5 is five inches. A luminous 
point lies on the axis so far away that it may be considered 
as being at infinity. Prove that its image formed by rays 
which are reflected at the first surface is 2.5 inches in front 
of the lens; the image formed by rays which are refracted 
twice at the first surface and reflected once at the second 
surface is on the other side of the lens at a distance of 1.25 
inches from it; and, finally, the image formed by rays which 
after being reflected twice at the second surface have emerged 
again into the surrounding air is 0.5 inch from the lens on 
the side away from the source. 

58. A concave mirror, of radius r, has its center at the 
optical center of a thin lens, of focal length /, and the axes 
of lens and mirror are in the same straight line. Rays com- 
ing from an axial object point at a distance u from the lens 
traverse the lens and after being reflected at the mirror 
pass through the lens again and emerge from it as a bundle 
of rays parallel to the axis. Prove that 

W=o. 

u r f 



CHAPTER VIII 

CHANGE OF CURVATURE OF THE WAVE-FRONT IN REFLEC- 
TION AND REFRACTION. DIOPTRY SYSTEM 

99. Concerning Curvature and its Measure. — Since the 
rays or lines of advance of the light-waves are always at 
right angles to the wave-surface (§ 7), one way of investi- 
gating the procedure of light is to study the form of the 
wave-surface; for, in general, the effect of reflection and re- 
fraction will be to produce an abrupt change of curvature 
of the wave-front. In this method attention is concen- 
trated primarily on the wave-surface rather than on the 
rays themselves; but in reality the only difference between 
it and the ray-method consists in a new point of view, which 
may, however, be serviceable. Thus, when a plane wave 
is incident on a lens, the wave-front on emergence will no 
longer be plane but curved in such fashion that the light- 
waves either converge to or diverge from a point in the second 
focal plane of the lens. The effect of the lens or optical 
system is to imprint a new curvature on the wave-front, 
and if the change of curvature which is thus produced can 
be ascertained, the final form of the wave can be determined 
by mere algebraic addition of the initial and impressed 
curvatures. It will be necessary, however, to explain pre- 
cisely what is meant by this term curvature and how it is 
measured. 

In passing along an arc of a plane curve from a point A 
(Fig. 131) to a point B, the total curvature of the arc AB is 
the change of direction of the curve between A and B, which 
is evidently measured by the angle between the tangents 
to the curve at these two places. This angle is equal to 
the angle at O between the normals AO and BO which are 

258 



99] 



Curvature of an Arc 



259 



perpendicular to the tangents at A and B. The mean curva- 
ture between A and B is the change of this angle per unit 
length of the arc AB. If, therefore, the length of the arc 
AB is denoted by a and the magnitude of the angle BOA 




Fig. 131. — Mean curvature of arc AB measured by 
<P/a, where a-denotes length of arc and <f> denotes 
angle between the normals AO and BO. 

by <p, the mean curvature between A and B is equal to (pja. 
And the limiting value of this quotient when the point B is 
infinitely near to A is the measure of the actual curvature 
at the point A or, as we say, the curvature at A. If the curva- 
ture at A is denoted by the capital letter R, then R is equal 
to the Umiting value of <p/a when the arc a is indefinitely 
small. 

In Fig. 132 the point B is supposed to be infinitely near 
to A; and the point of intersection C of the normals drawn 



260 



Mirrors, Prisms and Lenses 



[§99 



to the two contiguous points A, B on the curve passing 
through these two points is called the center of curvature 
of the curve at the point A; the circle described in the 
plane of the curve around this point C as center with radius 





Fig. 132, a and b. — Curvature of arc BAB at point A midway between 
B and B is measured by the sagitta AD. (a) Convex, (b) Concave arc. 

r= AC, which will coincide with the given curve throughout 
the infinitely small arc AB, is called the circle of curvature 
and its radius r is called the radius of curvature at the point 
A. Now since by definition the angle <p is equal to the arc 
BA divided by the radius r, that is, since <p =a/r, the 
curvature at A is equal to l/r; that is, the curvature at any 
point on a curve is equal to the reciprocal of the radius of curva- 
ture at that point, or 

r 
The sign of the curvature is the same as that of the radius 
of curvature. Accordingly, if the surface is convex with re- 
spect to the incident light, the curvature is to be counted as posi- 
tive, in accordance with our previous usage in this respect. 



§ 99] Measure of Curvature 261 

Thus, for example, when spherical waves spread out from 
a point-source, the wave-front at any instant is concave 
and its curvature is reckoned, therefore, as negative. If a 
convex lens is interposed at a distance from the point-source 
greater than its focal length, the light-waves will thereby 
be converged to a focus on the other side of the lens whence 
they will ultimately diverge again. While the wave-front 
is advancing from the lens to the focus, its curvature is pos- 
itive; at the focus itself the wave-front collapses into a point, 
the curvature of the wave at this place being infinite; and 
beyond the focus the curvature becomes negative. As long 
as the wave does not undergo any reflection or refraction, 
its curvature varies continuously; whereas a sudden change 
of curvature is imprinted on the wave when there is a transi- 
tion from one medium to another. 

Another method of measuring the curvature of a small 
arc BB (Fig. 132) is in terms of its bulge AD, where the 
points designated by A and D are the middle points of the 
arc and its chord. If the points A and B are so close to- 
gether that they may be regarded as lying on the circle of 
curvature corresponding to the point A, the ordinate DB = h 
will be a mean proportional between the two segments into 
which the diameter of the circle is divided by the point D, 
so that we have the proportion : 

XD:h = h:(2r~AD). 
Since the segment AD is always very small in comparison 
with the diameter of the circle of curvature, only a vanish- 
ingly small error will be introduced by writing 2r in place 
of (2r— AD) in the above proportion. Thus, we obtain: 

h 2 



or since R = 1/r, 



AD =2r' 



h 2 

ad- | a 



If the arc BB is not infinitely small, this equation contains 
a certain error which is more and more negligible in pro- 



262 



Mirrors, Prisms and Lenses 



99 



portion as the arc is taken smaller and smaller. For a small 
arc, therefore, we may say that the segment AD is propor- 
tional to the curvature (R) at the point A, and hence it 
may be said to measure the curvature at this place. This 
segment AD was called by Kepler the sagitta of the arc 
BB because of its resemblance to an "arrow" on a bow. 





Fig. 133. — Curvatures of arcs 
BAB and BKB are in same 
ratio as their sagittae AD 
and KD. 



Fig. 134. — Curvatures of 
arcs AP and AQ in 
same ratio as their 
sagittae VP and VQ. 



Obviously, it does measure the bulge or " sag " of the curve 
at A. In Fig. 133, where the straight line BDB is the com- 
mon chord of the small arcs BAB and BKB, the curvatures 
at A and K are evidently in the ratio of AD to KD. Or, 
again, consider Fig. 134, where the two arcs AP and AQ 
have a common tangent at A. If on this tangent a point V 
is taken very close to A, and if through V a straight line is 
drawn perpendicular to AV intersecting the two arcs in 



99] 



Spherometer and Lens-Gauge 



263 



the points designated by P and Q, the curvatures at A will 
be in the ratio of VP to VQ. 

In many optical problems (as has been explained in the 
last two chapters) we are concerned only with a very small 
portion of the reflecting and 
refracting surface (case of 
paraxial rays), and under 
such circumstances it is 
especially convenient and 
simple to measure the curv- 
atures of the wave-fronts 
before and after refraction 
or reflection and the curva- 
tures of the mirrors or 
lenses by means of their 
sagittae. In fact, the ordi- 
nary method of determining 
the curvature of an optical 
surface with an instru- 
ment called a spherometer 
(Fig. 135) consists essen- 
tially in employing a mi- 
crometer screw to measure the sagitta of the arc whose 
chord is equal to the diameter of the circle circumscribed 
about the equilateral triangle formed by the conical points 
of the tripod which supports the instrument on the curved 
surface to be measured. The simple lens-gauge (Fig. 136) 
used by opticians to measure the power of a spectacle lens 
is based on the same principle. In size and external ap- 
pearance it resembles a watch, except that on its lower side 
it has three metallic pins projecting from it in parallel lines 
which all lie in a plane parallel to the face of the gauge. The 
two outer pins are stationary and symmetrically placed so that 
when the instrument is held in a vertical plane with the pins 
pointing downwards, the straight line BB (Fig. 132) joining 
the conical points of the outer pins is horizontal; whereas 




Fig. 135. — Spherometer. 



264 



Mirrors, Prisms and Lenses 



[§99 



the other pin which is midway between the two outer ones 
is capable of being pushed upwards by a slight pressure so 
that its tip A which left to itself falls a little below the 
straight line BB can be made to ascend a little above this 

line. The vertical dis- 
placement of the tip A of 
the middle pin above or 
below the level of the 
chord BB, which is equal 
to the sagitta of the arc 
BAB whose curvature is 
to be measured, is regis- 
tered on the dial (see 
§ 108) by the angular 
movement of a light hand 
or pointer with which the 
movable pin is connected. 
If the circle is drawn 
which passes through the 
end-points of the three 
pins B, A and B, the 
diameter drawn through A will bisect the chord BB at a 
point D; and since the products of the segments of two 
intersecting chords of a circle are equal, we obtain imme- 
diately : 

AD (2r-AD)=h 2 , 
where r denotes the radius of the circle and 2h = chord BB. 
Hence, exactly as above, we obtain here also: 

h 2 
AD= — , approximately; 

thus proving again that the sagitta AD is proportional to 
the curvature l/r = R. In using the lens-gauge care must 
be taken to see that the plane of the instrument is not tilted 
out of the vertical, and this is one reason why a spherometer 
is more accurate. On the other hand, the lens-gauge, be- 
sides being more handy and convenient, possesses a de- 




Fig. 136. — Lens-gauge. 



100] 



Plane Refracting Surface 



265 



cided advantage over a spherometer supported on a tripod 
by reason of the fact that it can be used to measure the 
curvatures in different meridians of a non-spherical surface 
of revolution, for example, the curvatures of the normal 
sections (§ 111) of a cylindrical or of a toric surface (§ 112). 
How the lens-gauge is graduated will be explained presently 
(§ 108). 

100. Refraction of a Spherical Wave at a Plane Surface. 
— The whole duty of an optical system, therefore, whether 



z 






:-•'.'■'- ■' ^^^ 


B 


Pr<T*. • ' • 


AIR ^</ 


% ■ : * GLASS 




a\\ ; ; • 

>':"} V: A * ." . - . 

i- \.\ - ' 

:■■•' \ -\ ' ' 
■■■'■: 'A \ ' ■ • 

■;'■•' I > • * 


M'\ M\ A 


". K" ,'j . ,. 


•* \ 
\ 


:■'--/ i ' v • . 

& •'■•/• ' • • ' 

r -. : ■//*•' 

■ J i . ' 
■'•"• / / ■ ' • 
:••:/ /. .. • , 

'■'/ ' ' • ' • « ■ ' 


B 


fe>sv 


Z 





3fc 



Fig. 137, a. — Divergent spherical waves refracted at plane 
surface from air to glass. 

it be a single lens or mirror or a combination of such parts 
is to imprint a certain curvature on the surface of the in- 
cident wave; and if we consider only such portions of the 
wave-fronts as lie very close to the axis of symmetry of 
the instrument, it is evident that this method of investi- 
gating the change of curvature that is produced in the 
wave-front at the point where the axis meets it should lead 
to precisely the same results as have been found already 



266 



Mirrors, Prisms and Lenses 



100 



in the corresponding problems concerning the reflection 
and refraction of paraxial rays. In fact, according to this 
method, these results should be found to apply not merely 
to the case when the reflecting and refracting surfaces are 
plane or spherical, but equally also to the more general 
case when these surfaces have any form whatever, provided 
they are symmetrical around the optical axis. 





Z 


\1 : 




AIR 


/! 
/ 1 


•V^^ GLASS 

■■'■■ ■ \ \ 






ft 


;:•.•• • \ \ 






i 

/ i 

i 
i 
i 


v ' ■ . N " ^V^ 

• • - ; n -A. 

■■:•'-. • \ ^ 




J 




A • • . /M 


^V 




\ \ 








\ \ 


:•...•• • / ^s^' 






\ \ 


• s ^-^ 






\\ 


■■■.-• / ^y^ 






\\ 


>■ '• . / ^y^ • - - 






\\ 


!•: -. s^s^ 






M 


B: -;•/.:, ' 






z 







Fig. 137, b. 



-Convergent spherical waves refracted at plane surface from 
air to glass. 



We shall begin by investigating the simple case of the 
refraction of a spherical wave at a plane surface. 

In the diagrams (Fig. 137, a, b, c, and d) the straight line ZZ 
represents the trace in the plane of the paper of a plane re- 
fracting surface separating two media of indices n, n' . 
Around the point M as center spherical waves are supposed 
to be advancing in the first medium (n) towards the refract- 
ing surface, and at a certain instant when the disturbance 



100] 



Plane Refracting Surface 



267 



has begun to affect a point B on this surface the incident 
wave will be represented in the plane of the figure by the 
circular arc BJB described around M as center with radius 
equal to BM; the point designated by J lying on the arc 
midway between its two ends B, B, so that the straight 
line MJ is the perpendicular bisector at A of the chord BB. 
The two points M, J will be found to lie always on opposite 



GLASS 




Fig. 137, c. — Divergent spherical waves refracted at plane 
surface from glass to air. 

sides of the refracting plane. In Fig. 137, a and c, where the 
point M is shown as lying in front of the surface ZZ, the 
arc BJB is indicated by a dotted line, because it marks the 
position which the incident wave-front would have had 
if the refracting surface had not been interposed. But 
the waves travel faster in the rarer medium (air) than in 
the denser medium (glass); and, consequently, the vertex 
of the refracted wave-front instead of being at the point J 



268 



Mirrors, Prisms and Lenses 



[§100 



on the axis will be at a point K on this line, and therefore 
the position of the refracted wave-front at the moment 
when the disturbance arrives at B will be represented by 
the arc BKB of a circle whose center is at a point M' 
on the axis. If, for example, the waves are refracted from 
air to glass, that is, if n'>n t the velocity v in the first me- 







z 


GLASS •' 

/ 
* ' . .''1. 
■ '•/ 
".■. * • / 

" . 1 • 


■A 
if}. 

1 !■'■ 
1 "' ' 


B 

V. AIR 

\ \ 

\ ^ 


«!■■ 

■.»■ ..»v 


J : 

v\'J 

\\- 
■ x \ 


A / M ' ^"M 


' J^$ 


B 


* 




Z 



3C 



Fig. 137, d. — Convergent spherical waves refracted at plane surface from 

glass to air. 

dium will be greater than the velocity v' in the second me- 
dium, so that for this case AK will be shorter than AJ, and 
the effect of the retardation will be to flatten the wave- 
front, as shown in Fig. 137, a and b. On the other hand, if 
n'<n, then v'>v, so that now AK will be longer than AJ, 
and the effect of the refraction will be to increase the curva- 
ture or bulge of the wave-front, as shown in Fig. 137, c and d. 
Since (see § 31) 

AJ: AK = v: v' = n':n, 



101] 



Spherical Refracting Surface 



269 



it follows that 



.KA = n.JA. 



Now JA and KA are the sagittce (§ 99) of the small arcs 
BJB and BKB, respectively, and hence they are propor- 
tional to the curvatures of these arcs, that is, to l/JM and 
1/KM'. If the point B is infinitely near to A, we may put 
JM = AM = u, KM' = AM' = <- and thus we obtain: 



u u 
which will be recognized as the relation which we found 
for the refraction of paraxial rays at a plane surface (§41). 
101. Refraction of a Spherical Wave at a Spherical Sur- 
face. — Here the same method is employed as in the preced- 




Fig. 138, a. 



-Divergent spherical waves refracted at convex surface 
from air to glass. 



ing section. In each of the diagrams (Fig. 138, a, b, c, d, 
e, f, g, and h) the circular arc ZZ represents the trace in the 
plane of the paper of a meridian section of the spherical 
refracting surface with its vertex at A and center at C. The 
surface is convex in Fig. 138, a, b, c, and d and concave in 
Fig. 138, e, f, g, and h. The point M on the axis is the center 
of a system of spherical waves which are advancing in the 
first medium, of index n, towards the refracting surface. 
In Fig. 138, a, c, e, and g the point M lies in front of the re- 



270 



Mirrors, Prisms and Lenses 



101 



fracting surface, whereas in Fig. 138, b, d, f, and h this point 
is situated on the other side of the surface. The points 





y^T 


>b^l ' ' 




AIR 


jt/'l' 
f'/r 


■ />£* 


^ GLASS 




/'/ • 
/•■/• 

. / ' 

1 • 














a|;K 


• I • 

• \ ■ ■■ 


D 






\\ 




/• s' 




V \ - 














'•\\.\ • 




s- * 




^\\ 


/ ■"" 


' 




NA\ 







Fig. 138, b. — Convergent spherical waves refracted at convex 
surface from air to glass. 

marked B, B are two points on the arc ZZ very close to- 
gether but at equal distances on opposite sides of the op- 



glass 




Fig. 138, c. — Divergent spherical waves refracted at convex 
surface from glass to air. 

tical axis, so that the arc BJB described around M as center 
with radius equal to BM shows the position of the wave- 
front of the incident waves at the instant when the disturb- 



101] 



Spherical Refracting Surface 



271 



ance begins to affect the points B, B; the point where this 
arc crosses the optical axis being designated by J. 







NB^^Z 






. • • • ■ ::]yi 


*5^^ 






//] 


^^^^^ 






•'•'-;/// 


^^^n,,^ 




I glass . :■ 


•/ // 


X^\. AIR 






/ / ' 


V N ^^^\ 






/ 

l 








• .. a| 


J J 

\ 


K 


D C /'M' 


^^ >M 


■ • :'% 


\ 




s ^r^^ 






\ v \ 


'' J^f 




• • '* :." 


\ ^ \ 


** *^*^ 




; • 


•:\ \\ 


'' ^"'^^ 






•- : \ M 


,' ^**^ 






:% \\ 


,' -*^^ 








x\\ 


^^^ 





Fig. 138, d. — Convergent spherical waves refracted at convex 
surface from glass to air. 

When the waves enter the second medium, of index n', 
they will proceed with augmented or diminished speed ac- 



GLASS 




Fig. 138, e. — Divergent spherical waves refracted at 
concave surface from air to glass. 

cording as n is greater or less than n'. In the diagrams 
Fig. 138, a, b, e, /, the case is represented where n' >n; and 
in the diagrams Fig. 138, c, d, g, h the second medium is 



272 



Mirrors, Prisms and Lenses 



[§101 



supposed to be less highly refracting than the first {n'<n). 
The center of curvature of the refracted waves will lie at 



GLASS 




Fig. 138, /. — Convergent spherical waves refracted at concave 
surface from air to glass. 

a point M' on the axis, so that the wave-front in the second 
medium which passes through B, B will be represented by 





. ."V'-.^N 


3^- 






GLASS 


^"''"■y^.- 




AIR 


^*-«* 


""* j< .• 














„-*"" 








-""*''' y^ 








































M'~^-^ M \^ 


C . D 


K ! :• 


J 


A 


***"•**» ^\ 








"****»„ 


^v v -'■ 




*•/ 


** 


^**^JnJ 







Fig. 138, g. — Divergent spherical waves refracted at con- 
cave surface from glass to air. 

the arc BKB of a circle described around M' as center with 
radius equal to BM'; the point where this arc crosses the 
axis being designated by K. 

In each of the diagrams of Fig. 138 one of the two arcs 



101] 



Spherical Refracting Surface 



273 



BJB and BKB is shown by a dotted line, because, on ac- 
count of the interposition of the refracting surface ZZ, the 
part of one or the other of these wave-fronts which is com- 
prised between B, B does not actually materialize; but this 
circumstance does not in the least affect the geometrical 
relations. 

Thus, during the time the light takes to go in the first 
medium from J to A (or from A to J), it will travel in the 




Fig. 138, h. — Convergent spherical waves refracted at 
concave surface from glass to air. 



second medium from K to A (or from A to K). In other 
words, the optical lengths (§ 39) of the axial line-segments 
AJ and AK are equal, and therefore : 

n.AJ = n.AK. 

This shows how the position of the point IVT may be found, 
for we have only to lay off on the axis a piece 

AK=-,AJ, 

n 

and to locate the point M' at the place where the perpendic- 
ular bisector of the chord BK intersects the optical axis. 
Draw the chord BDB crossing the optical axis at right 



274 



Mirrors, Prisms and Lenses 



[§102 



angles at the point D; then, evidently, since 

AJ = AD+DJ = AD-JD, AE>AD+DK = AD-KD, 

we have: n(AD-JD) = n'(AD-KD). 

Now recalling the fact that the points B, B were assumed 
to be very close to the vertex A of the spherical refracting 
surface, we remark that the arcs whose summits are at A, J 
and K are all very small; and hence the segments AD, JD 
and KD may be regarded as the sagittce of these arcs and 
proportional to their curvatures (§ 99), viz., l/r, 1/u and 
1/u', respectively, where r = AC, w = AM = JM, u' = AM' 
= KM', approximately. Introducing these values in the 
equation above, we obtain the characteristic invariant re- 
lation for the case of the refraction of paraxial rays at a 
spherical surface, viz., 



(i_lW(I_l,) 

\r u) \r u / 



in the same form as was found in § 78. 

102. Reflection of a Spherical Wave at a Spherical Mir- 
ror. — The problem of reflection at a spherical mirror may 




Fig. 139, a. — Divergent spherical waves reflected at convex mirror. 

be investigated in the same way. In Fig. 139, a and b, 
the arcs BAB, BJB and BKB represent the traces of the 
mirror and of the wave-fronts of the incident and reflected 



102] 



Spherical Mirror 



275 



waves, respectively. In the case of reflection the condition 
evidently is : 

KA = AJ, 
because while the incident wave advances along the optical 
axis through the distance A J or J A, the reflected wave will 
travel in the opposite direction through an equal distance 




Fig. 139, b. — Divergent spherical waves reflected at concave mirror. 

KA or AK. Therefore the center M' of the reflected wave 
may be found by laying off AK = JA and locating the point 
where the perpendicular bisector of the chord KB inter- 
sects the axis. 

Here also the segments AD, JD and KD are to be re- 
garded as the measures of the curvatures of the small arcs 
BAB, BJB and BKB, respectively, and proportional, there- 
fore, to the reciprocals of the radii of curvature, viz., l/r, 
1/m and 1/V, where r = AC, u=AM=JM, u'=AM' = KM' 
in the limit when the arcs are infinitely small. Now 
KD = KA+ AD = AJ+ AD = AD+DJ+ AD, 



276 Mirrors, Prisms and Lenses [§ 103 

that is, 

JD+KD = 2AD; 

hence, substituting the symbols u, v! and r, we derive the 
abscissa-formula for the reflection of paraxial rays at a 
spherical mirror (§ 64), viz., 

2+1 = 2 . 

u u r 
which may be expressed in words by saying that the curva- 
ture of the mirror is the arithmetical mean of the curvatures 
of the incident and reflected waves at the vertex of the mirror; 
that is, 

R 2-' 

where U=l/u, U' = l/u' denote the curvatures of the in- 
cident and reflected waves, and R = l/r denotes the curva- 
ture of the mirror. Thus, for example, if an incident plane 
wave (U = 0) is advancing parallel to the axis of the mirror, 
the curvature of the reflected wave will be twice that of 
the mirror, and consequently, the center F of the reflected 
wave-front will lie midway between the vertex A and the 
center C of the mirror (§ 69). 

Of course, the condition KA = AJ might have been de- 
rived at once from the condition n.AJ = n'.XK, which was 
found in § 101, by putting in this equation n'=—n, in ac- 
cordance with the general rule given in § 75. 

103. Refraction of a Spherical Wave through an In- 
finitely Thin Lens. — Since, as has been shown (§ 89), a 
homocentric bundle of incident paraxial rays with its ver- 
tex at a point M on the axis of a thin lens is transformed 
into a homocentric bundle of emergent rays with its vertex 
at the conjugate point M', we know that if the waves are 
spherical before traversing the lens, they will issue from it 
as spherical waves, at least in the neighborhood of the axis. 

Each of the diagrams (Fig. 140, a and b) represents a 
meridian section of the lens which is convex in one figure 
and concave in the other. As a matter of fact the lens is 



§103] 



Infinitely Thin Lens 



277 



assumed to be infinitely thin, and perhaps it is well to call 
particular attention to this fundamental consideration, be- 
cause in the diagrams, in order to exhibit the relations by 
means of the sagittce, the lens-thickness is shown very much 
exaggerated. 




Fig. 140, a. — Divergent spherical waves refracted through thin convex lens. 

Take a point Bi on the first surface of the lens not very 
far from the vertex Ai of this surface, and around the axial 
object-point M as center with radius equal to BiM describe 
the circular arc BiJBi which is bisected by the axis of the 
lens in the point designated by J; evidently, this arc will 
represent the trace in the plane of the diagram of the wave- 
front of the incident waves at the moment when the dis- 
turbance reaches Bi. Now the disturbance which is propa- 
gated onwards from Bi will proceed across the lens to a 
point B2 on the second face of the lens, and since the lens 
is supposed to be infinitely thin, the distances of Bi, B 2 
from the axis are to be regarded as equal, that is, DiBi = 
D 2 B 2 , where Di, D 2 , designate the feet of the perpendicu- 
lars let fall from Bi, B 2 , respectively, to the axis of the 
lens. If, therefore, around the point M' conjugate to M 
an arc B 2 KB 2 is described with radius equal to B 2 M', which 
is bisected by the axis at the point designated by K, this 
arc will represent the trace in the plane of the diagram of 
the wave-front of the emergent waves at the same instant 



278 



Mirrors, Prisms and Lenses 



[§103 



that the arc B1JB1 shows the wave-front of the incident 
waves. 

With M, M' as centers and with any convenient radii 
describe also the arcs GH, SL intersecting the axis of the 




Fig. 140, b. — Divergent spherical waves refracted through thin concave lens. 

lens at G, S and meeting the straight lines BiM, B 2 M', in 
H, L, respectively; so that these arcs represent, therefore, 
successive positions of the wave-front before and after 
transmission through the lens. Now the optical length of 
the light-path from H to L is equal to that along the axis 
of the lens from G to S (§ 39); and, hence, if n, n' denote 
the indices of refraction of the two media concerned, we 
may write : 

n.HBi+n , .B 1 B 2 +n.B 2 L = n.GAi+n , .AiA2+n.A 2 S; 
and since 

n(MH+LM') =n(MG+SM0, 
we obtain by addition of these two equations : 

n(MB 1 -fB 2 M , )+n , .B 1 B 2 = n(MAi+A 2 M , )+n , .AiA 2 . 
Now MBi = MJ, B 2 M' = KM', B 1 B 2 = D 1 D 2 ; 

and therefore: 

w(M J- M Ai+ KM - A 2 M') = n'(Af A 2 - DiD 2 ) . 



§ 104] Reduced Distance 279 

Substituting in this equation the following expressions, viz. : 
MJ-MAi = AiM+MJ = AJ = AiDd-DiJ = A1D1-JD1, 
KM'-A 2 M' = KM'+M'A 2 = KA 2 = KD 2 +D 2 A 2 
= KD 2 -A 2 D 2 , 
AiA 2 = AiDi+DiD 2 +D 2 A 2 = AiDi+DiD 2 -A 2 D 2 , 
we obtain : 

n(AiDi-JD 1 +KD 2 -A 2 D 2 )=n , (AiDi-A 2 D 2 ); 
which may be put finally in the following form: 
w(KD 2 -JDi) = (n'-n) (AiDi-A 2 D 2 ). 
It has been assumed here that the lens is surrounded by 
the same medium (n) on both sides, but the same method 
would lead to a more general formula for which the initial 
and final media were different. 

Evidently, since the points Bi, B 2 are very near the verti- 
ces Ai, A 2 , the segments A1D1, JDi, A 2 D 2 , KD 2 may be re- 
garded as the sagittce of the small arcs B1A1B1, B1JB1, 
B 2 A 2 B 2 , B 2 KB 2 , respectively; and since these arcs all have 
equal chords, the reciprocals of the radii of curvature may be 
substituted in the equation above in place of the sagittce. 
Accordingly, if the radii of the lens-surfaces are denoted 
by n, r 2 , and if we put AM = JM = w, A 2 M / = KM'=w' J as 
is permissible in this case, we derive immediately the fa- 
miliar lens-formula for the refraction of paraxial rays (§ 89), 
viz.: 

W ul \ri r 2 / f 

where / denotes the primary focal length of the lens. 

104. Reduced Distance. — If P, Q designate the positions 
of two points lying both in the same medium of refractive 
index n, el distinction has already been pointed out (see § 39) 
between the actual or absolute distance of these points from 
each other and the so-called " optical length" of the seg- 
ment PQ of the straight line joining these points, which 
is obtained by multiplying the absolute length by the index 
of refraction of the medium, and which is equal therefore 
to n.PQ. A further distinction, due originally to Gauss, 



280 Mirrors, Prisms and Lenses [§ 104 

is to be made now by employing the term reduced distance 
between P and Q to mean, not the product, but the quo- 
tient of the distance PQ by the index of refraction of the medium 
in which the two points P and Q lie; that is, the reduced dis- 

PQ 

tance from P to Q is equal to — — . Thus, for example, if 

the medium is glass of index 1.5, and if the distance PQ = 
12 inches, the optical distance or equivalent light-path in 
air will be 18 inches, whereas the reduced distance will be 

8 inches. The reduced thickness of a lens is c = -, where 

n 

d = AiA 2 denotes the distance of the second vertex A 2 of 
the lens from the first vertex Ai and n denotes the index of 
refraction of the lens-substance. The optical distance is 
never less, and the reduced distance is never greater, than 
the actual distance. If the medium is air (n=l), the op- 
tical distance and the reduced distance are both equal to 
the absolute distance. Apparently, the first use of the term 
"reduced distance" in this sense in English occurs in 
Pendlebury's Lenses and systems of lenses, treated after 
the manner of Gauss (Cambridge, 1884). A distinct ad- 
vantage in the direction of simplification is usually gained 
in mathematical formulation by denoting a more or less 
complex function by a single symbol; and modern optical 
writers, notably Gullstrand and his disciples in Germany, 
have recognized the convenience of this idea of " reduced 
distance" and utilized it to express the relations between 
object and image in their simplest forms; as we shall show 
presently by several examples. 

In this connection the attention of the student needs to 
be called to a point which has been alluded to before (see 
§ 8), but which is not always clearly understood. Although 
two points P, Q may be situated physically in different 
media, they may be regarded as optically in the same me- 
dium. Thus, any point which is on the prolongation, in 
either direction, of the line-segment which represents the 



§ 105] Refracting Power 281 

actual path of a ray of light through a certain medium, may, 
and in fact generally must, be regarded as a point belonging 
to the medium in question, no matter what may be its ac- 
tual physical environment. No better illustration of this 
notion can be given than is afforded by considering the 
focal points on the axis of a spherical refracting surface. 
The points F and F' lie always on opposite sides of the ver- 
tex A, but no matter whether the first focal point F is 
on one side of A or on the other, it is to be considered 
always as a point in the first medium; and, similarly, 
the second focal point F' is to be considered always as 
a point in the second medium, so that the reduced dis- 
tance between F and F' is FA/n+AF'/n' both for a conver- 
gent and for a divergent system. The reduced focal lengths 

f f 

of a spherical refracting surface are - and — .; so that the 
^ n n 

f f 
reduced distance of F' from F is equal to - - — , . 

n n 

The boundary between two optical media is a "twilight 
zone," so to speak, which cannot be said properly to be- 
long to either medium; and hence linear magnitudes which 
refer specifically to the interface or surface of separation 
cannot be definitely assigned to one medium or the other. 
This applies, for example, to the radius of curvature of a 
mirror or of a refracting surface. Whether a surface which 
separates air from glass is convex or concave, we have no 
right to say that the radius of curvature lies in the air or 
in the glass; and thus we never speak of the "reduced ra- 
dius" of a reflecting or refracting surface. 

105. The Refracting Power. — In the w-form of the ab- 
scissa-equation which gives the relation between a pair of con- 
jugate points on the axis, we are concerned not so much with 
the linear magnitudes themselves, that is, with the abscissa, 
as with the reciprocals of these magnitudes, which, as we 
have seen, represent the curvatures of the surfaces of which 
these abscissa? are the radii. It is partly for this reason 



282 Mirrors, Prisms and Lenses [§ 105 

that many teachers of geometrical optics regard the so- 
called " curvature method" of studying these problems as 
both more natural and more direct than the "ray method." 
There is certainly much to be said in its favor, but the truth 
is, both methods have their advantages, and neither is to 
be preferred to the other. The student who desires to have 
more than a mere elementary knowledge of optics will find 
it necessary to be acquainted with both points of view; and 
when he has attained this position, he will realize that the 
two methods are perfectly equivalent and that the distinc- 
tion between them is more or less artificial. 

But whether we have the so-called "curvature method" 
in mind or not, it will evidently be a step in the direction of 
simplifying the abscissae-formula if we introduce symbols 
for the reciprocals of the abscissae, and thereby get rid of 
the fractional forms. Thus, instead of employing the re- 
duced focal length, it will be better to introduce a term for 
the reciprocal of this magnitude. Accordingly, the refract- 
ing power of an optical system is defined to be the reciprocal 
of the reduced primary focal length. These reciprocal mag- 
nitudes will be denoted by capital italic letters. For ex- 
ample, the refracting power of an optical system will be de- 
noted by F; that is, according to the above definition: 

The refracting power of a spherical refracting surface (see 
§79) is: 

F = - f =~j, = (n'-n)R, 

where R = - denotes the curvature of the surface. If the 
r 

first medium is air (n = l), then F=j. The refracting power 

of a spherical refracting surface is directly proportional to 
the curvature of the surface. 



§ 105] Refracting Power 283 

The reflecting power of a spherical mirror {n f = — n, /'=/) 
is defined in the same way, viz., 

F = - = -2n.R, 

where n denotes the index of refraction of the medium in 
front of the mirror. Thus, although the position of the 
focal point (F) and the magnitude of the focal length (/) of 
a curved mirror will not be altered by changing the medium 
in front of the mirror, its reflecting power will be affected; 
and this will be the case whether the mirror is concave or 
convex. If the focal length of a mirror is 8, its reflecting 
power will be one-eighth when the mirror is in contact with 
air (n=l), but it will be raised to one-sixth if the medium 
in front of the mirror is water (n = 4) . 

The refracting power of a lens surrounded by the same 
medium (ri) on both sides is 

F = ^=-^ 
f /'' 
If the curvatures of the two faces of an infinitely thin lens 
are denoted by Ri and R 2) that is, if 

Ri = — , R 2 = — , 

ri r 2 

then 

F=(n'-n) (Ri-R 2 ), 
where n' denotes the index of refraction of the lens-substance 
and n denotes the index of refraction of the surrounding 
medium. If either one of these media is changed, other 
things remaining the same, the refracting power of the lens 
will be altered. 

If F\, F 2 denote the refracting powers of the two surfaces 
of a lens, then 

F^tn'-^Ri, F 2 = (n-n')R 2 , 
and in place of the preceding equation we may write : 

F = F 1 +F 2 ; 
and thus it appears that the refracting power of an infinitely 



284 Mirrors, Prisms and Lenses [§ 106 

thin lens is equal to the algebraic sum of the refracting powers 
of the lens-surfaces. 

The refracting power of a lens depends, therefore, on 
the curvatures of both faces, but evidently a lens of given 
material and of prescribed refracting power may have very 
different forms. One of the minor problems of optical 
construction is to "bend" a lens, as the technicab phrase 
is, that is, being given the curvature of one face of the lens, 
to find the curvature of the other face so that the refracting 
power of the lens may have a given value. If, for example 
the magnitudes denoted by n, n' , R 2 and F are assigned, 
the curvature of the first face must be : 

F 

Ri = R 2 -\- , 

n —n 

If the media are different on the two sides of the lens, and if 
the indices of refraction of the three media in the order in 
which they are traversed by the light are denoted by n\, 
n<t and n 3 , we find easily the following formula for the re- 
fracting power of an infinitely thin lens : 

F= J= - J, =(n 2 -n 1 )R 1 +(n 3 -n 2 )R 2 = F l +F 2 , 

where the symbols have precisely the same meanings as 
before. 

It will be seen from these examples that one effect of in- 
troducing the term refracting power is a simplification in 
consequence of the fact that the two magnitudes denoted 
by / and /' are now expressed in terms of a single magni- 
tude F. 

106. Reduced Abscissa and Reduced " Vergence ". — The 
reduced abscissae of a pair of conjugate axial points M, M' 
are defined in exactly the same way as the reduced focal 
lengths. The point designated by M is to be regarded al- 
ways as lying in the first medium of the system, and, sim- 
ilarly, the point designated by M' is to be regarded as lying 
in the last medium, entirely irrespective of the question as 



§ 106] Reduced Abscissa and Reduced "Vergence" 285 

to whether either of these points is "real" or "virtual," 
as explained in § 104. 

By way of illustration, suppose that the optical system 
consists of a single spherical refracting surface separating 
two media ef indices n and n' . If the origin of abscissae 
is taken at the vertex A, so that w = AM, w' = AM', then 

the reduced abscissae will be -, — . The reciprocals of these 

n' n' 

magnitudes, denoted by U, U' are called the reduced "ver- 

gences," with respect to the point A; thus, 

U=-, U'=-,. 
u u 

These functions U, U' are the measures of the convergence 
or divergence of the bundles of object-rays and image-rays; 
and in this illustration these magnitudes are evidently pro- 
portional to the curvatures of the incident and refracted 
wave-fronts at the instant when the disturbance arrives 
at the surface of separation of the two media. 

Since (§ 79) the abscissa-formula for a spherical refracting 
surface may be written in the form : 

n f _n , n 
u~u~f' 
this relation may now be expressed in the elegant and con- 
venient form: 

U'=U+F. 

This same formula holds in the case of a spherical mirror, 
in which case U f = — n/u' ', where n denotes the index of re- 
fraction of the medium in front of the mirror. 

Moreover, the same formula U'=U-\-F is found to be 
applicable to the case of an infinitely thin lens. If the lens 
is surrounded by the same medium (n) on both sides, then 
we must put U = n/u, U' = n/u' and F = n/f, where n' de- 
notes the index of refraction of the lens-substance. Or in 
case the last medium (w 3 ) is different from the first medium 
(ni), then U = ni/u, U' = ns/u', and F = n\jf. In both cases 
the formula will be found to be identical in form with that 



286 Mirrors, Prisms and Lenses [§ 107 

given above. In fact, as we shall see in Chapter X, the 
formula U'=U-\-F is perfectly general and applicable to 
any optical instrument which is symmetrical about an axis. 
The advantage of a single formula which has such wide ap- 
plicability is obvious. It is easy to remember* that the re- 
duced vergence (W) on the image-side of the instrument is 
equal to the algebraic sum of the reduced vergence (U) on the 
object-side and the refracting power (F) . 

If the abscissas are measured from the focal points F, F', 
that is, if we put z = FM, x / --=F / M r , the magnitudes 

X = -, X' = -' 
x x 

are called the reduced focal point vergences; and the relation 

between X, X' is expressed by the equation : 

X-X'=-F\ 

107. The Dioptry as Unit of Curvature. — Obviously, the 

magnitudes which have been denoted above by capital 

italic letters, since they are all equal or proportional to 

the reciprocals of certain linear magnitudes, are essentially 

measures of curvature, and hence they must be described or 

expressed in terms of some unit of curvature, which will itself 

be dependent on the unit of length. Opticians guided by 

purely practical considerations were the first to recognize 

the need of a suitable optical unit for this purpose. The 

unit of curvature which is now almost universally used in 

spectacle optics and which is coming to be employed more 

and more in all other branches of optics is the curvature 

of an arc whose radius of curvature is one meter. To this 

unit the name dioptry* has been given. Originally, the 

* The name "dioptrie" was first suggested by Monoyer of France 
in 1872 (see Annates d'oculistique, LXV1II, 111), being derived from 
the Greek tcl o\o7rrpi/<a, whence came also the term "dioptrics" 
which was formerly much used by scientific writers as applying to the 
phenomena of refraction, especially through lenses. The word is 
generally written dioptre in French and Dioptrie in German. Etymo- 
logically, the correct English form would appear to be dioptry, and 
this spelling has been adopted by the American translators of both 



§ 107] Dioptry 287 

dioptry was defined as the refracting power of a lens in air 
of focal length one meter. Consequently, a lens whose focal 
length was 50 cm. or half a meter would have a refracting 
power of 2 dptr., whereas another lens of focal length 2 meters 
would have a refracting power of \ dptr. In general, if 
the focal length of a lens surrounded by air is / centimeters, 
its refracting power will be 100// dptr. But according to 
the definition which we have given, the dioptry is a unit 
not of refracting power only but of any similar magnitude 
of the nature of a curvature. Thus, for example, if the 
radius of a mirror or of a spherical refracting surface is half 
a meter, its curvature is 2 dptr. If the distances denoted 
by f> r , u > x ) etc., are expressed in meters, the magnitudes 
denoted by the corresponding capital letters F, R, U, X, 
etc., will be in dioptries. Dr. Drysdale has suggested 
that we introduce also the convenient terms millidioptry 
( = 0.001 dptr.), Hectodioptry ( = 100 dptr.) and Kilodioptry 
( = 1000 dptr.) corresponding, respectively, to the Kilo- 
meter, centimeter and millimeter as units of length. Thus, 
the refracting power of a lens of focal length 10 cm. might 
be variously described as equal to 100 millidioptries, to 
10 dioptries, to 0.1 Hectodioptry or to 0.01 Kilodioptry. 
But these terms have not come into general use. 

If the focal length of a lens in water (n = 1.3) is 13 cm., 
its refracting power will be the same as that of a lens in 
air (n = l) of focal length 10 cm., viz., 10 dptr. If the pri- 
mary focal point of a spherical refracting surface is situated 

Landolt's and Tscherning's books on physiological optics; notwith- 
standing the fact that the word is usually spelled and pronounced 
dioptre in England and diopter in America. Dr. Crew in his well known 
text-book of physics writes dioptric. The author has concluded that 
on the whole it is best to adopt the spelling used in the text. 

The usual abbreviation of dioptry is a capital D.; but as this letter 
is liable to be confused with the symbols of magnitude employed in the 
formulae, it seems preferable to follow the usage of Von Rohr and 
other modern writers on optics who have adopted the abbreviation 
dptr., although doubtless many will object to this long form. 



288 Mirrors, and Prisms Lenses [§ 108 

(optically) in air (n=l) at a distance of 1 meter from the 
vertex, the refracting power of the surface will be 1 dptr. 
and the radius of the surface will be equal to (n f — 1) meters, 
where ,n f denotes the index of refraction of the second me- 
dium. If the radius of curvature of a mirror is 50 cm., its 
reflecting power will be 4 dptr. if the reflecting surface is 
in contact with air (n = l), but it will be 5^ dptr. if the 
surface is in contact with water (w =-§-)• These examples 
are given merely to illustrate how the term dioptry is used. 

108. Lens-Gauge — The dial of the opticians' lens-gauge 
described in § 99 is usually graduated so as to give in di- 
optries the refracting power of the surface which is measured. 
The refracting power of a spherical refracting surface is 
proportional to its curvature, as we have seen (§ 105), but 
it is dependent also on the indices of refraction of the two 
media. If the first medium is air and if the index of re- 
fraction of the second medium is denoted by n, then F = 
(n—l)R. The gauge actually measures the curvature R, and 
the readings on the dial correspond to the values of R mul- 
tiplied by the factor (n— 1). Direct readings of the refract- 
ing power (F) imply, therefore, that the maker has assumed 
a certain value of the index of refraction n; and if the actual 
value of n is different from this assumed value, the readings 
will be erroneous. The value of n assumed by the maker is a 
constant of the instrument, which should be marked on it, 
although it may easily be determined empirically by com- 
paring the readings with the determination of the curvature 
as obtained with an ordinary spherometer. 

Suppose that this constant is denoted by c, and that we 
wish to use the gauge to measure the refracting power (F) 
of a lens of negligible thickness made of glass of index n. 
If the refracting powers of the two surfaces of the lens are 
denoted by F\ and F 2 and the curvatures by Rx and R 2 , 
then F = Fi+F 2 where Fi = (n-l)R h F 2 = — {n-l)R 2} the 
minus sign in front of the last expression being necessary 
because the refraction in this case takes place from glass 



§ 109] Lens-System of Negligible Thickness 289 

to air. But if the constant c has a value different from n, 
the readings of the instrument for the two faces of the lens 
will not give the correct values F h F 2 of the refracting powers. 
Suppose the readings are denoted by Fi, F 2 , so that 
iY = {c-\)Ri,F 2 '=-{c-l)R 2 . Then evidently 



and hence 



t\ — 7^1> ^2 7^2, 

c- 1 c — 1 



F=~ (Fi'+ftO. 

c— 1 



The gauge-readings must be multiplied therefore by the 
factor 

n-1 

c.-l 

in order to obtain the correct values of the refracting powers. 

Suppose, for example, that the graduations on the dial cor- 
respond to a value c=1.54 and that the index of the lens 
to be measured is n=1.52. Then the value of the factor 
is 0.963; so that if the lens-gauge gives for the refracting 
power F the value 6.25 dptr., the correct value is obtained 
by multiplying this value by 0.963, that is, the correct 
value will be 6.02 dptr. 

109. Refraction of Paraxial Rays through a Thin Lens- 
System. — Let Mi' designate the position of a point conju- 
gate to an axial object-point with respect to an infinitely 
thin lens of refracting power F\, and let the point where 
the axis crosses the lens be designated by Ai. If the lens 
is surrounded by air, and if we put Wi = AiMi, wi' = AiMi', 
tfi = l/wi, tfi' = l/wi', 
then 

£/i'=£7i+Fi. 
If now at a point A 2 on the axis of the lens beyond Ai (such 
that the distance d = AiA 2 is measured in the direction in 
which the light is going) another infinitely thin lens is set 
up with its axis in the same straight line with that of the 
first lens, then Mi' may be regarded as an axial object-point 



290 Mirrors, Prisms and Lenses [§ 109 

M 2 with respect to the second lens; and if M 2 ' designates 
the position of the point conjugate to M 2 (or Mi') with re- 
spect to this lens, then also (supposing that the second lens 
is surrounded by air and that its refracting power is denoted 

by F 2 ), 

U 2 ' = U2+F 2 , 

where £7 2 = l/w 2 , UJ = 1/1*2', w 2 = A 2 M 2 = A 2 M/, w' 2 = A 2 M 2 '. 
Obviously, the point M 2 ' is the image-point conjugate to 
the axial object-point Mi with respect to the two lenses; 
so that regarding the system as a whole, we may write M, M' 
in place of Mi, M 2 ' and U, U' in place of Ui, U2, respectively. 
Now let us impose the condition that the two thin lenses 
are in contact with each other or that they are as close together 
as possible; in other words, that the axial distance d between 
the lenses is negligible. If this is the case, the points Ai, A 2 
are to be regarded as a pair of coincident points, and hence 

w=u 2] 

and, therefore, we may write now : 

UJ = U+F U U'=Ui'+F 2 . 
Eliminating Ui', we obtain : 

U^U+^+Ft); 
and if we put 

we have finally: 

U'=U+F. 
Since this formula is seen to be identical in both form and 
meaning with the formula for a single thin lens, it appears 
therefore that a combination of two thin coaxial lenses in 
contact is equivalent to a single lens of refracting power F equal 
to- the algebraic sum of the refracting powers F\ and F 2 of the 
component lenses. 

Theoretically, this rule can be applied to a centered system 
of any number of thin lenses in contact. Thus, the total re- 
fracting power of a thin lens-system will be 

F=Fl +F 2 + . . . +F m , 



F=F 1 +F 2 , 



§ 110] Prismatic Power of Thin Lens 291 

where the total number of lenses is denoted by m. This 
formula may be written : 

i=m 

F= 2 P it 

where F\ denotes the refracting power of the ith. lens. 

In the case of actual lenses placed together in this fashion 
it will always be a question, How far are we justified in 
neglecting the total thickness of the system? Two adjacent 
lenses may be placed in actual contact, but a third lens can- 
not be in contact with the first. Moreover, even when 
there are only two lenses, their outward forms may be such 
that it will not be possible to place them in tangential con- 
tact at their vertices, although they can always be made 
to touch at two points symmetrically situated with respect 
to their common axis. Attention is directed to this ques- 
tion chiefly in connection with the method of neutraliza- 
tion of lenses which is practiced extensively in the fitting 
of spectacle glasses. Two infinitely thin lenses of equal and 
opposite refracting powers are said to "neutralize" each 
other, because when they are placed in contact their total 
refracting power (F1+F2) is equal to zero. Strictly speak- 
ing, the neutralization of a negative glass by a positive glass 
implies not only that the focal lengths are equal in magni- 
tude but also that the primary focal point of one lens shall 
coincide with the secondary focal point of the other. Both 
of these conditions are realized in a combination of a plano- 
concave with a plano-convex lens fitted together so as to form 
a slab with plane parallel sides. But even with the relatively 
thin lenses employed in spectacles sensible errors may be 
introduced by assuming, as is usually done, that the con- 
dition Fi+F 2 = is the -sole or even the main consideration 
for neutralization. 

110. Prismatic Power of a Thin Lens. — Only such rays 
as go through the optical center (§ 88) emerge from a lens 
without being deviated from their original directions. The 
prismatic power of a thin lens, which, like the power of a 



292 



Mirrors, Prisms and Lenses 



[§110 




Fig. 141, a and b. — Prismatic power of infinitely thin lens, (a) Convex, 
(b) Concave lens. 

thin prism (§ 70), is measured by the deviation of a ray in 
passing through it, depends not only on the refracting power 
of the lens but also on the place where the ray enters the 
lens. In the accompanying diagram (Fig. 141, a and b) 



§ 110] Prismatic Power of Thin Lens 293 

the point A designates the axial point of a thin lens of re- 
fracting power F. A ray RB incident on the lens at B passes 
out in the direction BS. If M, M' designate the points 
where the incident and emergent rays cross the axis, then 
ZM'BM= e is the angle of deviation; and if = Z AMB de- 
notes the slope of the incident ray and 0' = Z AM'B denotes 
the slope of the emergent ray, evidently we have the rela- 
tion: 

e= 0- $'. 
The distance /i = AB of the incidence-point B from the axis 
of the lens or the incidence-height of the ray is called by 
the spectacle-makers the decentration of the lens. Since 
the decentration of an ophthalmic lens is always compara- 
tively small, the ray RB may be regarded as a paraxial 
ray, and hence we can put 6 and 6' in place of tan0 and 
tan#' and write: 

6=--=-h.U, $'=-- = -h.U', 

u u 

where u = AM, m' = AM', U=l/u, U' = 1/u', since the lens 

is supposed to be surrounded by air (n=l). Accordingly, 

e = h(U'-U), 
the deviation-angle e being expressed in radians if h, u and 
v! are all expressed in terms of the same linear unit. But 

U'-U=F; 
and hence 

e = h.F radians. 
In this formula the decentration h must be expressed in 
meters if the refracting power F is given in dioptries. The 
above relation may be derived immediately also from 
Fig. 142, where the incident ray RB is drawn parallel to 
the axis of the lens, so that in this case 6' + e = 0; and 

since tan 6' = 6' = =rr-r = -tt, = — t = — h.F. we obtain, as above, 
F'A /' / 

e — h.F. If a screen is placed perpendicular to the incident 
light coming in the direction RB, a spot of light will be pro- 
duced on the screen at the point N where the straight line 



294 



Mirrors, Prisms and Lenses 



[§110 



RB meets the screen; and if now a lens is interposed at a 
certain known distance from the screen, the deviation e can 
easily be determined by measuring the distance NL through 
which the spot of light is deflected. 

However, both the radian and the meter are inconven- 
iently large units for expressing the values of the small mag- 

£ 




Fig. 142. — Prismatic power of infinitely thin lens; incident ray parallel 

to axis. 

nitudes denoted by e and h. Opticians measure the devia- 
tion in terms of the centrad or in terms of the prism-dioptry, 
which in the case of small angles, as we have seen (§ 70), 
is practically the same unit as the centrad. If the angle 
of deviation expressed in centrads or prism-dioptries is de- 
noted by p, while e denotes the value of this angle in ra- 
dians, then 

p = 100 €. 
Moreover, if the decentration h is given in centimeters in- 
stead of in meters, we obtain the following formula: 

p = h.F; 
that is, the deviation (p) in prism-dioptries (or centrads) pro- 
duced by a thin lens in any zone is equal to the product of the 
refracting power (F) of the lens in dioptries by the radius (h) 
of the zone in centimeters; or as the opticians usually express 
it, the prismatic power of a thin lens in prism-dioptries is 



Ch. VIII] Problems 295 

equal to the product of the refracting power of the lens in 
dioptries by the decentration in centimeters. For example, 
a spectacle glass of refracting power 5 dptr. must be de- 
centered about 0.4 cm. or 4 mm. in order to have a pris- 
matic power of 2 prism-dptr. 

If in Fig. 142 the distance AP of the screen from the lens 
is 1 meter, the deflection LN in centimeters of the spot of 
light will be equal to the prismatic power of a lens of focal 
length /= AF' decentered by the amount /i = AB. 

PROBLEMS 

1. How is the curvature of a wave affected by reflection 
at a plane mirror? How is the curvature of a plane wave 
affected by reflection at a spherical mirror? 

2. The distance between a luminous point and the eye 
of an observer is 50 cm. A plate of glass (n=1.5), 10 cm. 
thick, is interposed midway between the point and the eye 
with its two parallel faces perpendicular to the line of vision. 
Spherical waves spreading out from the luminous point 
are refracted through the plate and into the eye. Find the 
curvature of the wave-front: (a) just before it enters the 
glass, (6) immediately after entering the glass, (c) im- 
mediately after leaving the glass, and (d) when it reaches 
the eye. 

Ans. (a) -5 dptr.; (6) — 3| dptr.; (c)-3| dptr.; (d) 
-2| dptr. 

3. What is the refracting power of a spherical refracting 
surface of radius 20 cm. separating air (n = l) from glass 
(n' = 1.5)? 

Ans. +2.5 dptr. or —2.5 dptr., according as the surface 
is convex or concave, respectively. 

4. If the cornea of the eye is regarded as a single spheri- 
cal refracting surface of radius 7.7 mm. separating air (n=l) 
from the aqueous humor (n' = 1.336), what is its refracting 
power? Ans. 43.6 dptr. 



296 Mirrors, Prisms and Lenses [Ch. VIII 

5. Using the data of the preceding problem, find the re- 
fracting power of the cornea when the eye is under water 
(n = 1.33). Ans. Nearly 0.78 dptr. 

6. What is the reflecting power of a concave mirror of 
radius 20 cm. when the reflecting surface is in contact with 
(a) air (n=l) and (6) water (n=-J)? 

Ans. (a) 10 dptr.; (b) 13.33 dptr. 

7. A convex spherical surface of radius 25 cm. separates 
air (n=l) from glass (V = 1.5). Find the refracting power 
and the reflecting power of the surface. 

Ans. Refracting power is +2 dptr.; reflecting power is 
- 8 dptr. 

8. The reflecting power of a spherical mirror in contact 
with air is +2 dptr. Determine the form of the mirror. 

Ans. A concave mirror of radius 1 meter. 

9. A spherical mirror is in contact with a liquid of re- 
fractive index n. If the reflecting power of the mirror is 
+2 dptr., show that the mirror is a concave mirror of radius 
n meters. 

10. The index of refraction of carbon bisulphide is 1.629. 
What is the reflecting power of a concave mirror of radius 
25 cm. in contact with this liquid? Ans. +13.032 dptr. 

11. What is the refracting power of a thin symmetric 
convex lens made of glass of index 1.5, if the radius of cur- 
vature of each surface is 5 cm.? Ans. +20 dptr. 

12. The refracting power of a thin plano-convex lens 
made of glass of index 1.5 is 20 dptr. Find the radius of 
the curved surface. Ans. 2.5 cm. or nearly 1 inch. 

13. A thin convex meniscus lens is made of glass of in- 
dex 1.5. The radius of the first surface is 10 and that of the 
second surface is 25 cm. Assuming that the lens is sur- 
rounded by air (n= 1), find its refracting power. 

Ans. +3 dptr. 

14. If the lens in the preceding example were made of 
water of index ~, what will be its refracting power? 

Ans. +2 dptr. 



Ch. VIII] Problems 297 

15. If the first surface of the lens in No. 13 were in con- 
tact with water (fti = |) and the second surface in contact 
with air (n 3 = 1), what will be the refracting power? 

Ans. — ■§- dptr. 

16. If the first surface of the lens in No. 13 were in con- 
tact with air (fti=l) and the second surface in contact with 
water (n 3 = 4), what will be the refracting power? 

Ans. +4^ dptr. 

17. In examples 13, 14, 15 and 16 suppose the lens were 
reversed so that the opposite face was turned to the inci- 
dent light. What would be the answers to these problems 
then? 

Ans. The same answers would be obtained for Nos. 13 
and 14; but the answers for Nos. 15 and 16 would be inter- 
changed. 

18. Show that the lateral magnification in a spherical 
mirror, a spherical refracting surface or an infinitely thin lens 
is equal to the ratio of the reduced " vergences " U and U'. 

19. Describe the spherometer and the lens-gauge and 
explain their principles. 

20. Show how a plane wave is refracted through a thin 
lens, and derive from a diagram for this case the formula 
for the refracting power. 

21. Show how a plane wave is refracted through a thin 
prism, and derive the formula for the deviation in terms of 
the refracting angle of the prism and the relative index of 
refraction. 

22. The refracting power of a thin lens is +6 dptr. It 
is made of glass of index 1.5 and surrounded by air (w=l). 
If the radius of the first surface is +10 cm., what is the 
radius of the second surface? Ans. r 2 = — 50 cm. 

23. A convex lens produces on a screen 14.4 cm. from 
the lens an image which is three times as large as the object. 
Find the refracting power of the lens. Ans. 27.78 dptr. 

24. A lens-gauge graduated in dioptries for glass of in- 
dex 1.5 is used to measure a thin double convex lens made 



298 Mirrors, Prisms and Lenses [Ch. VIII 

of glass of index 1.6. The readings on the dial give +4 for 
both surfaces. Find the refracting power of the lens, assum- 
ing that its thickness is negligible. Ans. +9.6 dptr. 

25. Modern spectacle glasses are meniscus lenses with 
the concave surface worn next the eye. If the glass is to 
give the proper correction, it is very important for it to be 
adjusted at a certain measured distance from the eye. In 
determining this distance it is necessary to ascertain the 
" vertex depth" of the concave surface, that is, the perpen- 
dicular distance (t) of the vertex from the plane of the edge 
or contour of the surface. If the diameter of this contour 
expressed in millimeters is denoted by 2h, and if the refract- 
ing power of the surface next the eye, expressed in dioptries, 
is denoted by F 2 , and, finally, if the index of refraction of 
the glass is denoted by n, show that the vertex depth of the 
surface is approximately: 

t = _ 0.0005 -^4 millimeters. 
n—1 

26. What is the refracting power of a lem which is equiva- 
lent to two thin convex lenses of focal lengths 15 and 30 cm., 
placed in contact? Ans. 10 dptr. 

27. A concave lens of focal length 12 cm. is placed in 
contact with a convex lens of focal length 7.5 cm. Find 
the refracting power of the combination. Ans. 5 dptr. 

28. The refracting power of a thin concave lens is 5 times 
that of a thin convex lens in contact with it. If the focal 
length of the combination is 8 cm., find the refracting power 
of each of the components. Ans. — 15 § and +3 § dptr. 

29. Two thin lenses, made of glass of indices 1.5 and 1.6, 
are fitted together with the second surface of the first lens 
coincident with the first surface of the second lens (rz = r 2 ). 
The radii of the surfaces are all positive and equal to 4, 11 
and 6 cm. taken in the order named. Find the refracting 
power of the combination. Ans. 12.5 dptr. 

30. What is the prismatic effect of a lens of power +4 dptr. 
decentered 0.75 cm.? Ans. 3 prism-dioptries. 



Ch. VIII] Problems 299 

31. Two thin convex lenses have each a focal length of 
1 inch. Find the position of the second focal point of the 
combination of these two lenses when they are placed with 
their axes in the same straight line: (a) when they are in 
contact, (b) when they are separated by 1.5 inches, and 
(c) when they are separated by 3 inches. Draw a diagram 
for each case showing the path of a beam of light coming 
from a distant axial object-point. 

Ans. (a) Half an inch beyond the combination; (b) be- 
tween the lenses and 1 inch from second lens; (c) 2 inches 
beyond second lens. 

32. A convex lens of focal length 20 cm. and a concave 
lens of focal length 5 cm. are placed 16 cm. apart. Find the 
positions of the focal points of the combination. 

Ans. One of the focal points is 420 cm. from the convex 
lens and 436 cm. from the concave lens; and the other focal 
point is 36 cm. from the convex lens and 20 cm. from the 
concave lens. 

33. How much must a lens of 5 dptr. be decentered in 
order to produce a deviation of 3° 307 Ans. 1.22 cm. 

34. The radius of a spherical surface is measured by a 
spherometer and found to be 14.857 cm. Measured by a 
lens-gauge the reading is 3.5 dptr. What is the index of re- 
fraction of the glass for which the readings on the dial of the 
gauge have been calculated? Ans. 1.52. 

35. The radii of each surface of a thin symmetric double 
convex glass lens is 6 inches. The lens is supported with 
its lower face in contact with the horizontal surface of still 
water. Assuming that the sun is in the zenith vertically 
above the lens, and that its apparent diameter is 30', find 
the position and size of the sun's image. (Take the indices 
of refraction of air, glass and water equal to 1, f and f, 
respectively.) 

Ans. A real image 12 inches below the surface of the water, 
0.0785 inch in diameter. 



CHAPTER IX 

ASTIGMATIC LENSES 

111. Curvature and Refracting Power of a Normal Sec- 
tion of a Curved Refracting Surface. — The refracting power 
(F) of a spherical surface is proportional to the curvature 
(R) of the surface, that is, F=(?i'—ri)R, where n and n' 
denote the indices of refraction of the media on opposite 
sides of the surface (§ 105). A spherical surface has the 
same curvature in every meridian, and hence also its re- 
fracting power is uniform, so that the refracted rays in 
one meridian plane are brought to the same focus as those 
in another meridian plane. But the surfaces of a lens are 
not always spherical (§ 87), and therefore, in order to ascer- 
tain what happens when a narrow bundle of rays is inci- 
dent perpendicularly on a curved reflecting or refracting 
surface of any form, we must investigate the reflecting or 
refracting power in different sections of the surface; and 
this means that we must investigate the curvature of these 
sections. In general, this is a problem of some difficulty 
and involves a more or less extensive knowledge of the 
theory of curved surfaces and the methods of infinitesimal 
geometry. No attempt can be made to explain this theory 
here, but for the student who is not already familiar with 
it, certain general definitions and propositions of geometry 
which have a direct bearing on the optical problems to be 
treated in this chapter will be stated as succinctly as pos- 
sible. 

The normal to a curved surface at any point is a straight 
line drawn perpendicular to the tangent plane at that point. 
The curved line which is traced on the surface by a plane 
containing the normal at a point A of the surface is called 

300 



§111] 



Normal Sections of Curved Surface 



301 



a normal section through this point. The normal sections 
of a sphere, like the meridians of longitude of the earth (as- 
sumed to be a perfect sphere), are all great circles of the 
sphere, and their curvatures are equal. But, generally, 




Fig. 143. — Normal sections of curved surface: xAy and xAz planes of 
principal sections; xAP plane of oblique normal section. 

the curvatures of the normal sections through a point on 
a curved surface will vary from one section to the next; so 
that if we imagine a plane containing the normal to be turned 
around this line as axis, we shall find that for one special 
azimuth of this revolving plane the curved line which it 
carves out on the surface will have the greatest curvature, 
and that then as the plane continues to revolve the curva- 
ture of the section decreases and reaches its least value for 
an azimuth which is exactly 90° from that for which the 
curvature was greatest. Thus, for example, in a cylindri- 
cal surface the curvature at any point is least and equal to 
zero in a normal section whose plane is parallel to the axis 
of the cylinder, and it is greatest in a normal section made 



302 Mirrors, Prisms and Lenses [§111 

by a plane perpendicular to the axis. At each point A of a 
curved surface the normal sections of greatest and least curva- 
tures lie always in two perpendicular planes, which are called 
the planes of the principal sections of the surface at A. The 
lines of intersection of these planes with each other and 
with the tangent plane at A may be chosen as the axes 
of reference of a system of rectangular coordinates x, y, z 
whose x-axis is the normal Ax (Fig. 143). The centers of 
curvature of the principal sections made by the xy-plsme 
and the £2-plane will be designated by C y and C z , respec- 
tively; and the curvatures of the principal sections will be 
denoted by R y and R z , so that if r y = AC y and r z = AC z 
denote the principal radii of curvature of the surface at 
the point A, we must have here (§ 99) R y = l/r y and R 2 

Now there is a remarkable geometrical relation between 
the curvature of any normal section at A and the curvatures 
of the principal sections of the surface at this point which 
will be stated also without giving the proof. Let a plane 
containing the normal Ax intersect the tangent plane (or 
2/2-plane) in the straight line AP (Fig. 143) and put ZyAP 
= 0. The center of curvature of the normal section made 
by this plane lies also on the normal Ax at a point which 
may be designated as Gg, so that the radius of curvature 
is ACo = re, and the curvature itself is He=lfro. The con- 
nection between Re and the principal curvatures R y and 
R z is expressed by the following formula: 
Re = R y .cos 2 d+R z .sm 2 0, 
where denotes the angle which the normal section makes 
with the xy-pl&ne. 

In a normal section at right angles to the first we should 
have, therefore, 

Re+w = R y .cos 2 ( 0+9O°)+# z .sin 2 ( (9+90°), 
or, since cos( (9+90°) = - sin 0, sin( 0+90°) =cos 0, 

#0+9O° = -R y .sin 2 0+# z cos 2 0. 



§ ill] Normal Sections of Curved Surface 303 

Adding the curvatures Re and Rd+<d0°, we obtain the rela- 
tion: 

Rd+Re+9o°=Ry+R z ; 
that is, the algebraic sum of the curvatures of any two normal 
sections intersecting each other at right angles at a point on 
a curved surface has a constant value, which is equal to the 
algebraic sum of the principal curvatures at this point. 

These theorems concerning the curvatures of the normal 
sections at a point of a curved surface are due to the great 
mathematician Euler (1707-1783), who made notable con- 
tributions also to the theory of optics. 

Since, therefore, the curvature of a surface at the point A 
varies from one azimuth to another as has just been ex- 
plained, the power of a refracting surface will vary in 
exactly the same way. Accordingly, the principal sections 
for which the curvature of a refracting surface has its great- 
est and least values (R y , R z ) are also the sections at this 
place of greatest and least refracting powers (F y ,F z ), because 

F y = (n'-n)R y , F y =(n'-ri)R z . 
The refracting power at this place in an oblique normal 
section which is inclined to the xy-plsaie at an angle 6 will be : 

Fd={n f ~n)Re; 
and the relation between Fe and F y , F z is given by the 
formula: 

F0 = F y .cos 2 0+F z .sin 2 0; 
and moreover: 

Fe+Fe+90 o =F y +F z ; 
that is, the algebraic sum of the refracting powers in any two 
normal sections through a point on a curved refracting sur- 
face is constant and equal to the algebraic sum of the princi- 
pal refracting powers. 

For example, in Fig. 144, let A designate a point of a 
curved refracting surface, and let the normal at this point 
be represented by the straight line Ax, which in accordance 
with the preceding discussion is to be taken as the z-axis 
of a system of rectangular coordinates with its origin at A. 



304 



Mirrors, Prisms and Lenses 



[§111 



The y-Sixis is represented by a straight line drawn in the 
plane of the paper perpendicular to Ax. The plane of the 
paper represents the plane of one of the principal sections, 
whereas the £2-plane at right angles to this plane represents 




Fig. 144. — Chief ray of narrow bundle of rays normal to curved re- 
fracting surface: Principal sections xAy, xkz; tangent plane ykz. 

the plane of the other principal section. The tangent-plane 
at A is represented by the 2/2-plane perpendicular to the 
normal. Consider now a narrow bundle of rays which pro- 
ceeding from a point M on the normal are incident on the 
curved refracting surface at points which are all very close 
to A. This point M may be designated also by M y or by 
M z according as it is regarded as lying in the one or the other 
of the two principal sections; or it may be designated also 
by Me if it is to be considered as lying in an oblique normal 
section which is inclined to the ^-plane at an angle 6. The 
chief ray of the bundle is the ray which coincides with the 
normal to the surface at A and which proceeds therefore 
into the second medium without being deviated. A plane 
containing this chief ray will cut out from the bundle a pen- 
cil of rays which will be refracted at points of the surface 
which lie in a normal section. The pencil of rays proceed- 
ing from M y in the xy-pleme will be refracted to a point M y ', 
while the pencil of rays proceeding from M z will be refracted 
to a point M z '; and, in general, these points M y ' and M/ will 
be two different points on the normal Ax. Now if U yt U y ' 
denote the reduced "vergences" (§ 106) of the pair of conju- 



§ 112] Surfaces of Revolution 305 

gate points M y , M y ' in one principal section; and, similarly, 
if U z , U z denote the reduced '' vergences" of the pair of con- 
jugate points M z , M z ' in the other principal section, evi- 
dently we shall have the following relations: 

U y '=U y +F y , U Z '=U Z +F Z . 

Similarly, also, a pencil of rays proceeding from Me and 
meeting the refracting surface at points in an oblique nor- 
mal section will be refracted to a point M0' which will lie 
on Ax between M/ and M/, so that 
U0'=Ue+Fe. 

If the bundle of incident rays is homocentric, that is, if 
the points designated by M y , M z and M0 are all coincident, 
then U y =U z =Ud=U. The peculiarity of the imagery 
consists in the fact that instead of obtaining a single image- 
point M' corresponding to an object-point M, as in the case 
of a spherical refracting surface, we find here a whole se- 
ries of such points lying on the segment M y 'M z ' of the nor- 
mal Ax. This will be explained more fully in § 113. 

112. Surfaces of Revolution. Cylindrical and Toric 
Surfaces. — The curved reflecting and refracting surfaces 
of optical mirrors and lenses are almost without exception 
surfaces of revolution, that is, surfaces generated by the revo- 
lution of the arc of a plane curve around an axis in its plane. 
Accordingly, it is desirable to call attention to some of the 
special properties of these surfaces. The curve traced on 
a surface of revolution by a plane containing the axis of 
revolution is called a meridian section. The normals to the 
generating curve are also normals to the surface; and since 
the normal at any point of the surface lies in the meridian 
section which passes through that point, it follows that the 
normals to a surface of revolution all intersect the axis of 
revolution. 

The two principal sections at any point of a surface of 
revolution are the meridian section which passes through 
that point and the normal section which is perpendicular 
to the meridian section. The center of curvature of the 



306 



Mirrors, Prisms and Lenses 



112 



latter principal section lies on the axis of revolution at the 

point where the normal crosses it. 

Not only are the surfaces of mirrors and lenses generally 

surfaces of revolution, but usually they are very simple types 

of such surfaces. A spher- 
ical surface may be consid- 
ered as generated by the. 
revolution of a circle 
around one of its diame- 
ters. The other chief 
forms of reflecting and re- 
fracting surfaces are cyl- 
indrical and toric surfaces, 
which are also compara- 
tively easy to grind. 

A cylindrical surface of 
revolution is generated by 
the revolution of a straight 
line about a parallel straight 
line as axis, called the axis 
of the cylinder. A meridian 
section of a cylinder at a 
point A on the surface 
(Fig. 145) will be a straight 
line of zero curvature, 
whereas the other principal 
section at right angles to 

Fig 145 -Refracting power of cylin- th axig f th cylinder ^U 
drical surface: Principal sections ° 

made by planes Ay and Kz; oblique be the arc of a Circle 

section AP. whose curvature is R = l/r, 

where r denotes the radius of the cylinder. If the i/-axis 
is drawn parallel to the cylinder-axis, then R y = 0, R Z = R; 
and hence according to Euler's formula given in §111, 
the curvature in an oblique normal section AP inclined to 
the axis of the cylinder at an angle 6 will be 
Re=R.sm 2 6. 




112] 



Cylindrical Refracting Surface 



307 



This result may be obtained also independently by observ- 
ing that although the arcs Az and AP in Fig. 145 have the 
same sagitta (§ 99), their chords denoted by 2h and 2he are 
unequal in length, because h = hd.smd. Now the curva- 
tures of two arcs having the same sagitta are inversely pro- 
portional to the squares of their chords; consequently, 

R he*' 
and hence 

Re = R.sin 2 d, 
exactly as above. Moreover, in a normal section perpen- 




Fig. 146. — Principal sections of toric surface. 



dicular to the section AP, we find, by writing (0+90°) in 
place of 6, 

R0+CjQ° = R.cos 2 6; 
and therefore 

Re-\-Re+V0 o =R' 
Accordingly, in the case of a cylindrical refracting sur- 
face, if the maximum refracting power is denoted by F> 
the refracting power in an oblique section inclined to the 



308 



Mirrors, Prisms and Lenses 



[§112 



axis at an angle 6 will be F.sin 2 6, and in a section at right 
angles to this F.cos 2 6. The refracting power F of a cylin- 
drical refracting surface may, therefore, be considered as 
in a certain sense capable of resolution into a refracting 
power F.sin 2 6 in one oblique section and a refracting power 







Fig. 147, a and b. — Toric surfaces (reproduced from Prentice's Ophthalmic Lenses 
and Prisms by permission of the author) . 

F.cos 2 6 in a section at right angles to the first; and since 

F0+Fd+9O°=F, 
we can say that the algebraic sum of the refracting powers in 
any two mutually perpendicular sections of a cylindrical re- 
fracting surface is constant and equal to the maximum refract- 
ing power. 

A toric or toroidal surface (so-called from the architect- 
ural term torus applied to the molding at the base of an 
Ionic column) is a surface shaped like an anchor-ring which 



§ 112] Toric Surfaces and Lenses 309 

is generated by the revolution of a conic section around an 
axis which lies in the plane of the generating curve but does 
not pass through its center. The surface of an automobile 
tyre is a toric surface, being generated by the revolution 
of the circular cross-section of the tyre around an axis per- 



Fig. 148, a and b. — Principal sections of toric lenses (reproduced from 
Prentice's Ophthalmic Lenses and Prisms by permission of the author). 

pendicular to the plane of the wheel at its center. Toric 
refracting surfaces are generated always by the revolution 
of the arc of a circle (Fig. 146). The arcs of the two prin- 
cipal sections of a toric surface of a lens bisect each other 
at the vertex A of the surface, so that the normal Ax is an 
axis of symmetry. If the axis of revolution is parallel to 
the 2/-axis of the system of rectangular coordinates, the 
center of the meridian section through A is at the center 
C y of the generating circle, whereas the center of the other 
principal section at A is at the point of intersection C z of 
the normal Ax with the axis of revolution. 

The diagrams, Fig. 147, a and b (which are copied from 



310 



Mirrors, Prisms and Lenses 



[§H3 



the beautiful drawings of Mr. Prentice in his valuable 
and original essay on " Ophthalmic Lenses and Prisms' ' in 
the American Encyclopaedia of Opthhalmology) show the two 
principal forms of toric surfaces. The principal sections of 
some types of toric lenses are indicated in Fig. 148, a and b. 

A cylindrical surface of revolution may be considered as 
a special form of toric surface by regarding the segment of 
the generating straight line as the arc of a circle with an 
infinite radius. 

113. Refraction of a Narrow Bundle of Rays incident 
Normally on a Cylindrical Refracting Surface. Sturm's 
Conoid. — In order to obtain a clear idea of the character 
of a bundle of rays refracted at a cylindrical surface or 
through a thin cylindrical lens, suppose, by way of illustra- 




Fig. 149. — Chief ray of narrow bundle meets cylindrical refracting surface 
normally; astigmatic bundle of refracted rays. Principal sections xAy 
and xAz. 



tion, that we consider a special case of the problem which we 
had in § 111 in connection with Fig. 144, namely, the case in 
which a narrow homocentric bundle of incident rays, origi- 
nally converging towards a point M, is intercepted before it 
reaches this point by being received on a cylindrical refract- 
ing surface which is placed so that the chief ray of the bundle 
meets the surface normally at a point A and proceeds, there- 
fore, along the normal Ax (Fig. 149) without being deflected. 



§113] Astigmatic Bundle of Rays 311 

For convenience of delineation, the cylindrical surface is 
represented in the figure as the first surface of an infinitely 
thin piano-cylindrical lens, but the explanation is not es- 
sentially affected by the fact that it applies to a bundle of 
rays which have undergone also a second refraction at the 
plane face of the lens. The bundle of incident rays is not 
represented in the figure. The point where the chief ray 
meets the lens is designated by A. In the drawing this point 
A is marked on the second or plane face of the lens, but since 
the lens is supposed to be infinitely thin, this point may be 
regarded also as lying on the first face. The plane of the 
paper represents the meridian section of the cylindrical sur- 
face through the vertex A, and hence the axis of the cylinder 
is in this plane and parallel to the straight line Ay perpendic- 
ular to Ax in the meridian or xy-pl&ne. This meridian plane 
is one of the principal sections at the vertex A of the cylin- 
drical surface; whereas the other principal section is the 
zz-plane at right angles to the plane of the paper. The 
bundle of rays is cut by these principal sections in a pencil 
of meridian rays lying in the meridian xy-pl&ne and a pencil 
of sagittal rays (named by analogy with the so-called " sagittal 
suture" in anatomy) lying in the xz-pl&ne; the chief ray of 
the bundle being common to both of these pencils, since it 
is the line of intersection of the two principal sections of the 
bundle. Now the meridian rays traversing the infinitely 
thin cylindrical lens in a section containing the axis of the 
cylinder will be entirely unaffected in transit and will pro- 
ceed therefore to the point M just as though the thin piece 
of glass had not been interposed in the way; so that this 
point regarded now as the point of rendezvous, so to speak, 
of the meridian rays after they have passed through the 
lens may also be designated by M y ', as in fact it is marked 
in the diagram. On the other hand, the rays of the sagittal 
pencil meet the surface in points lying on the arc of the sec- 
tion made by the zz-plane, and the rays in this plane are 
refracted just as they would be through a piano-spherical 



312 Mirrors, Prisms and Lenses [§ 113 

lens of the same curvature as that of the cylinder; and ac- 
cordingly after passing through the lens they will be brought 
to a focus at a point M z ' on the chief ray Ax, which in the 
case here supposed will be between the lens and the point 
M y ', as represented in the figure. 

The bundle of rays after refraction is no longer homocentric, 
so that an object-point is not reproduced in a cylindrical 
lens by a single image-point or even by a pair of image-points, 
since only the meridian and sagittal image-rays intersect 
in the so-called image-points M y ' and M/, respectively. 
Under such circumstances, the bundle of image-rays is said 
to be astigmatic (or without focus), which, in fact, is the 
general character of a bundle of optical rays, as will be 
further explained in Chapter XV. 

Rays which are incident on the cylindrical surface in an 
oblique section made by a plane containing the normal Ax 
will be brought to a focus at a point lying between M y ' and 
M z ', as explained in § 111. But the two points M y ' and M z ' 
have a superior right to be regarded as the image-points of 
the astigmatic bundle of rays, not only because they are 
the image-points of the two principal pencils of the bundle, 
but also because the so-called image-lines of the astigmatic 
bundle of rays are located at these places, as we shall pro- 
ceed to show. 

Imagine a straight line drawn on the surface of the cylin- 
der parallel to the i/-axis and at a short distance from the 
zy-plane, and consider the pencil of rays which meet the 
surface in points lying along this line; these rays after pass- 
ing through the lens will meet in a point in the zz-plane a 
little to one side of the image-point M y '; and the assemblage 
of these image-points will form a very short image-line per- 
pendicular to the meridian section of the bundle of rays at 
the point M/; just as though the pencil of meridian rays 
had been rotated through a very small angle around an 
axis parallel to the y-a,xis and passing through M z .' And, 
similarly, if the pencil of sagittal rays is rotated slightly 



113] 



Sturm's Conoid 



313 



on both sides of the zz-plane around an axis parallel to the 
2-axis and passing through the image-point M/, the image- 
point M z ' will trace out a little image-line perpendicular to 
the sagittal section of the astigmatic bundle of rays. Thus, 
instead of a point-like image of a point-like object or point- 
to-point correspondence between object and image, that is, 
instead of the so-called punctual imagery which we have 
when paraxial rays are reflected or refracted at a spherical 
surface, we obtain here something essentially different; for 
in this case each point of the object is reproduced by two 



A 




l2 ^"~-~---«^^^ 




x 


y 


o 


Z 3 A 5""^ 

- ooQ 


_ 7 





Fig. 150. — Sturm's conoid. 

tiny image-lines, each perpendicular to the chief ray of the 
bundle, one in one principal sectio?i and the other in the other 
principal section; so that if one of the image-lines is vertical, 
the other will be horizontal. The image-line which passes 
through the image-point of the meridian rays lies in the 
plane of the sagittal section, and vice-versa. 

The case in which an object-point is reproduced by two 
short image-lines is the simplest form of astigmatism, and 
it is only under exceptionally favorable circumstances that 
it can be actually realized as described above. The astig- 
matic bundle of rays represented in Fig. 150, which is com- 
pletely symmetrical in the two principal sections is known 
as Sturm's conoid after the celebrated mathematician who 
appears to have been the first to make a systematic investi- 
gation (1838) of the characteristics of a narrow bundle of 



314 



Mirrors, Prisms and Lenses 



[§114 



optical rays. If the lens-opening is determined by a small 
circular stop in a plane at right angles to the optical axis 
(or rr-axis) and with its center on this axis, the transverse 
sections of the astigmatic bundle of refracted rays made by 
planes perpendicular to the chief ray (that is, parallel to 
the 2/2-plane) will be ellipses with their major axes parallel 
to the i/-axis in one part of the bundle and parallel to the 
2-axis in the other part. These elliptical sections become 
narrower and narrower as they approach either of the image- 
lines, at both of which places the elliptical section collapses 
into the major-axis of the ellipse. At some intermediate 
point between the two image-lines the section of the bundle 
will be a circle (the so-called " circle of least confusion"). 
114. Thin Cylindrical and Toric Lenses. — Optical lenses 
may now be classified in two principal groups, namely, 
anastigmatic (or simply stigmatic) lenses and astigmatic lenses, 
according as the imagery produced by the refraction of par- 




es, Concave. 




b, Convex. 
Fig. 151, a and b. — Piano-cylindrical lenses. 

axial rays through the lens is punctual imagery or not (§ 113). 
Anastigmatic lenses are single focus lenses, whereas astig- 
matic lenses may be said to be double focus lenses. The 
essential requirement is that the optical axis of the lens, 



114] 



Cylindrical Lenses 



315 



which is generally an axis of symmetry, shall meet both 
faces normally (§ 87) ; and another condition that must 
always be fulfilled in an actual lens is that the planes of the 
principal sections at the vertex of the first surface shall also 
be the planes of the principal sections at the vertex of the 
second surface. Astigmatic lenses are generally cylindrical 
or toric. 

Cylindrical lenses are made in three forms, namely, piano- 
cylindrical (one surface cylindrical and the other plane, 




Fig. 152. — Sphero-cylindrical lens. 




Fig. 153. — Sphero-cylindrical lens. 

Fig. 151, a and 6), cross-cylindrical (both surfaces cylindrical, 
the axes of the cylinders being at right angles), and sphero- 
cylindrical (one surface cylindrical and the other spherical, 
Figs. 152 and 153). All of these forms are quite common in 
modern spectacle glasses, but prior to 1860 cylindrical lenses 
were hardly employed at all. The first scientific use of a 
cylindrical lens seems to have been made by Fresnel 
(1788-1827) in 1819 for the purpose of obtaining a luminous 
line. In 1825 Sir George Airy (1801-1892), afterwards the 
distinguished astronomer-royal at Greenwich, employed a con- 



316 Mirrors, Prisms and Lenses [§ 114 

cave sphero-cylindrical glass to correct the myopic astigma- 
tism of one of his eyes. But it was not until Donders 
(1818-1889) published his treatise on astigmatism and cyl- 
indrical glasses in 1862 that their importance began to be 
recognized by ophthalmologists all over the world. 

In a toric lens usually only one of the surfaces is toric 
(§ 112), while the other is plane or spherical. The diagrams, 
Fig. 147, a and b, and Fig. 148 show the principal types of 
toric lenses. 

Let Fy t i, Fy t 2 and F Zi i, F Zy 2, denote the refracting powers 
of the two surfaces of an astigmatic lens in the xy-plsaie and 
zz-plane, respectively, which are the planes of the principal 
sections of the thin lens with respect to its optical center A. 
Now the total refracting power (F) of a thin lens was found 
(§ 105) to be equal to the algebraic sum (F1+F2) of the 
powers of the two surfaces of the lens; so that applying this 
formula to an astigmatic lens, we obtain for the refracting 
power in the two principal sections : 

F y = F y< i-\-Fy t 2, F z = F Zt \-\-F z ^ 

In each of the following special cases the lens is supposed 
to be surrounded by the same medium (n) on both sides, 
while the index of refraction of the lens itself is denoted by n'. 

(1) Consider, first, the case of a piano-cylindrical lens, 
which in a principal section containing the axis of the cylin- 
der acts, as was remarked (§ 113), like a slab of the same 
material with plane parallel faces; whereas in the other prin- 
cipal section the effect is the same as that of a piano-spherical 
lens of the same radius (r) as that of the cylinder. If the 
axis of the cylinder is parallel to the y-axis, and if the plane 
surface is supposed to be the second surface, we shall have 
in this case : 

F y ,i = F y< 2 = F z< 2 = 0, 
and, consequently: 

F y = 0, F z = F Ztl = F=(n'-n)R, 
where F denotes the maximum refracting power of the cylin- 
drical surface, and R = l/r denotes its curvature. 



§ 114] Cylindrical and Toric Lenses 317 

If M designates the position of an object-point lying on 
the optical axis (z-axis) of a thin piano-cylindrical lens, and 
if M#' designates the position of the corresponding image- 
point produced by the refraction through the lens of the 
rays which lie in the plane of a normal section inclined at 
an angle 6 to the axis of the cylinder; and if we put 

AM = u, AMe' = u', U = n/u, Ud' = n/u e ', 
then 

Ue'= U+Fe, where Fe = F. sin 2 6; 
and for the two principal sections: 

U y '=U, U Z '=U+F. 

(2) In a cross-cylindrical lens the axes of y and z are par- 
allel to the axes of the cylinders. Assuming that the cylin- 
drical axis of the first surface of the lens is parallel to the 
2/-axis, we have for a thin lens of this form : 

F y = F y , 2 =-(n f -n)R 2 , F z = F Zil =(n' '-n)R h 
F=(n'-n)(R 1 .sm 2 d -ft.cos 8 0); 
where R\, R 2 denote the maximum curvatures of the cylin- 
ders and Fe denotes the refracting power in a section in- 
clined at an angle 6 to the axis of the first surface. 

(3) In a thin sphero-cylindrical lens, if we suppose, for 
example, that the axis of the cylindrical surface is parallel 
to the ?/-axis and that this surface is also the first surface 
of the lens, then 

F y ,i = 0, F y , 2 =F z , 2 = F 2 , 

F y = F y , 2 =-(n f -n)R 2 , 
F z = F z , 1 +F y = (n f -n)(R l -R 2 ), 
Fe=(n'-n)(R 1 .sm 2 d-R 2 ); 
where R\, R 2 denote the maximum curvatures of the cylin- 
drical and spherical faces, respectively, and Fe denotes the 
refracting power of the combination in a plane inclined at 
an angle 6 to the axis of the cylinder. 

(4) Consider, finally, a thin toric lens, whose second face 
may be supposed to be spherical, so that if r 2 denotes 
the radius of this surface, its refracting power will be 



318 Mirrors, Prisms and Lenses [§ 115 

'Ft— — {n' — n)R 2} where R 2 = lfr 2 . Then if R yyh R z ,i denote 
the principal curvatures of the toric surface, the refracting 
powers of the lens will be 

F y =(n'-n) (R y , 1 -R 2 ) ) F z = {n'-n) (R z ,i-Rt), 
Fe= (n'-n) (# y ,i.cos 2 d-\-R z ,i.sin 2 0-R 2 ). 
115. Transposing of Cylindrical Lenses. — The orientation 
of a cylindrical refracting surface is described by assigning 
the value of the angle <p which the axis of the cylinder makes 
with a fixed line of reference. In a cylindrical spectacle 
glass this line of reference is a horizontal line usually imag- 
ined as drawn from a point opposite the center of the pa- 
tient's eye either towards his temple or towards his nose; 





18CT x \ y 1 0' isoj 

TEMPLE NOSE NOSE TEM.P.LE 

Fig. 154. — Mode of reckoning axis of cylindrical eye-glass. 

and the angle through which this line has to be rotated in 
a vertical plane in order for it to be parallel to the axis of 
the cylinder is the angle denoted by <p. In England and 
America it is customary to imagine the horizontal line of 
reference as drawn from the center of the glass towards that 
temple of the patient which is on the right-hand side of an ob- 
server supposed to be adjusting the glass on the patient's 
eye; so that for a glass in front of either eye the radius vector 
is supposed to rotate in a counter-clockwise sense from 0° 
to 180°, as represented in Fig. 154. A different plan was 
recommended by the international ophthalmological con- 
gress which met in Naples in 1909, whereby the angle cp was 
to be reckoned from an initial position of the radius vector 
drawn horizontally from a point opposite the center of the 
eye towards the nose. According to this plan, the sense of 



§ 115] Transposing of Cylindrical Lenses 



319 



rotation will be clockwise for one eye and counter-clockwise 
for the other eye, as represented in Fig. 155. 

A sphero-cylindrical glass is described in an ophthalmo- 
logical prescription by giving the refracting power P of 
the cylindrical component and the refracting power Q of 
the spherical component, together with the slope <p of the 
axis of the cylinder, in a formula which is usually written 
as follows: 

Q sph. 3 P cyl., slx.<p, 
where the symbol O means "combined with." 

Opticians speak of transposing a lens when they substi- 
tute a glass of one form for an equivalent glass of another 



180 





180° 



TEMPLE NOSE NOSE TEMPLE 

Fig. 155. — Mode of reckoning axis of cylindrical eye-glass. 



form. All that is necessary for this purpose is to see that 
the powers of the lens in the two principal sections remain 
the same as before. The following rules for transposing 
cylindrical lenses may be useful : 

(1) To transpose a sphero-cylindrical lens into another 
sphero-cylindrical lens or into a cross-cylindrical lens: 

A lens given by the formula Q sph. O P cyl., ax. <p is 
equivalent to either of the following combinations : 

a. Sphero-cylinder: (P+Q) sph. C -P cyl., ax. (<p ±90°) 

b. Cross-cylinder : (P+ Q) cyl. , ax. <p O Q cyl. , ax. ( <p ± 90°) . 
The power of the spherical component in the original com- 
bination is Q dptr. in both principal sections, and the power 
of the cylindrical component is P dptr. in the section which is 
inclined to the line of reference at an angle (<p =*= 90°) ; so that 
the combined power in this latter section is (P+Q) dptr. 



320 Mirrors, Prisms and Lenses [§ 116 

Accordingly, a spherical surface of power (P+Q) dptr. 
must be combined with a cylindrical surface of power 
— P dptr. and of axis-slope (^>=*=90°). With respect to the 
double sign in the expression v <£>=•= 90°), the rule is to select 
always that one of the two signs which will make the slope 
of the cylinder-axis positive ard less than 180°. Thus, for 
example, +8 dptr. sph. O +2 dptr. cyl.', ax. 20° is equiva- 
lent to +10 dptr. sph. O — 2 dptr. cyl., ax. 110° or to 
+ 10 dptr. cyl., ax. 20° C +8 dptr. cyl., ax. 110°. 

(2) To transpose a cross-cylindrical lens into a sphero- 
cylindrical lens: 

The combination P cyl., ax. <p O R cyl., ax. (<£>=*= 90°) 
is equivalent to either of the following: 

a. Sphero-cylinder: P sph. O (R—P) cyl., ax. (<p =*=90°), or 

b. Sphero-cylinder: R sph. O (P—R) cyl., ax. <p. 

Thus, +2 cyl., ax. 80° C +3 cyl., ax. 170° may be replaced 
by either +2 sph. C +1 cyl., ax. 170° or +3 sph. O 
-1 cyl., ax. 80°. 

(3) To transpose a spherical lens into a cross-cylinder: 

Q sph. is equivalent to Q cyl., ax. <p O Q cyl., ax. (<p =*= 90°), 
where the angle (p may have any value between 0° and 180°. 
For example, +5 sph. is equivalent to +5 cyl., ax. 10° O 
+5 cyl., ax. 100°. 

(4) The refracting powers of a toric surface in the prin- 
cipal sections are F y =(n f — n)/r y and F z = (n' — n)/r z . Let 
us suppose that the axis of revolution is parallel to the 
2/-axis. The toric refracting surface may be replaced by a 
sphero-cylindrical lens in either of two ways, as follows: 

.a. F z sph. O (F y —F z ) cyl., axis parallel to y-sads. 

b. F y sph. O (Fz — P y ) cyl., axis parallel to 2-axis. 

116. Obliquely Crossed Cylinders. — Oculists and optom- 
etrists sometimes prescribe a bi-cylindrical spectacle-glass 
with the axes of the cylinders crossed, not at right angles 
(as in the so-called cross-cylinder) , but at an acute or obtuse 
angle 7; and as it is not easy to grind a lens of this form, 
the optician prefers to make an equivalent sphero-cylinder 



116] 



Obliquely Crossed Cylinders 



321 



or a cross-cylinder, which will have precisely the same op- 
tical effect as the prescribed combination of obliquely crossed 
cylinders. His problem may be stated thus : 

Being given the refracting powers Fi, F 2 of the two sur- 
faces of the bi-cylindrical lens, and the angle y between 
the directions of the axes of the cylinders, it is required to 
calculate the refracting powers P and Q of the cylindrical 
and spherical components, respectively, of the equivalent 
sphero-cylindrical combination, together with the direction 
of the axis of the cylinder; that is, it is required to transpose 

Fi cyl., ax.<p C F 2 cyl., ax. (<p+y) 
into 

Q sph. O P cyl., ax. (<p-\- a). 

Simple working formula? for converting one of these lenses 
into the other were developed 
first by Mr. Charles F. 
Prentice. The following 
method is based on an ar- 
ticle " On obliquely crossed 
cylinders" by Professor S. P. 
Thompson published in the 
Philosophical Magazine (se- 
ries 5, xlix., 1900, pp. 316- 
324). 

In Fig. 156 the straight 
lines OA and OB are drawn 
parallel to the cylindrical 
axes of the bi-cylindrical lens, 
sothatZAOB = 7. Through 
O draw another straight line 
OC, and let ZAOC be de- 
noted by 6. In the sec- 
tion of the lens at right 
angles to OC the total r 
§112): 

Fi.cos 2 0+^2.cos 2 (y 




Fig. 



156. — Axes of obliquely crossed 
cylinders. 



efracting power will be (see 



ey, 



322 Mirrors, Prisms and Lenses [§ 116 

and in the section containing OC : 

Fi.sin 2 0+^2.sin 2 (7-0). 

The sum of these two expressions is equal to (F1+F2); and 
according to the theory of curved surfaces (§ 111), this sum 
must also be equal to the sum of the maximum and mini- 
mum refracting powers of the equivalent sphero-cylindrical 
lens. Now, obviously, (P+0) will be the maximum (or 
minimum) refracting power in a section of the latter lens 
at right angles to the axis of the cylinder, whereas Q will be 
the minimum (or maximum) refracting power in the sec- 
tion containing the axis of the cy Under; accordingly, first 
of all, we find that we must have: 

2Q+P = F 1 +F 2 . 

Now there is a certain value of the angle 6, say, 6 = a, 
for which the first of the two expressions above will be a 
maximum (or minimum) and the second a minimum (or 
maximum) ; and if we can determine this angle a, the prob- 
lem will practically be solved, because then we shall have: 
P+0 = ^i.cos 2 a+F 2 .cos 2 (7- a), 
Q = Fi.sin 2 a+F 2 .sin 2 (7- a); 

where (on the assumption that Q is the minimum refracting 
power in the section containing the axis of the cylinder) a 
denotes the angle between the cylindrical axis of the sphero- 
cylinder and the cylindrical axis of the cylinder whose refract- 
ing power is denoted by F\. Now in order to ascertain this 
angle a, all we have to do (as will be obvious to any one 
who is familiar with the elements of the differential calculus) 
is, first, to differentiate the expression 

Fi.cos 2 0+F 2 .cos 2 (7-0) 

with respect to 6, and then, after writing a in place of 0, 
to put the resultant expression equal to zero. Thus we ob- 
tain the following equation for finding the angle a in terms 
of the known magnitudes F h F 2 and 7 : 

— 2i^i.sina .cosa+2F 2 .sin(7— a).cos(7— a)=0; 



§116] 



Obliquely Crossed Cylinders 



323 



which may also be put in the following form: 
F\ _ F 2 
sin2(7— a) sin2a' 

Moreover, since P = (P*+Q) — Q, we find: 

P = Fi(cos 2 a-sin 2 a)+F 2 {cos 2 (7~ a)— sin 2 (y- a)} 
= Fi.cos2a+F 2 .cos2(7— a); 

and if in this formula we substitute the value 

sin2a 



F, = -. 



sin2(7 — a) 



Fi, 



we shall find : 



sin27 
sin2(7~ a) 



v*y 



*V 




a* 

Fig. 157. — Graphical mode of finding cylindrical component (P) of 
sphero-cylinder equivalent to two obliquely crossed cylinders of powers 
F\ and F2. 

Hence, 



F l 



sin2(7~ a) sin2a sin27* 



324 Mirrors, Prisms and Lenses [§ 116 

which at once suggests an elegant and simple graphical 
solution of the problem. For, evidently, according to the 
above relations, the magnitudes denoted by F h F 2 and P 
may be represented in a diagram (Fig. 157) by the sides of 
a triangle whose opposite angles are 2(7— a), 2a and 
(180°— 27), respectively. Hence the rule is as follows: 
On any straight line lay off a segment AB to represent, ac- 
cording to a certain scale, the magnitude of the refracting 
power Fi; and let X designate the position of a point on AB 
produced beyond B. Construct the ZXBC equal to twice 
the angle between the axes of the two given cylindrical com- 
ponents (ZXBC = 2 7); and along the side BC of this angle 
lay off the length BC to represent the magnitude of the re- 
fracting power F 2 . Then the straight line AC will repre- 
sent on the same scale the magnitude of the refracting power 
P of the cylindrical member of the equivalent sphero- 
cylindrical lens, and the Z B AC = 2 a will be equal to twice 
the angle between the cylindrical axes of the surfaces whose 
powers are denoted by F\ and P. For calculating the values 
of P, Q and a, we have by trigonometry the following sys- 
tem of formulae : 

P = + \ / F 2 1 +F 2 2 +2Fi.F2.cos2y, 
Q= FM-P 



tan2a 



2 

F 2 .sm2y 



^ 1 +F 2 .cos27' 

which will be found to be applicable in all cases, whether 
the signs of F h F 2 are like or unlike. 

There is, to be sure, another solution also, in which the 
cylindrical axis of the sphero-cylindrical lens is inclined to 
the cylindrical axis of the cylinder of power Fi at the angle 
(90°+ a). For if the refracting power Q of the spherical 
member is assumed to be the maximum (instead of the 
minimum) refracting power of the sphero-cylindrical com- 
bination, then (P-f-Q) will be the minimum power in a sec- 
tion at right angles to the axis of the cylinder; and in this 



§ 116] Obliquely Crossed Cylinders 325 

case the refracting power of the cylindrical component will 

be represented by the dotted line AC in Fig. 157 which is 

equal to AC in length but opposite to it in direction. In 

fact, in this case the formulae for P and Q will be as follows: 

P = - VFl-]-Fi+2Fi.F2.GOs2y, 

n _Fi±IW> 

Q 2 * 

This result could have been obtained from the first result by 

transposing; for, according to § 115, Q sph. O P cyl., ax. <f> 

is equivalent to (P+Q) sph. O — P cyl. ax. (<£=*= 90°), where 

the symbols P and Q denote here the powers of the first 

combination. 

Moreover, since Q sph. O P cyl., ax.<£ is equivalent also 
to (P+Q) cyl., ax.</> O Q cyl., ax. (<£=*= 90°), two obliquely 
crossed cylinders may be replaced by a cross-cylinder of 
powers (P+Q) and Q. In fact, since 

(P+Q)+Q=Pi+P 2 , 

(P+Q)~Q = VPi+Pi+2Pi.P 2 .cos2 y, 
it follows that : 

(P+Q)Q=F 1 .F 2 .sin' 2 y; 
so that this formula will give us the product of the powers 
of the equivalent cross-cylinder, and since their sum P+2Q 
= Fi+F 2 , the values of (P+Q) and Q may be obtained in- 
dependently, without first finding the value of P. 

The following numerical example will serve to illustrate 
the use of the formula; : 

Given a combination of obliquely crossed cylinders as 
follows : 

+4 cyl., ax. 20° C -2.75 cyl., ax. 65°; 
let it be required to find the equivalent sphero-cylinder 
and also the equivalent cross-cylinder. 

We must put Pi = +4, because Pi denotes the power of 
the cylinder whose axis-slope is the smaller of the two. Then 
F 2 = -2.75 and y =(65° -20°) =45°. Substituting these 
values, we find : 

P=+4.86, Q=-1.8, a=-17°16'. 



326 Mirrors, Prisms and Lenses [Ch. IX 

Accordingly, the given combination is equivalent to one of 
the three following: 

+4.85 cyl., ax. 2° 44' O- 1.8 sph. ; 
-4.85 cyl., ax. 92° 44' C +3.05 sph.; 
+3.05 cyl., ax. 2° 44' C - 1.8 cyl., ax. 92° 44'. 
If 7 = 90 o , then P = F 1 -F 2 , Q = F 2 and a = 0°, or 
P = F 2 -Fi, Q=F\ and a = 0°; so that we can write: 

Fi cyl., ax. <£ C F 2 cyl., ax. (<£±90°) 
is equivalent to 

Fi sph. C (F 2 -Fi) cyl., ax. (<£±90°) 
or 

F 2 sph. C (F1-F2) cyl., ax. cj>; 
exactly as found in § 115. 

PROBLEMS 

1. The radius of a convex cylindrical refracting surface 
separating air from glass (n = 1.5) is 8|- cm. What is its 
refracting power in a normal section inclined to the axis of 
the cylinder at an angle of 60°? Ans. +4.5 dptr. 

2. A curved refracting surface separates air and glass 
(n':n = 3: 2), and the radii of greatest and least curvature 
at a point A on the surface are r y = + 10 cm. and r z = 
+5 cm. Find the interval between the two principal image- 
points corresponding to an object-point lying on the normal 
to the surface at A in front of the surface and at a distance 
of 30 cm. from it. Ans. 67.5 cm. 

3. The principal refracting powers of a thin astigmatic 
lens surrounded by air are denoted by F y and F z . The prin- 
cipal image-points corresponding to an axial object-point 
M are designated by My and M z . If the optical center of 
the lens is designated by A, and if we put U=l/u, where 
u= AM, then 

M' M' = F y~ Fz . 



Ch. IX] Problems 327 

4. The refracting powers of a thin astigmatic lens in the 
two principal sections are +3 and +5 dptr. The lens is 
made of glass of index 1.5. Find the radii of the two sur- 
faces for each of the following forms: (a) Cross-cylinder; 
(6) Sphero-cylinder; c) Plano-toric. 

Ans. (a) Double convex cross-cylinder, radii 10 and 
16 -| cm.; (6) Double convex sphero-cylinder, radius of 
sphere 16 f cm., radius of cylinder 25 cm.; or convex me- 
niscus sphero-cylinder, radius of sphere 10 cm., radius of 
cylinder 25 cm. ; (c) Radii of toric surface 10 and 16 J cm. 

5 The principal refracting powers of a thin lens are +4 
and —5 dptr. If the refracting power in an oblique normal 
section is +2 dptr., what will be its refracting power in a 
normal section at right angles to the first? and what is the 
angle of inclination of the +2 section to the +4 section? 

Ans. -3 dptr.; 28° 7' 32". 

6. Two cylinders each of power +1.18 dptr. are com- 
bined with their axes inclined to each other at an angle of 
32° 3' 50". Show that the combination is equivalent to 
+0.18 sph. O +2 cyl., axis midway between the axes of 
the two given cylinders. 

7. Show that 

+2 cyl., ax. 0° C -3 cyl., ax. 53° 26' 14" 
is equivalent to 

-2.53 sph. C +4.06 cyl., ax. -22° 30'. 

8. Transpose 

-1.25 cyl., ax. 20° C +3.25 cyl., ax. 53° 41' 24.25" 
into the equivalent sphero-cylinder. 

, Ans. —0.5 sph. O + 3 cyl., ax. 65°, 
or + 2.5 sph. O — 3 cyl., ax. 155°. 

9. Transpose 

+9.5 cyl., ax. 0° C +10 cyl., ax. 57° 40' 45" 
into the equivalent sphero-cylinder. 

Ans. +4.53 sph. C +10.43 cyl., ax. 30°, 
or +14.96 sph. C - 10.43 cyl., ax. 120°. 



328 Mirrors, Prisms and Lenses [Ch. IX 

10. Find the sphero-cylindrical equivalent of 

+2 cyl, ax. 20° C +3 cyl., ax. 70°. 

Ans. +0.85 sph. C + 3.3 cyl., ax. 51° 42', 
or + 4.15 sph. C -3.3 cyl., ax. 141° 42'. 

11. Transpose 

-1.75 cyl., ax. 120° C +1.25 cyl., ax. 135° 
into the equivalent cross-cylinder. 

Ans. +0.207 cyl., ax. 98° 30' C -0.707 cyl., ax. 8° 30'. 

12. Transpose +4 cyl., ax. 80° C -2 cyl., ax. 120° into 
the equivalent cross-cylinder. 

Ans. +3.075 cyl., ax. 65° 50' C -1.075 cyl., ax. 155° 50'. 



/ 



CHAPTER X 

GEOMETRICAL THEORY OF THE SYMMETRICAL OPTICAL 
INSTRUMENT 

117. Graphical Method of tracing the Path of a Paraxial 
Ray through a Centered System of Spherical Refracting 
Surfaces. — Nearly all optical instruments consist of a com- 
bination of transparent, isotropic media, each separated 
from the next by a spherical (or plane) surface; the centers 
of these surfaces lying all on one and the same straight line 
called the optical axis of the centered system of spherical 
surfaces, which is an axis of symmetry. In a symmetrical 
optical instrument of this kind it is sufficient to investigate 
the procedure of paraxial rays in any meridian plane con- 
taining the axis. 

The indices of refraction of the media will be denoted by 
Tii, 712, etc., named in the order in which they are traversed 
by the light; so that if m denotes the number of refracting 
surfaces, the index of refraction of the last medium into 
which the rays emerge after refraction at the mth surface 
will be n m+1 . The indices of refraction of the two media 
which are separated by the A:th surface (where k denotes 
any integer between 1 and m, inclusive) will be n k and n k+1 . 
The vertex and center of the kth surface will be designated 
by A k and C k , respectively; and the radius of this surface 
will be denoted by r k = A k C k . Moreover, if M k , M k+ i 
designate the positions of the points where a paraxial ray 
crosses the axis before and after refraction, respectively, at 
the kth surface, these points will be a pair of conjugate axial 
points with respect to this surface; and the points Mi, M m+ i 
will, therefore, be a pair of conjugate axial points with respect 

329 



330 



Mirrors, Prisms and Lenses 



[§H7 



to the entire centered system of m spherical refracting sur- 
faces. 

The accompanying diagram (Fig. 158) represents a merid- 
ian section of an optical system of this kind. The straight 
line MiBi represents the path of a paraxial ray in the first 
medium (wi) which crossing the axis at Mi meets the first 
surface r (i/i) in the point marked Bi. Similarly, the path 




Fig. 158. — Path of paraxial ray through centered system of spherical re- 
fracting surfaces. 



of the ray from the first surface to the second surface is 
shown by the straight line BiB 2 which crosses the axis at M2. 
Thus, the entire course of the ray is shown by the broken 
line M1B1B2B3M4 which is bent in succession at each of the 
incidence-points Bi, B2, B 3 (supposing that m = 3, as repre- 
sented in the diagram) . 

The figure shows also the path of another paraxial ray, 
emanating from an object-point Qi near the optical axis but 
not on it and represented here as lying perpendicularly 
above Mi. This ray is the ray which leaves Qi along a straight 
line which passes through the center Ci of the first refracting 
surface and also through the point Q 2 which is conjugate to 
Qi with respect to this surface. This point Q2 can be lo- 
cated by determining the point of intersection of the straight 
line Q1C1 with the straight line M2Q2 drawn perpendicu- 
lar to the axis at M2. Similarly, the point Q3 conjugate to 
Q 2 with respect to the second refracting surface will be at 



§ 117] Centered System of Spherical Surfaces 331 

the point of intersection of the straight line Q2C2 with the 
straight line drawn perpendicular to the axis at M 3 ; and 
so on from one surface to the next. Provided, therefore, we 
know the path of one paraxial ray through the system, it 
is easy to construct the path of a second ray. 

But the best graphical method of tracing the path of a 
paraxial ray through a centered system of spherical refract- 
ing surfaces consists in applying the construction described 




Fig. 159. — Graphical method of tracing path of paraxial ray through cen-V 
tered system of spherical refracting surfaces. 

in § 76, as follows: If the straight line M1B1 (Fig. 159) rep- 
resenting the path of the ray in the first medium meets the 
perpendicular erected to the optical axis at the center Ci in 
the point Xi, and if on this perpendicular a second point X/ 
is taken such that C1X1 : CiXi' = n 2 : n h then the straight 
line BiXi' will determine the path BiB 2 of the ray in the 
second medium. Draw C2Y2 parallel to C1X1, and let Y 2 
designate the point of intersection of the straight lines 
BiB 2 and C 2 Y 2 ; and on C 2 Y 2 take a point Y 2 ' such that 
C 2 Y 2 : C 2 Y 2 ' = n 3 : n 2 , and draw the straight line Y 2 'B 2 meeting 
the third refracting surface in B 3 and intersecting in Z 3 the 
straight line drawn through C 3 parallel to C2Y2. If on C 3 Z 3 
a point Z/ is taken such that C 3 Z 3 : C 3 Z 3 ' = n 4 : n 3 , then the 
straight line B 3 Z 3 ' will determine the path of the ray after 
refraction at the third surface. This process is to be re- 
peated until the ray has been traced into the last medium. 



332 Mirrors, Prisms and Lenses [§ 118 

118. Calculation of the Path of a Paraxial Ray through 
a Centered System of Spherical Refracting Surfaces. — Ob- 
viously, just as in the case of a single spherical refracting 
surface (§ 80), any figure lying in a plane in the object-space 
perpendicular to the optical axis of a centered system of 
spherical refracting surfaces will be reproduced by means 
of paraxial rays by a similar figure in the image-space also 
lying in a plane perpendicular to the optical axis. 

Moreover, if we put 

A k M k = u k , A k M k+ i=w k ', 
the abscissa-formula (§ 78) for the kth surface may be writ- 
ten: 

n k+ i _ n k , n k+ i— n k 
u k u k r k 

If also we employ the symbol 

4 = A k A k+ i 
to denote the distance of the vertex of the (k-\-l)th surface 
from that of the A;th surface or the so-called axial thickness 
of the (7c+l)th medium, then, evidently: 

u k+ i = u k '—d k ; 
which enables us to pass from one surface to the next. 

If in these so-called recurrent formulae we give k in suc- 
cession the values k = l, 2, . . . , (ra— 1), and if also in the 
first formula we put finally k = m, we shall obtain (2m— 1) 
equations; and if the constants of the system are all known, 
that is, if the values of all the magnitudes denoted by n, r 
and d are given, together with the initial value ui, which 
denotes the abscissa of the axial object-point, these (2m— 1) 
equations will enable us to determine the value of each of 
the u's in succession. The position of the image point M m+ i 
conjugate to the axial object-point Mi will have been ascer- 
tained when we have found the value of the abscissa u m '. 

The secondary focal point of the system is the point F' 
where a paraxial ray which is parallel to the axis in the first 
medium crosses the axis in the last medium; and if we put 
Ui = oo , then u m ' = A m F' will be the abscissa of the second- 



§ 118] Lateral Magnification 333 

ary focal point with respect to the vertex of the last surface. 
Similarly, the primary focal point is the point F where a par- 
axial ray must cross the axis in the first medium if it is to 
emerge in the last medium in a direction parallel to the axis. 
In this case, therefore, we must put u m f = oo and solve for 
ui = AiF in order to obtain the abscissa of the focal point F 
with respect to the vertex of the first surface of the system. 

The focal planes are the planes at right angles to the axis 
at the focal points F, F'. 

Moreover, if we put 2/k = M k Q k , then according to the 
formula for the lateral magnification in a spherical refracting 
surface (§ 82), we can write for the kih surface: 
2/k+i _ ftk V. 
Vk n k+ \ Uk ' 
and if we give k all integral values from k = l to k = m, we 
shall obtain m equations, one for each surface, wherein the 
denominator of the ratio on the left-hand side of each of 
these proportions will be the same as the numerator of the 
corresponding ratio in the preceding one of the series. Hence, 
if we multiply together all of these equations, and if, finally, 
we put 

y=yi, y'=ym+i, ' n = n h n' = n m + h 
we shall obtain : 

y' nui.u 2 '. . . Um' 



n'u\.ui. 



which may be written also : 



k= 

TT 

y n' J - J -Wk ' 
k=i 



y' _n 1 TV 



where the symbol IT placed in front of an expression in this 
way means merely that the continued product of all terms 
of that type is to be taken. Thus having found the values 
of all the u's, both primed and unprimed, we can calculate 
by this formula the lateral magnification produced by the 



334 Mirrors, Prisms and Lenses [§ 119 

entire centered system of spherical refracting surfaces for 
any given position of the object-point. 

Moreover, for the kth surface the so-called Smith- 
Helmholtz formula (§ 86) will have the form: 

n k .y k . 6 k = n k+ i.y k+ i. d k + h 
where k = ZA k M k B k ; and if here also we give k all values 
in succession from k = 1 to k = m, we shall obtain : 

ni.yi.di = n 2 .y2-62= • • • = n m +i.y m +i. m +i; 
and finally: 

n'.y'. 6' = n.y. 6, 

where n, n r and y, y f have the same meanings as above, and 

6= 0i, 6'= m +i. 

119. The so-called Cardinal Points of an Optical System. 
The methods which have just been explained, although 
perfectly simple in principle, involve a more or less tedious 
process of tracing the path of a paraxial ray from one surface 
to the next throughout the entire system. We have now 
to explain the celebrated theory of Gauss (1777-1855) which 
was developed (1841) in order to avoid as much of this labor 
as possible, by keeping steadily in view the fundamental re- 
lations between the object-space and the image-space. It is 
easy to show that the imagery produced by a symmetrical op- 
tical instrument in the vicinity of the axis is completely de- 
termined so soon as we know the positions of the focal points 
and one pair of conjugate points on the axis, together with 
the ratio of the indices of refraction of the first and last media 
of the system. However, for this purpose certain pairs of 
conjugate axial points are distinguished above others on 
account of their simple geometrical relations; and of these 
the most important are the principal points and the nodal 
points. These two pairs of conjugate points, together with 
the focal points, are sometimes called the cardinal points of 
the optical system. We shall explain now how these points 
are defined. 

(1) The Focal Planes and the Focal Points. — In every 
centered system of spherical refracting surfaces there are 



§ 119] Principal Planes and Principal Points 335 

two (and only two) transversal planes at right angles to the 
axis which are characterized by the following properties: 

A bundle of paraxial object-rays which all meet in a point 
in one of these planes {called the primary focal plane) will 
emerge from the system as a cylindrical bundle of parallel 
image-rays; and, similarly, a cylindrical bundle of parallel 
object-rays will emerge from the system as a bundle of image- 
rays which all meet in a point in the other one of these planes 
{called the secondary focal plane) . The points in which these 
focal planes are pierced by the axis are the primary and sec- 
ondary focal points F and F', respectively. 

(2) The Principal Planes and the Principal Points. — Again, 
in every symmetrical optical system there is one (and only 
one) pair of conjugate transversal planes characterized by 
the property, that in these planes object and image are con- 
gruent; and, therefore, any straight line drawn parallel to the 
axis will intersect these planes in a pair of conjugate points. 
These are the so-called principal planes, one belonging to 
the object-space {the primary principal plane) and the other 
belonging to the image-space {the secondary principal plane). 
The points H, H' where the optical axis crosses the prin- 
cipal planes are the principal points of the system. Atten- 
tion was first directed to these points by Moebius in 1829, 
but it was Gauss who recognized their significance for the 
development of simple and convenient general formulae in 
the theory of optical imagery. 

In the principal planes the lateral magnification is unity, 
that is, y' — y. (And hence the principal planes and principal 
points are called also, especially by English writers, the unit 
planes and the unit points.) Consider, for example, the case of 
a single spherical refracting surface, for which we found (§ 85) 
y'_f _f'+W 

V f+u r ' 
If we put y'=y, we find u' = u = 0] which means that the 
principal points of a spherical refracting surface coincide with 
each other at the vertex of the surface (§ 81). We saw likewise 



336 



Mirrors, Prisms and Lenses 



[§H9 



that these points coincided with each other at the optical 
center of an infinitely thin lens (§ 94). 

A useful rule is as follows: 

To any ray in one region (object-space or image-space) 
which goes through the focal point belonging to that region, 







w 




w" 


















s 




X 


/ p 


N,^ 


H 








/y' 






?S 


V 












a 











w 


f 




W ^ 




F' H' 




H ^-.F 


X 


\\ ** — . 


V' 


\ 


V 




ss 





Fig. 160, a and b— Focal points (F,F') and principal points (H, H') of 
(a) convergent and (6) divergent optical system. 

there will correspond a ray in the other region which is par- 
allel to the axis, and the rectilinear portions of the path of 



119] 



Nodal Planes and Nodal Points 



337 



the ray in these two regions will intersect in a point lying in 
the principal plane of that region to which the focal point in 
question belongs; as is illustrated in the accompanying dia- 
grams at W and at V' (Fig. 160, a and b). 

(3) The Nodal Planes and the Nodal Points. — Finally, in 
every centered system of spherical refracting surfaces there 
is also a pair of conjugate transversal planes characterized 
by the property, that the angle between any pair of object- 











X 




U 


XT 


X 




A 




S 


X 




N 


H 


N* 


H' 



Fig. 161. — Principal points (H, H') and nodal points 
(N, N'). 

rays which intersect in a point lying in the so-called primary 
nodal plane will be exactly equal to the angle between the cor- 
responding pair of image-rays which meet in the conjugate 
point of the secondary nodal plane. The nodal points N, N' 
where the axis meets these planes were remarked first by 
Moser in 1844, but they were brought into prominence 
through the work of Listing (1845) with whose name there- 
fore they are generally associated. The distinguishing fea- 
ture of this pair of conjugate axial points is that object-ray 
and image-ray cross the axis at the nodal points at exactly the 
same slope. For example, if the straight line NU (Fig. 161) 
represents the path of an object-ray which crosses the axis 
at the primary nodal point and meets the primary principal 
plane in the point marked U, the path of the corresponding 
image-ray will be represented by a straight line N'U' which 
is drawn parallel to NU and which meets the secondary prin- 



'338 Mirrors, Prisms and Lenses [§ 119 

cipal plane in the point marked U', so that if ZHNU= 0, 
ZH'N'U'= 0',then0' = 0. 

Obviously, the quadrilateral NUU'N' is a parallelogram, 
and hence H'N' = HN; that is, the step from one of the prin- 
cipal points to the corresponding nodal point is identical with 
the step from the other principal point to its corresponding nodal 
point. The nodal points, therefore, lie always on the same 
side of the corresponding principal points and at equal dis- 
tances from them. If the primary nodal point and principal 
point coincide, the same will be true of the secondary nodal 
point and principal point. Moreover, since NN / = UU , = 
HH', the interval between the nodal planes is precisely the 
same as the interval between the principal planes. 

If in the Smith-Helmholtz formula (§ 118) we put 0' = 
0, we find for the lateral magnification in the nodal planes 
of a centered system of spherical refracting surfaces 

y l= — 

y n" 
where n and n' denote the indices of refraction of the first 
and last media, respectively. Applying this result to the 
case of a single spherical refracting surface, we obtain for 
the nodal points N, N' the conditions u r = u = r, that is, 
AN' = AN = AC. Consequently, the nodal points of a spher- 
ical refracting surface coincide with each other at the center 
C of the surface; as might have been inferred at once from 
the fact that a central ray is not deviated by refraction at a 
spherical surface. 

(4) Various writers on optics have distinguished other 
pairs of conjugate axial points besides the principal points 
and nodal points, but none of these can be said to have 
achieved a permanent place in the literature of the subject. 
We may mention the so-called negative principal points, in- 
troduced by Toepler in 1871, which are characterized by 
the fact that for this pair of points the lateral magnification is 
equal to —I; that is, y'=—y, so that the image is inverted 
and of same size as object. Professor S. P. Thompson, hav- 



§120] 



Construction of Image 



339 



ing this property in view, has re-named them much more 
happily the symmetric points of the optical system. 

120. Construction of the Image-Point Q' conjugate to an 
Extra-axial Object-Point Q. — If the principal planes and 
focal planes have been determined, it will not be necessary 
to trace the path of a ray in the interior of the system. Sup- 




Fig. 162. — Construction of image-point Q' conjugate to object-point Q 
in an optical system. 

pose, for example, that Q (Fig. 162) designates the position 
of an object-point not on the axis; the position of the point 
Q' conjugate to Q may be constructed as follows: 

Through Q draw a straight' line QV parallel to the axis 
meeting the secondary principal plane in the point marked 
V and also another straight line QF meeting the primary 
principal plane in the point marked W. The required point 
Q' will be found at the point of intersection of the straight 
line V'F' with the straight line WQ' drawn parallel to the 
axis. The feet of the perpendiculars let fall from Q, Q' on 
to the axis will locate also a pair of conjugate axial points 
M, M'. The construction is seen to be entirely similar to 
that given in §§71, 81 and 92. The case represented in the 
figure is that of a convergent optical system, in which parallel 
object rays are converged to a real focus at a point in the 
secondary focal plane. The student should draw for him- 



340 



Mirrors, Prisms and Lenses 



121 



self the corresponding diagram for the case of a divergent 
optical system. 

121. Construction of the Nodal Points N, N\ — Having de- 
termined the position of the point Q' conjugate to Q, we can 
easily locate the positions of the nodal points N, N'. For 
example, on the straight line WQ' (Fig. 162) take a point Z 
such that ZQ' = HH', and draw the straight line QZ meeting 
the primary principal plane in the point U. Draw UU' par- 
allel to the axis meeting the secondary principal plane in 
the point U'. Evidently, the straight lines QU and QTJ' will 




Fig. 163. 



-Construction of nodal points (N, N'), and proof of 
relation I'F' = FR. 



be parallel, and the points where they cross the axis will be 
the nodal points N, N' (§ 119). 

A simpler way of constructing the nodal points N, N' is 
as follows : 

Through the primary focal point F draw a straight line 
FW meeting the primary principal plane in the point marked 
W, and through W draw a straight line parallel to the axis 
meeting the secondary focal plane in a point marked I' in 
Fig. 163. This point I' is the image-point of the infinitely 
distant point I of the straight line FW. The straight line 
drawn through I' parallel to FW will meet the axis in the 
secondary nodal point N' ; and the position of the other nodal 
point N can be found immediately. 

The diagram shows also that 

FH = N'F'; 



§121] 



Construction of Image 



341 



whence it follows (§ 119) that 

F'H' = NF. 
Accordingly, the step from one nodal point to the correspond- 
ing focal point is identical with the step from the other focal 
point to its corresponding principal point. In fact, the three 
segments of the axis FF', HN' and H'N all have a common 
half-way point. 

Incidentally, another useful relation may be seen at a 
glance in Fig. 163. Let R designate the point where the ray 
IH which passes through the primary principal point crosses 



a 






V 


V' 








X 














F 


fco^^ 


H 


H' \, 


F' 


X 










"-—~4sj^ 
















Y' \ 










W 


w 




a' 



Fig. 164. — Construction of image-point Q' conjugate to object-point Q in 
an optical system. 

the primary focal plane; the corresponding image-ray will 
pass through the secondary principal point IT and cross the 
secondary focal plane at I'; and, obviously, since FRHW 
and HWI'F' are both parallelograms, 

I'F' = FR; 
Consequently, a pair of conjugate rays passing through the 
principal points H, H' will cross the focal planes at equal dis- 
tances from the axis, but on opposite sides thereof. 

This result may be utilized in the construction of the 
point Q' (Fig. 164) conjugate to the object-point Q. Let 
X designate the point where the straight line QH crosses 
the primary focal plane ; and take a point Y' in the secondary 
focal plane such that F' Y' = XF. Then the required point Q' 



342 Mirrors, Prisms and Lenses [§ 122 

will be at the point of intersection of the straight line H'Y' 
with either of the straight lines W'Q' or V'F' shown in the 
figure. 

122. The Focal Lengths f, f . — Let us employ the symbols 
co, co' to denote the slopes of a pair of conjugate rays which 
pass through the principal points H, H'; thus, in Fig. 164 
ZFHX= co, ZF / H / Y / = co'; and since in the case of paraxial 
rays we may write co andco' in place of tanco and tana/ 
(see § 63), we have: 

FX = _ F^ = ' , 

FH W ' F'H' ° ) ' 

Accordingly, dividing one of these equations by the other, 

and taking account of the fact that F'Y' = XF (§ 121), we 

obtain : 

FH _&/ 
F'H' co ' 

Since the lateral magnification in the principal planes is 
equal to +1, that is, since y' ' — y (§ 119), the Smith-Helm- 
holtz formula (§ 118) for the pair of conjugate points 
H,H' takes the form: 

n'.co'=n.co, 
where n and n' denote the indices of refraction of the first 
and last media of the optical system. 

If, therefore, the focal lengths of the optical system are de- 
fined as the abscissa? of the principal points with respect to their 
corresponding focal points, that is, if we put /=FH, /' = F'H', 
where /and/' denote the primary and secondary focal lengths, 
respectively, then combining the relations found above so 
as to eliminate the angles co and co', we find: 

/' n' ' 

which may be put in words as follows: The focal lengths of 
a centered system of spherical refracting surfaces are propro- 
tional to the indices of refraction of the first and last media, 
and are opposite in sign; except in the single case when the 
optical system includes an odd number of reflecting surfaces f 



122] 



Focal Lengths of Optical System 



343 



in which case the focal lengths will have the same sign (that is, 
in this exceptional case, ///' = +w/n'). 
It appears, therefore, that the formula, 
n'.f+n.f'=0, 
which was found (§§ 79 and 96) to hold for a single spherical 
refracting surface and for an infinitely thin lens, expresses, 
in fact, a perfectly general relation which is true of any 
centered system of spherical refracting surfaces. Consider, 
for example, the optical system of the human eye in which 
the first medium is air (n = l) and the last medium is the 



To I at cc 



To E at oo- 




To J'ata? 



Fig. 165. — Focal lengths (/,/') of an optical system. 



so-called vitreous humor whose index of refraction is n' = 
1.336. In Gullstrand's schematic eye (see § 130) the 
primary focal length is found to be /= + 17.055 mm., 
whence, according to the above formula, the secondary 
focal length is /'= -22.785 mm. 

In particular, when the media of object-space and image- 
space are identical (n r = n) , the focal lengths are equal in mag- 
nitude, but opposite in sign ( /' = — /) . This is the case with 
most optical systems, since they are usually surrounded by 
air. According to the definitions of the focal lengths given 
above, it follows from § 121 that 

FH = N'F' =/, F'lT = NF =/'; 

and hence we see that the nodal points (N, N') of an optical 

system surrounded by the same medium on both sides coincide 

with the principal points (H, H') ; for when n f = n, then 

FH=/= -/' = FN, F'H'=/' = -/=F'N'. 



344 Mirrors, Prisms and Lenses [§ 123 

The focal lengths of a centered system of spherical re- 
fracting surfaces may be defined also exactly as in §§ 83 
and 95. If in Fig. 165 we put ZHFW= 0,ZH'F'V' = 0', 
we can write: 

HW H'V 



tan ' * tan 0' ' 

and since HW = FT, H'V' = FJ, tan0 = 0, tan0' = 0', we 
have: 

Accordingly, we may also define the focal lengths as follows : 
The focal length of the object-space (f) is equal to the ratio of 
the linear magnitude of an image formed in the focal plane 
of the image-space to the apparent (or angular) magnitude of 
the correspondingly infinitely distant object; and, similarly, the 
focal length of the image-space (/') is equal to the ratio of the 
linear magnitude of an object lying in the focal plane of the 
object-space to the apparent (or angular) magnitude of its in- 
finitely distant image. 

The focal lengths may be said, therefore, to measure the 
magnifying power of the optical instrument, for if the appara- 
tus is adapted to an emmetropic eye (§ 153), the image will 
be formed at infinity, and the magnifying power will be deter- 
mined by the ratio of the apparent size of the image to the 
actual size of the object (see Chapter XIII). 

123. The Image-Equations in the case of a Symmetrical 
Optical System. — The image-equations are a system of re- 
lations which enable us to find the position of an image- 
point Q' (Fig. 162) conjugate to a given object-point Q. 
The position of the point Q will be given by its two co- 
ordinates referred to a system of rectangular axes in the 
object-space in the meridian plane in which the point Q lies. 
Naturally, the optical axis will be selected as the axis of 
abscissae and either the primary focal point F or the primary 
principal point H as the origin. Thus, if we put 
FM = z ; HM=u, MQ=2/, 



§ 123] The Image-Equations 345 

the object-point Q will be the point (x, y) or the point (u, y), 
according as we take the origin at F or H, respectively. 
Similarly, in the image-space, if we put 

F'M' = :r/, H'M' = < M'Q' = y', 
the coordinates of Q' will be denoted by (x f , y') or (u f , y') 
according as the origin of this system of axes is at F' or H', 
respectively. 

a. The image-equations referred to the focal points F, F'. — 
The following proportions are obtained from the two pairs 
of similar triangles FHW, FMQ and F'H'V, F'M'Q': 

HW = FH M'Q^F'M'. 
MQ FM' H'V' F'H'' 
and since 

HW = M'Q' = 2/' f H , V' = MQ = 2/, FH=/, F'H'=/', 
we find immediately : 

y x f" 

whence the coordinates x', y' can be found in terms of the 
given coordinates x, y and the focal lengths /, /'. 

These formulae, which were obtained formerly for cer- 
tain simple special cases (§§ 69, 85 and 93) are seen, there- 
fore, to be entirely general and applicable always to any 
symmetrical optical system. The so-called Newtonian 
form of the abscissa-relation, viz., 

x.x'=ff, 
shows that the product of the focal-point abscissae is constant. 

b. The image-equations referred to the principal points 
H,H'. — Again, the following proportions are derived from 
the two pairs of similar triangles FHW, QVW and F'H'V, 
Q'W'V: 

WV = VQ = HM VW = W^' = ITM'. 

HW FH FH ' H'V F'H' FTT ' 
and since WV = WH+HV = Q'M'+MQ =-(?/-?/) and 
V'W' = V'H'+H'W' = QM+M'Q' = (?/-?/), we find: 
y'—y _ _ u y'-y _u f 

y' f y /'* 



346 Mirrors, Prisms and Lenses [§ 123 

These relations give the following expressions for the lateral 
magnification : 

y'_ f _f'+W_ f u^ 

y f+u f f'u ' 

Clearing fractions, we obtain : 

f.u f +f.u+u.u' = 0, 
and dividing through by u.u' ', we have the well-known 
abscissa-relation : 

£+4+1=0; 

u u 

which may also be obtained directly by substituting x =/+w, 
x'=f'-\-u' in the equation x.x' =}.}'. 

By means of these formulae, the coordinates v! , y f may 
be found in terms of the given coordinates u, y and the 
focal lengths/,/'. 

Since n'.f+n.f = (§ 122), we have also another expres- 
sion for the lateral magnification, viz., 

y' _n.u f m 
y n'.u ' 
winch has likewise been obtained already in the special case 
of a single spherical refracting surface (§ 82). 

A simple and convenient method of locating the positions 
of pairs of conjugate axial points is suggested by the ab- 
scissa-relation 

£+4+1=0; 
u u 

which may be put in the following form : 
HF , H'F' 
u u 
Suppose, therefore, that the axial line segment H'F r is shoved 
along the optical axis until the secondary principal point H' 
is brought into coincidence with the primary principal point 
H, and that then the optical axis in the image-space (x r ) is 
turned about H until it makes a finite angle with the op- 
tical axis in the object-space (x), as represented, for example, 
in Fig. 166. Through the focal points F and F' draw the 



§123] 



The Image-Equations 



347 



straight lines FS and F'S parallel to H'F' and HF, respec- 
tively, and let S designate their point of intersection. Then 
any straight line drawn through S will intersect x and x' in 
a pair of conjugate axial points M, M'; for if we put w = HM 
and w' = H'M' in the equation above, the equation will 




Fig. 166. — Construction of point M' conjugate to 
axial object-point M in an optical system. 

evidently be satisfied. The vertex S of the parallelogram 
HF'SF is the center of perspective of the two point-ranges 
x and x'. 

c. The image-equations referred to any pair of conjugate 
axial points 0, O'. 

If the origins of the two systems of rectangular axes are 
a pair of conjugate axial points 0, 0' whose distances from 
the focal points F, F' are denoted by a, a', respectively, so 
that FO =a ; F'O' = a'; and if we put 

OM = z, 0'M' = z', 
then 

x = a-\-z, x' = a'-\-z'; 
and if these values of x and x' are substituted in the equa- 
tions 






348 Mirrors, Prisms and Lenses [§ 123 

we obtain : 

y'_ f _a'+z' 

Since a.a' =/./', the relation between z and z r may be put in 
the form: 

^'+1=0, 

Z Z 

where the constants are now a and a' instead of / and/'. 

Suppose, for example, that the pair of conjugate axial 
points 0, O' is identical with the pair of nodal points N, N'; 
then 

a = FO = FN = -/', a' = F'O' = F'N' = -/; 
so that the image-equations referred to the nodal points 
will have the following forms : 

1+1-1 = t = J-= z ^l 

z z f y z-f /' ' 

where z = NM, 2'=N'M'. 

d. The image-equations in terms of the refracting power 
and the reduced vergences (see §§ 105 and 106). 

The refracting power of the optical system is defined 
(§ 105) by the relations: 



/ T 

where n, n' denote the indices of refraction of the first and 
last media. Similarly, the reduced vergences (§ 106) with 
respect to the principal points are : 

u= n - w-%. 

U U 

If, therefore, in the image-equations referred to the prin- 
cipal points we eliminate /, /' and u, u' by means of these 
two pairs of formulae, we obtain the image-equations in the 
following exceedingly useful and convenient form: 

v' U' 

U' = U-\-F — = — . 

If the linear magnitudes are measured in terms of the meter 



§ 124] Magnification-Ratios 349 

as unit of length, the magnitudes denoted here by U, JJ' 
and F will all be expressed in dioptries (§ 107). 

124. The Magnification-Ratios and their Mutual Re- 
lations. — (a) The lateral magnification y. This has al- 
ready been defined as the ratio of conjugate line-segments 
lying in planes at right angles to the optical axis. The fol- 
lowing expressions were obtained for this ratio in § 123: 
_y' _f_x' _ f J'+v! _J.v! _n.u' _XJ . 
U y x /' J+u f f'.u n'.u U''' 

whence we see that the lateral magnification is a function 
of the abscissa of the object-point, and that in any optical 
system it may have any value from — oo to + oo depending 
on the position of the object. 

(b) The axial magnification or depth-ratio x. If x, x' de- 
note the abscissae with respect to the focal points of a pair 
of conjugate axial points, and if x-\-c, x f -\-c' denote the ab- 
scissae of another pair of such points immediately adjacent 
to the former, then, since 

x.x'=f.f = (x+c) (x'+c f ), 

and since moreover the product c.c' is a small magnitude of 

the second order as compared with either of the small factors 

c or c', and is therefore negligible, we find : 

c.x'-\-c.x = 0. 

The ratio c' : c of small conjugate segments of the axis 
is called the axial or depth-magnification. If this ratio is 
denoted by the symbol x, then, according to the equation 
above : 

C X x 2 ' 

so that, whereas the lateral magnification is inversely pro- 
portional to the abscissa x, the depth-magnification is inversely 
proportional to the square of x. In fact, the relation between 
the axial magnification and the lateral magnification may 
be expressed as follows : 

U 2 f n ' 



350 



Mirrors, Prisms and Lenses 



124 



The axial magnification or " depth-elongation' ' of a small 
object is proportional to the square of its lateral magnification. 
If, therefore, we take a series of ordinates, 1, 2, 3, 4, etc. 
(Fig. 167), all of equal height and at equal intervals apart 




Fig. 167. — Relation between axial or depth-magnification and lateral 

magnification. 

(like a row of telegraph poles), their images will be of un- 
equal heights and at unequal distances apart; but the in- 
tervals between the successive images will increase or di- 
minish far more rapidly than the corresponding changes in 
their heights. Accordingly, the image of a solid object can- 
not, in general, be similar to the object, but will be distorted, 
since the dimension parallel to the axis of the optical system 
is altered very much more than the dimensions at right angles 
to the axis. This uneven distribution of the images of ob- 



J 










^^^ 




r 


-"""^l 


"\^ 


Te 5 "^ 


"M ^ 


F H 




H' F' 


M'" 



Fig. 168. — Angular magnification or convergence-ratio. 

jects at different distances explains "the curious effect no- 
ticeable in modern binocular field-glasses of high power, 
but seen also in opera-glasses and telescopes, in which the 
successive planes of landscapes seem exaggerated, and flat- 
tened almost like the flat scenery of the theater. Thin trees 
and hedges, for example, seem to occupy definite planes; and 



Ch. X] Problems 351 

the more distant objects appear to be compressed up toward 
those in front of them " (Professor S. P. Thompson). 

(c) The angular magnification or so-called convergence- 
ratio z. If the slopes of conjugate rays are denoted by 0, 0', 
that is, if we put = ZFMJ, 0' = ZF'MT (Fig. 168), 
where M, M' designate the points where the ray crosses the 
axis in the object-space and image-space, respectively, and 
J and V designate the points where it crosses the primary 
and secondary focal planes, then evidently: 

tan0= lS' tan(? ' = 5FF- 

But the focal lengths are denned by the equations (§ 122): 
FT FJ 

' tan0' J tan0" 
and therefore : 

tan = —z- , tan 0' = -p- . 

Eliminating the intercepts FJ and FT, we obtain: 
= tan0_' = _x_ = _/ 
*~tan0~ /' x" 
where the ratio z=tan0' : tan (or 0' : 0) is called the 
angular magnification or the convergence-ratio. It is directly 
proportional to the abscissa x of the object-point M. 

The three magnification-ratios jc, y and z are connected 
by the following relation: 

JL =1 . 

x.z 



PROBLEMS 

1. Taking the index of refraction of water = |, show 
that the sun's rays passing through a globe of water, 6 inches 
in diameter, will be converged to a focus 6 inches from the 
center of the sphere. 

2. A small object is placed at a distance u from the nearer 
side of a solid refracting sphere of radius r and of refractive 



352 Mirrors, Prisms and Lenses [Ch. X 

index n. Show that the distance of the image from the other 
side of the sphere is 

,_ 2r(u — r) — n.u.r 
U ~2(n-l)u-(n-2)r' 
and find the lateral magnification. 

3. A luminous point is situated at the first focal point of 
an infinitely thin symmetric double convex lens made of 
glass (of index 1.5) and surrounded by air. The radius of 
each surface is 15 cm. Show that the image formed by rays 
which have been twice reflected in the interior of the lens 
before emerging again into the air will be on the other side 
of the lens at a distance of 2.5 cm. from it. 

4. An optical system is composed of two equal double 
convex lenses. The index of refraction of the glass is n = 
1.6202, and the radii, thicknesses, etc., are as follows: 

ri =-r 4 = 47.92243; r 3 = -r 2 = 9.39617; 
^ = ^3 = 0.2; d 2 = 2.4287. 
If an incident paraxial ray crosses the axis at a distance 
u\—— 7.31101 from the vertex of the first surface, show 
that the emergent ray will cross the axis at a distance u\ — 
33.65725 from the vertex of the last surface. 

5. A. Gleichen in his Lehrbuch der geometrischen Optik 
gives the following data of P. Goerz's "double anastigmat" 
photographic objective, composed of three cemented lenses, 
the first being a positive meniscus of crown glass, the second 
a double concave flint glass lens, and the third a double con- 
vex crown glass lens : 

Indices of refraction: 

m = n b = l; n 2 = 1.5117; w 3 = 1.5478; n 4 = 1.6125 
Radii: 
n = - 0. 128965 ; r 2 = - 0.049597 ; r 3 = +0. 196423 ; 
r 4 = -0.1266629 
Thicknesses: 
d l= +0.01277; d 2 = +0.00664; d,= +0.02114. 
Show that the second focal point of this system is at a dis- 
tance of +1.111095 from the vertex of the last surface. (See 



Ch. X] Problems 353 

scheme for calculation of paraxial ray through a centered 
system of spherical refracting surfaces, § 181). 

6. Define the nodal points N, N' and show that FN= -/', 
F'N' = — /, where F, F' designate the positions of the focal 
points and /, /' denote the focal lengths of the optical system. 
Under what circumstances are the nodal points identical 
with the principal points? 

7. Derive the image-equations referred to the principal 
points. 

8. Given the positions on the optical axis of the principal 
points and of the focal points; construct the nodal points. 
Also, construct the point Q' conjugate to a given object- 
point Q. Draw diagrams for convergent and divergent 
systems. 

9. Prove that 

n'./+w./' = 0, 
where / and /' denote the focal lengths of the optical system, 
and n and n' denote the indices of refraction of the first and 
last media. 

10. A small cube is placed on the axis of a symmetrical 
optical instrument with one pair of its faces perpendicular 
to the axis. Find the two places where the image of the cube 
will also be a cube. (Assume that the instrument is sur- 
rounded by the same medium on both sides.) 

Ans. At the points for which the lateral magnification is 
+ lor -1. 

11. An object is placed 3 inches in front of the primary 
focal plane of a convergent optical system. Show that the 
image will be one-and-a-half times as large as it was at first 
if a plate of glass (n = 1.5) of thickness 3 inches is interposed 
in front of the object. 

12. Show that the axial magnification at the nodal points 
has the same value as the lateral magnification in the nodal 
planes. 

13. A symmetrical optical instrument is surrounded by 
the same medium on both sides. If the images of two small 



354 Mirrors, Prisms and Lenses [Ch. X 

objects A and B on the axis are formed at A' and B', show 
that the ratio of A'B' to AB is equal to the product of the 
lateral magnifications for the pairs of conjugate points A, A' 
and B, B'. 

14. Show that in a symmetrical optical instrument there 
are two pairs of conjugate points on the axis for which an 
infinitely small axial displacement of the object will cor- 
respond to an equal displacement of the image ; and that the 
focal points are midway between these points. 

15. Show that in a symmetrical optical instrument sur- 
rounded by the same medium on both sides there are two 
points on the axis where object and image will be in the same 
plane; and that if a denotes the distance between the prin- 
cipal planes, the distance between t hese two points will be 

Va(a+4f). 

16. In a centered system of m spherical refracting surfaces 
the vertex of the &th surface is designated by A k . A par- 
axial ray crosses the axis before refraction at the first surface 
at a point Mi which coincides with the primary focal point F 
of the optical system. Before and after refraction at the 
fcth surface this ray crosses the axis at M k and M k+ i, re- 
spectively. If we put w k = A k M k , w k ' = A k M k+ i, show that 

u 2 .u s . . . u m „ A 

U\.U<i. . . W m -1 

where / denotes the primary focal length of the optical 
system. 

17. If the symbols w k , w k , employed in the same sense as 
in the preceding problem, refer to a paraxial ray which is 
incident on the first surface of the system in a direction 
parallel to the optical axis, show that 

71 u\.U 2 . . . U^ ,_ U V U 2 . ■ » ^m 

n' U2.U3. . . U m ' U2.U3. . . u m ' 

where /, /' denote the focal lengths of the system and n, n' 
denote the indices of refraction of the first and last media. 

18. Employing the formulae of No. 17, determine the focal 
lengths of a hemispherical lens of glass of refractive index 



Ch. X] Problems 355 

1.5; and find the positions of the principal planes and the 
focal planes. 

Ans. If r denotes the radius of the curved surface, and if 
distances are measured from the vertex of this surface, the 
distances of the focal points are — 2r and +7r/3, and the 
distances of the principal points are and +r/3. The focal 
length is twice the length of the radius. 

19. If a paraxial ray, proceeding originally in a direction 
parallel to the axis of a centered system of spherical refract- 
ing surfaces (as in No. 17), crosses the axis in the medium of 
index n k at a point M k whose distance from the vertex of 
the kih surface is w k = A k M k (U k = n k /u k ), show that 

Fhk = Fhk-i (U k +F k ) ( — — — — — — -), 

Vc7k-i+/'k-i n k I 

where F k denotes the refracting power of the A;th surface, 

Fi,k denotes the refracting power of the system of surfaces 

bounded by the 1st and kth. inclusive {F\,\ = Fi and Fi, = 0), 

and d k _i = A k _i A k denotes the axial thickness between the 

surfaces bounding the medium of index n k . 



CHAPTER XI 

COMPOUND SYSTEMS. THICK LENSES AND COMBINATIONS 
OF LENSES AND MIRRORS 



125. Formulae for Combination of Two Optical Systems 
in terms of the Focal Lengths. — Suppose that the optical 
system consists of two parts I and II, each composed of 
a centered system of spherical refracting surfaces with their 
optical axes in the same straight line. On a straight line 
parallel to this common optical axis take two points P, P' 
(Fig. 169), which we shall assume to be a pair of conjugate 
points with respect to the compound system (I +11); and 



X' 



P Vi' 


I 


V' K z 


JL 


Kl P 




\^\ H-, 




HI \ / 




yf 




x I 


I f\ f,\^ 




F{\ /F ? H 2 




H^F'/F' H' 


\ w, 








<H 7 

L' 2 



Fig. 169. — Combination of two optical systems. Letters with subscripts 
refer to component systems; letters without subscripts refer to com- 
pound or resultant system. 

since these points are on the same side of the optical axis 
and at equal distances from it, evidently, they must lie in 
the principal planes of the compound system (§ 119). Ac- 
cordingly, the feet of the perpendiculars drawn from P, P' 
to the optical axis will be the pair of principal points H, H' 
of the compound system. 

On the optical axis select a point Fi for the position of 

356 



§ 125] Combination of Two Optical Systems 357 

the primary focal point of system I ; and select also the posi- 
tions of the principal points Hi, Hi' and H 2 , H 2 ' of systems 
I and II, respectively. Through Fi draw the straight line 
PWi meeting the primary principal plane of system I in the 
point Wi; take Hi'Wi' = HiWi, and draw the straight line 
Wi'G 2 parallel to the axis meeting the primary principal 
plane of system II in the point G 2 ; take H 2 'G 2 ' = H 2 G 2 , and 
draw the straight line G 2 / P / , which must necessarily cross 
the optical axis at the secondary focal point F 2 ' of system II. 

Let the straight line drawn through P parallel to the op- 
tical axis meet the primary and secondary principal planes 
of system I in the points designated by Vi and Vi', respec- 
tively; and select a point on the optical axis for the position 
of the secondary focal point F/ of system I. Through Fi' 
draw the straight line V/Fi' meeting the primary principal 
plane of system II in L 2 ; take H 2 'L 2 ' = H 2 L 2 , and draw the 
straight line L 2 'P', which will cross the optical axis in the 
secondary focal point F' of the compound system. 

Let the straight line drawn through P' parallel to the op- 
tical axis meet the primary and secondary principal planes 
of system II in the points K 2 and K 2 ', respectively ; and let 
O designate the point of intersection of the pair of straight 
lines W/G 2 and V/L 2 . The point where the straight line K 2 
crosses the optical axis will be the position of the primary 
focal point F 2 of system II. Let the straight line K 2 F 2 meet 
the secondary principal plane of system I in the point T/, 
and take HiTi = Hi'T/; then the straight line PTi will cross 
the optical axis at the primary focal point F of the com- 
pound system. 

The diagram constructed according to the above direc- 
tions represents a perfectly general case. The focal lengths 
of the component systems are: /i = FiHi, /i / = Fi / H/ and 
/ 2 = F 2 H 2 , / 2 / =F 2 / H 2 / ; and the focal lengths of the compound 
system are: /=FH, /'=F'H'. The step from the secondary 
focal point of the first system to the primary focal point of 
the system will be denoted by the symbolA; thus, A = Fi'F 2 . 



358 Mirrors, Prisms and Lenses [§ 125 

Now if we know the positions on the optical axis of the 
focal points Fi, Fi'and F 2 , F 2 ' of the two component systems, 
together with the values of the focal lengths f h // and / 2 , / 2 ', 
it is easy to calculate the positions of the focal points F, F' 
and the values of the focal lengths /, /' of the compound 
system; as will now be shown. 

The position of the primary focal point F of the compound 
system may be found from the fact that F and F 2 are a pair 
of conjugate axial points with respect to system I, and hence 
(§123, a); 

FiF. FiTi-A/i'. 
And, similarly, the position of the secondary focal point F' 
may be found from the fact that Fi' and F' are a pair of con- 
jugate points with respect to system II, so that 

F/"EV 77* T7I / i* J* / 
2 r .r 2 r i =J 2 .J 2 . 

Accordingly, the positions of the focal points F, F' with re- 
spect to the known points Fi, F 2 ', respectively, are given by 
the following f ormulse : 





FxF^f 1 ', 


F 2 'F'= 


hU 






A 




A 




In order to finti the focal lengths 


/, /', we may 


proceed 


as follows : 










In the similar triangles FHP, FHil 


\ we have: 






FH 


HP 








FHi 


"HiTi ' 






and since 












HP=H 2 K 2 , 


HiTi- 


-HxT/, 





the proportion above may be written: 

FH H 2 K 2 

FHi Hi'Ti'' 
Now from the similar triangles F 2 H 2 K 2 , F 2 H 1 , T/ we have 



H 2 K 2 F 2 H 2 
H?T7 = F 2 Hi' ; 



§ 125] Combination of Two Optical Systems 359 

and hence: 

F H=g|.FH, 

Now FH 1 =FF 1 +F 1 H 1 =-^+/ 1 =-£(/ 1 '-A); 

and F 2 H 1 '=F 2 F 1 , -fFi , H 1 '=/ 1 , -A. 
Accordingly, putting FH=/, F 2 H 2 =/ 2 , we obtain: 

f /i >h 

} ~ ~a~' 

whereby the primary focal length of the compound system 
may be calculated. 

Similarly, from the figure we obtain the relations: 
FTT _ IFF H/Vi' ^ F/H/ . 
F , H 2 , ~H 2 , L 2 ,_ H 2 L 2 ~ Fi'H 2 ' 
and since F'H' =/', F^H/ =//, 

, FH a / -FF,'+F, / H,'-^+/» / -^(/«+A) l 

A A 

F/H 2 = Fi F 2 -r-F 2 H 2 =/ 2 -f-A, 
we obtain an analogous expression for the secondary focal 
length of the compound system, as follows : 

J A ' 

By varying the interval A, which is the common denom- 
inator of all these expressions, it is obvious that it is possible 
with two given component systems to obtain combinations 
of widely different optical effects. In particular, when 
Fi' coincides with F 2 , so that the interval A vanishes, the 
focal points F, F' will be situated both at infinity, so that 
the focal lengths /, /' will be infinite also. This is the case, 
for example, with the optical instrument known as the tele- 
scope; and, accordingly, any optical system which trans- 
forms a cylindrical bundle of parallel rays into another 
cylindrical bundle of parallel rays is called a telescopic (or 
afocal) system. The simplest illustration of such a system 
is afforded by a single plane refracting surface or by a plane 
mirror. 



360 Mirrors, Prisms and Lenses [§ 126 

126. Formulae for Combination of Two Optical Systems 
in terms of the Refracting Powers. — Although the formulae 
derived in the preceding section are very simple and con- 
venient, Gullstrand's system of formulae in terms of the 
refracting powers possesses certain advantages and is even 
more useful. The latter formulae may be derived immedi- 
ately from the former, as will now be shown. 

In Gullstrand's system the interval between the two 

component optical sj^stems is expressed, not by A, but by 

the reduced distance (§ 104) c of the primary principal point 

H 2 of system II from the secondary principal point H/ of 

system I. Thus, if n h n 2 and n 2 , n^ denote the indices of 

refraction of the first and last media of systems I and II, 

respectively, then 

H/H 2 
c = . 

n 2 

The connection between the two magnitudes c and A is 
easily obtained; for since 

F 1 T 2 =F 1 , H/+H 1 , H 2 +H 2 F 2 , 
we find immediately: 

A=/i'+n 2 .c-/ 2 . 
Now let us introduce the following symbols: 

/i /i h h J J 

where F\, F 2 denote, therefore, the refracting powers of the 
component systems and F denotes the refracting power of 
the compound system (§§ 105 and 123, d). Hence, since 



/l== ~Fi J /2= F 2 



we may write: 



Now if this value of A is substituted in either of the formulae 
J ~ "A" J J " A ' 



§ 126] Combination of Two Optical Systems 361 

and if the focal lengths are expressed in terms of the refract- 
ing powers, we find : 

F=F l +F 2 -c.F l .F 2 ; 
which is Gullstrand's formula for the refracting power 
of the compound system in terms of the refracting powers 
of the two component systems and of the interval c between 
them. 

Likewise, if in the formulae 

-p -p _ /i*/i -p /-p/ _ _ h-h 
A ' 2 A 

F 

we eliminate /i, // and f 2 , // and put A= — n 2 ri „ , we ob- 

t \.t 2 

tain for the reduced steps FiF and F 2 / F / the following ex- 
pressions: 

FiF = J^_ FVF = F± 
m ~F.Fi 7i3 " F.F 2 

The positions of the focal points F, F' of the compound sys- 
tem with respect to Hi, H 2 ', respectively are obtained as 
follows : 

H X F = H1F1+F1F = FiF - m/Fi, 
H 2 'F' = H 2 , F 2 , +F 2 , F / =F 2 'F , +n 3 /> 2 ; 
and if herein the values of FiF and F 2 'F' are substituted, 
and if also we note that 

F-F^F^l-c.Fi), F-F 2 =Fi(l-c.F 2 ), 
we obtain finally: 

HiF l-c.F 2 H 2 'F' 1-c.ffi 
m F ' m ~ F ' 

Moreover, since 

H 1 H = H 1 F+FH = HiF+m/F, 
H 2 'H' = H 2 'F'+F'H' = WF'+rh/F, 
the Gullstrand system of formulae for the combination of 
two optical systems may be written as follows : 
HiH F 2 H 2 H F\ 

~W = ~F' C} ~^~ = ~F X ' 
F=F l +F 2 -c.F l .F 2 . 
Accordingly, if the positions of the principal points Hi, H/ 



362 Mirrors, Prisms and Lenses [§ 127 

and H 2 , H 2 ' of the two component systems, the refracting 
powers F\y F 2 and the indices of refraction m, n 2 and n 3 are 
known, we can calculate the reduced interval c and find the 
refracting power F of the compound system and the posi- 
tions of the principal points H, H'. We shall see numerous 
applications of these formulae in the succeeding sections of 
this chapter. 

127. Thick Lenses Bounded by Spherical Surfaces. — 
When a centered system of spherical refracting surfaces con- 
sists of two surfaces, it constitutes a spherical lens involving 
three media, viz., the medium of the incident rays (ni), the 
medium comprised between the two spherical surfaces, 
sometimes called the lens-medium (n 2 ), and the medium of 
the emergent rays (n 3 ), which is generally but not necessarily 
the same as that of the incident rays. Usually, a lens is de- 
scribed by assigning the values of the three indices of re- 
fraction and the positions of the centers Ci, C 2 and the ver- 
tices Ai, A 2 on the optical axis; the usual data being the 
radii ri=AiCi, r 2 = A 2 C 2 and the thickness d=A\A 2 . The 
lens may be regarded, therefore, as a combination of two 
spherical refracting surfaces whose refracting powers Fi, F 2 
are given by the formulae (§ 105) 

v fh-ni „ n 3 -n 2 
r \ = , r 2 = . 

n r 2 

Since the principal points of a spherical refracting surface 
coincide with each other at the vertex of the surface (§§ 81 

and 119), the interval c= — - — -= * * , and therefore 

n 2 n 2 

d 
c= — . 

n 2 

Accordingly, if, by way of abbreviation, we introduce the 
special symbol 

N = n 2 \ (n 2 -ny)r 2 -(n 2 -n 3 )ri}+(n 2 -n 3 )(n 2 -ni)d 

to denote a constant of the lens, we obtain, by substituting 



§ 127] Thick Lens Formulae 363 

the values of F h F 2 and c in the formula F=Fi+F 2 — c.Fi.F 2 , 
the following expression for the refracting power F of a lens: 

n2.r1.r2 

where the value of F will be given in dioptries in case the 

distances n, r 2 and d are all measured in meters (§ 107). 

The positions of the principal points (H, H') of a lens are 

determined in the same way by the formulae : 

AiH n 2 -n 3 , A 2 H' n 2 -ni , 
= ^7— n.d, = ^r— r 2 .d; 

and the positions of the focal points (F, F') may likewise be 
calculated from the following expressions: 



— = -^{712.7-24 



A 2 F' r 2 



= — \n 2 .r\ — (712 — n{) d 



713 N % 

When, as is usually the case, the lens is surrounded by the 
same medium on both sides, we may put 

m = n 3 = n, n 2 =n'; 
and then the above formulae become : 

N = (n' - n) { n' (r 2 - r 1) + (n' - n) d } ; 

N 



F = 



n .n.r2 



AiH n' — n , A 2 H' n f — n 1 

= Tf- n.d, = - - 1 r f - r 2 .d; 

n N n N 

AiF n\ , ft NJ ] A 2 F' r 2 \ , f , NJ 

- ^ n'.r 2 +(n f -n)d\, — — =~\ n'.n- {n'—n) d 



n N{ j n N [ 

The nodal points (N, N r ) of a lens surrounded by the same 
medium on both sides coincide with the principal points 
(§ 122). 

The positions of the focal points and principal points 
may be exhibited in the case of a thick convergent lens in 
the following manner, as described in Grimsehl's Handbuch 
der Physik: 

Two thin piano-lenses, each 4 cm. in diameter, are ce- 
mented with Canada balsam to the opposite faces of a glass 
























\ 








\ 






F 




_A^ 


H 


T^^— 1 


A 2 ' 






(b) / 




















i== ^ z ^_ 














F ^ 




_Ai| 


H 




H' 


A2^-— 


— ^F 










(c)/ 


















Fig. 170, a, 6, c, and d. — Double convex lens: (a) Location of second focal 
point (F') and principal point (H') ; (6) Location of first focal point (F) 
and principal point (H) ; (c) Location of focal points (F, F') and principal 
points (H, H'). (d) Meniscus convex lens: location of principal points 
and focal points, showing their unsymmetrical positions with respect to 
the surfaces of the lens. 



§128] "Vertex Refraction" of Lens 365 

cube of edge 4 cm. and made of the same glass, so as to form 
a thick symmetric double convex lens, as represented in 
Fig. 170, a, b and c. A diaphragm with three parallel horizon- 
tal slits is placed in the path of a cylindrical beam of parallel 
rays so as to separate it into three smaller beams, and the 
lens is adjusted so that the middle beam proceeds along 
the axis of the lens. The paths of the rays in air can be 
rendered visible by tobacco-smoke and may be photo- 
graphed. In this way figures will be obtained similar to 
those shown in the diagrams. The position of the second- 
ary focal point F' is shown by the point of convergence of 
the rays on emergence (Fig. 170, a). A point in the second 
principal plane of the lens may be located by rinding the 
point of intersection of an incident ray parallel to the axis 
with the corresponding emergent ray (§ 119), as indicated 
by the dotted lines in the figure; and the second principal 
point H' will be at the foot of the perpendicular dropped 
from this point on to the axis. If the rays are sent through 
the lens from the opposite side (that is, from right to left in 
the drawing, Fig. 170, 6), they will intersect on emergence 
in the primary focal point F; and the position of the primary 
principal point H may be found in exactly the same way 
as above. The two diagrams Figs. 170, a and b, are com- 
bined in one in Fig. 170, c. In Fig. 170, d, the lens is con- 
cave towards the incident light and convex when viewed 
from the other side; and this figure shows very clearly how 
the focal points F, F' and the principal points H, H' may be 
both unsymmetrically placed with respect to the lens, al- 
though here also we have, as before, FH = H'F'. 

128. So-called " Vertex Refraction " of a Thick Lens — 
The step from the second vertex (A 2 ) of a lens to the second 
focal point (F'), which may be denoted by v, is sometimes 
called the "back focus" of the lens; that is, v=A 2 ~F'. If 
the lens is surrounded by the same medium (n) on both 
sides, then v/n = (l — c.Fi)/F, where F denotes the refract- 
ing power of the lens, F\ denotes the refracting power of 



366 Mirrors, Prisms and Lenses [§ 129 

the first surface, and c=d\n' denotes the reduced thickness. 
The reciprocal of this magnitude v/n is called the vertex re- 
fraction of the lens ( — = VJ and its relation to the re- 
fracting power is given by the formula: 

F F 



V = 



1-c.Fi n'—n d 



ri n 
If F is given in dioptries, the values of d and n must be ex- 
pressed in meters; and then the expression above will give 
the value of V in dioptries. The importance of this function 
V in the theory of modern spectacle lenses has been pointed 
out by Von Rohr; it is measured from the second face of the 
lens because that is the side next the eye. When a lens 
(with spherical surfaces) is reversed by turning it through 
180° around any line perpendicular to its axis, the refracting 
power F remains the same, whereas the vertex refraction V 
will be different unless the lens is a symmetric lens or in- 
finitely thin, in which latter case d = and V—F. Thus, 
whereas the refracting power of a lens is the same whether 
the light traverses it from one side or the other, the vertex 
refraction depends essentially on which side of the lens is 
presented to the incident rays. 

129. Combination of Two Lenses. — Let us take the sim- 
plest case, and suppose that the system is composed of two 
infinitely thin co-axial lenses, each surrounded by air. Let 
Ai and A 2 designate the points where the optical axis meets 
the two lenses, and let the interval between them be denoted 
by c; that is, put c = AiA 2 . Since the principal points of 
an infinitely thin lens coincide with each other at the 
point A where the axis crosses the lens, and since the inter- 
vening medium is assumed to be air of index unity, this 
distance c has here the same meaning as the reduced in- 
terval c = Hi'H 2 /n 2 in the general formulae of § 126. Ac- 
cordingly, we may write immediately the following system 
of formulae for a combination of two thin lenses of refracting 



129] 



Combination of Two Lenses 



367 



powers Fi, F 2 , surrounded on both sides by air and sepa- 
rated by the distance c: 

F=Fl +F 2 - c .F 1 .F 2 ; 

AiH=-^ r> A 2 H'=— -^-; 



AiF=- 



1-c.Fo 



A.F- 1 "^ 1 



F ' "" F 

These formulae may also be expressed in terms of the focal 
lengths /i and/ 2 , as follows: 

f _ Mi 

J /1+/2-C' 

AM-U A&—U A lF =-^p^, A 2 F-^p). 

J2 /l J2 /l 

The positions of the focal points F, F' and the princi- 
K J L 




Fig. 171, a. — Combination of two thin lenses. Graphical method 
of determining the positions of the first focal point (F) and 
principal point (H) : Case when both lenses are convex. 

pal points H, H' of a combination of two infinitely thin 
lenses surrounded by air may be constructed geometrically 
as follows : 



368 



Mirrors, Prisms and Lenses 



[§129 



Draw a straight line to represent the common axis of the 
pair of thin lenses, and mark the points Ai and A 2 (Fig. 171, 
a, b, and c) where the axis crosses the lenses, and also the 
positions of the primary focal points Fi and F 2 . Through 
F 2 draw a straight line perpendicular to the axis, and take 
on it a point K such that F 2 K = F 2 A 2 =/2; this point K lying 




Fig. 171, b. — Combination of two thin lenses. Graphical method 
of determining the positions of the first focal point (F) and 
principal point (H) : Case when first lens is concave and second 
lens convex. 

above or below the axis according as the second lens is con- 
vex or concave, respectively. Through K draw a straight 
line parallel to the axis and through Ai a straight line per- 
pendicular to the axis; and let L designate the point where 
these two lines intersect. Moreover, let P designate the 
point of intersection of the pair of straight lines LFi and KAi. 
The foot of the perpendicular let fall from P on to the axis 
will be the primary focal point F of the compound system; 
and the ordinate FP will be equal to the primary focal length 
f of the compound system; and hence if the quadrant of a 



129] 



Combination of Two Lenses 



369 



circle is described around F as center with radius FP, it will 
cut the axis at the primary principal point H, which lies to 
the right or left of F according as the point P falls above or 
below the axis. 

According to this construction, the points P and K are 
a pair of conjugate extra-axial points with respect to the 




Fig. 171, c. — Combination of two thin lenses. Graphical method of de- 
termining the positions of the first focal point (F) and principal point 
(H) : Case when first lens is convex and second lens concave. 

first lens; so that the construction really consists in locating 
the object-point P which is imaged by the first lens in the 
point K. This will help the student to remember the con- 
struction. 

In order to show that the construction is correct, let J 
designate the point of intersection of the pair of straight 
lines FP and LK. Then since JP and FP are corresponding 
altitudes of the similar triangles PLK and PFiAi, we have: 
JP = L K ^ AiF 2 = AiA 2 +A 2 F 2 ^ c-/ 2 
FP"FiArFiAi" FiAi /i 

Now JP=JF+FP=KF 2 +FP = FP-/ 2 , and therefore: 
FP-/ 2 c-/ 2 
FP /i ' 
and if this equation is solved for FP, we find : 



FP 



/1+/2-C 



■/, 



370 Mirrors, Prisms and Lenses [§ 130 

in agreement with the formula found above. Moreover , in 
the similar triangles AiFP and AiF 2 K, 

AiF = A x F 2 . 

FP ~F 2 K ; 
and since AiF 2 = c-/ 2 , F 2 K=/ 2 , FP=/, we find: 

which is likewise in agreement with the formula found above. 

Similarly, mark the positions of the secondary focal points 
F/ and F 2 ', and through F/ draw a straight line perpendic- 
ular to the optical axis, and take on it a point O such that 
Fi'O = F/Ai =//. Through O draw the straight line OR par- 
allel to the axis, and through A 2 a straight line perpendicular 
to the axis; and let R designate the point where these two 
lines intersect. Then if Q designates the point of intersec- 
tion of the straight lines F 2 'R and A 2 0, that is, if Q is the 
image of O in the second lens, the secondary focal point F' of 
the combination will be at the foot of the perpendicular 
drawn from Q to the optical axis, and the secondary prin- 
cipal point H' will lie on the axis at a distance F'H^F'Q. 
This construction may be proved in a manner entirely an- 
alogous to the proof given above. 

130. Optical Constants of Gullstrand's Schematic Eye.— 
As a further illustration of the use of the formulae for the 

CORNEA , ^^ , 




AQUEOUS / / CORE \ \VITREOUS 




■"3 \ txA tic, 



Fig. 172. — Schematic eye. 




combination of two optical systems, let us apply them to 
the calculations of the refracting power (F) of the human 
eye, together with the positions of the principal points (H,H') 



130] 



Constants of Schematic Eye 



371 



and the focal points (F, F'). For this purpose we shall use 
the data of Gullstrand's schematic eye (in its passive 
state, accommodation entirely relaxed) which are given 
in the third edition of Helmholtz's Handbuch der physiolo- 
gischen Optik, Bd. I (Hamburg u. Leipzig, 1909), pages 300 
and 301, as follows (see Fig. 172) : 

Indices of refraction: 

Cornea n 2 = 1.376 

Aqueous and vitreous humors n 3 = n 7 = 1.336 

Lens n 4 = n 6 = 1 .386 

Lens-core n 5 = 1.406 

Position of ssur faces: 

mm. 



Posterior surface of cornea : 


AiA 2 = 0.5 


Anterior surface of lens : 


AiA 3 = 3.6 


n 


u 


" lens-core: 


AiA 4 =4.146 


Posterior 


a 


" lens-core: 


AiA 5 = 6.565 


a 


u 


" lens: 


AiA 6 = 7.2 


Radii of surfaces: 






Anterior surface of cornea : 


ri=+ 7.7 


Posterior 


a 


a a 


r 2 = + 6.8 


Anterior 


a 


lens: 


r 3 =+10.0 


it 


a 


lens-core: 


r 4 =+ 7.911 


Posterior 


a 


u a a 


r 5 =- 5.76 


a 


it 


lens: 


r 6 =- 6.0 



mm. 



Consider, first, the cornea-system composed of the an- 
terior and posterior surfaces of the cornea. The refracting 
power of the anterior surface is : 

n 2 -ni 



Fi = 



n 



+48.831 dptr.; 



and that of the posterior surface is : 



F 2 = 



n% — n<i 
r 2 



= -5.882 dptr. 



The reduced interval between the two surfaces is: 

AxA 2 0.0005 



Cl = 



m 



1.376 



372 Mirrors, Prisms and Lenses [§ 130 

Hence, if Fn denotes the refracting power of the cornea- 
system, where 

Fi2=Fi+F 2 — C1.F1.F2, 
we find: 

F12 = +43.053 dptr. 
The positions of the principal points of the cornea-system 
are given by the formulae: 

AiHi2 _ Ci.F a A 2 Hi 2 ' g.Fi # 

m ~ Fu ' n 3 Fn ' 

whence we find: 

A1H12 = - 0.0496 mm., AiHi 2 ' = - 0.0506 mm. 
The lens-system is composed of four refracting surfaces. 
The first two surfaces form the so-called anterior cortex and 
the last two surfaces the posterior cortex. The refracting 
power of the anterior surface of the lens is : 

ft .*=*.. +5 dptr.; 

and that of the anterior surface of the lens-core is : 

F J*Zl** +2.528 dptr. 

r 4 

The reduced interval between these two surfaces is 
A3A4 ^ 0.000546 
C3 ~ ru ~ 1.386 ' 
Hence, if F 34 denotes the refracting power of the combina- 
tion, that is, if 

Fs^=F 3 +F4 — C3.F3.F4, 
we find: F 34 =4-7.523 dptr. 

If the principal points of the anterior cortex are designated 
by H34, H 34 ', then 

A3H34 C3.F4 A 4 H 34 / = C3.F3 . 
m F34 ' nh F 3 4 ' 

whence we obtain: 

AiH 34 = +3.777 mm., A1H34' = +3.778 mm. 
so that the principal points of the anterior cortex are coin- 
cident with each other, 



§ 130] Constants of Schematic Eye 373 

Proceeding in the same way with the posterior cortex, we 
have: 

F 5 =^^- 5 = +3.472 dptr., F 6 =^^- 6 = +8.333 dptr., 
rb r& 

_ A 5 A 6 _ 0.000635 . 

C5 ~ n 6 1.386 ' 

and hence if 

F 66 = F 5 +F 6 ~C & .F 5 .F 6) 

we find : F b6 = + 1 1 .792 dptr. 
Moreover, since 

rib FbQ ' rn Fb& 

we have finally for the positions of the principal points of 
the posterior cortex: 

AiH 56 = +7.0202 mm., AiH 56 ' = +7.0198 mm. ; 
so that H 56 and H 56 ' may also be regarded as coincident. 

If the refracting power of the lens-system as a whole is 
denoted by L, then 

L =7^34+^56^5.^34-^56, 

where 

_ H 84 H 56 / __ 0.0032422 . 
S ~ rib ""1.406 
and if P, P' designate the principal points of the lens-system, 
then 

B^jJF^ Hse'P' ^ 5.^34 

7i3 L ni L 

Accordingly, we find: 

L= +19.110 dptr.; 
AiP= +5.6780 mm., AiP' = +5.8070 mm. 
Lastly, combining the cornea-system and the lens-system, 
we obtain for the refracting power of the entire optical sys- 
tem of the eye : 

F=F l2 +L-c.F 12 .L, 
where 

== Hi2 / P ^ 0.0057285 
C ~ m 1.336 ' 



374 Mirrors, Prisms and Lenses [§ 131 

Also, 

Hi2H = cX P / H / = _c.Fi 2 
m F m F ' 

where H, H' designate the positions of the principal points of 
the eye. Thus, we find : 

F =+58.64 dptr.; 
AiH= +1.348 mm., AiH' = +1.602 mm. 
If the focal lengths of the eye are denoted by / and /', then, 
since f= rii/ F and /' = - n 7 /F, we obtain : 

/= +17.055 mm., /' = -22.785 mm. 
The focal points F, F' are located as follows : 

AiF = - 15.707 mm., AiF' = +24.387 mm. 
In Gullstrand's schematic eye the length of the eyeball 
is taken as 24 mm., and therefore the second focal point F' 
is not on the retina but 0.387 mm. beyond it; so that the 
schematic eye is not emmetropic but hypermetropic (see 
§ 153) to the extent of 1 dptr. 

131. Combination of Three Optical Systems. — It is fre- 
quently the case, especially in problems connected with 
physiological optics, that we desire to find the resultant of 
three co-axial optical systems of known refracting powers 
Fi 9 Fi and F s separated by given intervals c h c 2 , where 

Hi H2 Ho H3 

Ci = , c 2 = , 

the principal points of the component systems being desig- 
nated by Hi, Hi'; H 2 , H 2 '; and H 3 , H 3 '. The indices of re- 
fraction of the first and last media of system I are denoted 
by nit n 2 ; °f system II by ti 2 , ^3,' and of system III by n 3 , n 4 . 

Here let us employ the symbol D to denote the refracting 
power of the compound system (I +11), and the letters G, G' 
to designate the positions of the principal points of this par- 
tial combination. Evidently, according to the formulae 
derived in § 126, we may write: 

D=F 1 +F 2 -c 1 .F 1 .F 2 ; 
HiG = c ± F 2 H 2 / G^ c1.F1 

Til D ' • 713 D 



§ 131] Combination of Three Optical Systems 375 

Now let F denote the refracting power of the combination 
of systems I, II and III, and let H, H' designate the posi- 
tions of the principal points of this compound system. Then 
if the reduced interval between (I +11) and III is denoted 
by k, that is, if 

G'H 3 



k = 



nz 



then also 



Since 



we find: 



F = D+F z -k.D.F 3 , 

GH k.F 3 H S 'K' k.D 

m F ' m " F 

Gr H3 Cj H2 H2 H 3 

nz nz nz 

k= c 1 .F 1 +c 2 .D 



D 

If now these equations are combined so as to eliminate D 
and k, the following system of formulae for the combination of 
three optical systems will be obtained finally : 

F=Fi(l - c 2 .F 3 ) +F 2 (1 - c1.F1) (1 - c 2 F,) +F 8 (1 - C1.F1) ; 
HiH ci C2.F3-C1.F1 . 

m 1-ci.Fi F(l-ci.Fi)' 
H 3 / H / = _ c 2 c 2.F3-c1.F1 

7i4 I-C2.F3 F(1-C2.F3) ' 

In the special case when the compound system is symmet- 
rical with respect to system II, that is, when n 3 = n 2 and n± = n\ 
and c 2 = ci = c and F 3 =Fi, the formulae above will be simpli- 
fied as follows: 

F=(l-c.F0 (2Fi+F 2 -c.Fi.F 2 ), 
HiH = HTV = c 
ni n\ 1 - c.Fi ' 

Thus, if an optical system is symmetrical with respect to 
a middle component part of the system, the principal 
points (H, H') will be symmetrically placed, and their posi- 
tions will be independent of the refracting power F 2 of the 



376 Mirrors, Prisms and Lenses [§ 132 

middle system. These latter formulae should be compared 
with the formulae for a " thick mirror" to be developed in 
the following section. 

132. " Thick Mirror." — The general formulae which have 
been derived in this chapter are applicable also when 
the centered system of spherical surfaces includes one or 
more reflecting surfaces, provided that reflection is treated 
as a special case of refraction, according to the method ex- 
plained in § 75. Thus, for example, if the rays are reflected 
at the kth surface of the system, we must put n k+1 = -n k ; 
and, consequently, if the reflecting power of this surface is 
denoted by F k , we shall have F k = n k /f k =n k /f k , in accord- 
ance with the characteristic requirement that the focal 
lengths of a spherical mirror are identical, that is, /=/' 
(see §77). 

A special case of much interest and practical importance 
occurs when the last surface of the system acts as a mirror, 
the rays of light arriving there being reflected back through 
the system as so to emerge finally at the first surface into 
the medium of index n\ where they originated. For ex- 
ample, this happens always in the case of an ordinary glass 
mirror which is silvered at the back. The rays return into 
the air in front of a mirror of this kind after having twice 
traversed the thickness of the glass, and the failure to take 
account of the refractions from air to glass and from glass 
to air is sometimes responsible for serious errors in the 
measurement of the focal length of a glass mirror silvered 
at the back. The image produced by rays which have been 
partially reflected from the second surface of an ordinary 
lens is often very disturbing, although the intensity of the 
reflected fight is usually comparatively feeble unless the 
second surface of the lens has been silvered. 

The name "thick mirror" has been applied by Dr. Searle* 
to any combination of centered spherical refracting surfaces 

* G. F. C. Searle: The determination of the focal length of a thick 
mirror. Proc. Cambr. Phil. Soc, xviii, Part iii, 1915, 115-126. 



132] 



" Thick Mirror 



377 



wherein the rays are supposed to be reflected at the last sur- 
face and to return through the system # in the opposite sense. 
It may easily be shown that a " thick mirror" as thus de- 
fined acts exactly like a single spherical reflecting surface 
(or "thin mirror," as we may calhit, having in mind a cer- 
tain analogy which exists here between lenses and mirrors), 
whose vertex and center have perfectly definite and calcu- 
lable positions depending on the constants of the "thick 
mirror." This is proved by Dr. Searle in a simple manner 
as follows : 

In Fig. 173 the system is represented as consisting of 
three spherical surfaces, the first two forming a thick lens 




LENS 



MIRROR 



Fig. 173. — Diagram of "thick mirror" system. 

and the last surface being a spherical mirror with its vertex 
at a point A on the axis of the lens. Draw the straight line 
QV parallel to the axis of the system to represent the path of 
an incident ray; which after traversing the lens and being 
reflected at the mirror will again emerge from the lens and 
cross the axis at the secondary focal point (F') of the system. 
The point V designates the point of intersection of the in- 
cident ray QV and the corresponding emergent ray VF', 
and hence this point must lie in the secondary principal 
plane of the system (§ 119). Consequently, the foot of the 
perpendicular let fall from V on to the axis will be the sec- 
ondary principal point H'. But by the principle of the re- 



378 Mirrors, Prisms and Lenses [§ 132 

versibility of the light-path (§ 29), if the straight line F'V is 
regarded as an incident ray, then VQ will be the path of the 
corresponding emergent ray, and since in this case the emer- 
gent ray is parallel to the axis, the corresponding incident 
ray F'V must cross the axis at the primary focal point F, 
so that the two focal points F and F' will be coincident. 
Moreover, the point V must lie in the primary principal 
plane, and hence the two principal planes are coincident. 
But these are the characteristics of a spherical mirror, and 
it is evident that the " thick mirror" is equivalent to a "thin 
mirror" with its vertex at H (or IF) and its center at a point 
K such that HK=2HF. 

The four images of Puekinje are the catoptric images 
formed in the eye by reflection at the anterior and posterior 
surfaces of the cornea and the crystalline lens; which are of 
fundamental importance in determining the curvatures and 
positions of the refracting surfaces in the optical system of 
the eye. The first image is produced by direct reflection at 
the anterior surface of the cornea, but the optical systems 
which give rise to the three other images are more or less 
complicated. However, according to the above explanation, 
each of these systems may be reduced to a single reflecting 
surface of appropriate radius with its center at a certain 
definite place to be ascertained by the conditions of the 
problem. One of these cases will be investigated presently, 
as soon as the formulae for a thick mirror have been devel- 
oped. 

The radius and positions of the vertex and center of the 
equivalent "thin mirror" may easily be calculated by means 
of the general formulae which were obtained in the preced- 
ing section for a combination of three optical systems. Here 
the first system (I) of refracting power F\ may be regarded 
as composed of the entire lens-system lying in front of the 
reflecting surface; while the mirror itself of reflecting power 
F 2 may be regarded as the second system (II). In this case 
the third system (III) will be the lens-system reversed, and 



§132] " Thick Mirror " 379 

its refracting power will be the same as that of system I, 
that is, Fz = F 2 ; but the principal points H 3 and H 3 ' of system 
III will coincide with the principal points HY and Hi, re- 
spectively, of system I. Above all we must impose here 
the conditions that 

7i 3 = —ri2=—n f , ft 4 = — fti=— ft, 
where n denotes the index of refraction of the medium of 
the object-space and n' denotes the index of refraction of- 
the medium in contact with the reflecting surface. These 
conditions take account of the fundamental fact that the 
sense of propagation of the light is reversed by the mirror. 
The principal points H 2 , H 2 ' of the mirror coincide with each 
other at its vertex which will be designated here by the 
letter A'. If therefore Ci, Oi denote the reduced intervals 
between the first system and the mirror and between the 
mirror and the third system, we have : 

H/A' A'H 3 Hi'A' 
Ci = — — , c 2 = = — — , 

n m n 

and hence 

ci = c 2 = c, say. 

Moreover, if the radius of the reflecting surface is denoted 

2n' 
by r', then F = ——. Introducing these relations in the 

general formulae for the combination of three optical sys- 
tems (§ 131), we obtain the following expressions for finding 
the reflecting power (F u ) and the positions of the principal 
points H13, Hi 3 ' of a "thick mirror": 

Fi 3 = (l-c.F 1 ) (2F 1 +F 2 -c.F 1 .F 2 ) 

= a-c.F 1 ){2F 1 -'^-(l-c.F 1 )}; 

H,H 13 H^,/ c 



ft ft 1 — c.Fi 

Accordingly, we see not only that the principal points of 
a " thick mirror" are coincident with each other, but that 
the position of the vertex Hi 3 of the equivalent "thin mirror " 
is entirely independent of the power F 2 or the curvature of 



H 13 K 



380 Mirrors, Prisms, and Lenses [§ 132 

the actual mirror. The position of Hi 3 does depend on the 
position of the vertex A' of the actual mirror; but for any 
mirror placed at A' the vertex of the equivalent " thin mir- 
ror" will be at the same point Hi 3 . It may be noted that 
the formula for the reflecting power of a " thick mirror" is 
identical in form with the expression for the refracting power 
of a compound system which is symmetric with respect to 
a middle member (see end of § 131). 

If the center of the equivalent "thin mirror" is designated 
by K, then its radius will be 

2n 

and hence 

HiK _ c.F 2 -2 
n 2F1+F2-C.F1.F2' 

If the surface of the mirror (II) is plane, then F 2 =0, and 
in this case the formulae for the equivalent "thin mirror" 
become : 

F 1 ,=2F 1 (l-c.F 1 ), Mi! = Sl5il = c H 1 K = _1 

n n 1-c.Fi n Fi 

The distinguishing characteristic of the imagery in a 
spherical mirror is that a pair of conjugate axial points M, M' 
is harmonically separated by the vertex H and the center K 
of the mirror, that is, (KHMM') = -1(§ 68). An interest- 
ing special case occurs when one of the points K or H is at 
infinity; for in that case the reflecting power of the mirror 
vanishes (F=0). When the center of the mirror is at an 
infinite distance from it, the mirror lies midway between 
object and image (MH = HM') and the lateral magnifica- 
tion is equal to +1 (y' = y); which is the case of an ordinary 
plane mirror. But, on the other hand, if the mirror itself 
is at an infinite distance, while the center K remains in the 
region of finite space, it is the center of the mirror in this 
case that is always midway between object and image, that 
is, MK = KM', and now the lateral magnification will be 
equal to — 1, that is, the image will be of the same size as 



§ 132] " Thick Mirror" 381 

the object but inverted {y f =—y). Both of these special 
cases may be realized by a "thick mirror"; for the condi- 
tion that the reflecting power of the equivalent "thin mirror" 
shall vanish (Fu=0) requires that either 

2^1+^2-0.^1.^2=0, 

or 

1-0.^1=0. 

In the former case the center of the mirror (K) is at infinity, 
and in the latter case the vertex of the mirror (Hi 3 ) is at 
infinity. If therefore the distance between the anterior lens- 
system and the final reflecting surface of a "thick mirror" 
is c = l/F h the system will produce an inverted image of the 
same size as the object, no matter where the object is placed. 
As an illustration of the use of the formulae for a "thick 
mirror," consider the optical system in the eye which pro- 
duces the third of the so-called Purkinje images, to which 
allusion was made earlier in this section. The third image is 
formed by rays which coming from an external source enter 
the eye, and after having traversed the cornea system and 
the aqueous humor are reflected at the anterior surface of the 
crystalline lens; whence returning through the same media 
in the reverse order they issue again into the air. In order 
to find the "thin mirror" which is equivalent to this system, 
we shall employ the constants of Gullstrand's schematic 
eye as given in § 130. The vertex of the anterior surface of 
the cornea will be designated by A x and the principal points 
of the cornea-system by Hi and H/. We found that AiHi = 
-0.0496 mm. and AiH/= -0.0506 mm.; also, Fi = +43.05 
dptr., where F\ denotes the refracting power of the cornea- 
system. The reflecting power of the anterior surface of the 
lens is given by the formula: 

r 3 
where n 3 = 1.336 and r 3 = +0.010 m.; accordingly, we find: 
F 2 = ~ 267.2 dptr. 



382 Mirrors, Prisms and Lenses [§ 132 

The reduced distance between the cornea-system and the 

first surface of the lens is : 

H/A 3 
c = , 

n z 

where A 3 designates the vertex of this surface; AiA 3 = 
0.00036 m. Thus, we obtain : 

c= 0.0027325. 
Substituting these numerical values in the system of for- 
mulae for a "thick mirror, " we find for the reflecting power of 
the equivalent "thin mirror" in this case: 
F 13 =~ 132.062 dptr.; 
and for the positions of its vertex Hi 3 and its center K: 
HiH 13 = +3.0968 mm., HiK= +18.2412 mm. 
Accordingly, the system that produces the third of the 
Purkinje images in Gullstrand's passive schematic eye 
is equivalent to a convex mirror of radius 15.14 mm. with 
its vertex at a distance of 3.047 mm. from the vertex of the 
anterior surface of the cornea. 

Formulas for calculating the reflecting power Fu of a 
"thick mirror" may also be obtained in terms of different 
data from those employed in the expressions which have 
been deduced above. Suppose, for example, that we are 
given the refracting power (F) of a centered system of 
spherical refracting surfaces, the positions of the principal 
points of the system (H, H')> and the indices of refraction 
of the first and last media (n, n'); together with the radius 
(r') and the position of the vertex (A') of the last surface; 
and that it is required to determine in terms of these data the 
characteristics of the imagery produced by light which pro- 
ceeding from the object-space through the system is partially 
reflected at the last surface and again partially refracted at 
the first surface into the original medium. In order to solve 
this problem in the simplest way, it is convenient to employ 
a mathematical artifice which will be found to be serviceable 
in other optical problems. The refracting power of an in- 
finitely thin concentric lens is equal to zero, and it is easy to 



§132] " Thick Mirror" 383 

show that such a lens may be inserted anywhere in an opti- 
cal system without affecting at all the resultant imagery 
(see § 90). Let us suppose, therefore, that the given optical 
system is terminated by an infinitely thin layer of material of 
index n', bounded by two concentric spherical surfaces, the 
first of which coincides with the last surface of the given 
system. Under these circumstances the resultant system may 
be considered as compounded of three component systems, 
namely, (1) the given system of refracting power Fi = F, 

2n' 
(2) a mirror of reflecting power F 2 = , and (3) the given 

system reversed (F 3 = F). Hence, if 

H'A' 

the following formulae will be obtained in the same way as 
above : 



F^il-c.F) 

HH13 HHi,/ 



2F- £5.(1- c.F) 



n n l-c.F ' 

which are similar in form to the previous expressions, but 
c here has a different meaning and F denotes the refracting 
power of the entire lens-system and not merely of that part 
of the system which is in front of the reflecting surface. 

A problem of considerable interest, especially in connec- 
tion with the optical system of the human eye, is the inves- 
tigation of the procedure of the light which after being par- 
tially reflected at the last surface of the system (as in the 
case above) is also partially reflected at the first surface, so 
that it emerges finally into the last medium of index n' . The 
imagery in this case may be determined by adding a second 
infinitely thin concentric lens, which is assumed to be made 
of material of index n and whose second surface coincides 
with that of the first surface of the system. Accordingly, 
now we shall have five systems in all, namely, the first three 
systems whose reflecting power Fu was obtained above, 



384 Mirrors, Prisms and Lenses [Ch. XI 

a fourth system consisting of the first surface of the lens- 

2n 
system acting as a mirror, whose reflecting power is F 4 = — , 

r 

where r denotes the radius of this surface, and a fifth system 
of refracting power F & = F. The entire system, whose refract- 
ing power may be denoted by Fu, and whose principal points 
may be designated by H i5 , Hi 5 ', may, therefore, be considered 
as compounded of 3 systems of powers Fu, 2n/r and F, sep- 
arated by the intervals Ci and c 2 , where (if A designates the 
vertex of the first surface of the lens-system) 

AH 13 AH 

ci= » c 2 = . 

n n 

Accordingly, by substituting Fu in place of F h 2n/r in place 

of F 2 , and F in place of F 3 in the formulae of § 131 for the 

combination of three optical systems, we obtain here: 

Fn=F u (1- C2 .F)+^ (I-C1.F13) (l-c.F)+F (1-d.Fu); 

H13H15 d c2.F-c1.Fu . 



n 1-ci.Fn Fu(l-ci.Fi3) ' 

ITHil = ___C2_ c^.F-d.Fu 
n' 1-C2.F + Fn(l-C2.F)' 

Being given the magnitudes denoted by n, n', r, r', and F 
and the positions of the points designated by A, A' and H, H', 
and having found by means of the previous formulae the 
magnitude denoted by Fu and the position of the point 
designated by Hi 3 (or Hi/), we can introduce these data and 
results in the expressions above and thus determine the re- 
fracting power Fit and the distances AH15, A'His' of the prin- 
cipal points H15, Hi 5 ' from the vertices A, A' of the first and 
last surfaces, respectively. 

PROBLEMS 

1. Find the refracting power and the positions of the focal 
points and principal points of each of the following glass 
lenses surrounded by air (n = l, n' = 1.5); and make an ac- 



Ch. XI] Problems 385 

curate sketch of each lens, marking the positions of the 
points mentioned. 

(a) Double convex lens of radii 10 cm. and 15 cm. and of 
thickness 3 cm. 

(&) Double concave lens with same data as above. 

(c) Meniscus lens for which ri=+5cm., r 2 =+10cm., 
and d=+3 cm. 

(d) Meniscus lens for which ri=+6 cm., r 2 =+3 cm., and 
d=+2.52 cm. 

(e) A plano-convex lens with its curved surface, of radius 
5 cm., turned towards the incident light; d = +0.5 cm. 

(/) Symmetric convex lens, the radius of each surface 
being 5 cm.; d = +0.5 cm. 

(g) Symmetric concave lens with same data as above. 

(h) A meniscus lens with radii ri=+5cm., r 2 =+8cm., 
and thickness d=+0.5 cm. 

(i) A meniscus lens with radii ri=+8cm., r 2 = +5cm., 
and thickness d= +0.5 cm. 

(j) A meniscus lens with radii ri = +8cm., r 2 =+7cm., 
and thickness d= +3 cm. 

(k) A plano-convex lens with its curved surface, of radius 
5 cm., turned towards the incident light; d=+5 cm. 

Answers : 



F 

in clptr. 


AiF 
in cm 


A 2 F' 
in cm. 


AiH 
in cm. 


A 2 H' 
in cm. 


(a) + 8.000 


-11.667 


+ 11.250 


+0.833 


-1.250 


(b) - 8.667 


+ 12.307 


-12.692 


+0.769 


-1.154 


(c) + 6.000 


- 18.333 


+ 13.333 


-1.667 


-3.333 


(d) - 6.000 


+21.333 


- 14.333 


+4.667 


+2.333 


(e) +10.000 


- 10.000 


+ 9.667 


0.000 


-0.333 


(/) +19.666 


- 4.915 


+ 4.915 


+0.170 


-0.170 


(g) -20.335 


+ 5.082 


- 5.082 


+0.164 


-0.164 


(h) + 3.959 


-25.789 


+24.421 


-0.526 


-0.842 


(i) - 3.542 


+ 29.176 


-27.647 


+0.941 


+0.588 


(j) 0.000 


00 


00 


00 


00 


(k) +10.000 


-10.000 


+ 6.667 


0.000 


-3.333 



2. In a symmetric lens (n = —r<i=r) surrounded by the 



386 Mirrors, Prisms and Lenses [Ch. XI 

same medium (index n) on both sides, show that we have 

the following sj^stem of formulae: 

N=( : n , -n) \ (n'-n) d-2n'.r } ; 

„ N AiF A 2 F' r\ ( , w , 

F=—— r - ; = — = — —\ (n'—n) d—n' 

n'r* n n N { 

AiH__A 2 H / _ _ n'-n , 

n n N 

3. If the first face of a lens is plane, and if the radius of 
the curved face is denoted by r, show that 

r n n 

AiF_ r d m A2F r = r 

n n'—n n' ' n n'—n ' 

And if the second face of the lens is plane, 

„ n'—n . TT . A 2 H' d 

F= ; AiH=0; — — =— -.; etc. 

r n n 

If either face of a lens is plane, the refracting power of the 
lens is equal to that of the curved surface and is entirely in- 
dependent of the thickness of the lens; and, moreover, one 
of the principal points coincides with the vertex of the curved 
face. 

4. If the radii n and r 2 of the two surfaces of a lens are 
both positive, and if r 2 is greater than n, show that the lens 
is convergent, provided the lens-medium is more highly re- 
fracting than the surrounding medium. 

5. A "lens of zero-curvature" is a crescent-shaped menis- 
cus for which r 2 = ri=r. Show that such a lens is always 
convergent unless it is infinitely thin; and that this is the 
case whether the lens-medium is more or less highly refract- 
ing than the surrounding medium. 

6. Show that a meniscus lens for which 

ri>r 2 >0 and n'>n 

is divergent provided its thickness is less than 

n'(ri—r 2 ) 
n' — n 



Ch. XI] Problems 387 

7. Show that in any meniscus lens surrounded by air at 
least one of the principal points must lie outside the lens. 

8. A so-called concentric lens is one for which the centers 
of curvature of the two faces are coincident (d = r\ — r 2 ). It 
may be double convex or meniscus. Show that the refract- 
ing power of a concentric lens surrounded by the same me- 
dium on both sides is 

w _ n(n'-n) (l _1_\ 
n' \n rj' 
and that the principal points coincide at the common center 
of the two surfaces. 

9. Find the refracting power and the positions of the focal 
points and principal points of each of the following concentric 
glass lenses (ft' = 1.5) surrounded by air (ft = l); and draw 
accurate sketch of each lens showing the positions of the 
points named : 

(a) Double convex lens with radii n. = + 10 cm., r 2 = — 2 cm. 

(b) Meniscus lens with radii n = +5 cm., r 2 = +2 cm. 

Ans. (a) F=+20 dptr., AiF=+5cm., A 2 F=+3 cm., 
AiH = +10 cm., A 2 H' = -2 cm.; (b) F=-10 dptr., A]F = 
+15 cm., A 2 F'=-8 cm., AiH = +5 cm., A 2 H'=+2 cm. 

10. Find the focal length and the positions of the prin- 
cipal points of a concentric glass lens surrounded by air 
(ft = l, n' = 1.5), with radii ri=+8 cm., r 2 = +5 cm. 

Ans. /=— 40 cm., AiH = +8 cm., A 2 H'=+5 cm. 

11. What is the refracting power of a concentric glass 
meniscus lens surrounded by air (n = l, ft' = 1.5), the radii 
being 5 cm. and 3 cm.? Ans. F= — 4.44 dptr. 

12. The radius of the second surface of a concentric glass 
lens surrounded by air (ft = l, ft' = 1.5) is +3 cm., and its re- 
fracting power is — 2 dptr. Determine its thickness. 

Ans. 6.59 mm. If it were not too heavy, this would 
be a fairly good form of spectacle glass for a near-sighted 
person. 

13. If the two principal points of a lens surrounded by 
the same medium on both sides coincide with each other at 



388 Mirrors, Prisms and Lenses [Ch. XI 

a point midway between the two vertices, what is the form 
of the lens? Ans. A solid sphere. 

14. The refracting power of a symmetric glass lens sur- 
rounded by air (n = l, w' = 1.5) is +10 dptr., and its thick- 
ness is 0.5 cm. Determine the radius of the first surface. 

Ans. +9.916 cm. 

15. A solid sphere is a symmetric concentric lens. If the 
radius is denoted by r (r = AiC), show that we have the fol- 
lowing system of formulae for a solid sphere surrounded by 
the same medium (ri) on both sides : 

2n(n'-n) . H 'A -r • A,F' FA, S 2n ~ n '> 

16. If the plane surface of a glass hemisphere, of index n' 

and surrounded on both sides by a medium of index n, is 

turned towards the incident light, and if r denotes the radius 

of the curved surface, show that 

n'—n . TT n.r . TT , _ A „ n 2 r 
F=- , AiH= T , A 2 H' = 0, AiF = 



n 
A 2 F' 



n'in' -n) ' 



n.r 
n'-n 



17. An object is placed in front of the plane surface of 
a glass hemisphere, of index 1.5 and radius 3 inches, at a 
distance of 10 inches from this surface. Find the position, 
nature and size of the image. 

Ans. A real, inverted image, of same size as object, will 
be formed at a distance of 25 inches from the object. 

18. What is the refracting power of a glass sphere (n' = 
1.5), 16| cm. in diameter, (a) surrounded by air (w=l), 
and (b) surrounded by water (n=|)? 

Ans. (a) +8 dptr.; (b) +3| dptr. 

19. The radius of each surface of a symmetric convex 
glass lens (n' = 1.5) is 10 cm., and the thickness of the lens 
is 5 mm. What is its refracting power (a) when the thick- 
ness is neglected, and (6) when the thickness is taken into 
account? Ans. (a) +10 dptr.; (b) +9|| dptr. 

20. The radii of a convex meniscus glass lens (n' = 1.5) 



Ch. XI] Problems 389 

surrounded by air (n = l) are 2.5 cm. and 5 cm. (a) If the 

lens is infinitely thin, what is its refracting power? (6) If 

the thickness of the lens is 1 cm., what is its refracting power? 

Ans. (a)F=+10dptr.; (6) F=+ll£ dptr. 

21. Determine the focal length (/) of a glass lens of in- 
dex 1.5 surrounded by air for which n=+10, r 2 =+9, 
(1) when thickness d = 0, and (2) when thickness d= +1. 

Ans. (1) /=-180; (2) f=-270. 

22. A plane object is placed at right angles to the axis of 
a plano-convex lens at a distance of 8.77 cm. in front of its 
curved surface. The lens is made of glass of index 1.52, 
and the thickness of the lens is 0.5 cm. The radius of the 
curved surface is 4.56 cm. Show that the image will be at 
infinity, and that, in order to see distinctly the image of a 
point in the object which is 2 cm. from the axis, an eye be- 
hind the lens must look in a direction inclined to the axis 
of the lens at an angle of nearly 12° 51'. 

23. The refracting power of a meniscus spectacle glass is 
+6 dptr., and r 2 =2r h d = 6 mm. The index of refraction 
is 1.5. Find the radii n and r 2 and the vertex refrac- 
tion V. 

Ans. ri = +4.36 cm., r 2 = +8.72 cm., 7= +6.29 dptr. 

24. The thickness of a spectacle glass is 4.75 mm., and 
the index of refraction is 1.5. The refracting power of the 
first surface is +15.4 dptr., and that of the second surface 
is —9.1 dptr. Find the refracting power of the lens and its 
vertex refraction. Ans. F= +6.74 dptr.; V =+7.09 dptr. 

25. A paraxial ray is incident on the cornea of Gull- 
strand's schematic eye (§ 130) in a direction parallel to the 
axis. Trace the path of this ray through the eye and de- 
termine the position of the secondary focal point F' (see 
calculation-scheme, § 181) ; and calculate the focal lengths 
/, f according to the formulae derived in problem No. 17 
at the end of Chapter X. 

Ans. Distance of F' from the vertex of the cornea is 
24.387 mm.; /= +17.055 mm., f= -22.785 mm. 



390 Mirrors, Prisms and Lenses [Ch. XI 

26. The reduced thickness of a symmetric spectacle glass 
is denoted by c. If V denotes its vertex refraction, show 
that 

r V(Vl+c 2 .V 2 -c.V) 

? ' 

27. A hollow globe of glass is filled with water. The di- 
ameter of the water sphere is 8.5 inches and the thickness of 
the glass shell is 0.25 inch. Show -that a narrow beam of 
parallel rays directed towards the center of the globe will be 
converged to a point 4.68 inches from the outside surface, 
the indices of refraction of glass and water being # and 
-|, respectively. 

28. What is the focal length of a combination of two thin 
convex lenses, each of focal length /, placed at a distance 
apart equal to 2//3? Ans. 3//4. 

29. An optical system is composed of two thin convex 
lenses of refracting powers +10 dptr. and +6§ dptr., 
the stronger lens being towards the incident light. Find 
the refracting power of the combination and the positions 
of the principal points and focal points when the dis- 
tance between the lenses is: (a) 5 cm.; (b) 25 cm.; and 
(c) 40 cm. 

Ans. (a) Convergent system: F = 1S~ dptr.; AiH = 
+2.5 cm. ; A 2 H' = - 3.75 cm. ; AiF = - 5 cm. ; A 2 F' = +3.75 cm. ; 
(b) Telescopic system: F = 0; focal and principal points 
all at infinity; (c) Divergent system: F= — 10 dptr.; AiH = 
-26| cm.; A 2 H , = +40 cm.; AiF=-16f cm.; A 2 F' = 
+30 cm. 

30. An optical system is composed of two thin lenses, 
namely, a front concave lens of power — 10 dptr. and a rear 
convex lens of power +6| dptr. Find the refracting power 
of the combination and the positions of the focal points and 
principal points, when the interval between the lenses is: 
(a) 2.5 cm.; (b) 5 cm.; (c) 6.25 cm.; (d) 20 cm. 

Ans. (a) Divergent system : F = — 1| dptr. ; AiH = — 10 cm. ; 
A 2 H'=-15 cm.; AiF =+50 cm.; A 2 F'=-75 cm.; (6) Tele- 



Ch. XI] Problems 391 

scopic system: F=0, focal and principal points all at 
infinity; (c) ^=+| dptr.; AiH = +50 cm.; A 2 H'=+75 cm.; 
AiF=— 70 cm.; A 2 F' = +195 cm.; (d) Convergent system: 
^=+10 dptr.; AiH = +13^ cm.; A 2 H' = +20 cm.; AiF = 
-f3^cm.;A 2 F' = +30cm. 

31. Two thin convex lenses of focal lengths /i and / 2 are 
separated by an interval equal to 2/ 2 . If /i = 3/ 2 , what is 
the focal length of the combination? 

Ans. Convergent system of focal length 3/ 2 /2. 

32. Two lenses, one convex and the other concave, are 
separated by an interval 2a. The convex lens is the front 
lens, and its focal length is a, while that of the concave lens 
is —a. Find the focal length of the combination and the 
positions of the principal points and focal points. 

Ans. /=a/2; AiH = A 2 H' = -a; AiF = 3A 2 F' = -3a/2. 

33. Where are the principal planes of a system of two thin 
convex lenses of focal lengths 2 inches and 6 inches, separated 
by an interval of 4 inches? 

Ans. The principal planes coincide with the focal planes 
of the stronger lens. 

34. The objective of a compound microscope may be re- 
garded as a thin convex lens of focal length 0.5. inch. The 
ocular may also be regarded as a thin convex lens of focal 
length 1 inch. The distance between the two lenses is 6 
inches. Where must an object be placed in order that its 
image may be seen distinctly by a person whose distance of 
distinct vision is 8 inches? 

Ans. If inch in front of the objective. 

35. The focal lengths of the objective and ocular of a com- 
pound microscope are 0.5 inch and 1 inch, respectively. If 
the distance of distinct vision is 12 inches, find the distance 
between the objective and ocular when the object viewed is 
0.75 inch from the objective. Ans. 2.42 inches. 

36. A thin convex lens, of focal length 5 inches, is placed 
midway between two thin concave lenses each of focal length 
10 inches. The distance between the first lens and the second 



392 Mirrors, Prisms and Lenses [Ch. XI 

is 5 inches. Find the focal length of the system and the po- 
sitions of the principal points. 

Ans. /=+6| inches; the principal points are 3^- inches 
from the outside lenses. 

37. In the preceding problem, suppose that the two 
outside lenses are concave, everything else remaining the 
same. 

Ans. /=+6| inches. The principal points are on op- 
posite sides of the middle lens and If- inches from it. 

38. A thin convex lens, of focal length 10 inches, is placed 
in front of a concave mirror of focal length 5 inches, the dis- 
tance between them being 5 inches. The light traverses the 
lens, is reflected at the mirror, and again passes through the 
lens. Find the focal length of this so-called " thick mirror " 
and the positions of the principal points. 

Ans. /=+6§ inches; the principal points coincide with 
each other at a point 5 inches behind the vertex of the mirror. 

39. In the preceding problem, suppose that the lens is 
concave, everything else remaining the same. 

Ans. /= +6| inches; the principal points coincide with 
each other at a point between the lens and the mirror and 
3| inches from the former. 

40. In front of each of the systems described in Nos. 35, 
36, 37 and 38, an object, one inch high, is placed at a distance 
of 5 inches from the first member of the system. Find the 
position, size and nature of the image in each case. 

Ans. In No. 35: A real, inverted image, 2 inches beyond 
the third lens and 0.8 in. high. In No. 36: A real, inverted 
image, 30 inches beyond the third lens and 4 inches high. 
In No. 37: A real, inverted image, 2 inches in front of the 
lens and 0.8 in. high. In No. 38: A real, inverted image, 
30 inches in front of the lens and 4 inches high. 

41. The center of a concave mirror, of radius r, coincides 
with the optical center of a thin lens, of focal length /, and 
the axes of lens and mirror are in the same straight line. The 
light traverses the lens, is reflected at the mirror, and again 



Ch. XI] Problems 393 

traverses the lens. Show that the system is equivalent to a 
thin mirror of radius r.f/(r+f), with its center at the same 
place as the center of the given mirror. 

42. A centered system of lenses (I) is placed in front of 
a spherical mirror (II), and the whole constitutes a "thick 
mirror/' as explained in § 132. Show that the vertex A and 
the center C of the actual mirror are the images of the vertex 
H and the center K, respectively, of the equivalent "thin 
mirror/' which are produced by the lens-system I in the 
medium of index n 2 between systems I and II. 

43. A "thick mirror" consists of a thin lens of focal length 
/i and a spherical mirror of focal length / 2 placed co-axially 
so that the focal point of the mirror coincides with the opti- 
cal center Ai of the thin lens. Show that the focal length of 
the equivalent "thin mirror" is 

t /l-/2 . 

Si- ft' 

and that the positions of the vertex H and the center K are 
given by the following expressions : 

Al H = -A4, A 1 K=-/^4. 

/l-/ 2 ' /1+/2 

Does it make any difference whether the lens is convex or 
concave? 

44. At each of the focal points of a thin convex lens of 
focal length / 2 is placed a thin lens of focal length /1. Find 
the focal length of the combination of the three lenses and 
the positions of the principal points. Does it make any dif- 
ference whether the two equal outside lenses are convex or 
concave? 

/l/2 . A XT XT/ A /l-/2 



Ans./=-^ ; AiH = H'A 3 =- 



Jl—fl /1-/2 

45. A thin convex lens of focal length 10 cm. is placed in 
front of a plane mirror at a distance of 8 cm. from it. Find 
the radius of the equivalent "thin mirror" and the position 
of its vertex H. 



394 Mirrors, Prisms and Lenses [Ch. XI 

Ans. The equivalent "thin mirror" is a concave mirror of 
radius 50 cm. with its vertex 32 cm. behind the plane mirror. 

46. The axes of three thin convex lenses are all in the 
same straight line, the interval between the first and second 
lenses being one inch and the interval between the second 
and third lenses being half an inch. The focal lengths of the 
first, second and third lenses are f, -^ and f inch, re- 
spectively. A plane object is placed at right angles to the 
axis of the lens-system; show that an inverted image of the 
same size as the object will be formed in the plane of the 
object. 

47. A plano-concave flint glass lens of index 1.618 is ce- 
mented to a double convex crown glass lens of index 1.523. 
The radii and thicknesses are as follows : r\ = oo , r<i = 
+50.419 mm., r 3 = — 74.320 mm.; c?i = +2.15 mm., d 2 = 
+4.65 mm. Find the focal length of the combination and 
the positions of the principal points. 

Ans. /= +192.552 mm.; distances of principal points from 
the plane surface, +5.466 and +7.908 mm. 

48. A plano-concave flint glass lens of index 1.618 is ce- 
mented to a double convex crown glass lens of index 1.523. 
The radii and thicknesses are as follows: r\= +22.00 mm., 
r 2 =-19.65 mm., r 3 = oo ; di=+2.60 mm., d 2 = +2.00 mm. 
Find the focal length of the combination and the positions of 
the principal points. 

Ans. /= +52.26 mm.; distances of principal points from 
plane surface, —5.03 and —3.36 mm. 

49. The radii and thickness of a symmetric double convex 
lens are 10 cm. and 1 cm., respectively. The lens is made of 
glass of index 1.5 and surrounded by air of index unity. 
A portion of the light which enters the lens will be reflected 
at the second surface and partially refracted at the first 
surface from glass back into the air. Find the radius, re- 
flecting power and position of the vertex of the equivalent 
"thin mirror." 

Ans. Concave mirror of radius —53.050 mm., reflecting 



Ch. XI] Problems 395 

power +37.7 dptr., with its vertex +6.90 mm. from the ver- 
tex of the first face of the lens. 

50. In the case of the lens in the preceding problem, as- 
sume that the light is reflected internally twice in succession 
and issues finally at the second face into the air. Find the 
refracting power and the positions of the principal points 
for the imagery produced by these rays. 

Ans. Refracting power, +53.08 dptr.; distances of prin- 
cipal points from vertex of first surface of the lens, +10.94 
and — 0.94 mm. 

51. In Gullstrand's schematic eye in its state of maxi- 
mum accommodation the crystalline lens consists of an outer 
symmetric double convex lens of index ^ 4 = n 6 = 1.386 (see 
§ 130), enclosing an inner symmetric double convex "core" 
lens of index n 5 = 1.406; the inner portion being symmetri- 
cally placed with respect to the surrounding outer part. 
The radii of the surfaces are as follows : 

Outer portion: r 3 =A 3 C 3 = +5.3333 mm.= -r 6 = C 6 A 6 ; 

Inner portion: r4 = A 4 C 4 = +2.6550 mm.= -r 5 = C 5 A 5 . 
Moreover, 

A 3 A4 = A3C5 = A 5 A 6 = C4A 6 = 0.6725 mm.; 
A4A5 = C5C4 = A4C4 = C5A5 = 2.6550 mm. 
The entire lens is surrounded by a medium of index n 3 = 
717 = 1.336. Show (1) that the refracting power of the inner 
portion or "core" lens is F 4 b = +14.959 dptr., and that its 
principal points are 1.9905 mm. from the anterior and pos- 
terior surfaces. Moreover, employing the formulae of § 131, 
show (2) that the refracting power of the entire lens in case 
of maximum accommodation is F 3Q = +33.056 dptr. and that 
A 3 H 36 = H 36 'A 6 = +1.9449 mm. 

52. Using the data of the preceding problem, find the 
refracting power (F) and the positions of the principal points 
(H, H') of Gullstrand's schematic eye in its state of maxi- 
mum accommodation: being given, according to the results 
of § 130, that the refracting power of the cornea system is 
Fn =43.053 dptr. and that AiHi 2 = -0.0496 mm., AiH'i 2 = 



396 Mirrors, Prisms and Lenses [Ch. XI 

— 0.0506 mm., aifld also that for maximum accommodation 
AiA 3 =+3.2 mm. 

Ans. ^=+70.575 dptr.; AiH = + 1.772 mm.; AiH^ 
+2.086 mm. 

53. Two thin lenses of focal lengths /i and/2 are placed on 
the same axis with the second focal point (F/) of the first 
lens coincident with the first focal point (F 2 ) of the second 
lens, so as to form an afocal or telescopic system. Show 
that the lateral magnification is constant and equal to 
— /2//1, and that the angular magnification is likewise con- 
stant and equal to the reciprocal of the lateral magnification. 

54. If (as in Huygens's ocular) two thin lenses are placed 
on the same axis with their second focal points in coincidence, 
show that the second focal point of the combination is mid- 
way between this common focal point and the second lens, 
and that the deviation produced by the second lens is twice 
that produced by the first (assuming that the angles are 
small) . 



CHAPTER XII 

APERTURE AND FIELD OF OPTICAL SYSTEM 

133. Limitation of Ray-Bundles by Diaphragms or Stops. 
— The geometrical theory of optical imagery which has been 
developed in Chapter X was based on the assumption of 
punctual correspondence between object-space and image- 
space, whereby each point of the object is reproduced by 
one point, and by one point only, in the image; and on this 
hypothesis simple relations in the form of the so-called 
image-equations (§ 123) were obtained for determining the 
position and size of the image in terms of the focal lengths 
of the optical system. When we attempted to realize the 
imagery expressed by these equations, we were obliged to 
confine ourselves to the so-called paraxial rays comprised 
within the narrow cylindrical region immediately surround- 
ing the axis of symmetry or optical axis of the centered sys- 
tem of spherical refracting or reflecting surfaces. Based on 
the same assumptions, certain rules were given for con- 
structing the image-point Q' corresponding to a given object- 
point Q. For example, a pair of straight lines was drawn 
through Q (Fig. 174), one parallel to the optical axis and 
meeting the second principal plane of the system in a point 
V, and the other going through the primary focal point F 
and meeting the first principal plane in a point W. The 
required point Q' was shown to lie at the point of intersec- 
tion of the straight line V'Q', drawn through the second 
focal point F', with the straight line WQ' drawn parallel to 
the axis. The position of the point Q' having been located, 
the problem was considered as solved, and we were not par- 
ticularly concerned with inquiring whether the straight 
lines used in the construction represented the paths of ac- 

397 



398 



Mirrors, Prisms and Lenses 



133 



tual rays that formed the image at Q\ As a matter of fact, 
the pair of geometrical lines which is employed here will 
generally not belong to the bundle of optical rays by which 
the imagery is actually produced; and a glance at the dia- 
gram will show how the diameter of the lens and the size of 




Fig. 174. — Effective rays as distinguished from rays used in making geo- 
metrical constructions. 

the object control the selection of the rays that are really 
effective in producing the image. 

In Chapter 1 attention was called to the fact that every 
optical instrument is provided with some means of cutting 
out such portions of a bundle of rays as for one reason or 
another are not desirable; which is usually accomplished, 
as has been explained, by interposing in the paths of the 
rays at some convenient place a plane opaque screen at 
right angles to the axis containing a circular aperture with 
its center on the axis. There may, indeed, be several such 
diaphragms or stops disposed at various places along the 
axis of the instrument. A perforated screen of this kind is 
called a front stop, a rear stop or an interior stop, according 
as it lies in front of, behind or within the system, respect- 
ively. The rims and fastenings of the lenses act in the same 
way as the diaphragms to limit the ray-bundles. The stops 
have various duties to perform, their chief functions being 



§ 134] Aperture-Stop 399 

to cut off the view of indistinct parts of the image (limita- 
tion of the field of view) , to cut out such rays as would tend 
to mar the perfection of that part of the image which is to 
be inspected (limitation of the aperture of the system), and, 
finally, to nullify injurious reflections from the sides of the 
tube or other parts of the instrument. 

134. The Aperture-Stop and the Pupils of the System. — 
To an eye looking into the instrument from the side of the 
object, a front stop (which may be the rim of the first lens 
of the system) will be the only one that will be visible di- 
rectly. Any other stop or lens-rim will be seen only by means 
of the real or virtual image of it that is cast by that part of 
the optical system which is between it and the eye. Simi- 
larly, if the eye is directed towards the instrument from the 
image-side, an interior stop or a front stop may be seen by 
means of the image of it that is produced by the part of the 
system that lies between it and the eye. Now these impal- 
pable stop-images, whether visible or not, are just as effect- 
ive in cutting out the rays as if they were actual material 
stops; because, obviously, any ray that goes through an 
actual stop must necessarily pass either really or virtually 
through the corresponding point of the stop-image; whereas 
a ray that is obstructed by a stop will not go through the 
opening in the stop-image. 

That one of the stops which by virtue of its size and po- 
sition with respect to the radiating object is most effective 
in cutting out the rays is distinguished as the aperture-stop 
of the system (§ 11), and in order to determine which of the 
several stops performs this office, it is necessary, first of all, 
to assign the position of the axial object-point M, without 
which the aperture of the system can have no meaning. Ac- 
cordingly, we must suppose that the instrument is focused 
on some selected point M on the axis, which is reproduced 
by an image at the conjugate point M'. The transversal 
planes at right angles to the axis at M and M' will be a pair 
of conjugate planes, for it is assumed here that the imagery 



400 Mirrors, Prisms and Lenses [§ 134 

is ideal and of the same character as that produced by par- 
axial rays. Now this pair of conjugate planes plays a very 
important role in the theory of an optical instrument, so 
that hereafter we shall refer to the object-plane as the focus- 
plane (or the plane which is in focus on the screen) and to 
the conjugate plane in the image-space as the screen-plane. 

Now if the eye is supposed to be placed on the axis at the 
point M and directed towards the instrument, the stop or 
stop-image whose aperture subtends the smallest angle at M 
is called the entrance-pupil of the system. All the effective 
rays (§11) in the object-space must be directed towards 
points which He within the circumference of the circular 
opening of the entrance-pupil. In general, the entrance- 
pupil is the image of the aperture-stop as seen by looking into 
the instrument in the direction of the light coming from the 
object; but if the aperture-stop is a front stop, it will also 
be the entrance-pupil. 

On the other hand, when the eye is placed on the axis at 
the point M' so as to look into the instrument through the 
other end, the stop or stop-image which subtends the smallest 
angle at M' is called the exit-pupil, and all the effective rays 
when they emerge from the instrument must go, really or 
virtually, through the opening of the exit-pupil. In this 
statement it is tacitly assumed that M' is a real image of M; 
otherwise, it would not be possible for the eye placed at M' 
to look into the instrument through the end from which the 
rays emerge. But in any case the exit-pupil is the stop or 
stop-image which subtends the smallest angle at M'. Gen- 
erally, the exit-pupil will be the image of the aperture-stop 
as seen by looking into the instrument from the image-side; 
but if the aperture-stop is a rear stop, it will be itself the 
exit-pupil. 

Since the effective rays enter the system through the 
entrance-pupil in the object-space and leave it through the 
exit-pupil in the image-space, it is evident that the exit- 
pupil is the image of the entrance-pupil, so that the pupil- 



§ 135] Entrance-Pupil of Eye 401 

centers, designated 03- and 0', are a pair of conjugate axial 
points with respect to the entire system. 

The apertures of the ray-bundles in the object-space are 
determined by the entrance-pupil of the system; and the 
exit-pupil has a similar office in the image-space. Each of 
the pupils is the common base of the cones of effective rays 
in the region to which it belongs. 

135. Illustrations. — The name "pupil" applied to these 
apertures by Abbe was suggested by an analogy with the 
optical system of the eye. The pupil of the eye is the con- 
tractile aperture of the colored iris, the image of which 
produced by the cornea and the aqueous humor is the en- 
trance-pupil of the eye corresponding to what is popularly 
called the " black of the eye," because it looks black on the 
dark background of the posterior chamber of the eye. Since 
the center O of the entrance-pupil is the image of the center 
K of the iris-opening formed by rays that are refracted from 
the aqueous humor through the cornea into the air, then, 
by the principle of the reversibility of the light-path, we 
may also regard K as the image of O formed by rays which 
are refracted from air (n = l) through the cornea into the 
aqueous humor (n' = 1.336). The apparent place of the eye- 
pupil varies slightly in different individuals and in the same 
individual at different ages. If we assume that the point O 
is 3.03 mm. from the vertex (A) of the cornea, that is, if we 
put u = 0.00303 m., then U = n/u = 330 dptr. And if we take 
the refracting power of the cornea asF = 42 dptr. (§ 130), 
then, since U'=U+F, we find U' = 372 dptr. and consequently 
u' = AK = n'/U' = 0.0036 m.; so that with these data the 
plane of the iris is found to be at a distance of 3.6 mm. from 
the vertex of the cornea. Thus we see that the entrance- 
pupil of the eye is very nearly 0.6 mm. in front of the iris. 

As a simple illustration of these principles, consider an 
optical system which consists of an infinitely thin convex 
lens, with a stop placed a little in front of it. In the dia- 
gram (Fig. 175) the straight line DG perpendicular to the 



402 



Mirrors, Prisms and Lenses 



[§135 



axis of the lens represents the diameter of the lens which 
lies in the meridian plane of the paper. The diameter of 
the stop-opening is shown by the straight line BC parallel 
to DG. The centers of the lens and stop are designated by 




Screen 
Plovxc 



Exit Pupil 

Fig. 175. — Optical system composed of thin convex lens with front 

stop. 



A and K, respectively. The position of the focus-plane is 
determined by the axial object-point M, which in the figure 
is represented as lying in front of the lens beyond the pri- 
mary focal plane. The solid angle subtended at M by the 
opening in the stop is supposed to be smaller than that 
subtended by the rim of the lens; that is, as here shown, 
Z AMC<Z AMG; and, consequently, the front stop acts here 
both as aperture-stop and entrance-pupil, so that the center K 
of the aperture-stop is likewise the center O of the entrance- 
pupil. Looking through the lens from the other side, one 
will see at 0' a virtual, erect image B'C of the aperture-stop 
BC, and hence this image is the exit-pupil of the system. 
The angle BMC is the aperture-angle of the cone of rays 
that come from the axial object-point M in the focus-plane; 
after passing through the system, these rays meet at M' in 
the screen-plane, the aperture-angle of the bundle of rays 



135] 



Pupils of Optical System 



403 



in the image-space being ZB'M'C. The effective rays 
coming from a point Q in the focus-plane are comprised 
within ZBQC in the object-space and ZB'Q'C in the 
image-space. If the object-point does not he in the 
focus-plane, and yet not too far from it, the opening BOC will 
act as entrance-pupil for this point also. Thus, for example, 
in order to construct the point R' conjugate to an object- 
point R which does not he exactly in the focus-plane, we 
have merely to draw the straight lines RB, RO, RC until 
they meet the lens, and connect these latter points with B', 
0', C, respectively, by straight lines which will intersect in 
the image-point R'. 

Again, consider a system composed of two equal thin 
convex lenses whose centers are at Ai and A 2 (Fig. 176), 




Fig. 176, 



-Optical system composed of two equal thin convex lenses with 
interior stop placed midway between the two lenses. 



with a stop UV placed midway between them; if the center 
of the stop is designated by K, then AiK = KA 2 . The image 
of the stop as seen through the front lens is BOC, and its 
image as seen by looking through the other lens in the op- 
posite direction is B'O'C; these images being equal in size 
and symmetrically situated with respect to the stop itself. 
The image of the rim of each lens cast by the other lens 
should also be constructed, but for the sake of simplicity 
these images are not drawn in the figure, because the di- 



404 Mirrors, Prisms and Lenses [§ 136 

ameters of the lenses are taken sufficiently large as com- 
pared with the diameter of the stop interposed between 
them at K to insure that the latter acts as aperture-stop 
with respect to the axial object-point M on which the in- 
strument is supposed to be focused. Consequently, since 
the stop-image BC subtends at M an angle less than that 
subtended by the rim of the front lens or by the image of 
the rim of the second lens, it will be the entrance-pupil of 
the system; and, similarly, B'C which is the image of BC 
formed by the system as a whole will be the exit-pupil. 
Thus, in order to construct the image-point M' conjugate 
to the axial object-point M, we have merely to draw the 
straight line MC and to determine the point where this line 
meets the first lens; and from the latter point draw a straight 
line through the point V in the edge of the stop to meet the 
second lens; and, finally, draw the straight line which joins 
this latter point with the point C in the edge of the exit- 
pupil; this fine will cross the axis at the required point M' 
in the screen-plane. Similarly, drawing from the object- 
point Q the three rays QB, QO and QC, we can continue 
the paths of these rays from the first lens to the second 
through the points U, K and V, respectively, in the stop- 
opening; and since the rays must issue from the second lens 
so as to go through B', 0' and C, respectively, in the exit- 
pupil, their common point of intersection in the image- 
space will be the point Q' conjugate to Q. In the diagram 
the point Q is taken in the focus-plane; but the same con- 
struction will apply also to determine the position of an 
image-point R' conjugate to an object-point R which does 
not lie in the focus-plane. 

136. Aperture- Angle. Case of Two or More Entrance- 
Pupils.— The angle OMC=t? (Figs. 175 and 176) subtended 
at the axial object-point M by the radius OC of the entrance- 
pupil is called the aperture-angle of the optical system. If 
we put OC = p (where p is to be reckoned positive or neg- 
ative according as the point C lies above or below the axis) 



§ 136] System with Two Entrance-Pupils 405 

and OM=2, then tsn\r)=—p/z. In like manner, if r)' = 
Z O'M'C denotes the angle subtended at the point M' con- 
jugate to M by the corresponding radius of the exit-pupil 
(0'C' = p'), and if also O'MW, then tan 77'= -p'/z'. 

The pupils of an optical system depend essentially, as has 
been stated, on the position of the axial object-point M on 




Fig. 177. — Case of two entrance-pupils. 

which the instrument is focused. In the diagram (Fig. 177) 
I and II represent a pair of stops or stop-images as seen 
by an eye looking into the front end of the instrument. Join 
one end of the diameter of one of these openings by straight 
lines with both ends of the diameter of the other opening; 
and let the points where the straight lines cross the axis be 
designated by X and Y. The two apertures subtend equal 
angles at these points, and hence if the object-point M co- 
incides with either X or Y, the entrance-pupil of the system 
may be either I or II ; in fact, for these two special positions 
of M there will be two entrance-pupils, and, of course, also 
two exit-pupils. If the object-point M lies between X and 
Y, then in the case represented in the figure the opening II 
will subtend a smaller angle at M than the opening I so 



406 Mirrors, Prisms and Lenses [§ 137 

that the former will act as the entrance-pupil. But for any- 
other position of the axial object-point M besides those 
above mentioned the opening I will be the entrance-pupil. 

137. Field of View. — The limitation of the apertures of 
the bundles of effective rays is not the only office of the 
stops and lens fastenings. One of their most important 
functions is to define the extent of the object that is to be 
reproduced in the instrument as has been pointed out in 
several simple illustrations in the earlier pages of this book 
(see §§ 9, 16, 73 and 98). In the adjoining diagram 
(Fig. 178), where the entrance-pupil of the system is repre- 
sented by the opening BC, the other stops or stop-images in the 
object-space act like circular windows or port-holes through 
which the rays that are directed from the various parts of 
the object towards points in the open space of the entrance- 
pupil will have to pass if they are to succeed in getting 
through the instrument without being intercepted on the 
way. Evidently, that one of these openings which subtends 
the smallest angle at the center O of the entrance-pupil will 
limit the extent of the field of view in the object-space. This 
opening which is represented in the figure by GH is called 
the entrance-port; and the material stop or lens-rim which 
is responsible for it is called the field-stop (§9). 

Let the straight line CH drawn through the upper extrem- 
ities of the diameters BC and GH of the entrance-pupil and 
entrance-port meet the optical axis in the point designated 
by L and the focus-plane in the point designated by U. If 
this straight line is revolved around the axis of the instru- 
ment, the point U will describe a circle in the focus-plane 
around the axial object-point M as center; and it is obvious 
that any point in this plane within the circumference of this 
circle, or, indeed, any object-point contained inside the 
conical surface generated by the revolution of the straight 
line passing through C and H, may send rays to all parts 
of the entrance-pupil which will not be intercepted any- 
where in the instrument. Thus, the entire aperture of the 



137] 



Field of View 



407 



entrance-pupil will be the common base of the cones of ef- 
fective rays emanating from sources which lie within this 
region of the object-space. 




Entr-auce 
Pupil 



Fig. 178. 



-Field of view of optical system on side of object, determined 
by the entrance-pupil and the entrance-port. 



Again, the straight line OH drawn through the center of 
the entrance-pupil and the upper edge of the entrance-port 
will determine a second limiting point V in the focus-plane 
which is farther from the optical axis than the first point U; 



408 Mirrors, Prisms and Lenses [§ 137 

and in case of object-points lying in the focus-plane between 
U and V the sections of the bundles of effective rays made 
by the plane of the entrance-pupil will have areas that are 
comprised between the entire area of the opening of the 
entrance-pupil and half that area; and this will be true like- 
wise with respect to all those points in the object-space that 
are contained between the two conical surfaces generated 
by the revolution of the straight lines CH and OH around 
the axis of symmetry. Such points will not lie outside the 
field of view, but although they can utilize more than half 
the opening of the entrance-pupil, they are not in a position 
to take advantage of the entire opening. 

Finally, the straight line BH drawn through the lower 
edge of the entrance-pupil and the upper edge of the entrance- 
port, which crosses the optical axis at the point marked J, 
will determine an extreme point W in the focus-plane which 
is more remote from the axis than the point V; and it is evi- 
dent from the figure that object-points in the focus-plane 
which lie in the annular space between the two circles de- 
scribed around M as center with radii MV and MW are 
even more unfavorably situated for sending rays into the 
entrance-pupil, because they cannot utilize as much as half 
of the pupil-opening. In fact, the effective rays which come 
from the farthest point W pass through the circumference 
of the pupil, and any point lying beyond W will be wholly 
invisible, that is, entirely outside the field of view of the 
instrument. 

Thus, we see that the focus-plane is divided into zones by 
three concentric circles of radii MU, MV and MW. Object- 
points lying in the interior central zone send their light 
through the entrance-pupil without let or hindrance on the 
part of the field-stop; so that this is the brightest part of 
the field. But in the two outer zones there is a gradual fad- 
ing away of light until we reach finally the border of complete 
darkness. The three regions of the field of view in the object- 
space are usually defined by the angles 271, 27, and 272 



§ 138] Lens and Eye 409 

whose vertices are on the optical axis at the points L, O and 
J, respectively; so that yi = ZSLH, 7 = ZS0H and 72 = 
ZSJH. If the radii of the entrance-pupil and entrance- 
port are denoted by p=OC and 6=SH, and if the distance 
of the entrance-pupil from the entrance-port is denoted by 
c= SO, then 

b — p b b+p 

tan7i=— — -, tan7=--, tan7 2 =— — -. 
c c c 

The field of view in the image-space is determined in like 
manner. The image of the entrance-port GH with its center 
at S, which is produced by the entire optical system, is the 
exit-port G'H' with its center at S'; and by priming all the 
letters in the expressions above a similar system of equations 
will be obtained for defining the three regions, 27/, 27' 
and 272', of the field of view in the image-space. Generally, 
the edge of the field is considered as determined by the cen- 
ter of the pupil, that is, by the angle 27 in the object-space 
and the angle 27' in the image-space. 

138. Field of View of System Consisting of a Thin Lens 
and the Eye. — A simple but very instructive illustration of 
the principles explained in the foregoing section is afforded 
by an ordinary convex lens used as a magnifying glass. In 
order to obtain a virtual, magnified image with a lens of 
this kind, the distance of the glass from the object must not 
exceed the focal length of the lens, and then when the image 
is viewed through the glass, the iris of the observer's eye 
will act as the aperture-stop of the system, no matter where 
the eye is placed, provided the diameter of the pupil of the 
eye is less than that of the lens, as is practically nearly al- 
ways the case. Moreover, since the pupil of the eye is the 
common base of the bundles of rays which come to it from 
the various parts of the image, it is the exit-pupil of the 
system, and its image in the glass is, therefore, the entrance- 
pupil. If the eye is placed on the axis of the lens between 
the lens and its second focal point (Fig. 179), the entrance- 
pupil will be a virtual image of the pupil of the eye and will 



410 



Mirrors, Prisms and Lenses 



138 



lie on the same side of the lens as the eye; if the eye is placed 
at the second focal point of the glass, the entrance-pupil will 
be at infinity (see § 144) ; and, finally, if, as represented in 




Fig. 179. — Field of view of thin convex lens when the eye 
is between the lens and its second focal plane. 

Fig. 180, the eye is placed at a point O' beyond the second 
focal point of the convex lens GH, the center of the entrance- 
pupil will be at a point O on the same side of the lens as the 
object MQ. The distance between the eye and the second 
focal point of a convex lens used as a magnifying glass is 
never very great, and, consequently, the distance of the cen- 
ter O of the entrance-pupil from the first focal point is rela- 
tively always quite large. The rim of the glass acts as the 
field-stop, and it is at the same time both the entrance-port 
and the exit-port of the system; and hence the field of view 
exposed to the eye in the image-space is entirely analogous 
to the field which would be seen by an eye looking through 
a circular window of the same form, dimensions and position 
as the lens. Since the exit-port is represented here as being 



138] 



Lens and Eye 



411 



at a considerable distance from the exit-pupil, the field of 
view will appear vignetted, that is, the border will not be 
sharply outlined, but the field will fade out imperceptibly 



M' 







-".;>. 


H 






\?^ 








""]].■■'.■ 


":: :! * 


N. 




^\^~^^ 


C r 
D' 




M 




A \ 


B' 








1 





Fig. 180. — Field of view of thin convex lens when the eye 
beyond the second focal plane. 



placed 



towards the edges. If the diameter of the lens is denoted 
by 2b, and if the distance of the eye from the lens is denoted 
by c = AO', then tany=— 5/c, where y' = ZAO'H. The 
extent of the field as measured by the angle 2 7' is indepen- 
dent of the size of the pupil of the eye. If the focus-plane 
coincides with the first focal plane of the magnifying glass, 
the diameter of the visible portion of the object will be 2y = 
-2/.tan7 / . 

In a compound microscope or in an astronomical telescope 
the object-glass produces a real inverted image of the object, 
and this image is magnified by the ocular, which is essen- 
tially a convergent optical system on the order of a convex 
lens used as a magnifying glass. In the interior of the in- 
strument between the object-glass and the ocular, at the 
place where the real image is cast by the object-glass, 
there is usually inserted a material stop, which cuts off the 



412 



Mirrors, Prisms and Lenses 



[§138 



" ragged edge" of the field of view, so that only the central 
portion which sends complete bundles of rays through the 
instrument is visible to the eye. 

In the Dutch telescope the ocular is a divergent optical 
system which may be represented in a diagram by a con- 




Fig. 181. — Ocular system of Galileo's telescope represented in the dia- 
gram by a thin concave lens. Diagram shows how the rays, after 
having passed through the object-glass, enter the .pupil of the observ- 
er's eye B'C. Inverted image of distant object in the object-glass of the 
telescope is formed at MQ; M'Q' is the image of MQ in the ocular. 
G'H' is the image of the rim of the object-glass in the ocular. B'C is 
the image of BC in the ocular. 

cave lens (Fig. 181) which is placed between the object- 
glass and the real image of the object in the object-glass; 
so that so far as the ocular is concerned, this image is a vir- 
tual object, shown in the figure by the line-segment MQ. 
The eye in this case is usually adjusted very close to the 
concave lens. The pupil of the eye is represented in the 
figure by the opening B'C' with its center on the axis at 0'; 
its image in the lens is BC. Here also, just as in the case of 
a convergent ocular, the pupil of the eye will act as the exit- 
pupil unless the diameter of the lens is so small that the 
lens-rim itself performs this office. The image of MQ is 
M'Q', which latter will be erect if MQ is inverted, and since 
MQ is always inverted in the simple telescope, the final 
image in the Dutch telescope is erect. In the case of the 
Dutch telescope the rim of the ocular lens does not limit 



§ 139] Chief Rays 413 

the field of view, but this is limited by the rim of the object- 
glass, which is the entrance-port of the telescope. Hence, 
the image of the object-glass in the ocular is the exit-port. 
This image (called the " eye-ring," § 159) is represented in 
the diagram by the opening G'H' with its center on the 
axis at S'. The object-point Q, as shown in the figure, is 
just at the edge of the field, because the image-ray coming 
from Q' which is directed towards the center O' of the exit- 
pupil is made to pass through the edge of the exit-port (7' = 
ZS'OTT). 

139. The Chief Rays. — Every bundle of effective rays 
emanating from a point of the object contains one ray which 
'in a certain sense is the central or representative ray of the 
configuration and which may therefore be distinguished as 
the chief ray (see § 11). The ray which is entitled to this 
preeminence is evidently that one which in traversing the 
medium in which the aperture-stop lies passes through the 
center K of this stop. If the optical system is free from the 
so-called aberrations, both spherical and chromatic (as is 
assumed in the present discussion), the chief ray of the 
bundle may also be defined as that ray which in the object- 
space passes through the center of the entrance-pupil; 
but the first definition is preferable because it is applicable 
to actual as well as to ideal optical systems. 

The totality of the chief rays coming from all parts of the 
object constitute, therefore, a homocentric bundle of rays 
in the medium where the aperture-stop lies, and these rays 
proceed exactly as though they had originated from a lu- 
minous point at K. 

If the aperture-stop is very narrow, comparable, say, with 
the dimensions of a pin-hole, the apertures of the bundles of 
effective rays will be correspondingly small; and in the limit 
when the opening in the stop may be regarded as reduced to 
a mere point at its center K, the ray-bundles will have col- 
lapsed into mere skeletons, so to speak, each one represented 
by its chief ray. It is because the chief rays are the last 



414 



Mirrors, Prisms and Lenses 



[§140 



survivors of the ray-bundles that it is particularly impor- 
tant in nearly all optical problems to investigate the pro- 
cedure of these more or less characteristic rays. 

140. The so-called " Blur-Circles » (or Circles of Dif- 
fusion) in the Screen-Plane. — Now if the cardinal points of 
the optical system are assigned, the image-relief correspond- 
ing to a three-dimensional object may be constructed point 
by point, according to the methods which have been ex- 
plained. But, as a matter of fact, the image produced by 




Fig. 182. — Diagram showing how object-relief and image-relief are pro- 
jected in focus-plane and screen-plane from entrance-pupil and exit- 
pupil, respectively; and the "blur circles" in these planes. 

an optical instrument, instead of being left, as it were, 
floating in space, is almost invariably received on a surface 
or screen of some kind, as, for example, the ground-glass 
plate of a photographic camera. In case the image is vir- 
tual, as in a microscope or telescope, it is intended to be 
viewed by the eye looking into the instrument, so that here 
also in the last analysis the image is projected on the sur- 
face of the retina of the observing eye. This receiving sur- 
face is called technically the "screen," which affords also 
an explanation of the name screen-plane (§ 134) as applied 
to the plane conjugate to the focus-plane. 

In the diagram (Fig. 182) the screen-plane is placed at 



§140] "Blur-Circles" 415 

right angles to the axis at the point marked M' which is 
conjugate to the axial object-point M, so that this point is 
seen sharply focused on the screen. Evidently, however, 
the optical system cannot be in focus for all the different 
points of the object-relief at the same time, because the 
screen-plane is conjugate to only one transversal plane of 
the object-space, namely, the focus-plane perpendicular to 
the axis at M. Thus, for example, the reproduction of a 
solid object such as an extended view of a landscape on the 
ground-glass plate of a camera is not an image at all in the 
strict optical sense of the term, inasmuch as it is not con- 
jugate to the entire object with respect to the photographic 
objective. Only such points of the object as lie in the focus- 
plane will be reproduced by sharp clear-cut image-points 
in the screen-plane (as, for example, the point marked 1 in 
the figure); whereas object-points situated to one side or 
the other of the focus-plane will be depicted more or less in- 
distinctly on the screen-plane by small luminous areas which 
are sections cut out by this plane from the cones of image- 
rays emanating originally from points of the object such as 
those marked 2, 3 in the diagram. These little patches of 
light on the screen, which are usually elliptical in form, and 
whose dimensions depend on obvious geometrical factors, 
such as the diameter and position of the exit-pupil, etc., 
are the so-called circles of diffusion or "blur-circles ," in 
consequence of which details of the image as projected on 
the screen are necessarily impaired to a greater or less 
degree. 

It is a simple matter to reconstruct the object-figure 
which is optically conjugate to this configuration of image- 
points and " blur-circles" in the screen-plane, which will 
obviously be a similar configuration of object-points and 
"blur-circles" all lying in the focus-plane. Moreover, since 
the exit-pupil is conjugate to the entrance-pupil, the cones 
of rays in the object-space corresponding to those in the 
image-space may be easily constructed by taking the points 



416 



Mirrors, Prisms and Lenses 



141 



of the object-relief as vertices and the entrance-pupil as the 
common base of these cones. The tout ensemble of the sec- 
tions of all these bundles of object-rays made by the focus- 
plane will evidently be the figure in the object-space that 
corresponds to the representation on the screen, and ac- 
cording to the theory of optical imagery these two plane 
configurations will be similar. This " vicarious" object in 
the focus-plane is sometimes called the projected copy of the 
object-relief, because it is obtained by projecting the points 
of the object from the entrance-pupil on the focus-plane. 

141. The Pupil-Centers as Centers of Perspective of 
Object-Space and Image-Space. — It hardly needs to be 
pointed out that the " blur-circles " which arise from this 




Fig. 183. — Projection of object-relief and image-relief in focus-plane and 
screen-plane from the centers of entrance-pupil and exit-pupil, respectively. 

mode of reproducing a solid object on a plane (or curved) 
surface are due to no faults of the optical system itself, but 
are necessary consequences of the mode of representation, 
having their origin, in fact, in the object-space by virtue of 
the process employed. The only possible way of diminish- 
ing or eliminating the indistinctness or lack of detail in the 
reproduction of parts of the object that do not lie in the 
focus-plane consists in reducing the diameter of the aperture- 
stop, or in " stopping down" the instrument, as it is called. 



§ 142] Distance of Photograph 417 

If the stop-opening is contracted more and more until finally 
it is no larger than a fine pin-hole, the pupils likewise will 
tend to become mere points at their centers O, O' (Fig. 183), 
and the " blur-circles " both in the focus-plane and in the 
screen-plane will diminish in area pari passu and ultimately 
collapse also into the points where the chief rays cross this 
pair of conjugate planes. The points marked I, II, III, 
etc., where the chief rays belonging to the object-points 1, 
2, 3, etc., cross the focus-plane, and which are the centers of 
the so-called " blur-circles" in this plane, are obtained, 
therefore, by projecting all the points of the object from the 
center of the entrance-pupil on to the focus-plane. This 
mode of representing a three-dimensional object is, however, 
in no wise peculiar to the optical system itself, but is the 
old familiar process of perspective reproduction by central 
projection on a plane. Thus, the pupil-centers O, 0' are to 
be regarded as the centers of perspective of the object-space 
and image-space, respectively. 

142. Proper Distance of Viewing a Photograph. — These 
principles explain why it is necessary to view a photograph 
at a certain distance from the eye in order to obtain a cor- 
rect impression of the object which is depicted. Suppose, 
for example, that O, 0' (Fig. 184) designate the centers of 
the pupils of a photographic lens, and that an object NR is 
reproduced in the screen-plane by the perspective copy 
M'Q' whose size is one kth. of that of the projection MQ of 
the object in the focus-plane. Now if the picture is to pro- 
duce the same impression as was produced by the original 
itself on an observer with his eye placed at O, the photo- 
graph must be held in front of the eye at a place P such that 
the visual angle KOP which it subtends at the center of 
rotation of the eye shall be equal to the angle QOM ; that is, 
the distance PO in the figure must be equal to one kth. of 
the distance of the center of the entrance-pupil from the 
focus-plane, or PO = MO//c. If (as is usually the case with 
a landscape lens) the focus-plane is at infinity, then PO will 



418 



Mirrors, Prisms and Lenses 



[§142 



be equal to the focal length (/) of the objective. Generally 
speaking, we may say, therefore, that the correct distance 
for viewing a photograph of a distant object is equal to the 
focal length of the objective, this distance being measured 



Screen 
Plane 




Fig. 184. — Correct distance of viewing photograph. 



from the picture to the center of rotation of the observer's 
eye. Accordingly, if the focal length is less than the dis- 
tance between the near point of the eye and the center of 
rotation, which in the case of a normal emmetropic eye of 
an adolescent is about 10 or 12 cm., it will be impossible to 
see the picture distinctly with the naked eye and at the same 
time under the correct visual angle. Moreover, even if the 
focal length of the photographic lens were not less than this 
least distance of distinct vision, the effort of accommodation 
which the eye has to make in order to focus the image sharply 
on the retina under the correct visual angle will superinduce 
an illusion which will be different from the impression of 
reality which it is the purpose of the picture to convey. In 
the case of a photograph made by an objective of very short 
focal length it is possible indeed to make an enlarged copy 
which may be viewed at the correct distance, but this is 
always more or less troublesome and expensive. Dr. Von 
Rohr has invented an instrument called a verant which is 
ingeniously designed to oyercome as far as possible the dif- 
ficulties above mentioned; so that viewed through this ap- 



143] 



Perspective Elongation of Image 



419 



paratus the photograph is seen more or less exactly as the 
object appeared. 

143. Perspective Elongation of Image. — If the screen- 
plane is not focused exactly on the image-point R' (Fig. 185), 




Fig. 185. — Perspective elongation of image. 

this point will be shown on the screen by a "blur-circle' 7 
whose center will be at the point Q' which is the projection 
of R' from the center O' of the exit-pupil. Let e = L'M' 
denote the distance of the screen-plane M'Q' from the image- 
plane L'R', where L', M' designate the feet of the perpen- 
diculars dropped from R', Q', respectively, on the axis. 
From the diagram we obtain the proportion : 

M'Q' ^ 0'M' _ Q'M' 

TTW'MZ ~0'M'+M'L' ; 
which may be written: 

y'ly'^z'Kz'-e), 
where y' = M'Q', y" = L'R' and z' = 0'M'. Moreover, since 
e may be regarded as small in comparison with z f , we obtain : 

y' — y" =—y" i approximately. 

The difference (y' —y") is the measure of the perspective 
elongation due to imperfect focusing. 

If the exit-pupil is at infinity, then R'Q' will be parallel 
to the axis and y f = y" ; and under these circumstances, the 
perspective reproduction in the screen-plane will be of the 
same size as the image, no matter how much it is out of focus. 



420 



Mirrors, Prisms and Lenses 



;§ 144 



144. Telecentric Systems. — A common laboratory use of 
an optical instrument is to ascertain the size of an inacces- 
sible or intangible object from the measured dimensions of 
its image as determined by means of a scale on which the 




Fig. 186. — Telecentric optical system: Case of a thin convex lens with front 
stop in first focal plane. Object represented by LR; blurred image 
M'Q' appears of the same size as sharp image L'R'. 

image is projected; but, in general, unless the scale is exactly 
in the same plane as the image, there will be a parallax error 
in the measurement of the image due to its perspective 
elongation. However, if the chief rays in the image-space 
are parallel to the axis, which may be effected by placing the 
aperture-stop so that the entrance-pupil lies in the primary 
focal plane of the instrument, as illustrated in Fig. 186, the 
perspective elongation vanishes (y'—y" = 0, as explained 
in § 143); and, consequently, the image y" = L'R' will ap- 
pear of the same size as its projection ?/ = M'Q', no matter 
whether it lies in the same plane as the scale or not. 

Similarly, if the aperture-stop is placed so that the en- 
trance-pupil is at infinity and the exit-pupil lies therefore in 
the secondary focal plane, the chief rays in the object-space 
will then all be parallel to the optical axis. 

Systems of this description in which one or other of the 
two projection centers 0, 0' is at infinity are said to be 



§ 144] Keratometer 421 

telecentric. This is the principle of nearly all systems for 
micrometer measurements of optical images. 

A simple illustration of a device of this kind that is tele- 
centric on the side next the object is afforded by the oph- 
thalmic instrument called a keratometer, which, as the name 
implies, is intended primarily to measure the diameter of the 
cornea or the apparent diameter of the eye-pupil. It is used 
also to measure the distance of a correction-glass (§ 154) 
from an ametropic eye (§ 153), which is an important factor 
in the prescription of spectacles. The instrument consists 
essentially of a long narrow tube, near the middle of which 
is mounted a convex lens of low power adjusted so that its 
second focal point F' coincides exactly with the center of a 
small aperture in a metal disk placed at the end of the tube 
where the observer puts his eye. At the opposite end of the 
tube a scale graduated in half-millimeters is mounted so that 



eye of 
Patient; 




r^i ; 



Spectacle Glass 

Scale 

Fig. 187. — Diagram of instrument called keratometer, as used to measure 
the distance of spectacle glass from the cornea of the eye. 

its upper edge coincides with a horizontal diameter of the 
tube at this place. The upper part of this end of the tube 
is cut away in order to admit sufficient light to illuminate the 
scale. 

When the keratometer is used to measure the distance 
between the vertex of the cornea and the vertex of the cor- 
rection-glass, it is placed with its axis at right-angles to the 
line of sight of the patient, as represented in the diagram 
(Fig. 187), the scale being brought as near as possible to 



422 



Mirrors, Prisms and Lenses 



l§ 144 



the patient between his eye and the spectacle-glass. The 
distance AB to be measured is projected on the scale by 
rays that are parallel to the axis of the lens, so that when 
the observer looks through the instrument he can read off 
this distance on the image of the scale. 

Practically the same principle is employed also in Badal's 
optometer for measuring the visual acuity of the eye. It 




Fig. 188. — Badal's optometer, with second focal point (F)' of 
convex lens at first nodal point of patient's eye; forming in 
conjunction with the eye a telecentric system. 

consists of a single convex lens mounted at one end of a long 
graduated bar which is provided with a movable carrier 
holding a test-chart of some kind. If the lens, which usually 
has a refracting power of about 10 dioptries, is adjusted 
about 9 cm. in front of the cornea so that its second focal 
point F' coincides with the nodal point of the eye (Fig. 188), 
a ray meeting the lens in a direction parallel to the axis will 
emerge from it so as to go through the nodal point of the 




Fig. 189. — Badal's optometer, with second focal point (F') of 
convex lens at first focal point of patient's eye; forming in 
conjunction with the eye a telescopic system. 

eye and thence to the retina without change of direction. 
Accordingly, just as though a narrow aperture were placed 



Ch. XII] Problems 423 

at the nodal point of the eye, the size of the retinal image 
will not be altered whether the object or chart on the bar be 
far or near; whereas the distinctness with which the details 
of the object are seen, which affords the measure of the visual 
acuity, will depend on the distance of the object. 

Another method of using this optometer is to place the 
lens about 2 cm. farther from the eye, as shown in Fig. 189, 
so that now its second focal point lies in the anterior focal 
plane of the eye. Under these circumstances an incident 
ray proceeding parallel to the axis will emerge from the lens 
and cross the axis at the anterior focal point of the eye, so 
that after traversing the eye-media it will again be parallel 
to the axis. Consequently, here also the image formed on 
the retina will be of the same size no matter where the object 
is placed on the bar in front of the lens, just as if there were 
a narrow stop at the anterior focal point of the eye. In this 
latter adjustment the lens and the eye together constitute 
an optical system which is telecentric on both sides, that is, 
a telescopic system (§ 125). 

PROBLEMS 

1. A cylindrical tube, 2 cm. in diameter and 10 cm. long, 
is closed at one end by a thin convex lens of focal length 4 cm. 
If this end of the tube is pointed towards a distant object, 
what will be the position and diameter of the entrance- 
pupil? Ans. 6| cm. in front of the lens; diameter, 1^ cm. 

2. In the preceding problem, where would the object have 
to be in order that the lens itself might act as entrance- 
pupil? 

Ans. In front of the lens, not more than 20 or less than 
4 cm. away. 

3. If in No. 1 the other end of the tube is closed by a thin 
eye-lens whose focal length is such that when the combina- 
tion is pointed at an object 24 cm. from the object-glass, the 
bundles of rays issuing from the eye-lens are cylindrical, find 



424 Mirrors, Prisms and Lenses [Ch. XII 

the positions of the pupils of the system and the focal length 
of the eye-lens. 

Ans. Entrance-pupil 6| cm. in front of object-glass; 
exit-pupil coincides with eye-glass; focal length of eye-glass, 
5.2 cm. 

4. In the preceding problem what will be the answers on 
the supposition that the object is 12 cm. from the object- 
glass? 

Ans. Entrance-pupil coincides with object-glass; exit- 
pupil is 6| cm. beyond eye-glass; focal length of eye- 
glass, 4 cm. 

5. A real inverted image of an extended object is formed 
by the object-glass of a simple astronomical telescope in the 
primary focal plane of the eye-glass. The focal lengths of 
the object-glass and eye-glass are 2 feet and 1.5 inches, re- 
spectively, and their diameters are 6 inches and 1 inch, 
respectively. If the distance of the object from the object- 
glass is 240 feet, find the position and diameter of the en- 
trance-port and the diameter of the portion of the object 
that is completely visible through the telescope. 

Ans. Entrance-port is 30.21 feet from object-glass, and 
its diameter is 1.175 feet; diameter of visible portion of ob- 
ject, 5.865 feet. 

6. A thin convex lens of focal length 10 cm. and diameter 
4 cm. is used as a magnifying glass. If an eye adapted for 
parallel rays is placed at a distance of 5 cm. from the lens, 
what will be the diameter of the portion of the object that 
can be seen distinctly? Ans. 8 cm. 

7. The diameter of a thin convex lens is 1 inch, and its 
focal length is 10 inches. The lens is placed midway between 
the eye and a plane object which is 10 inches from the eye. 
How much of the object is visible through the lens? 

Ans. 1| inch. 



CHAPTER XIII 

OPTICAL SYSTEM OF THE EYE. MAGNIFYING POWER 
OF OPTICAL INSTRUMENTS 

145. The Human Eye. — The organ of vision is composed 
of the eye-ball, wherein the visual impulses are produced by 
the impact of light; the optic nerve which transmits these 
excitations to the brain; and the visual center in the brain 
where the sensation of vision comes to consciousness. 

The eye-ball (Fig. 190) lying in a bony socket on a cushion 
of fat and connective tissue, in which it is free to turn in all 
directions with little or no friction, consists of an almost 
spherical dark chamber, filled with transparent optical media 
which form the optical system of the eye (Fig. 191). The 
outer protecting envelope of the eye-ball is the tough, white 
membrane called, from its hardness, the sclerotic coat or sclera, 
popularly known as the " white of the eye." This opaque 
membrane is continued in front by a round opening or win- 
dow called, on account of its horny texture, the cornea. The 
cornea is beautifully transparent, and its mirror-like surface 
forms a slight protuberance shaped something like a watch- 
glass or a prolate spheroid. In the interior of the eye the 
sclerotic coat is overlaid with the dark-colored choroid which 
contains the blood-vessels that nourish the eye and also a 
layer of brown pigment acting to protect the dark chamber 
of the eye from diffused light. Behind the cornea lies the 
anterior chamber filled with transparent fluid called the 
aqueous humor. This anterior chamber is limited behind the 
iris, which, rich in blood-vessels, imparts to the eye its char- 
acteristic color. This is an opaque screen or curtain which 
contains a central hole, the pupil, which is circular in the 
human eye. The aperture of a bundle of rays entering the 

425 



426 



Mirrors, Prisms and Lenses 



L § 145 



eye from a luminous point, in proportion to the dimensions 
of the eye, is enormous as compared, for example, with the 
same magnitude in a telescope; and the office of the pupil is 










to stop down this aperture to suitable proportions. The 
pupil contracts or dilates involuntarily and regulates the 
quantity of light that is admitted to the eye. In the struc- 
ture of the iris there are two sets of fibers, the circular and 



§145] 



Human Eye 



427 



the radiating; when the circular fibers contract, the pupil 
contracts, and when the radiating fibers contract, the pupil 




dilates. In the front part of the eye the choroid lining is 
bordered at the edge of the cornea by a kind of folded drapery 
the so-called ciliary body, which is hidden from without be- 



428 Mirrors, Prisms and Lenses [§ 145 

hind the iris and which contains the delicate system of 
muscles which control the mechanism of accommodation. 
The crystalline lens composed of a perfectly transparent 
substance is indirectly attached to the ciliary body by a 
band which surrounds the edge of the lens like a ring and 
which is disposed in radial folds somewhat after the manner 
of a neck-frill. This band is the suspensory ligament or zonule 
of Zinn. The lens itself is double convex, the posterior sur- 
face being more strongly curved than the anterior surface. 
The substance of the lens consists of layers of different in- 
dices of refraction increasing towards the center or core of the 
lens. The entire space behind the lens is filled with a trans- 
parent jelly-like substance called the vitreous humor, which 
has the same index of refraction as the aqueous humor, 
namely, 1.336. 

The light-sensitive retina lying on the inside of the choroid 
is exceedingly delicate and transparent. In spite of its 
slight thickness which nowhere exceeds 0.4 mm., the struc- 
ture of the retina is very complicated, and no less than ten 
layers have been distinguished (Fig. 192). The layer next 
the vitreous humor is composed of nerve-fibres spreading 
out radially from the optic nerve. This layer is connected 
with the following layer containing the large ganglion or 
nerve-cells, and this in turn is connected by an apparatus 
of fibers and cells with the peculiar light-sensitive elements 
of the retina, the so-called visual cells which form the "bacil- 
lary layer." These visual cells consist of characteristic 
elongated bodies which are distinguished as rods and cones. 
The rods are slender cylinders, while the cones or bulbs are 
somewhat thicker and flask-shaped. They are all disposed 
perpendicularly to the surface of the retina, closely packed 
together, so as to form a mosaic layer at the back of the 
retina. 

Near the center of the retina at the back of the eye, a little 
to the temporal side, is located the yellow spot or macula lutea, 
where the visual cells are composed mostly of cones. This 



145] 



Human Eye 



429 



is the most sensitive part of the retina, especially the minute 
pit or depression at the center of this area, called the fovea 
centralis, which consists entirely of cones densely crowded 
together. 

As compared with an artificial optical instrument, the 

2. &€nu <**** £>***** 




^flfe J &****•* Store J*y 



7 - J'wWw^ J^« **»-(« 



A WVu*. 3l&>* *<^8-r 
Fig. 192. — Structure of the retina of the human eye. 

field of view of the immobile eye is very extensive, amounting 
to about 150° laterally and 120° vertically. The diameter 
of the fovea centralis corresponds in the field of vision of the 
eye to an angular space which may be covered by the nail 
of the fore finger extended at arm's length. In this part of 
the field vision is so acute that details of an object can be 



430 Mirrors, Prisms and Lenses J§ 145 

distinguished as separate provided their angular distance is 
not less than one minute of arc (cf. §10). If the apparent size 
of an object is so small that its image formed on the retina at 
the fovea centralis covers only a single visual cell, the object 
ceases to have any apparent size at all and cannot be dis- 
tinguished from a point. The size of the retinal image corre- 
sponding to an object whose apparent size is one minute of 
arc is found by calculation from the known optical constants 
of the eye to be 0.00487 mm. Anatomical measurements 
give a similar value for the diameter of a visual cell. 

The inverted image cast on the retina of the eye has been 
compared to a sketch which is roughly outlined in the outer 
parts, but which is more and more finely executed in towards 
the center, until at the fovea centralis itself the details are 
exquisitely finished. Thus, only a comparatively small 
portion of an external object can be seen distinctly by the 
eye at any one moment. If all the parts of the field of view 
were portrayed with equal vividness at the same time and 
came to consciousness at once, the spectator would be com- 
pletely bewildered and unable to concentrate his attention 
on a particular spot or phase of the object. 

The ends of the rods next the choroid contain a coloring 
matter which is sensitive to light, the so-called visual purple, 
which is bleached white by exposure to bright light, but 
which is renewed in darkness by the layer of cells lying be- 
tween the choroid and the retina. The light-disturbance 
arriving at the retina penetrates it as far as the bacillary 
layer of rods and cones, and the stimulus is transmitted back 
through the interposed apparatus to the layer of nerve- 
fibers and thence conducted to the optic nerve in communi- 
cation with the brain. 

Not far from the center of the retina, a little to the nasal 
side, the optic nerve pierces the eye-ball through the sclera 
and choroid. Here the retina is interrupted, so that any light 
which falls on the optic nerve itself cannot be perceived. 
This is the place of the so-called blind-spot (punctum ccecum) 



§ 146] Optical Constants of Eye 431 

of the eye. Corresponding to the area of the blind spot, 
there is a gap in the field of vision of the eye amounting to 
about 6° horizontally and 8° vertically. The dimensions of 
the blind spot are great enough to contain the retinal im- 
ages of eleven full moons placed side by side. The optic 
nerve leaves the eyeball through a bony canal and passes 
thence to the visual center of the brain. 

The mobility of the eye is produced by six muscles, the 
four recti and the two oblique muscles (Fig. 190). The recti 
originate in the posterior part of the socket and are attached 
by their tendons to the sclera so as to move the eye up or 
down and to the right or left. The procedure of the oblique 
muscles is more complicated. The superior oblique, which 
also arises in the posterior part of the socket, passes in the 
front of the eye through a loop or kind of pulley lying on 
the upper nasal side of the socket and then turns downwards 
to attach itself to the sclera. The inferior oblique muscle has 
its origin on the front lower nasal side of the eye-socket, 
and passes to the posterior surface of the eye-ball, being at- 
tached to the sclera on the temporal side. The superior ob- 
lique turns the eye downwards and outwards, and the inferior 
oblique turns it upwards and outwards. 

The motor muscles of the two eyes act together so that 
both eyes turn always in the same sense, to the right or to 
the left, up or down. It is impossible to turn one eye up 
and the other down at same time, so as to look up to the 
sky with one eye and down at the earth with the other. 

146. Optical Constants of the Eye. — The optical axis of the 
eye may be defined as the normal to the anterior surface of 
the cornea which goes through the center of the pupil. This 
line passes approximately through the centers of curvature 
of the refracting surfaces. The schematic eye (see § 130) is 
a centered system of spherical refracting surfaces symmetric 
with respect to the optical axis. The point where the optical 
axis meets the anterior surface of the cornea is called the 
cornea vertex or anterior pole of the eye and is designated 



432 Mirrors, Prisms and Lenses [§ 146 

by A ; and the point where the optical axis meets the retina 
is called the posterior pole of the eye and is designated by B. 
In Gullstrand's schematic eye the distance from A to B 
is equal to 24 mm., therefore somewhat less than an inch. 

The motor muscles of the eye (§ 145) , acting in pairs, turn 
the eye-ball around axes of rotation which all pass through 
a fixed point or pivot called the center of rotation of the eye 
and designated by Z. This point may be considered as 
lying also on the optical axis in the medium of the vitreous 
humor about 13 or 14 mm. from the vertex of the cornea or 
about 10.5 mm behind the pupil. All the excursions of the 
eye are performed around this point. 

The object-point which is sharply imaged on the retina at 
the fovea centralis (§ 145) is called the point of fixation, and the 
straight line which joins the point of fixation with the centre 
of rotation is called the line of fixation. This line indicates 
the direction in which the eye is looking. The field of fixa- 
tion is measured by the greatest angular distance through 
which the line of fixation can be turned; which amounts to 
about a right angle both vertically and horizontally. 

In Gullstrand's schematic eye, as was shown in § 130, 
the primary focal point F lies in front of the eye at a dis- 
tance of 15.707 mm. from the anterior vertex of the cornea, 
while the secondary focal point F' lies on the other side of 
the cornea at a distance of 24.387 mm. The principal points 
(H, H') lie in the aqueous humor slightly beyond the cor- 
nea system at distances AH = +1.348 mm., AH' = +1.602 
mm. Thus the focal lengths are: /= +17.055 mm. /' = 
— 22.785 mm.; the ratio between them being equal to 1.336, 
which is therefore the value of the index of refraction (n') 
of the vitreous humor. Accordingly, the refracting power 
of Gullstrand's schematic eye is F= 58.64 dptr. The 
nodal points (N, N') lie close to the posterior vertex of the 
crystalline lens, on opposite sides of it, at the following dis- 
tances from the vertex of the cornea: AN = +7.078 mm., 
AN'= +7.332 mm. The straight line which joins the point 



147] 



Accommodation of the Eye 



433 



of fixation with the anterior nodal point of the eye is called 
the visual axis. It is parallel to the straight line which joins 
the posterior nodal point with the fovea centralis. Since the 
nodal points are so close together, for many problems con- 
nected with the eye they may be regarded as coincident; so 
that then the visual axis may be defined as the line drawn 
from the point of fixation to the fovea centralis. The visual 
axis meets the cornea a little to the nasal side of the anterior 
vertex and slightly above it, forming with the optical axis an 
angle between 3° and 5°. 

The above values are all given for the passive, unaccommo- 
dated eye. By the act of accommodation the positions of 
the focal points, principal points and nodal points are all dis- 
placed, and accordingly the focal lengths and the refracting 
power of the eye can be varied within certain limits depend- 
ing on the power of accommodation, as will be explained in 
the following section. 

147. Accommodation of the Eye. — When the eye is at rest, 
as when one gazes pensively into space, it is adapted for far 




Fig. 193. — Accommodation of the human eye; 
indicating how the crystalline lens is changed 
from far vision to near vision. 



vision, so that in order to see distinctly objects which are close 
at hand, an effort has to be made which will be greater in 



434 Mirrors, Prisms and Lenses [§ 148 

proportion as the object fixed is nearer to the eye. This proc- 
ess whereby the normal eye is enabled to focus on the retina 
in succession sharp images of objects at different distances 
is called accommodation, and it is this marvelous adapt- 
ability of the human eye, together with its mobility, which 
perhaps more than any other quality entitles it to superior- 
ity over the most perfectly constructed artificial optical in- 
struments. The power of accommodation is achieved by 
changes in the form of the crystalline lens, consisting chiefly 
in a change in the convexity of the anterior surface, produced 
through the mechanism of the ciliary muscle. According 
to the generally accepted theory, so long as the eye is passive, 
the elastic substance of the lens is held flattened in front by 
the suspensory ligament; but in the act of accommodation 
the ciliary muscle contracts, and this is accompanied by a 
relaxation of the ligament of the lens, which is thereby 
permitted to bulge forward by virtue of its own elasticity 
(Fig. 193). 

148. Far Point and Near Point of the Eye. — The far point 
of the eye {punctum remotum) is that point (R) on the axis 
which is sharply focused at the posterior pole of the eye 
when the crystalline lens has its least refracting power; it 
is the point which is seen distinctly when the accommodation 
is entirely relaxed. On the other hand, the near point (or 
punctum proximum) is that point (P) on the axis which is 
seen distinctly when the crystalline lens has its greatest re- 
fracting power, that is, when the accommodation is exerted 
to the utmost. The region of distinct vision within which an 
object must lie in order that its image can be sharply fo- 
cused on the retina of the naked eye is comprised between 
two concentric spherical surfaces, the far point sphere and 
the near point sphere, described around the center of ro- 
tation of the eye (Z) with radii equal to ZR and ZP, re- 
spectively. If the far point lies at infinity, as is the case in the 
normal eye, the far point sphere is identical with the infinitely 
distant plane of space {cf. § 83), as represented in Fig. 194; 



149] 



Presbyopia 



435 



whereas the near point sphere will be real and at a finite 
distance in front of the eye. In such a case the eye can be 
directed towards any point in the field of fixation (§ 146) 
lying on or beyond the near point sphere and accommodate 




Neat- Point 
Sphere 



Fig. 194. — Region of accommodation of emmetropic eye. 

itself to see this point distinctly. In a near-sighted eye both 
far point and near point are real points lying at finite dis- 
tances in front of the eye; but the far point of a far-sighted 
eye is a " virtual" point lying at a finite distance behind the 
eye, and hence an unaided far-sighted eye cannot see dis- 
tinctly a real object without exerting its accommodation to 
a greater or less degree. 

149. Decrease of the Power of Accommodation with 
Increasing Age. — The faculty of accommodation is greatest 
in youth and diminishes rapidly with advancing years. 
The near point of the eye gradually recedes farther and far- 
ther away, which is commonly supposed to be due to a pro- 
gressive diminution of the elasticity of the crystalline lens. 
Thus, at the ages of 10, 20 and 40 years the punctum proxi- 
mum of a normal eye, according to Donders, is in front of 
the eye at distances from the primary principal point equal 
to 7.1, 10 and 22.2 cm., respectively. When the near point 
has retreated to a distance of 22 cm., so that it is no longer 
possible to read or write or do "near work" conveniently 
without the aid of spectacles, the condition of presbyopia 
or old-age vision has begun to set in. Meantime, while the 



436 Mirrors, Prisms and Lenses [§ 150 

power of accommodation of the eye thus continually dimin- 
ishes as the near point recedes farther and farther away, 
the position of the far point remains practically fixed until 
well after middle life; but between the ages of 55 and 60 
years it too begins to separate farther from the eye, and 
thereafter both the near point and the far point travel out- 
wards along the axis of the eye, the former, however, con- 
stantly gaining on the latter; until at last in extreme old age 
the near point actually overtakes the far point, and from 
that time until death they remain together, the power of 
accommodation having been entirely lost. Both points are 
displaced along the axis always in the same direction, that is, 
opposite to that of the incident light. For example, the far 
point of a normal eye is infinitely distant up to about 55 
years of age, whereas ten years later, according to Donders, 
this point will be about 133 cm. behind the eye, having 
moved out through infinity, so to speak, and approached 
the eye from behind. At the sanie age, namely, 65 years, 
the near point will also be behind the eye at a distance of 
400 cm. At 75 years of age the two points will be together 
at a distance of 57.1 cm. behind the eye. Various theories 
have been advanced to account for the senile recession of 
the far point of the eye. It is probably due to a combina- 
tion of causes, partly to a change in the form of the lens pro- 
duced by the increased resistance of the enveloping coat of 
the eye-ball and the decreased pressure of the surrounding 
tissue, and partly also to senile changes in the lens-substance 
itself whereby the " total index" of the lens is lowered in 
value. 

150. Change of Refracting Power in Accommodation. — 
It was remarked above (§ 146) that the positions of the car- 
dinal points of the optical system of the eye are all altered 
in the act of accommodation. Thus, for example, in Gull- 
strand's schematic eye, which is calculated for an adoles- 
cent youth, the near point is at a distance AP = — 10.23 cm. 
from the vertex of the cornea; and for this state of maxi- 



§ 151] Amplitude of Accommodation 437 

mum accommodation the positions of the focal points and 
principal points are found to be as follows : 

AF = - 12.397 mm., AF ' = +21.016 mm., 
AH = + 1.772 mm., AH' = + 2.086 mm.; 
and, accordingly, the focal lengths and the refracting power 
are: 

i=+14.169mm., /'= -18.930 mm., F= +70.57 dptr. 
It will be observed that, whereas the focal points have un- 
dergone considerable displacements from their positions in 
the passive eye, the corresponding displacements of the 
principal points are less than half a millimeter; and since in 
most physiological measurements half a millimeter is within 
the limit of error, we can usually afford to neglect altogether 
the accommodative displacement of the principal points of 
the eye, that is, we may regard the positions of the princi- 
pal points H, H' as practically fixed and independent of 
the state of accommodation. This is one reason, among 
others, why the principal points of the eye have super- 
seded the other cardinal points as points of reference. Their 
proximity to the cornea is another advantage, inasmuch as 
measurements referred to them are easily related to an ex- 
ternal, visible and tangible point of the eye. In the so- 
called "reduced eye," which consists of a single spherical 
refracting surface separating the outside air from the vitre- 
ous humor and so placed that its vertex lies at the primary 
principal point of the schematic eye, the two principal points 
are, in fact, coincident with each other on the surface of this 
simplified cornea. 

151. Amplitude of Accommodation. — The far point dis- 
tance(a) and the near point distance (b) are the distances of 
the far point and near point, respectively, measured from 
the primary principal point of the eye; thus, a = HR, b = 
HP; it being tacitly assumed here that the position of the 
point H remains sensibly stationary during accommodation, 
as was explained above. Each of these distances is to be 
reckoned negative or positive according as the point in ques- 



438 Mirrors, Prisms and Lenses [§ 151 

tion lies in front of the eye or behind it, respectively. The 
reciprocals of these magnitudes, namely, A = l/a, B = l/b, 
are termed the static refraction (A), or the refraction of the 
eye when the accommodation is completely relaxed, and the 
dynamic refraction (B), or the refraction of the eye when 
the accommodation is exerted to the highest degree. If the 
distances a and b are given in meters, the reciprocal magni- 
tudes will be expressed in dioptries, as is generally the case. 

The range of accommodation is denned to be the distance 
of the near point from the far point, that is, RP = b — a; 
whereas the amplitude of accommodation is the value obtained 
by subtracting algebraically the magnitude of the dynamic 
refraction from that of the static refraction, thus: 
Amplitude of Accommodation = A— B. 
Imagine a thin convex lens placed in the primary principal 
plane of the eye with its axis in the same line as the optical 
axis of the eye, and of such strength that it produces at the 
far point of the eye an image of the near point; according to 
the above definition, the amplitude of accommodation of 
the eye is equal to the refracting power of this lens. For ex- 
ample, in the normal eye at 30 years of age, a= oo, b = 
— 14.3 cm., so that the amplitude of accommodation in this 
case amounts to 7 dptr.; whereas at 60 years of age a = 
+200 cm., b =—200 cm., and hence the amplitude of ac- 
commodation will have been reduced to 1 dptr. 

The distance from the secondary principal point (H') to 
the posterior pole (B) where the optical axis meets the retina 
may be regarded as a measure of the length of the eye-axis, 
especially since the position of H' is sensibly independent 
of the state of accommodation, as has been explained, (§ 150). 
If this distance is denoted by a', that is, if we put a' '= 
H'B, and if also we put A! ' = n'/a' ', where n' denotes the 
index of refraction of the vitreous humor, then we may write : 

A'=A+F, 
where F denotes here the refracting power of the passive, 
unaccommodated eye. Similarly, if the symbol F a is em- 



§153] Emmetropia and Ametropia 439 

ployed to denote the refracting power of the eye in its state 
of maximum accommodation, we shall have: 

A' = B+F & . 
Consequently, we may also say that the power of accommo- 
dation (A — B) is equal to the difference (F^ — F) between 
the greatest and least values of the refracting power of the 
eye. 

152. Various Expressions for the Refraction of the Eye. 
— The refraction of the eye in a given state of accommoda- 
tion is measured by the reciprocal of the distance from the 
eye of the axial object-point M for which the eye is accom- 
modated. Thus, if w=HM, x = FM denote the distances of 
M from the primary principal point and the primary focal 
point, respectively, the magnitudes U = l/u and X = l/x, 
usually expressed in dioptries, are the measures of the prin- 
cipal point refraction and the focal point refraction. The 
relation between U and X may be given in terms of the re- 
fracting power of the eye (F) when it is accommodated for 
the object-point M, as follows: 

TJ _ F.X v _ F.U 

U ~F-X' F+U' 

If an arbitrary point O on the axis of the eye is selected 
as the point of reference, and if we put OM = 2, the refrac- 
tion of the eye, referred to the point O, will be measured by 
Z = 1/z. If the distances of the points H and F from are 
denoted by b and g, that is, if 6 = OH, g = OF, then since 
z = u+b = x+g, we can obtain also the following useful re- 
lations between U, X and Z in terms of b and g: 
Z X 



x= 



l-b.Z l-(b-g)X' 

Z u 

1-g.Z l+(b-g)U' 

U X 



1+b.U 1+g.X' 
153. Emmetropia and Ametropia. — When the static re- 
fraction of the eye is equal to zero (A=0), that is, when 



440 Mirrors, Prisms and Lenses [§ 153 

the far point (R) is infinitely distant, the eye is said to be 
emmetropic. If in the equation A'=A+F, we put ^4 = 0, 
we obtain A' = F, which therefore may be said to be the 
condition of emmetropia. Here F denotes the refracting 
power of the eye when accommodation is entirely relaxed. 
In emmetropia, therefore, the second focal point (F') of 
the passive eye lies on the retina at the posterior pole (B) ; 



To Rat co 




Fig. 195. — Diagram of emmetropic eye. 

so that in a passive emmetropic eye incident parallel rays 
are converged to a focus on the retina, as represented in 
Fig. 195, and the length of the eye-axis is a' = — /'. The 
normal position of the far point is to be regarded as at in- 
finity; and in this sense an emmetropic eye is a normal eye, 
although, strictly speaking, an emmetropic eye may at the 
same time be abnormal in various ways. 

On the other hand, if the static refraction of the eye is 
different from zero (A^O), that is, when the far point (R) 
is not infinitely distant, the eye is said to be ametropic 
Thus, the condition of ametropia may be said to be charac- 
terized by the fact that the refracting power (F) of the 
unaccommodated eye is not equal to A' } which is equiva- 
lent to saying that the length of the eye-axis (a') is numer- 
ically different from the value of the second focal length 
(/'). In other words, the second focal point (F') of an 
ametropic eye in a state of repose does not fall on the retina. 

Two general divisions of ametropia are distinguished de- 
pending, on whether the far point (R) lies on one side or the 



153] 



Myopia and Hypermetropia 



441 



other of the primary principal point (H). Thus, if A<Q, 
that is, if the far point lies at a finite distance in front of the 
eye, the ametropia in this case is called myopia (Fig. 196). 
In a myopic eye in a state of repose the second focal point 




Fig. 196. — Ametropic eye: myopia. 

(F') lies in front of the retina (in the vitreous humor), so 
that parallel incident rays will be brought to a focus be- 
fore reaching the retina. On the other hand, if A>0, the 
far point will lie at a finite distance beyond (or behind) the 




Fig. 197. — Ametropic eye: hypermetropia. 

eye, and this form of ametropia is known as hypermetropia 
(Fig. 197). In a hypermetropic eye in a state of repose the 
second focal point (F') falls beyond the retina, so that in- 
cident parallel rays arrive at the retina before coming to a 
focus. A myopic eye cannot focus for a distant object with- 
out the aid of a glass, and it lacks therefore an important 
part of the capacity of an emmetropic eye. On the other 
hand, a hypermetropic eye must make an effort of accom- 
modation each time in order to focus on the retina the image 



442 Mirrors, Prisms and Lenses [§ 153 

of a real object; which frequently produces various troubles, 
sometimes very annoying. Accordingly, both conditions 
included under the general name of ametropia are disad- 
vantageous for practical vision. 

Theoretically, ametropia may be considered as due to 
some abnormality in the values of one or of both of the mag- 
nitudes denoted by A' and F' on which the value of the 
static refraction (^4) depends; so that the following cases 
are possible: 

(1) The length of the eye-ball (a') may be too great 
(axial myopia, a f > —/') or too small (axial hypermetropia, 
a' < — /'), whereas the refracting power (F) is normal. This, 
by far the most common, type is known as axial ametropia. 

(2) On the other hand, while the length of the eye-ball 
may be normal, the magnitude of the refracting power (F) 
may be abnormally great or small. In general, this form of 
ametropia, which is comparatively rare, is due to abnormal 
curvatures of the refracting surfaces {curvature ametropia). 
Or the indices of refraction of the eye-media may have ab- 
normal values (indicial ametropia). Here also may be men- 
tioned the condition known as aphakia produced by the 
extraction of the crystalline lens in the operation for cataract. 

(3) Finally, it may happen that the refracting power and 
the length of the eye-ball are both abnormal. In fact, these 
two anomalies might exist together in exactly the degree 
necessary to counteract each other, so that, in spite of its 
abnormalities, the eye in such a case would be emmetropic. 

In the case of axial ametropia, the relation between the 
static refraction (A) and the length (I) of the eye-ball is 
given by the following formula: 

<= AB = AH+ ^ 

and if the values for Gullstrand's schematic eye (§ 146) 
are substituted in this formula, it may be written as follows: 

1 = 1.602+— — — — - millimeters. 
^4 +58.64 



§154] 



Correction Eye-Glasses 



443 



According to this formula, the length of the eye varies from 
about 21.07 mm. in extreme axial hypermetropia (A = 
+ 10 dptr.) to about 36.18 mm. in case of the highest degree 
of axial myopia (A =—20 dptr.). The length of an axially 
emmetropic eye (.4=0) is 24,38 mm. The length of Gull- 



tnttt 












35 

30 
25 








^~ 
















20 


















Mi 




Hyper 


mctropia 










UJ 







-90 



-lb 



-10 



-5 



+ 5 



+10 



Fig. 198. 



-Curve showing connection between the length of 
the eye-axis and the static refraction. 



strand's schematic eye is 24.01 mm., and hence this eye 
has 1 dptr. of hypermetropia (A=+l dptr.). The accom- 
panying diagram (Fig. 198) exhibits graphically the relation 
between the magnitudes denoted by I and A. The heights 
of the ordinates indicate the axial length of the eye-ball in 
millimeters for values of the static refraction of the eye com- 
prised between —20 and +10 dioptries. 

154. Correction Eye-Glasses. — When a spherical spectacle 
lens is placed in front of the passive, unaccommodated eye, 
with the axis of the lens in the same straight line as the opti- 
cal axis of the eye, there will be a certain axial point M whose 
image in the lens will fall at the far point (R) of the eye; 
and hence the eye looking through the lens will see distinctly 
the image of an object placed at M. If the positions of the 



444 Mirrors, Prisms and Lenses [§154 

principal points of the lens are designated by Hi and Hi', 
and if we put 

tH = 1/Ui = HiM, Mi' = 1/Ui' = Hi'R, 
then 

Ui'-U+F h 

where F\ denotes the refracting power of the lens. Let the 
distance of the primary principal point (H) of the eye from 
the secondary principal point (Hi 7 ) of the lens be denoted 
by c, that is, c = Hi'H; then since a = ui'—c, where a de- 
notes the far point distance of the eye, the following ex- 
pression for the static refraction (A = l/a) may be derived 
immediately: 

A Ui+Ft 
l-c(Ui+Fi) ' 

In case the axial object-point M is infinitely far away, the 
lens is called a correction-glass, because it enables the pas- 
sive ametropic eye to see distinctly a very distant object 
on the axis of the lens, so that to this extent the lens inter- 
posed in front of the eye endows it with the characteristic 
faculty of an unaccommodated, naked, emmetropic eye. 
The condition that M shall be infinitely distant is Ui = 0; 
and hence the relation between the static refraction of the 
eye and the refracting power of a correction-glass is given as 
follows : 

a.* Fl= , A 



1-c.Fi 1+c.A 

If the distance c between the correction-glass and the eye is 
neglected entirely, then Fi = A, that is, the power of the 
correction-glass is approximately equal to the static refrac- 
tion of the eye. The distance c, which must be expressed 
in meters in case the magnitudes denoted by F\ and A are 
given in dioptries, is always a comparatively small magni- 
tude, which in actual spectacle glasses is comprised between 
0.008 and 0.016 m.; so that if, without neglecting c entirely, 



§ 154] Vertex Refraction of Spectacle Lens 445 

we neglect only the second and higher powers thereof, the 
formulae above may be written in the following convenient 
approximate forms: 

A = F 1 (l+c.F 1 ), F^Ail-cA); 
which for nearly all practical purposes will be found to be 
sufficiently accurate. 



f^ 




Fig. 199. — Correction of myopia with concave spectacle-glass. 

The condition that a spectacle-lens shall be a correction- 
glass may be expressed simply by saying that the second focal 
point (Fi r ) of the glass must coincide with the far point (R) of 
the eye. Thus, in case of a myopic eye the correction-glass 



Fig. 200. — Correction of hypermetropia with convex spectacle-glass. 

will be concave (Fig. 199) and in case of a hypermetropic 
eye it will be convex (Fig. 200). 

Instead of describing the power of a spectacle glass by 
means of its refracting power, it is really more convenient 
and logical to express it in terms of its vertex refraction (V), 
as defined in § 128. If the vertex of the lens which lies next 
the eye is designated by L, and if the distance of the eye from 
the glass is denoted by k, that is, if we put fc = LH, then, 
since the points designated by Fi' and R must be coincident, 



446 Mirrors, Prisms and Lenses [§ 155 

v = a+k, where a denotes the "back focus" of the lens, that 
is, v=l/7 = LF/ = LR; and hence: 

v , v- A 



l-k.V 1+k.A' 

or approximately: 

A = V(l+k.V), V=A(l-kA). 
It may be seen from the above formulae how the power of a 
correction-glass depends essentially on the location of the 
glass in front of the eye. The distance k, being referred to a 
tangible, external point of the glass, is more easily measured 
than the interval denoted by c. 

155. Visual Angle. — The apparent size of an object, as 
was explained in § 10, is measured by the visual angle co 
which it subtends at the eye; thus, if the vertex of this angle 
is designated by and if ?/ = MQ denotes a diameter of the 
object at right angles to the line of vision, the apparent size 
of the object in the direction of this dimension is co = Z MOQ. 
Accordingly, if the distance of the object from the eye is de- 
noted by z, that is, if 2 = 0M, then tana? =y/z. As the im- 
mobile eye looking in a fixed direction can see distinctly 
only that comparatively small portion of the object whose 
image falls on the sensitive part of the retina in the immedi- 
ate vicinity of the fovea centralis (§ 145), the rays concerned 
in the production of the retinal image in this so-called case 
of " indirect vision" may be regarded as paraxial rays. Ac- 
cordingly, the value of the angle co in radians may be sub- 
stituted here for the tanco, so that we may write: 

cc = y/z = y.Z, 
where Z = l/z. On the assumption that y is reckoned as 
positive, a negative value of the angle co indicates that the 
object is real and therefore in front of the point O where the 
eye is supposed to be. 

The exact meaning to be attached to the visual angle co 
will depend, of course, on the precise location with respect 
to the eye of the vertex of this angle. To be sure, so long as 
the object is quite remote from the eye, as is often the case, 



§ 155] Visual Angle 447 

it will not generally be necessary to define particularly the 
position of the vertex O of the visual angle. For example, 
to take a somewhat extreme instance, the apparent size of 
the moon will not be sensibly altered by removing the ver- 
tex of the visual angle as much as a mile or more away from 
the eye. And, in general, provided the object is not less 
than, say, 10 meters away, it will be sufficient to know that 
the vertex of the visual angle is in the eye without specifying 
its position more exactly. On the other hand, especially 
when the eye has to exert its power of accommodation in 
order to focus the object, it is sometimes a matter of much 
importance to define the visual angle with the utmost pre- 
cision. In such a case several meanings of this term are to 
be specially distinguished. For example, when the vertex 
of the visual angle is at the primary principal point of the 
eye, it is called the principal point angle (coh = ^MHQ), so 
that we may write : 

cc H = ij!u = y.U, 
where w=l/[/=HM denotes the distance of the object from 
the primary principal point. Similarly, the so-called focal 
point angle (co F = ZMFQ) is the angle subtended by the 
object at the primary focal point of the eye; and hence: 

u F = y/x = y.X, 
where x=l/I = PM denotes the distance of the object from 
the primary focal point of the eye. 

According to the definitions of these angles and the rela- 
tions between the magnitudes denoted by X, U and Z, as 
given in § 152, we may write therefore: 
co : cor- : co F = Z : U : X 

= 1 :(l+b.U) :(l+g.X) 
= (1-6.Z) :1 :(1-X/F) 
= (l-g.Z):(l+U/F):l; 
where F denotes here the refracting power of the eye when 
it is accommodated for the point M. 

The apparent size of an object may be measured also at 
other points of the eye, for example, at the center of the 



448 Mirrors, Prisms and Lenses [§ 156 

entrance-pupil, at the anterior nodal point, at the center of 
rotation, etc. The center of rotation or eye-pivot is the 
point of reference in the estimate of the apparent size of an 
object in the case of ordinary so-called "direct vision" with 
the mobile eye, when the gaze is directed in quick succession 
to the different parts of an extended object. Especially, in 
viewing an image through an optical instrument, it is nearly 
always desirable, if practicable, to adjust the eye in such a 
position that the center of rotation coincides with the center 
of the exit-pupil of the instrument, so as to command as 
large an extent of the field of view of the image-space as 
possible. Anyone who has ever tried to look through a key- 
hole in a door will realize how the field of view would have 
been widened if the eye could have been placed in the hole 
itself. 

156. Size of Retinal Image. — If the eye is accommodated 
to see an object y situated at a distance u ( = 1/U) from its 
primary principal point, the size of the image (y f ) formed on 
the retina is given by the relation: 
y.U=y'.A', 

where A' = n'/a' denotes the reciprocal of the reduced length 
of the eye-axis measured from the secondary principal point 
of the eye. Since 2/.J7=co H (§155), the above equation 
may be put in the following form : 

co h n' ' 

Since the positions of the principal points remain sensibly 
stationary in the act of accommodation (§ 150), the reduced 
length of the eye-axis (a'jn f ) may be considered as constant 
in the same individual. And hence the peculiar significance 
of the principal point angle consists in the fact that, ac- 
cording to this formula, this angle (co H ) may be taken as 
a measure of the size of the retinal image {y') which is in- 
dependent of the state of accommodation of the eye. Thus, 
for a given individual, all objects which have the same ap- 



157] 



Apparent Size of Image 



449 



parent size as measured at the principal point of the eye will 
produce retinal images of equal size. 

On the other hand, since y'.F=y.X=ai F (§ 155), it ap- 
pears that, for a given value of the refracting power (F), 
the size of the image on the retina of the eye is proportional 
to the focal point angle. And since the variations of the re- 
fracting power are, generally speaking, independent of axial 
ametropia (§ 153), the focal point angle will be particularly 
useful in comparing the apparent size of an object as seen 
by different individuals under the same external conditions. 

157. Apparent Size of an Object seen Through an Op- 
tical Instrument. — Let the principal points of the optical 
instrument be designated by H, IT (Fig. 201); and for the 




Fig. 201. — Apparent size of object seen through an optical 
instrument. 

sake of simplicity, let us assume that the instrument is sur- 
rounded by air so that the straight lines HQ, H'Q' joining 
the principal points with corresponding points of object and 
image will be parallel; and let ?/ = MQ, ?/' = M'Q' denote the 
linear magnitudes of object and image, respectively. Let 
the distance of the image from the eye be denoted by z= 
O'M', where O' designates the position of the eye on the 
axis. Then the apparent size of the image will be 

co=2/'.Z, 
where co = ZM'0'Q' (expressed here in radians) andZ = l/z. 
The angle co may be increased by reducing the distance be- 



450 Mirrors, Prisms and Lenses [§ 157 

tween the image and the eye, that is, by increasing Z; but 
this distance cannot be diminished below the near point 
distance of the eye, because then distinct vision would not 
be possible for the naked eye. 

If the distances of object and image from the principal 
points are denoted by u and u', that is, if w=HM, w' = H'M', 
then 

y'.U'=y.U, 
where U=l/u, U' = l/u'; and hence 

In general (except when the rays undergo an odd number 
of reflections) , the sign of Z as here defined will be negative, 
and therefore the sign of co will depend on the sign of the 
ratio U : U'. Accordingly, if object and image lie on the 
same side of their corresponding principal points, the sign 
of co will be negative, that is, the image will be erect. 

Let the distance of the eye from the instrument be de- 
noted by c = H'0 / ; then since u' = c+z, we may write: 

U' = Z . 

1+c.Z 

Accordingly, if the refracting power of the instrument is 
denoted by F, so that U=U f —F,we may write also: 

F-Z(l-c.F) 
1+c.Z 
Introducing these expressions, we obtain therefore the fol- 
lowing formula for the apparent size of the image: 
a>=-y\F-Za-c.F)\ . 
Thus, we see that the apparent size of the image may be 
varied in one of two ways, either by changing the position 
of the eye (that is, by varying c) or else by displacing the 
object so that Z is varied. There are two cases of special 
practical importance, namely: (1) When the eye is adjusted 
so that l — c.F = 0, and (2) When the object is focused so 
that Z = 0. In both cases the second term inside the large 
brackets vanishes, and hence oo = —y.F. The condition 



§ 157] Apparent Size of Image 451 

c = — l/F means that the eye is placed at the second focal 
point (F') of the instrument (which might easily be practi- 
cable if the optical system were convergent) ; so that under 
such circumstances the apparent size of the image would be 
the same for all positions of the object, because evidently 
the highest point (Q') of the image will always lie on the 
straight line which crosses the axis at the second focal point 
at the constant angle 6= —y.F. On the other hand, the 
condition Z = means simply that the object lies in the 
first focal plane of the instrument. Now this is the natural 
adjustment for a normal, unaccommodated, emmetropic 
eye, because then the rays flow into the eye in cylindrical 
bundles. This is the reason why the image produced by 
the object-glass of a telescope or microscope is usually fo- 
cused in the primary focal plane of the eye-piece or ocular. 
Accordingly, when Z = 0, the apparent size of the image 
will be independent of the position of the eye. 

An experienced observer who wishes to obtain the best 
results with an optical instrument will ordinarily adjust it 
to his eye in such a way that the image can be seen distinctly 
without his having to make an effort of accommodation. 
This will be the case if the image is formed at the far point 
(R) of the eye (§ 148). If, therefore, the static refraction 
of the eye is denoted by A (§151), then (assuming that the 
point O' in Fig. 201 is coincident with the anterior principal 
point of the eye) we may put Z = A; and hence the apparent 
size of an object as seen in an optical instrument by an eye 
with relaxed accommodation is given by the expression: 

co K =-y\F-A(l-c.F)\ . 
Thus, it is evident how the apparent size of the image de- 
pends not only on the refracting power of the instrument, 
but essentially also on the adjustment and idiosyncrasies 
of the eye of the individual who looks through it. 

It may be remarked that these formulae have been derived 
on the tacit assumption that the eye is at rest, and conse- 
quently only a small portion of the external field is sharply 



452 Mirrors, Prisms and Lenses [§ 158 

in focus at the sensitive part of the retina. Otherwise, we 
should have had to write tanco instead of co; nor should we 
have been justified in assuming that the effective rays were 
paraxial. If the eye turns in its socket to inspect the image, 
the apparent size of the image will depend essentially on 
the angular movement of the eye, and in this case the visual 
angle must be measured at the center of rotation of the eye. 
These are considerations that are too often overlooked in 
discussions of this kind. 

158. Magnifying Power of an Optical Instrument Used 
in Conjunction with the Eye. — An object may be so remote 
that its details are indistinguishable, or, on the other hand, 
it may be so close to the eye that not even by the greatest 
effort of accommodation can a sharp image of it be focused 
on the retina. Under such circumstances one has recourse 
to the aid of a suitable optical instrument whereby the ob- 
ject is magnified to such an extent that the parts of it which 
were obscure or entirely invisible to the naked eye will be 
revealed to view. The magnifying power is usually expressed 
by an abstract number M, which in the case of an optical 
instrument on the order of a microscope is defined to be the 
ratio of the apparent size of the image as seen in the instrument 
to the apparent size of the object as it would appear at the so- 
called "distance of distinct vision." This latter term is a 
somewhat unfortunate form of expression for several rea- 
sons, not only because the distance at which an object is 
ordinarily placed in order to be seen distinctly is different 
for different persons, but because the same person, accord- 
ing to the extent of his power of accommodation, usually 
possesses the ability of seeing distinctly objects at widely 
different distances. The expression appears to have arisen 
from a confusion of ideas, and its origin may probably be 
traced to the fact that even nowadays many people have 
difficulty in conceiving how the eye can be " focused for 
infinity," although, indeed, as has been explained, that is 
to be regarded as the natural state of the normal eye in re- 



§ 158] Magnifying Power of Optical Instrument 453 

pose. However, the phrase has become too deeply rooted 
in optical literature ever to be eradicated, and no harm will 
be done by continuing to use it, provided it is not taken 
literally, but is considered merely as the designation of a 
more or less arbitrary conventional projection-distance. 
Accordingly, if the so-called " distance of distinct vision" 
is denoted by I, the apparent size of the object (y) as seen 
at this distance from the eye will be —y/l, and hence if the 
apparent size of the image in the instrument is denoted by 
co, the magnifying power, as above defined, will be: 

y 

The actual value of this conventional distance I is usually 
taken as 10 inches or 25 centimeters, which is large enough 
for the convenient accommodation of most human beings 
who are not already past the prime of life and yet not so 
large that the size of the image on the retina differs much 
from its greatest dimensions. If distances are all measured 
in meters, the conventional value of the magnifying power 
will be given, therefore, by the formula: 

M= " 

The explanation of the minus sign in front of the fraction 
is to be found in the mode of reckoning the visual angle co, 
which, as we have pointed out (§ 157), is negative in case 
the image of the object y is erect, as, for example, with an 
ordinary convex lens used as a magnifying glass. Thus, 
according to the above formula, a positive value of the mag- 
nifying power means magnification without inversion. Or- 
dinarily, what is meant by the magnifying power of an op- 
tical instrument is the value of this abstract number M; 
which gives the ratio of the sizes of the retinal images when 
an emmetropic eye views one and the same object, first, in the 
instrument without effort of accommodation, and then with- 
out the instrument with an accommodation of four dioptries. 
If the expression for the visual angle co which was ob- 



454 Mirrors, Prisms and Lenses [§ 158 

tained in § 157 is introduced here, we shall derive, therefore, 
the following formula for the magnifying power (M) in terms 
of the refracting power (F) of the instrument, the distance 
(c) of the eye from the instrument, and the distance (2= 
1/Z) of the image (y f ) from the eye: 

M = l\F-Z(l-c.F)\. 
This expression is really a measure of the individual mag- 
nifying power, since it involves not merely the instrument 
itself but the characteristic peculiarities of the eye of the 
observer. In order to obtain a measure of the absolute mag- 
nifying power of the instrument, the second term inside the 
large brackets must be made to vanish. Thus, if the object 
is placed in the primary focal plane, so that the image is 
infinitely distant, then Z = 0, and now M = l.F denotes the 
absolute magnifying power. If 1 = 0.25 meter, then F = 4M; 
and usually, therefore, when we say that the magnifying 
power of a lens or microscope is M, this means simply that 
its refracting power is equal to 4M dioptries. 

If the image in the instrument is formed at the " distance 
of distinct vision" (I), then Z=—l/l. and 

M = l + (l-c)F. 
The distance (c) between the instrument and the eye is usu- 
ally small in comparison with I, so that it is often entirely 
neglected. Assuming that (l—c) is positive, we may say 
that in a convergent optical system (F>0), the object will 
appear magnified (M>1); whereas in a divergent optical 
system (F<0), the object appears to be diminished in size 
(M<1). 

In order to avoid the use of an arbitrary projection- 
distance, (Z), Abbe proposed to define the magnifying power 
as the ratio of the apparent size ( 00) of the image in the instru- 
ment to the actual size (y) of the object (compare with Abbe's 
definition of focal length, § 122) ; so that if this ratio is de- 
noted by P, then 

^ CO 

p=--. 

y 



§ 159] Magnifying Power of Telescope 455 

This measure of the magnifying power is not an abstract 
number like M, but a quantity of the same physical dimen- 
sions as the refracting power of the instrument. The two 
definitions are connected by the simple relation 

M = Z.P; 
so that if we put 1 = 0.25 m., the value of P will be obtained 
by multiplying M by the number four (P = 4M). Thus, for 
example, in the case of a convex lens of refracting power F 
used as a magnifying glass, if the object is placed in the first 
focal plane, we have P = F. 

159. Magnifying Power of a Telescope. — In the case of 
a telescope, which is an instrument for magnifying the ap- 
parent size of a distant object, neither of the definitions of 
magnifying power given in the foregoing section is appli- 
cable. An infinitely distant object (like the moon, for ex- 
ample) can be seen distinctly by an emmetropic eye without 
any effort of accommodation, but its apparent size may be 
so minute that the distinguishing features cannot be made 
out by the naked eye. This same eye looking at the object 
through a telescope will see an infinitely distant image of 
it, but presented to the eye under a larger visual angle, so 
that it appears magnified. Essentially, a telescope may be 
regarded as a combination of two optical systems, one of 
which — the part pointed towards the object — is a con- 
vergent system, generally of relatively long focus and large 
aperture (so as to intercept a large quantity of light), called 
the object-glass; while the other, composed of the lenses 
next the eye, and called therefore the ocular or eye-piece, 
may be a convergent or divergent system depending on the 
type of telescope. The object-glass which is at one end of 
a large tube forms a real inverted image of the object in its 
second focal plane or not far from it; and this image is in- 
spected through the ocular, which is usually fixed in a smaller 
tube inserted in the larger one so that the focus can be ad- 
justed to suit different eyes and different circumstances. 
A simple schematic telescope may be regarded as composed 



456 



Mirrors, Prisms and Lenses 



[§ 159 



of two thin lenses, one of which, of focal length /i (refracting 
power Fi) acts as the object-glass while the other, of focal 
length f 2 (refracting power F 2 ) performs the part of the oc- 
ular. When the telescope is adjusted for an emmetropic, 
unaccommodated eye, the second focal point (F/) of the ob- 
ject-glass coincides with the first focal point (F 2 ) of the ocular; 
and hence the focal length of the entire system is infinite 
(/= oo or F = 0), that is, the system is afocal or telescopic 
(§125). In this case the telescope is said to be in normal 
adjustment. 

The first telescope appears to have been invented by one 
of two Dutch spectacle-makers named Zacharias Jansen 
and Franz Lippershey (circa 1608). Galileo (1564- 
1642), having heard of this Dutch toy, was led to experiment 



To J at oc 




To J'atco 



Fig. 202. — Diagram of simple Dutch or Galilean telescope. 

with a combination of two lenses and he soon succeeded 
(1609) in making a telescope with which he made a number 
of renowned astronomical discoveries. The so-called Dutch 
or Galilean telescope, represented schematically in Fig. 202, 
consists of a large convex object-glass (Ai) combined with 
a small concave eye-piece (A2), which intercepts the con- 
verging rays before they come to a focus and adapts them to 
suit the eye of the observer. The other type of telescope 
(Fig. 203) is composed of two convex lenses. It is called the 
astronomical telescope or Kepler telescope, because the idea 



159] 



Magnifying Power of Telescope 



457 



occurred first to John Kepler (1611); but the first instru- 
ment of this kind was made by the celebrated Jesuit father, 
Christian Scheiner (1615), who also conceived the idea 
of using a third lens to erect the image as is done in the so- 
called terrestrial telescope. 

If the telescope is in normal adjustment, then from each 
point J of the infinitely distant object there will issue a bundle 




Fig. 203. — Diagram of simple astronomical telescope. 

of parallel rays whose inclination to the axis of the telescope 
may be denoted by 6. Falling on the object-glass, these 
rays are converged to a focus at a point P lying in the com- 
mon focal plane of object-glass and eye-piece; and conse- 
quently they will emerge from the eye-piece and enter the 
eye as a bundle of parallel rays proceeding from the infi- 
nitely distant image-point J' in a direction which makes an 
angle 6' with the axis. The slope-angles 6 and 6' have 
a constant relation to each other, as may easily be shown; 
for from the right triangles F/AiP and F 2 A 2 P (Figs. 202 
and 203), where AiF/ = AiF 2 =/i, ZFi'AiP = 0, and F 2 A 2 = 
Fi'A 2 =/ 2 , ZF 2 A 2 P= 6', we obtain immediately: 

tan 0' /i 

— = — - = constant. 

tan Q j 2 



458 Mirrors, Prisms and Lenses [§ 159 

Now the angles denoted here by 6 and 6 f are the measures 
of the apparent sizes of corresponding portions of the in- 
finitely distant object and image, and the ratio of these 
angles (or of their tangents) is defined to be the magnifying 
power of the telescope; so that if this ratio is denoted by M, 
we shall have: 

h 
Accordingly, the magnifying power of a telescope focused 

on an infinitely distant object and adjusted for distinct 
vision for an unaccommodated, emmetropic (or corrected 
ametropic) eye is measured by the ratio of the focal lengths 
of the objective and ocular. In the astronomical telescope 
/i and / 2 are both positive, and consequently the ratio M is 
negative, which means that the image is inverted ; whereas in 
the Dutch telescope /i is positive and f 2 is negative, and 
hence *M is positive, that is, the final image is erect. 

Another convenient expression for the magnifying power 
of a telescope, as defined above, may easily be obtained. 
All the effective rays which fall on the object-glass will after 
transmission through the instrument pass through a certain 
circular aperture called the eye-ring (or Ramsden circle), 
which is the image of the object-glass in the ocular. If the 
object-glass is brightly illuminated (for example, if the tele- 
scope is pointed towards the bright sky), this image appears 
as a luminous disk floating in the air not far from the ocular 
and can easily be perceived by placing the eye at a suitable 
distance. In the astronomical telescope the eye-ring is a 
real image which can be received on a screen, and in this 
instrument it usually acts as the exit-pupil (§ 134). In the 
case of the Dutch telescope the eye-ring is a virtual image 
on the other side of the ocular from the eye; and generally 
its effect is to limit the field of view in the image-space, that 
is, its office is that of the exit-port of the system (§§ 137, 
138). Now if the telescope is in normal adjustment, then 
the distance of the ocular from the object-glass is equal to 



§ 159] Magnifying Power of Telescope 459 

the algebraic sum (/1+/2) of the focal lengths of the two 
components ; and it may easily be shown that 
M _ /1 _ diameter of object-glass 
J2 diameter of eye-ring 
The advantage of this latter form of expression is to be 
found in the fact that even if the telescope is not in normal 
adjustment, it may still be considered in a certain sense as 
a measure of the magnifying power of the instrument. Sup- 
pose, for example, that the optical system is not telescopic, 
so that the interval between the second focal point (Fi') of 
the object-glass and the first focal point (F 2 ) of the ocular 
is not negligible, as frequently happens in focusing the eye- 
piece to suit the eye of the individual, especially if the object 
itself is not infinitely distant. Consider a ray which is di- 
rected originally from the extremity of the object towards 
a point O on the axis of the telescope and which emerges 
so as to enter the eye at the conjugate point O'. If the angles 
which the ray makes with the axis at O and O' are denoted 
by 6 and 6' ', respectively, then the ratio tan#' : tan# will 
be a measure of the magnifying power of the telescope for 
this adjustment and position of the eye. But according to 
the Smith-Helmholtz formula (§§ 86 and 118), since the 
telescope is surrounded by the same medium on both sides, 
we shall have here : 

tan#' : tand = y : y', 
where y and y' denote the linear magnitudes of an object and 
its image in conjugate transversal planes at O and O' (the 
planes of the pupils). Now if the point O' is at the center 
of the eye-ring, the point O will lie at the center of the object- 
glass, and the ratio y : y' will be equal to the ratio of the 
diameters of object-glass and eye-ring. Hence, provided 
the eye is placed at the eye-ring, the magnifying power of the 
telescope will be 

lvr _ diameter of object-glass 
diameter of eye-ring 
In an astronomical telescope the best adjustment for com- 



460 Mirrors, Prisms and Lenses [§159 

manding a wide extent of the field of view is to place the eye 
with its center of rotation at the center of the eye-ring, but 
in a Dutch telescope this is not practicable, because the eye- 
ring is not accessible. 

In order to obtain a general formula for the magnifying 
power of a telescope, let us fix our attention on the inverted 
image of the object which is formed by the object-glass. 
If u=l/U denotes the distance of the object from the object- 
glass and if q denotes the linear size of the image, the appar- 
ent size of the object as seen from the center of the object- 
glass will be 

tan0 =q(U+Fi), 

where F\ denotes the refracting power of the object-glass. 
On the other hand, according to the formula deduced in 
§ 157, the apparent size of the image seen in the telescope 
will be , 

ttmd'=-q{F 2 -Z(l-c.F 2 )} , 

where F 2 denotes the refracting power of the ocular, z—\\Z 
denotes the distance of the image in the ocular from the eye, 
and c denotes the distance of the eye from the ocular itself 
(or from its second principal point). Accordingly, we obtain 
the following expression for the magnifying power of the 
telescope: 

tanfl^ F 2 -Z(\-c.F 2 ) 

tanfl U+Fi 

which is applicable to all cases. If the object is infinitely 
distant, then £7 = 0; and if the telescope is in normal adjust- 
ment, then the image is also infinitely distant, that is, Z = 0, 
andM=-F 2 /Fi. 



Ch. XIII] Problems 461 

PROBLEMS 

1. If the refracting power of a correction spectacle-glass 
is +10 dptr., and if the distance of the anterior principal 
point of the eye from the second principal point of the glass 
is 12 mm., find the static refraction of the eye. 

Ans. +11.36 dptr. 

2. Take the refracting power of the eye equal to 58.64 dptr., 
the distances of the principal points from the vertex of the 
cornea as 1.348 and 1.602 mm., and the index of refraction 
of the vitreous humor equal to 1.336. If the refracting power 
of a correction spectacle-glass, whose second principal point 
is 14 mm. from the anterior principal point of the eye, is 
+ 5.37 dptr., show that the total length of the eye-ball is 
26.5 mm. 

3. In Gtjllstrand's schematic eye, with accommodation 
relaxed, the distance from the vertex of the cornea to the 
point where the optical axis meets the retina is 24 mm. The 
other data are the same as those given in No. 2 above. Find 
the position of the far point and determine the static refrac- 
tion. 

Ans. The far point is 99.34 cm. from the vertex of the 
cornea, and the static refraction is + 1 .008 dptr. 

4. In Gullstrand's schematic eye in its state of maxi- 
mum accommodation the distances of the principal points 
from the vertex of the cornea are 1.7719 and 2.0857 mm., and 
the refracting power is 70.5747 dptr. The length of the eye- 
ball is 24 mm., as stated in No. 3. Find the position of the 
near point and determine the dynamic refraction of the 
eye. 

Ans. The near point is 10.23 cm. from the vertex of the 
cornea; the dynamic refraction is —9.609 dptr. Accordingly, 
with the aid of the result obtained in No. 3, we obtain for 
the amplitude of accommodation 10.62 dptr. 

5. Taking the refracting power of the eye as equal to 
59 dptr., show that the size of the retinal image of an object 



462 Mirrors, Prisms and Lenses [Ch. XIII 

1 meter high at a distance of 10 meters from the eye will be 
1.7 mm. 

6. The apparent size of a distant air-ship is one minute of 
arc. Taking the refracting power of the eye as equal to 
58.64 dptr., show that the size of the image on the retina 
will be 0.00495 mm. 

7. What is the magnifying power of a convex lens of focal 
length 5 cm.? Ans. 5. 

8. A myope of 10 dptr. uses a convex lens of focal length 
5 cm. as a magnifying glass. Find the individual magnify- 
ing power, neglecting the distance of the eye from the glass. 

Ans. 7|. 

9. In the preceding example, what will be the individual 
magnifying power of the same glass in the case of an hyper- 
metrope of 10 dptr.? Ans. 2£. 

10. A certain person cannot see distinctly objects which 
are nearer his eye than 20 cm. or farther than 60 cm. Within 
what limits of distance from his eye must a concave mirror 
of focal length 15 cm. be placed in order that he may be able 
to focus sharply the image of his eye as seen in the mirror? 

Ans. In order to see a real image of his eye, the distance 
of the mirror must be between 43.23 cm. and 78.54 cm.; in 
order to see a virtual image, the distance of the mirror must 
be between 6.97 cm. and 11.46 cm. 

11. The magnifying power of a telescope 12 inches long 
is equal to 8: determine the focal lengths of object-glass 
and eye-glass (1) when it is an astronomical telescope and 
(2) when it is a Galileo's telescope. 

Ans. (1) /i = +10|, / 2 =+l| inches; (2)/i=+13|, 
/ 2 = — ly inches. 

12. The focal lengths of the object-glass and eye-glass of 
an astronomical telescope are /1 and / 2) and their diameters 
are 2hi and 2h 2) respectively. Show that the radius of the 
stop which will cut off the " ragged edge" (§ 138) is equal to 

M2—M1 



Ch. XIII] Problems 463 

13. A telescope is pointed at an infinitely distant object, 
and the eye-piece is focused so that the image is formed at 
the distance I of distinct vision of the eye. If the distance of 
the eye from the eye-piece is neglected, show that the mag- 
nifying power is M= — /i(7+/ 2 )/Z./ 2 , where /i, f 2 denote the 
focal lengths of the object-glass and eye-glass. 

14. A Ramsden ocular consists of two thin convex lenses 
each of focal length a separated by an interval equal to 2a/3. 
Show that the magnifying power of an astronomical tele- 
scope furnished with a Ramsden ocular is 4/i/3a, where /i 
denotes the focal length of the object-glass. 

15. The object-glass of an astronomical telescope has a 
focal length of 50 inches, and the focal length of each lens 
of the Ramsden ocular is 2 inches. The distance between 
the two lenses in the ocular is ^ inch. Show that the dis- 
tance between the object-glass and the first lens of the oc- 
ular is 50.5 inches, and that the magnifying power is equal 

to i r- 

16. If a Galileo's telescope is in normal adjustment, 
show that the angular diameter of the field of the image as 
measured at the vertex of the concave eye-glass is 2tanY' = 
— 2/ii/(/i+/ 2 ), where hi denotes the radius of the object-glass 
and /i, / 2 denote the focal lengths of object-glass and eye- 
glass. 

17. The focal length of the object-glass and eye-glass of 
an astronomical telescope are 36 and 9 inches, respectively. 
If the object is infinitely distant and if the eye is placed in 
the eye-ring at a distance of 9 inches from the image, show 
that the magnifying power is equal to 3. 

18. 'The magnifying power of a simple astronomical tele- 
scope in normal adjustment is M, and the focal length of the 
object-glass is /i. Show that if the eye-glass is pushed in a 
distance x and the eye placed in the eye-ring, the magnifying 
power will be diminished by x.M/fi. 

19. An astronomical telescope is pointed towards the sun, 
and a real image of the sun is obtained on a screen placed 



464 Mirrors, Prisms and Lenses [Ch. XIII 

beyond the eye-lens at a distance d from it. If the diameter 
of this image is denoted by 26, and if the apparent diameter 
of the sun is denoted by 2 6, show that the magnifying power 
of the telescope is M = 6. cot 6/d. 

20. The eye is placed at a distance c from the eye-glass of 
a Galileo's telescope in normal adjustment. The length 
of the telescope as measured from the object-glass to the 
eye-glass is denoted by d, the radius of the object-glass is 
denoted by hi, and the radius of the pupil of the eye is de- 
noted by g (it being assumed that g is less than the radius of 
the eye-glass). Show that the semi-angular diameters of 
the three portions of the field of view on the image-side are 
given by the following expressions : 

hi— gM , h , hx+gM 

where M denotes the magnifying power of the telescope. 



CHAPTER XIV 



DISPERSION AND ACHROMATISM 



160. Dispersion by a Prism. — When a beam of sunlight is 
admitted into a dark chamber through a small circular hole A 
(Fig. 204) in the window shutter, a round spot of white light 
will be formed on a vertical wall or screen opposite the win- 
dow, which will be, indeed, an image of the sun of the same 
kind as would be produced by a pinhole camera (§ 3) ; its 





Fig. 204. — Prism dispersion: Newton's experiment. 



angular diameter, therefore, being equal to that of the sun, 
namely, about half a degree. In the track of such a beam 
Newton inserted a prism with its refracting edge horizontal 
and at right angles to the direction of the incident light; 
whereupon the white spot on the screen vanished and in its 
stead at a certain vertical distance above or below the place 
that was first illuminated there was displayed an elongated 

465 



466 Mirrors, Prisms and Lenses [§ 160 

vertical band or spectrum, exhibiting the colors of the rain- 
bow in an endless variety of tints shading into each other by 
imperceptible gradations. This spectrum was rounded at 
the ends and its vertical dimension, depending on how the 
prism was tilted, was about 4 or 5 times as great as its hori- 
zontal dimension, the latter being equal to the diameter of 
the spot of white light that was formed on the screen before 
the interposition of the prism. For convenience of descrip- 
tion, Newton distinguished seven principal or " primary" 
colors arranged in the following order from one end of the 
spectrum to the other, namely, red, orange, yellow, green, 
blue, indigo,* and violet; of which the violet portion of the 
spectrum is the longest and the orange the shortest. The 
red end of the spectrum was the part of the image on the 
screen that was least displaced by the interposition of the 
prism. 

This phenomenon was explained by Newton on the as- 
sumption that ordinary sunlight is composite and consists 
in reality of an innumerable variety of colors all blended 
together; and that the index of refraction (n) of the prism, 
instead of having a definite value, has in fact a different 
value for light of each color, being greatest for violet and 
least for red light and varying between these limits for light 
of other colors. 

The resolution of white light into its constituent colors 
by refraction is called dispersion. If a puff of tobacco-smoke 
is blown across the beam of light where it issues from the 
prism, only the outer parts of the beam will show any very 
pronounced color, because the central parts at this place will 

* There has been much discussion as to what Newton understood 
by the color which he named " indigo" and which lies somewhere be- 
tween the blue and the violet. Indigo, as we understand it, is more 
nearly an inky blue rather than a violet blue, more like green than like 
violet; and hence it has been suggested that Newton's color vision 
may have been slightly abnormal. In this connection see article en- 
titled "Newton and the Colours of the Spectrum" by Dr. R. A. Hous- 
toun, Science Progress, Oct. 1917. 



160] 



Monochromatic Light 



467 



not have been sufficiently dispersed to exhibit their individ- 
ual effects. At some little distance away from the prism the 
entire section of the beam will be brilliantly colored. 

Having pierced a small hole through the screen at that 
part of it where the spectrum was formed (Fig. 205) , Newton 
was able by rotating the prism around an axis parallel to 




Fig, 205. — Newton's experiment with two prisms; showing that light of 
a definite color traverses the second prism without further dispersion. 



its edge to transmit rays of each color in succession through 
the opening to a second prism placed with its edge parallel 
to that of the first prism; and, agreeably to his expectations, 
he found that while these rays were again deviated in tra- 
versing the second prism, there was no further dispersion of 
the light. This experiment demonstrated that the single 
colors of the spectrum were irreducible or elementary and 
not a mixture of still simpler colors, and that the light which 
had been separated in this fashion from the beam of sun- 
light was monochromatic light. 

If all the various components of the incident light which 
has been resolved by the prism are re-united again, the effect 
will be the same as that of the light before its dispersion. 



468 



Mirrors, Prisms and Lenses 



160 



The simplest way to achieve this result is to cause the rays 
to traverse a second prism precisely equal to the first, but 
inverted so that the dihedral angle between the planes of 
the adjacent faces of the two prisms is equal to 180°, the 
edges of the prisms being parallel. Indeed, if the two prisms 
were placed in contact in this way, they would form a slab 
of the same material throughout with a pair of plane parallel 
faces, for which the resultant dispersion is zero; because the 
colored rays would all emerge in a direction parallel to that 
of the incident ray which was the common path of all these 




Fig. 206. — Light is not dispersed in traversing a plate with 
plane parallel faces surrounded by same medium on both 
sides. 



rays before they were separated by refraction at the first 
face of the plate (Fig. 206). 

Another and essentially different way of re-uniting the 
colored rays is to converge them to a single point by means 
of a so-called achromatic lens, as represented diagrammati- 
cally in the accompanying drawing (Fig. 207); so that the 
effect at the focus C where the colored rays meet is the same 
as that of light from the source. Beyond C the rays sepa- 
rate again, so that if they are received on a screen the same 
succession of colors will be exhibited as before, only in the 
reverse order. If some of the rays are intercepted before 
arriving at C, the color at C will be the resultant effect of 
the residual rays. The point B where the rays are separated 
on entering the prism and the point C where they are re- 
united by the lens are a pair of conjugate points with re- 
spect to the prism-lens system. 



§160] Spectrum 469 

The solar spectrum which Newton obtained in his cele- 
brated prism-experiments, described in 1672, had one serious 
defect, due to the fact that the colors in it were not in reality 
pure but consisted of a blending of two or more simple colors. 
When the light passes through a round hole before falling 
on the prism, the spectrum on the screen will be composed 
of a series of colored disks, each one overlapping the one next 
to it. The colors, therefore, are partly superposed on each 
other, and the eye is so constituted with respect to color 
vision that it cannot distinguish the separate effects and 




Fig. 207. — Achromatic lens used to re-unite the colored light after it has 
been dispersed by prism. 

analyze them but obtains only a general resultant impression 
of the whole. 

Wollaston's experiments in 1802 differed essentially 
from Newton's only in the form and dimensions of the beam 
of sunlight that was dispersed by the prism, but this simple 
modification represented a distinct advance in the mode of 
investigation of the spectrum. Wollaston admitted the 
sunlight through a narrow slit * whose length was parallel to 

* Dr. Houstoun, in the article already referred to, calls attention 
to the fact that in some of his prism-experiments Newton also em- 
ployed an opening in the form of a narrow slit, and was aware of its 
advantages with respect to the purity of the spectrum; for Newton 
states that "instead of the circular hole," "it is better to substitute an 
oblong hole shaped like a long Parallelogram with its length Parallel 
to the Prism. For if this hole be an Inch or two long, and but a tenth 



470 



Mirrors, Prisms and Lenses 



160 



the prism-edge; and in order to diminish still more the di- 
vergence of the incident beam, a screen with a second slit 
parallel to the first was interposed in front of the prism, as 
represented in the accompanying diagram (Fig. 208). The 
spectrum formed in this way is far purer than that obtained 
with a round opening in the shutter. But a difficulty that 





Fig. 208. — Pure spectrum obtained by causing sunlight to pass through two 
narrow slits before traversing prism. 

inheres in both methods arises from the fact that the image 
formed by a prism is always virtual, and therefore a homo- 
centric bundle of monochromatic divergent rays will nec- 
essarily be divergent after traversing a prism, so that if 
they are received on a screen they will illuminate a certain 
area on it which is the cross-section of the ray-bundle and 
not in any strict sense an optical image of the original source. 

or twentieth part of an Inch broad or narrower; the Light of the Image, 
or spectrum, will be as Simple as before or simpler, and the Image will 
become much broader, and therefore more fit to have Experiments 
tried in its Light than before." The fact that Newton did not dis- 
cover the Fraunhofer lines of the solar spectrum (§ 161) is probably 
to be explained on the supposition that his prisms were of an inferior 
quality of glass and that possibly also the surfaces were not as highly 
polished as they might have been. 



§ 160] Spectrum 471 

Consequently, if the source sends out light of different colors, 
the effect on the screen will correspond to the sections of all 
the bundles of colored rays, and since these sections will 
overlap each other to a greater or less extent, the spectrum 
will not be pure. The narrower the apertures of the bundles 
of rays and the farther the screen is from the prism, the less 




s 

Fig. 209. — Pure spectrum obtained by slit, prism and achromatic lens. 

will be the overlapping of the adjacent colors, and therefore 
the purer the spectrum; but on the other hand, the less also 
will be the illumination. 

A much more satisfactory method consists in making these 
divergent bundles of rays convergent by means of an achro- 
matic convex lens, as represented in Fig. 209; whereby the 
blue rays proceeding apparently from a virtual focus at B 
are brought to a real focus on the screen at B', and, similarly, 
the red rays are united at R\ The plane of the diagram 
represents a principal section of the prism. The light orig- 
inates in a luminous line or narrow illuminated slit at S par- 
allel to the prism-edge, and the spectrum R'B' on the screen 
consists of a series of colored images of this slit and is ap- 
proximately pure, except in so far as the slit must necessarily 
have a certain width. Moreover, in the case of a very nar- 
row slit, there are certain so-called diffraction-effects (§ 7) 
which are indeed of very great importance in any thorough 
scientific discussion of the condition of the purity of the 
spectrum. 



472 Mirrors, Prisms and Lenses [§ 162 

161. Dark Lines of the Solar Spectrum. — Wollaston 
himself observed that the spectrum of sunlight was not ab- 
solutely continuous, but that there were certain narrow gaps 
or dark bands in it parallel to the slit. Fraunhofer (1787- 
1826), with his rare acumen and experimental skill, was able 
to obtain spectra of far higher purity than any of his prede- 
cessors, and he discovered, independently, that the solar spec- 
trum was crossed by a very great number of dark lines, the 
so-called Fraunhofer lines, from which he argued that sun- 
light was deficient in light of certain colors. Fraunhofer 
counted more than 600 of these lines, but there are now 
known to be several thousand. One great advantage of this 
remarkable discovery, which Fraunhofer was quick to 
realize, consists in the fact that these lines are especially 
suitable and convenient for enabling us to specify particular 
regions or colors of the spectrum, because each of them cor- 
responds to a certain degree of refrangibility, that is, to a 
perfectly definite color of light. An explanation of the origin 
of the dark lines of the solar spectrum may be found in 
treatises on physics and physical optics. 

The dark lines are distributed very irregularly over the en- 
tire extent of the solar spectrum. In some cases they are 
sharp and fine and isolated; some of them are exceedingly 
close together so as to be hardly distinguishable apart; others 
again are quite broad and distinct. In order to describe 
their positions with respect to each other, Fraunhofer se- 
lected eight prominent lines distributed in the different 
regions of the spectrum, which he designated by the capital 
letters A (dark red), B (bright red), C (orange), D (yellow), 
E (green), F (dark blue), G (indigo), and H (violet). This 
notation is still in use, and has since been extended beyond 
the limits of the visible spectrum. 

162. Relation between the Color of the Light and the 
Frequency of Vibration of the Light- Waves.— According to 
the undulatory theory of light, a luminous body sets up 
disturbances or " vibrations'' in the ether which are prop- 



§ 162] Light-Waves and Color 473 

agated in waves in all directions with prodigious velocities. 
The velocity of light in the free ether is about 300 million 
meters per second. When a train of light-waves traverses 
a rectilinear row of ether-particles all lying in the same me- 
dium, the distance between one particle and the nearest one 
to it that is in precisely the same phase of vibration is called 
the wave-length; and the number of waves which pass a 
given point in one second or the frequency of the undulation 
will be equal to the velocity of propagation of the wave 
divided by the wave-length. The reciprocal of the frequency 
will be the time taken by a single wave in passing a given 
point, which is called the period of the vibration. If the 
wave-length is denoted by X, the velocity of propagation 
by v, the frequency by N, and the period by T=l/N, the 
relations between these magnitudes is expressed as fol- 
lows: 

\ = v /N = v.T. 
When ether- waves fall on the retina of the eye, they may 
excite a sensation of light provided their frequencies are 
neither too small nor too great, the limits of visibility being 
confined to waves whose frequencies lie between about 392 
and 757 billions of vibrations per second. Just as the pitch 
of a musical note is determined by its frequency, so also the 
sensation which we call color appears to be more or less in- 
explicably associated with the frequency of the vibrations 
of the luminiferous ether; so that to each frequency between 
the limits named there corresponds a perfectly definite kind 
of light or color. Absolutely monochromatic light due to 
ether-waves of one single frequency of vibration is difficult 
to obtain. In general, the light which is emitted by a lumi- 
nous body is more or less complex, and the sensation which 
it produces in the eye is due to a variety of impulses. The 
yellow light which is characteristic of the flame of a Bunsen 
burner when a trace of common salt is burned in it is a sen- 
sation excited by the impact of two kinds of ether-waves 
corresponding to the double D-line of the solar spectrum 



474 Mirrors, Prisms and Lenses [§ 162 

which have frequencies of about 509 and 511 billions of vi- 
brations per second. Red light corresponds to the lowest 
and violet light to the highest frequency. 

It is known that the velocity of light of a given color de- 
pends on the medium in which the light is propagated; and 
it has also been established that the velocity of light in a given 
medium depends on the color of the light. However, appar- 
ently light of all colors is transmitted with equal velocities 
in vacuo; and also in air, on account of its slight dispersion, 
there is practically no difference in the velocity of propaga- 
tion of light of different colors.* 

One reason for inferring that the frequency of the ether- 
vibrations is the physical explanation of the phenomenon of 

* "When white light enters a transparent medium, the long red waves 
forge ahead of the green ones, which in their turn get ahead of the blue. 
If we imagine an instantaneous flash of white light traversing a re- 
fracting medium, we must conceive it as drawn out into a sort of linear 
spectrum in the medium, that is, the red waves lead the train, the 
orange, yellow, green, blue, and violet following in succession. The 
length of this train will increase with the length of the medium traversed. 
On emerging again into the free ether the train will move on without 
any further alteration of its length. 

"We can form some idea of the actual magnitudes involved in the 
following way. Suppose we have a block of perfectly transparent glass 
(of ref. index 1.52) twelve miles in thickness. Red light will traverse 
it in 1/10000 of a second, and on emerging will be about 1.8 miles in 
advance of the blue light which entered at the same time. If white 
light were to traverse this mass of glass, the time elapsing between the 
arrival of the first red and the first blue light at the eye will be less than 
1/6000 of a second. Michelson's determination of the velocity of light 
in carbon bisulphide showed that the red rays gained on the blue in 
their transit through the tube of liquid. The absence of any change of 
color in the variable star Algol furnished direct evidence that the blue 
and red rays traverse space with same velocity. In this case the dis- 
tance is so vast, and the time of transit so long, that the white light 
coming from the star during one of its periodic increases in brilliancy 
would arrive at the earth with its red component so far in advance of 
the blue that the fact could easily be established by the spectro- 
photometer or even by the eye."— R. W. Wood: Physical Optics, 
Second Edition (New York, 1911), page 101. 



§ 162] Wave-Lengths of Light 475 

color is found in the fact that the color of monochromatic 
light remains unaltered when the light passes from one me- 
dium into another; and since the vibrations in the second 
medium are excited and forced by those in the first medium, 
it is natural to suppose that the vibration-frequency is the 
same in both media. 

Accordingly, it is the ratio 

that remains constant in the transmission of monochromatic 
light through different media. And hence if the velocities of 
light in two media are denoted by v, v' , and if the wave-lengths 
in these two media are denoted by X, X', then v/\ = v'/\' 
or X/X' = ^/V/ that is, the wave-length of light of a given color 
varies from medium to medium, and is proportional to the ve- 
locity of propagation of light of that color in the medium in 
question. Thus, the wave-length of yellow light is shorter 
in glass than it is in air, because light travels more slowly in 
glass than in air. 

Generally, therefore, when we speak of the wave-length of 
a given kind of light, w T e mean its wave-length measured in 
vacuo. The lengths of waves of light are all relatively very 
short, the longest, corresponding to the extreme red end of 
the spectrum, being less than one 13-thousandth of a centi- 
meter, and the shortest, belonging to the extreme violet end 
of the visible spectrum, being less than one 25-thousandth 
of a centimeter. These magnitudes are usually expressed in 
terms of a special unit called a " tenth-meter" which is one 
10-billionth part of a meter (10~ 10 meter) or in terms of a 
"micromillimeter" which is equal to the millionth part of 
a millimeter and for which the symbol fxfji is employed 
(l/x/x = lCT 6 mm.). Thus, the wave-lengths of light cor- 
responding to the red and violet ends of the visible spectrum 
are about 767/x/x and 397/^/x, respectively. The Fraun- 
hofer line A is a broad, indistinct line at the beginning of the 
red part of the spectrum, wave-length 759.4 fifi; the B-line 



476 Mirrors, Prisms and Lenses [§ 163 

in the red part corresponds to light of wave-length 686.7/x/x; 
the C-line in the orange corresponds to light of wave-length 
656.3/x/x; the D-line in the yellow is a double line, cor- 
responding to light of wave-lengths 589.6/x/a and 589.0/x/a; 
the E-line in the green corresponds to light of wave- 
length 527. OfJLfx; the F-line in the blue corresponds to light 
of wave-length 486.1^/a; the G-line in the indigo corre- 
sponds to light of wave-length 430.8ju/x; and the H-line, 
consisting of two broad lines in the violet, corresponds to 
light of wave-lengths 396.8/x/x and 393.3/a/x. 

163. Index of Refraction as a Function of the Wave- 
Length. — Now according to the wave-theory of light, the 
absolute index of refraction (n) of a medium for light of a 
definite color is equal to the ratio of the velocity of light 
in vacuo (V) to its velocity (y) in the medium in question 
(§33); that is, 

V 

n = — . 

v 

Strictly speaking, therefore, the index of refraction of a me- 
dium, without further qualification, is a perfectly vague ex- 
pression, because each medium has as many indices of re- 
fraction as there are different kinds of monochromatic light. 
When the term is used by itself, it is generally understood 
to mean the index of refraction corresponding to the D-line 
in the bright yellow part of the solar spectrum, which is 
characteristic of the light of incandescent sodium vapor. 
Hence, 

velocity of yellow light in vacuo 



n D = 



velocity of yellow light in the medium in question 
wave-length of j^ellow light in vacuo 



wave-length of yellow light in the given medium 

In the following table the values of the indices of refraction 
of several transparent liquids are given for light correspond- 
ing to the Fraunhofer lines A, B, C, D, E, F, G, and H. 



164] 



Irrationality of Dispersion 



477 





A 


B 


C 


D 


E 


F 


G 


H 


Wave-length 
in fifi 


759.4 


686.7 

1.360 
1.495 
1.616 
1.331 


656.3 


589.0 


527.0 


4£6.1 


430.8 


396.8 


Alcohol 
Benzene 
Sulphuric Acid 
Water 


1.359 
1.493 
1.610 
1.329 


1.361 
1.497 
1.620 
1.332 


1.363 
1.503 
1.629 
1.334 


1.365 
1.507 
1.642 
1.336 


1.367 
1.514 
1.654 
1.338 


1.371 
1.524 
1.670 
1.341 


1.374 
1.536 
1.702 
1.344 



It may be remarked that, in general, the shorter the wave- 
length, the greater will be the index of refraction of a sub- 
stance. But the exact relation between the index of refrac- 
tion and the wave-length of the light has to be determined 
empirically for each substance. There is, indeed, a certain 
group of substances which form an exception to the general 
statement made above, and which yield refraction-spectra 
with the order of the colors partially or entirely reversed. 
This phenomenon is called anomalous dispersion. 

164. Irrationality of Dispersion. — Other things being 
equal, the length of the spectrum or the interval between 




laau tola h m 



Fig. 210. — Irrationality of dispersion. 

a given pair of Fratjnhofer lines depends essentially on the 
nature of the refracting medium, so that, in general, as shown 
by the table in the preceding section, the dispersion of two 
colors will be found to be different for different substances. 



478 



Mirrors, Prisms and Lenses 



[§164 



For example, the dispersion of glass is greater than that of 
water, and the dispersion of so-called flint glass is higher 
that of so-called crown glass. In Fig. 210 are exhibited the 
relative lengths of the different regions of the solar spectra 
cast on the same screen under precisely the same cir- 
cumstances by prisms of equal refracting angles made of 
water, crown glass and flint glass. The length of the spec- 
trum may be increased by shifting the screen farther from 



■asassa 




Fig. 211. — Irrationality of dispersion. 

the prism, and Fig. 211 shows the relative positions of the 
Fraunhofer lines B, C, D, E, F, G and H, when the lengths 
of the' spectra of the crown glass prism and the water prism 
have been elongated in this manner until their lengths are 
both equal to the length of the spectrum of the flint glass 
prism for the interval between the Fraunhofer lines B and 
H. The other lines in the three spectra do not coincide at all. 
Moreover, it appears that the dispersion of water for the 
colors towards the red end of the spectrum is relatively high, 
whereas the dispersion of the flint glass is relatively high 
towards the blue end. In the spectrum of flint glass the in- 
terval between G and H, and in the spectrum of water the in- 
terval between B and F, is greater than it is in either of the 
other spectra. If the law of the variation of the index of re- 



§ 165] Dispersive Power 479 

fraction with the color of the light has been found empirically 
for one substance, this will not afford any clue to the corre- 
sponding law in the case of another substance. Diamond, for 
example, is very highly refracting but shows comparatively 
little dispersion, whereas flint glass which has a much lower 
index of refraction gives a much higher dispersion; on the 
other hand, fluorite has a low index of refraction and at the 
same time a low dispersion. This phenomenon which is 
characteristic of refraction-spectra is known as the irration- 
ality of dispersion. 

165. Dispersive Power of a Medium. — In the case of a 
prism of small refracting angle fi the deviation is given by 
the formula e = (n— 1)/3, as was explained in §60. Let 
the letters P and Q be used to designate two colors, and let 
n F and n Q denote the indices of refraction of the prism- 
substance for these colors. If the angles of deviation are 
denoted by e P and €q then e Q — e F = (nQ— n F )/3, and, con- 
sequently, for a thin prism the angular magnitude of the 
interval in the spectrurn between the colors P and Q is pro- 
portional to the difference of the values of the indices of re- 
fraction. This difference (wq — n F ) is called the partial dis- 
persion of the substance for the spectrum-interval P, Q. 
Thus, in the brightest part of the spectrum comprised be- 
tween the Fratjnhofer lines C and F, the partial dispersion 
is (n F —n c ). The deviation of a prism of small refracting 
angle /3 for light corresponding to the D-line which lies 
between C and F is € D = (n D — 1)/3, and since e F — €c = 
(n F — n c ) fi, we obtain : 

€f~ e c = n F —n c 
€d n D -l 
This ratio of the angular dispersion of two colors to their 
mean dispersion is called the dispersive power or the relative 
dispersion of the substance for the two colors, which are usu- 
ally red (C) and blue (F); so that the dispersive power of 
an optical medium with respect to the visible spectrum may 
be* defined to be the quotient of the difference (n F — nc) 



480 Mirrors, Prisms and Lenses [§ 165 

between the indices of refraction for red and blue light by 
(n D — 1), where n D denotes the index of refraction for yellow 
light. The values of the dispersive powers of the various 
kinds of optical glass that are of chief practical importance 
in the construction of optical instruments vary from about 
~ to about ~; although there are compositions of glass 
with values of the dispersive power not comprised within 
these limits. Instead of assigning the value of the dispersive 
power of a substance, it is more convenient to adopt Abbe's 
method and employ the reciprocal of this function, which is 
denoted by the Greek letter v, and which is known, there- 
fore, as the i>-value of the substance; thus, 

n?-nc 
If the rvalue of one substance is less than that of another, 
the dispersive power of the former will be correspondingly 
greater than that of the latter. 

It is this constant v that is the essential factor to be con- 
sidered in the selection of different kinds of glass suitable to 
be used in making a so-called achromatic combination of 
lenses or prisms. Curiously enough, Newton persisted in 
maintaining that the dispersion of a substance was propor- 
tional to the refraction, which is equivalent to saying that 
the dispersive powers of all optical media are equal; and, 
consequently, he despaired of constructing an achromatic 
combination of lenses which would refract the rays without 
at the same time dispersing the constituent colors. This 
condition, however, is an essential requirement in the object- 
glass of a telescope, and it was just because Newton and his 
followers believed that a lens of this kind was in the nature 
of things unattainable that they expended their efforts in 
the direction of perfecting the reflecting telescope in which 
the convex lens was replaced by a concave mirror. On the 
other hand, from the assumption that the optical system of 
the human eye is free from color-faults (which is by no means 
true), it was argued, notably by James Gregory in England 



§ 166] Optical Glass 481 

(about 1670) and long afterwards by Euler in Germany 
(1747), that Newton's conclusions as to the impossibility 
of an achromatic combination of refracting media were er- 
roneous. In fact, an English gentleman named Hall suc- 
ceeded in 1733 in constructing telescopes which yielded 
images free from serious color faults. Klingenstierna in 
Sweden in 1754 demonstrated the feasibility of combining 
a pair of prisms of different kinds of glass and of different re- 
fracting angles so as to obtain, in one case, deviation without 
dispersion and, in another case, dispersion without deviation. 

But in its practical results the most important advance 
along this line was achieved by the painstaking and original 
work of the English optician John Dollond. Impressed 
by the force of Klingenstierna's demonstration, he care- 
fully repeated Newton's crucial experiment in which a glass 
prism was inclosed in a water prism of variable refracting 
angle; and having found that the results of this experiment 
were exactly contrary to those stated by Newton, he was 
led also to the opposite conclusion. After much persever- 
ance Dollond had succeeded by 1757 in making achromatic 
combinations of several different types, which produced a 
more or less colorless image of a point-source on the axis of 
the system. In its original form the combination consisted 
of a double convex " crown glass" lens cemented to a double 
concave " flint glass" lens. As a rule, the focus of the blue 
rays will be nearer a convex lens and farther from a concave 
lens than the focus of the red rays; and hence by combining 
a convex crown glass lens of relatively lower refractive index 
(shorter focus) and less dispersive power with a concave flint 
glass lens of higher refractive index and higher dispersive 
power, a resultant system may be obtained which still has 
a certain finite focal length and in which at the same time 
the opposed color-dispersions for two colors, say, red and 
blue, are compensated. 

166. Optical Glass.— Newton's error in supposing that for 
all substances the dispersion was proportional to the index 



482 Mirrors, Prisms and Lenses [§ 166 

of refraction retarded the development of technical optics 
for a long time to come. Although Dollond's achievement, 
mentioned above, was one of far-reaching importance for the 
practical construction of optical instruments, the great diffi- 
culty in the way of utilizing and applying the principle was 
to be found in the fact that the actual varieties of optical 
glass at the disposal of the optician were exceedingly limited 
in number; although from time to time systematic efforts 
were made, notably by Fraunhofer (about 1812) in Ger- 
many and by Faraday (1824), Harcourt (1834) and Stokes 
(about 1870) in England, to remedy this deficiency, by dis- 
covering and manufacturing new compositions of glass suit- 
able for optical purposes. For a long time after Fraun- 
hofer' s epoch the art of making optical glass was confined 
almost exclusively to France and England. It was a for- 
tunate coincidence that just about the time when E. Abbe 
had reached the conclusion that no further progress in op- 
tical construction could be expected unless totally new va- 
rieties of optical glass were forthcoming, 0. Schott was 
already beginning to experiment with new chemical combina- 
tions and processes of manufacture in his glass works at Jena. 
Thanks to the systematic and indefatigable efforts of these 
two collaborators, who were also encouraged by the Prus- 
sian government, the obstacle which had stood so long in the 
way of the improvement and development of optical instru- 
ments was at length triumphantly overcome by the successful 
production of an entire new series of varieties of optical glass 
with properties in some instances almost beyond the highest 
expectations. The first catalogue of the Glastechnisches 
Laboratorium at Jena was issued in 1885; which marked 
the beginning of the manufacture of the renowned Jena glass, 
to which more than to any other single factor the remarkable 
development of modern optical instruments is due. From 
that time to the present the great province of applied optics 
may almost be said to have become a German territory. 
The earlier so-called " ordinary" varieties of optical 



§ 166] Jena Glass 483 

glass were silicates in which the basic constituents were 
lime (crown glass) or lead (flint glass) combined with soda 
(Na 2 C0 3 ) or potash (K 2 C0 3 ) or both. The newer kinds of 
optical glass have been produced by employing a much 
greater variety of chemical substances, including, in addi- 
tion to those named above, hydrated oxide of aluminum 
(A1 2 3 ,H 2 0), barium nitrate (BaN 2 6 ), zinc oxide (ZnO), 
etc., and boric acid (H 3 B0 3 ) or phosphoric acid which to a 
greater or less extent replace the silica (Si0 2 ) in the older 
types. Some of the new compounds have been found to 
have slight durability, and for this and other reasons cer- 
tain products formerly listed in the Jena glass catalogue 
have been discontinued. At present, besides the old " or- 
dinary" silicate crown and flint, the chief varieties are ba- 
rium and zinc silicate crown, boro-silicate crown, dense 
baryta crown, baryta flint, antimony flint, borate glass and 
phosphate glass. The table on the following page contains 
a list of certain varieties of Jena glass arranged in the order 
of their ^-values. In the Jena glass catalogue the values of 
the dispersion are given also for the spectrum-intervals 
^d — ^a'> n F~ %>> nG'—n F (where A' and G' are the lines 
corresponding to the wave-lengths 768 and 434yuju, re- 
spectively) , together with the values of the so-called relative 
partial dispersions obtained by dividing each of these num- 
bers by the value of (n F — n c ). 

It has recently been proposed to describe an optical 
glass by means of two numbers of 3 digits each, separated 
by an oblique line. The first number gives the first three 
figures after the decimal point in the value of n D , .while the 
second number is equal to 10 times the value of v. 
Thus, for example, the second glass in the table would be 
described as crown glass No. 559/669. 



484 Mirrors, Prisms and Lenses (§ 166 

SELECTED VARIETIES OF JENA GLASS 





Index of 


Mean 


n — 1 


Description 


Refraction 


Dispersion 


"d . x 




n D 


n F -?i G 


n F -n c 


Light phosphate crown 


1.5159 


0.007 37 


70.0 


Medium phosphate crown 


1 . 5590 


0.008 35 


66.9 


Boro-silicate crown 


1.5141 


0.008 02 


64.1 


Boro-silicate crown 


1.5103 


0.008 05 


63.4 


Silicate crown 


1.5191 


0.008 60 


60.4 


Silicate crown 


1.5215 


0.008 75 


59.6 


Silicate crown 


1.5127 


0.008 97 


57.2 


Densest baryta crown 


1.6112 


0.010 68 


57.2 


Barium crown 


1 . 5726 


0.009 95 


57.5 


Dense' baryta crown 


1.6130 


0.010 87 


56.4 


Dense baryta crown 


1.6120 


0.010 98 


55.7 


Baryta flint 


1.5664 


0.010 21 


55.5 


Borate flint 


1.5503 


0.009 96 


55.2 


Baryta flint 


1.5489 


0.010 25 


53.6 


Baryta flint 


1.5848 


0.011 04 


53.0 


Antimony flint 


1.5286 


0.010 25 


51.6 


Boro-silicate flint 


1.5503 


0.011 14 


49.4 


Extra light flint 


1.5398 


0.011 42 


47.3 


Baryta flint 


1 . 5825 


0.012 55 


46.4 


Ordinary light flint 


1 . 5660 


0.013 19 


42.9 


Silicate flint 


1.5794 


0.014 09 


41.1 


Baryta flint 


1.6235 


0.015 99 


39.1 


Heavy borate flint 


1.6797 


0.017 87 


38.0 


Silicate flint 


1.6138 


0.016 64 


36.9 


Silicate flint 


1.6489 


0.019 19 


33.8 


Dense silicate flint 


1.7174 


0.024 34 


29.5 


Densest silicate flint 


1.9626 


0.048 82 


19.7 



In recent years in France, England and the United States 
much attention has been bestowed on the study of the com- 
position and manufacture of optical glass, and according to 
the 1916-17 report of the British Committee of the Privy 
Council for Scientific and Industrial Research (summarized 
in Nature, Vol. 100, pp. 17-20), Professor Jackson in England 
"has succeeded in defining the composition of the bath 
mixtures necessary for the production of several glasses 
hitherto manufactured exclusively in Jena, including the 
famous fluor-crown glass," and, moreover, "he has also 
discovered three completely new glasses with properties 
hitherto unobtainable," However, it seems improbable 



§ 166] -Manufacture of Optical Glass 485 

that any essential changes in the optical properties of glass 
are to be obtained by the use of materials that have not al- 
ready been tried. The index of refraction of all glasses at 
present available are comprised between 1.45 and 1.96. The 
mineral fluorite (calcium fluoride), which is used in the best 
modern microscope objectives, has an index of refraction of 
1.4338 and a rvalue of 95.4, so that in both respects it 
lies beyond the limits attainable with glass. Other crystal- 
line transparent minerals, notably rock crystal or quartz, 
have already been employed in lens-systems, and any es- 
sential improvement in the range of optical instruments 
in the future is more likely to come from an adaptation of 
these mineral substances than from the production of new 
kinds of glass. 

The difficulties involved in the manufacture of high-grade 
optical glass are very great, and the utmost care has to be 
exercised throughout every stage of the process. Not only 
must the raw materials themselves be free from impurities as 
far as possible, but the physical and chemical nature of the 
fireclays used in the pots or crucibles also requires the most 
painstaking care and preparation. The empty crucible is 
dried slowly and then heated gradually for several days until 
it comes to a bright red glow. Fragments of glass left over 
from a previous melting and of the same chemical composi- 
tion as the glass which is in process of making are introduced 
into the pot and melted. The raw materials, pulverized and 
mixed in definite proportions, are placed in the pot in layers 
little by little at a time, and the pot, which is covered to 
protect the contents from the furnace gases is maintained 
at a sufficiently high temperature (between about 800 and 
1000° C.) until the contents are all melted together. The 
molten mass is usually full of bubbles of all sizes, and the 
temperature must be raised until these are all gotten rid 
of as far as possible. This entire process takes a longer or 
shorter time depending on circumstances, say, from 24 to 
36 hours or more. After skimming off the impurities on the 



486 Mirrors, Prisms and Lenses [§ 166 

surface, the mixture is allowed to cool gradually, and at the 
same time it is kept constantly stirred in order to make the 
glass as homogeneous as possible. This part of the process 
requires constant care. When the glass in cooling has be- 
come quite viscous, so that it is no longer possible to con- 
tinue the stirring, it is allowed to cool very slowly over a 
period of days or even weeks. Usually at the end of the 
cooling process the solid contents of the pot will be found 
to be broken into irregular fragments of optical glass in the 
first stage of its manufacture. These fragments are care- 
fully examined to see whether they are homogeneous and 
above all free from striae; but the broken surfaces are so 
irregular that this preliminary examination is necessarily 
very imperfect. The pieces which pass muster in this way 
are selected for molding and annealing. The lumps of glass 
are placed in suitable molds made of iron or fireclay and 
heated until the glass becomes soft like wax, so that it takes 
the form of the mold usually with the aid of external press- 
ure. The molded pieces are then annealed by being cooled 
gradually for a week or longer. They are in the form of 
disks or rectangular blocks of approximately the right size 
for being made into lenses and prisms. At this stage the 
glass has to be subjected to the most rigid testing to see if 
it is really suitable for optical purposes. Two opposite faces 
on the narrow sides are ground flat and parallel and polished 
so that the slab can be inspected in the direction of its greatest 
diameter. If any striae or other imperfections are found, the 
piece will have to be rejected and melted over again. Even 
in case there are no directly visible defects, there may be in- 
ternal strains which will be revealed by examination with 
polarized light. Slight strains are not always serious, but even 
these will impair the image in a large prism or lens. These 
strains can be gotten rid of by heating the glass to a tempera- 
ture between 350 and 480° C, depending on the composition, 
and then cooling very slowly and uniformly over a period of 
about six weeks. It is very difficult to obtain pieces of op- 



§ 167] Achromatism 487 

tical glass which do not contain minute bubbles, and indeed 
they are often to be found in the best kinds of glass. 

Of course, the process as above described varies in details 
according to the special nature of the glass, but enough has 
been said to enable the reader to form some idea of the pa- 
tience and skill which are required in the manufacture of 
optical glass. A yield of 20 per cent, of the total quantity of 
glass melted is considered good. The glass to be used for 
photographic lenses has to fulfill the most exact requirements 
and must be of the highest quality. 

167. Chromatic Aberration and Achromatism. — Since the 
index of refraction varies with the color of the light, and since 
this function enters in one form or another in all optical cal- 
culations, it is obvious, for example, that the positions of 
the cardinal points of a lens-system will, in general, be differ- 
ent for light of different colors; and that there will be a whole 
series of colored images of a given object depending on the 
nature of the light which it radiates, these images being all 
more or less separated from each other and of varying sizes. 
This phenomenon is called chromatic aberration, and unless 
it is at least partially corrected, the definition of the resultant 
image is very seriously impaired. In an optical system which 
was absolutely free from chromatic aberration all these 
colored images would coalesce into a single composite image 
which, so far as the quality of the light was concerned, would 
be a faithful reproduction of the object. But nothing at all 
comparable to this ideal condition of achromatism can be 
achieved in the case of any actual lens-system. In fact, the 
term achromatism by itself and without any further explana- 
tion is entirely vague, for an optical system may be achro- 
matic in one sense without being at all so in other senses. For 
example, the images corresponding to different colors may 
all be formed in the same plane and yet be of different sizes, 
or vice versa. Fortunately, however, the fact that it is im- 
possible to achieve at best more than a partial achromatism 
is not such a serious matter after all. The kind of achromat- 



488 Mirrors, Prisms and Lenses [§ 167 

ism which is adapted for one type of optical instrument may 
be entirely unsuited to another type. Thus, it is absolutely 
essential that the colored images formed by the object-glass 
of a telescope or microscope shall be produced as nearly as 
possible at one and the same place (achromatism with re- 
spect to the location of the image), whereas, since the images 
in this case do not extend far from the axis, the unequal 
color-magnifications are comparatively unimportant. On 
the other hand, in the case of the ocular systems of the same 
instruments, the main consideration will be a partial achro- 
matism with respect to the magnification or the apparent 
sizes of the colored images. The object-glass of a telescope 
must be achromatic with respect to the position of its focal 
point, and the ocular must be achromatic with respect to its 
focal length. 

An optical system which produces the same definite effect 
for light of two different wave-lengths, no matter what that 
special effect may be, is to that extent an achromatic system. 
A combination which is achromatic, even in its limited sense, 
for a certain prescribed distance of the object will, in general, 
not be achromatic when the object is placed at a different 
distance. No lens composed of two kinds of glass only can 
be achromatic for light of all different colors. It can be con- 
structed, for example, so that it will bring the red and violet 
rays accurately to the same focus at a prescribed point on 
the axis; but then the yellow, green and blue rays will, in 
general, all have different foci, some of which will be nearer 
the lens than the point of reunion of the red and violet light 
while others will lie farther away. Accordingly, when achro- 
matism has been attained in the case of two chosen colors, 
there will usually remain an uncorrected residual dispersion 
or so-called secondary spectrum, which under certain circum- 
stances may impair the definition of the image to such a 
degree as to be very injurious and annoying. It is neces- 
sary to abolish the secondary spectrum in the object-glass of 
a microscope. This may be done by using more than two 



§ 16S] Optical and Actinic Achromatism 489 

kinds of glass. There is also the possibility of diminishing the 
secondary spectrum try employing two kinds of glass whose 
relative partial dispersions (§ 166) are very nearly the same 
for all the spectrum-intervals; and, in fact, one of the prin- 
cipal items in the Abbe-Schott programme for the manu- 
facture of optical glass was the production of various pairs 
of flint and crown glass suitable for such combinations, so 
that the dispersions in the different regions of the spectrum 
should be, for each pair, as nearly as possible proportional. 
This purpose was satisfactorily accomplished, and we have 
now achromatic lenses of a far more perfect kind than could 
be made out of the older kinds of glass. This higher degree 
of achromatism is called apochromatism. An apochromatic 
photographic lens is absolutely essential in the three-color 
process of photography in which the three images taken 
through light-filters on a plate of medium or large size must 
be superposed as exactly as possible. In most ordinary op- 
tical systems, however, the secondary spectrum is relatively 
unimportant, and achromatism with respect to two prin- 
cipal colors will usually be found to be sufficient. 

168. " Optical Achromatism " and " Actinic Achromat- 
ism." — The character and extent of the secondary spectrum 
(§ 167) of an achromatic combination of lenses will evidently 
depend essentially on the choice of the two principal colors for 
which the achromatism is to be achieved. This choice will 
be determined by the purpose for which the instrument is 
intended and the mode of using it. Thus, if the system is 
to be an optical instrument in the strict literal sense of the 
word, that is, if it is constructed to be used subjectively in 
conjunction with the eye, we shall be concerned primarily 
with the physiological action of the rays on the retina of the 
human eye; whereas in the case of a photographic lens which 
is used to focus an image on a prepared sensitized plate, it 
is important to have achromatism with respect to the so- 
called actinic rays corresponding to the violet and ultra- 
violet regions of the spectrum, because these are the rays 



490 Mirrors, Prisms and Lenses [§ 168 

which are most active on the ordinary bromo-silver gelatine 
plate. 

The retina of the human eye is most sensitive to the kind 
of light which is comprised within, the interval between the 
lines C and F, with a distinct maximum of visual effect cor- 
responding to wave-lengths lying somewhere between the 
lines D and E. Accordingly, in an optical instrument which 
is to be applied to the eye, it is usually desirable to unite the 
red and blue rays as nearly as possible at the focus of the 
yellow rays. If, for example, the system is assumed to be 
a convergent combination of two thin lenses in contact (as 
in the case of the object-glass of a telescope), it will be found 
that the focal points corresponding to the colors (say, green 
and yellow) between C and F will lie nearer the lens and the 
focal points corresponding to the other colors (dark red, 
dark blue and violet) will lie farther from it than the com- 
mon focal point of the two principal colors C and F. More- 
over, the residual color-error or secondary spectrum in this 
case will be least for some color very nearly corresponding to 
the D-line, which is a favorable circumstance, since, as above 
stated, this is the region of the brightest part of the visible 
spectrum. Achromatism with respect to the colors C and F 



omatism. 



I v = — ) is sometimes called optical achr< 

\ riF-nc/ 

On the other hand, in the construction of a photographic 
lens a kind of compromise must be effected between the con- 
vergence of the visual rays and the so-called actinic rays, 
because the image has to be focused first on the ground glass 
plate by the eye and afterwards it has to be received on the 
sensitized plate or film which is inserted for exposure in the 
camera in the place of the translucent focusing screen. Ac- 
cordingly, for ordinary photographic practice an exact co- 
incidence of the "optical" and " actinic" images is de- 
manded. Here it is found that the best results are obtained 
by uniting the colors corresponding to the D-line and the 
violet band in the spectrum of hydrogen, which, since it is 



§ 169] Achromatic Prism 491 

not far from the G-line, may be designated by G' (434^/0- 
This is sometimes called actinic or photographic achromat- 
ism for which the function v has a special value, namely : 

_ = flD-1 



riG'-nj) 

If the photographic lens is a combination of two thin 
lenses in contact, which is achromatic for the colors D and 
G', the focus of the rays corresponding to the blue-green 
region of the spectrum will be nearer the lens than the com- 
mon focus of the two principal colors and the focus of the 
bright red rays will be farther from the lens. In an achro- 
mat of this kind the residual dispersion will usually be quite 
large for both the " optical" and the "actinic" image, but 
for most practical purposes the definition of the image in 
either case is good enough. In astrophotography the focus 
of the camera is determined once for all, and a lens for stellar 
photography is usually designed to have an entirely actinic 
achromatism, the two principal colors in this case correspond- 
ing to the F-line (486/f/x) and the violet line in the mercury- 
spectrum (405 fifi). The rays belonging in these two colors 
are made to unite as nearly as possible at the fociis of the 
rays corresponding to the G'-line, which is approximately 
the place of maximum actinic action. In a photographic 
achromat of this kind the foci of the green, yellow and red 
rays will lie beyond the actinic focus in the order named. 

169. Achromatic Combination of Two Thin Prisms. — 
Two prisms of different substances may be combined so as 
to obtain achromatism in the sense that rays of light cor- 
responding to a definite pair of colors will issue from the 
system in parallel directions, as represented in Fig. 212. 
When an object is viewed through the combination, the red 
and blue rays, for example, will be fused or superposed and 
the residual color-effect will be comparatively slight. By 
employing a greater number of prisms a more perfect union 
of colors could be obtained, but usually two prisms are suf- 
ficient. 



492 Mirrors, Prisms and Lenses [§ 169 

The problem is simplified by assuming that the refracting 
angles of the prisms, denoted here by /3 and 7, are both 
small; so that the deviation produced by each prism may be 
considered as proportional to its refracting angle, accord- 
ing to the approximate formula deduced in § 60. Usually, 



Fig. 212. — Achromatic combination of two thin prisms. 

the two prisms are cemented together with their edges par- 
allel but oppositely turned, as shown in the diagram, so that 
the thicker portions of one prism are adjacent to the thinner 
portions of the other; accordingly, the total deviation (e) 
will be equal to the arithmetical difference of the deviations 
produced by each prism separately. 

Let P, Q and R designate three elementary colors, the 
color Q being supposed to lie between P and R in the spec- 
trum; and let the indices of refraction for these three colors 
be denoted by n P ', Uq and n R ' for the first prism and by 
n ~p", n Q f and n R " for the second prism. The total devia- 
tions for the three colors will be: 
€ P = (V - 1) j8 - (V -1)7, e Q = (V -1)0- ( V - 1) y, 

€ R =(n R '-l)i3-(n R "-l)7. 
Now if the system is to be achromatic with respect to the 
colors P and R, the condition is that e P = e R , which, there- 
fore, is equivalent to the following : 

/3_ n R "-y . 

7 n R '-n F ' ' 
that is, the refracting angles of the prisms must be inversely 



€Q = (W R / -Wp / ) 



P- 



§ 170] Direct Vision Prism 493 

proportional to the partial dispersions of the two media for 
the two given colors. 

Moreover, the deviation of the rays of the intermediate 
color Q will be : 

' V-l _ n Q "-l) * 
ftR'-ttp' WR // -Wp / ' 

Actually the colors P, Q and R are usually chosen to cor- 
respond to the Fraunhofer lines C, D and F, respectively, 
in which case the combination will be achromatic with re- 
spect to C (red) and F (blue) . Thus, the fractions inside the 
large brackets are the ^-values of the two kinds of glass. 
Accordingly, for an achromatic combination of two thin 
prisms for which the deviation € D has a finite value, whereas 
the dispersion ( ec — €p ) is abolished, we have the following 
formulae : 

l= n *"7 nc " , £D= (n F '-n c ') {v>-v")P. 
7 riF-nc 

Consider, for example, a combination of two kinds of Jena 

glass as follows: 

n D n F — uq v 

Light Phosphate Crown 1.5159 0.007 37 70.0 

Borate Flint 1.5503 0.009 96 55.2 

Assuming that the angle of the crown glass prism is (3 = 20°, 

we find: 7 = 14.8°, € D = 2.18°. Generally speaking, those 

pairs of glasses in which the partial dispersions are more 

nearly equal will be .found to be best adapted for achromatic 

combinations. 

170. Direct Vision Combination of Two Thin Prisms. — 

In the case of an ordinary prism-spectroscope the rays are 

deflected in passing through the system, so that in order to 

view the spectrum the eye has to be pointed not directly 

towards the luminous source, but in some oblique direction; 

which is sometimes inconvenient, especially in astrophysi- 

cal observations. Accordingly, various prism-systems have 

been proposed which are designed so that rays corresponding 

to some definite standard color are finally bent back into 



494 Mirrors, Prisms and Lenses [§ 170 

their original direction, with the result that there is disper- 
sion without deviation, which is an effect precisely opposite 
to that which is obtained with an achromatic prism. In 
these so-called direct vision prisms (prismes a vision directe) 




Fig. 213. — Direct vision prism combination (dispersion without deviation). 

the spectrum of an illuminated slit will be seen in the same 
direction as the slit itself. The condition that the light cor- 
responding, say, to the Fraunhofer D-line shall emerge 
from the system in the same direction as it entered is € D = 0. 
Assuming that the combination is composed, as before, of 
two thin prisms juxtaposed in the same way (Fig. 213), and 
employing the same symbols (§ 169), we derive immedi- 
ately the following f ormulse : 

jg_ n D "-l 

7 nW-l ' 

CO" €f=Od'-1) \y--) ' 

Consider, for example, the following combination : 

n D v 

Light Phosphate Crown 1.5159 70.0 

Heavy Silicate Flint 1.9626 19.7 

the difference of the ^-values here being very great. If 
we put = 20°, we find: 7 = 10.72°, € C - e F = 22.56'. 

It will be profitable for the student to satisfy himself by 
several examples that two kinds of glass which are best 
adapted for a direct vision prism combination are on the 



171] 



Direct Vision Prism 



495 



contrary not very suitable for an achromatic prism, and vice 
versa; as might naturally be expected, since the effects are 
opposite in the two cases. Generally speaking, the two kinds 
of glass used for a direct vision prism should have very dif- 
ferent ^-values, as in the illustration given above. 

In the case of prisms of large refracting angles, the formula? 
here and in § 169 are hardly to be considered as even ap- 
proximate. 

171. Calculation of Amici Prism with Finite Angles. — 
Accurate formulae for the calculation of an achromatic or 
direct vision prism-system may easily be derived when the 




Fig. 214. — Direct vision prism combination. Diagram represents one-half 
of so-called Amici direct vision prism. 



system consists of only two prisms. As an illustration of 
the method in the case of a direct vision prism, let us em- 
ploy here the symbols n\ and ni to denote the indices of re- 
fraction of the crown glass prism and the flint glass prism, 
respectively, for light of some standard wave-length; and 
let j8 and y denote their refracting angles. We shall sup- 
pose also that the two prisms are cemented along a common 
face, as represented in Fig. 214. A ray of the given wave- 
length is incident on the crown glass prism at an angle d 
and is refracted into this medium at the angle 0', so that 

tti.sin 0'=sin 6. (1) 

If the angles of incidence and refraction at the surface of 



496 Mirrors, Prisms and Lenses [§171 

separation of two kinds of glass are denoted by ^ and ^', 
then 

ni.sin^ = 7i2.sin^ r , (2) 

0' = $-+; (3) 

the angles here being all reckoned as positive. If, finally, it 
is assumed that the ray meets the second face of the second 
prism normally and issues again into the air in the same di- 
rection as it had originally, then also : 

f = 7, (4) and d=/3-y. (5) 

The problem consists in determining the angle of one of the 
prisms when the angle of the other is given. Suppose, for 
example, that an arbitrary value is assigned to the acute 
angle 7, and it is required to find the magnitude of the 
angle fi. Substituting in (1) the values of 6, d' as given 
in (3) and (5), we obtain: 

tti.sin(/3-^)=sin(/3- 7), 
whence we derive: 

a fti.sin^ — sin 7 
tan p = -. . 

Eliminating ^' from (2) and (4), we find: 
ni.sin ^=712. sin 7, 

and consequently also: 

Tii. cos ^ = \Zn\— nl.sui'y. 
Accordingly, the value of /5 in terms of m, ni and 7 is given 
by the formula: 

tMn p. («.-!) JE T . 

-y/nl — n^sin 2 7 - cos 7 
If, on the other hand, the value of the angle /3 has been 
chosen arbitrarily, the calculation of 7 will be found to be 
trigonometrically a little more difficult. It is left as an ex- 
ercise for the studen t to show that : 

fanT _ ^2-1+ V^(tt 2 -l) 2 + (n 2 i-l) (n 2 2 -^)tan 2 ^ tsing 
7 (ri 2 -nDtan 2 /3+(n 2 -l) 2 

If it is desired that the emergent ray shall not only be par- 
allel to the incident ray but that its path shall be along the 



172] 



Direct Vision Prism 



497 



same straight line, it is necessary to add to the above another 
combination identical with it and placed so that the two 
flint glass prisms constitute in reality one single prism of re- 
fracting angle 27 inserted between two equal crown glass 
prisms each of refracting angle j8, as shown in Fig. 215; and, 




Fig. 215. — Amici direct vision prism. 

in fact, this is the actual construction of the common form 
of the Amici prism. Suppose, for example, that the angle 
7=45° and that the two kinds of glass are those described 
in the Jena catalogue as " light phosphate crown" and 
"heavy silicate flint" with indices ni = 1.5159 and n 2 = 1.9626 
corresponding to the D-line; then we find that the angle /? = 
98° 7.4'. 

172. Kessler Direct Vision Quadrilateral Prism. — One 
of the principal objections to a train of prisms is the loss of 
light by reflection at the various surfaces and also by ab- 
sorption in traversing the successive media. Partly with a 
view to diminishing these losses and partly also on account 
of other advantages, many forms of direct vision prism have 
been proposed which are made of one piece of glass with four 
or more plane faces; in all of which, however, the principle 
is the same, namely, by means of a series of total internal 
reflections to bend the rays corresponding to some standard 
intermediate color back finally into their original direction. 
The simplest of all these devices is the four-faced prism 
ABCD (Fig. 216) proposed by Kessler, a principal section 
of which has the form of a quadrilateral with perpendicular 
diagonals. The ray of standard wave-length enters the prism 



498 Mirrors, Prisms and Lenses [§ 172 

and leaves it in a direction parallel to the diagonal BD; it is 
totally reflected twice, first at the face BA and again at the 
face AD, the path of the ray between these reflections being 
parallel to its direction at entrance and emergence. More- 




A 

Fig. 216. — Kessler direct vision prism. 

over, in virtue of the symmetry of the prism, the path of the 
emergent ray will be a continuation of the rectilinear path of 
the incident ray. If the angles at A, B and C are denoted by 
a, $ and 7, respectively, then 

a+2/3+7 = 360 o ; (1) 

and if the angles of incidence and refraction at the face BC 
are denoted by 6, 6', then 

o-\, 0'=|-<s; (2) 

and, finally, if the index of refraction is denoted by n, 

n.sin0' = sin0. (3) 

Consequently, eliminating the angles 6, 6' by means of 
(2) and (3), we obtain: 

n.sin(-^-/3)=sim| ; (4) 

so that if the value of one of the angles a, /? and 7 is chosen 
arbitrarily, the other two angles can be determined by means 
of equations (1) and (4). 

If the principal section of a Kessler prism has the form 
of a rhombus (Fig. 217), parallel incident rays may be re- 



173] Achromatic System of Lenses 499 

c 



A 

Fig. 217. — Rhomboidal form of Kessler prism. 

ceived on both faces BA and BC. In this case the angles 
a and y are equal, and hence /3+7 = 180°, and therefore 

= | , 0' = ^-18O°, 



so that 

7i.sin(^-180°)=sin^, 
whence we obtain : 



• 
sm^ = cos 



2 V \n 



2 2 V 4n 

For example, if n = 1.64, we find = 36° 24', 7 = 143° 36'. 
173. Achromatic Combination of Two Thin Lenses. — 

The positions of the principal and focal points of a lens- 
sj'stem vary for light of different colors, and if the system is 
to be used as a magnifying glass or as the so-called ocular 
of a microscope or telescope, a chief consideration will be 
that the apparent sizes of the colored virtual images which 
are presented to the eye shall all be the same, that is, that 
the red and blue images, for example, shall subtend the 
same angle at the eye, no matter whether their actual sizes 
and positions are different or not. But the apparent size 
of the infinitely distant image of an object lying in the 
primary focal plane of the lens-system is measured by the 



500 Mirrors, Prisms and Lenses [§ 173 

refracting power of the system (§ 122) ; and hence the condi- 
tion of achromatism in this case is that the refracting powers 
(or focal lengths) of the system shall be equal for the two 
colors in question. (Achromatism with respect to the focal 
length; see § 167.) 

Let us assume that the system is composed of two thin 
lenses whose refracting powers for light of a certain definite 
wave-length X are denoted by Fx and F 2 ; then the refract- 
ing power of the combination will be F=Fi+F 2 —c.Fi.F 2) 
where c denotes the air-interval between the two lenses. 
For a second color of wave-length X+AX (where AX de- 
notes a small variation in the value of X), the refracting 
powers of the lenses will be slightly different, and the re- 
fracting power of the combination for this color will be: 

F + AF=(F 1 +AF 1 )+(F 2 +AF 2 )-c(F 1 +AF 1 ) (F 2 +AF 2 ). 
Subtracting these two equations, at the same time neglect- 
ing the term which involves the product of the small varia- 
tions A,Pi and AF 2 , we obtain : 

AF=AF X +AF 2 - (F 2 .AFi+Fi.AF 2 )c. 
Evidently, the condition that the system shall be achromatic 
with respect to its refracting power is AF=0; which, there- 
fore, is equivalent to the following: 

_ F 2 .AFi+F lm AF 2 
C ~ AFi+AF 2 ' 
Now if n\ denotes the index of refraction of the first lens for 
light of wave-length X, then 

ft-(m-l)Ki, 

where Ki denotes a constant whose value depends simply on 
the form of the infinitely thin lens, that is, on the curvatures 
of its surfaces. Similarly, for light of wave-length X+AX, 
we have: 

Fi+AFi=(m+Ani-l)Ki; 



and hence 



AFx = Ki.Am=Fi- A 



fti— 1 



§173] 



Achromatic Lens-System 



501 



But Am/(ni—l)=l/Pi is the expression for the dispersive 
power of the material of the first lens (§ 165), and accordingly 
we may write : 

and, analogously, for the second lens: 

v 2 

Introducing these expressions for AFi and AF 2 in the equa- 
tion above, we find, therefore, as the condition that a pair of 
thin lenses shall be achromatic with respect to the refracting 




Fig. 218. — Hutgens's ocular. 



power of the system, the requirement that the distance be- 
tween the two thin lenses shall satisfy the following equa- 
tion: 

v 2 .Fi+vi.F 2 



c = 



(Vl+V2)Fl-F 2 ' 



or 



_^l./l + ^2-A 
• C j , 

where fi = l/Fi and f 2 = l/F 2 denote the focal lengths of the 
lenses. 



502 



Mirrors, Prisms and Lenses 



174 



If both lenses are made of the same glass, then v 1 = v 2 , so 
that in this case the condition of achromatism becomes : 

./1+/2 



c = 



Thus, for example, Huygens's ocular (Fig. 218) is composed 
of two plano-convex lenses made of the same kind of glass, 
the curved face of each lens being turned away from the 
eye and towards the incident light. The first lens is called 
the " field-lens" and the second lens is called the " eye-lens." 
In this combination /1 = 2/ 2 (although in actual systems this 




Fig. 219. — Ramsden's ocular. 

condition is usually only approximately satisfied) and c = 
3/2/2, or f 2 :c :/i = 2 :3 : 4. Ramsden's ocular (Fig. 219) 
consists likewise of two plano-convex lenses of the same kind 
of glass, but with their curved faces turned towards each 
other and in this combination f 1 =f 2 =f=c. Both of these 
types satisfy, therefore, the above condition of achromatism 
and yield images that are free from color-faults not only in 
the center but at the border of the field. 

174. Achromatic Combination of Two Thin Lenses in 
Contact. — If the two lenses are in contact (c = 0), the con- 
dition of achromatism, as found in the preceding section, 
becomes : 

Vifi+V 2 .f2==0, 

or 

^+^ =0. 
The quotient of the refracting power of a lens by the dis- 



§ 174] Achromatic Lens-System 503 

persive power of the glass of which it is made, namely, the 
magnitude F/v, is sometimes called the dispersive strength 
of the lens; so that according to the above equation we may 
say that the condition of achromatism of a combination of 
two thin lenses in contact is that the algebraic sum of their 
dispersive strengths shall vanish. Accordingly, it appears 
that such a system can be achromatic only in case the sub- 
stances of which the two lenses are made are different. More- 
over, while one of the lenses must be convex and the other 
concave, their actual forms are of no consequence so far as 
the mere correction of the chromatic aberration is concerned. 
It is to be remarked also that in an achromatic lens of neg- 
ligible thickness achromatism with respect to the focal lengths 
implies also achromatism with respect to the positions of 
the focal points and principal points, so that such a lens will 
be achromatic for all distances of the object. 

If F denotes the prescribed refracting power of the com- 
bination then, since, 

^ = ^1+^2, 

we find: 

Fi=^-F, F 2 =--^—F. 

Vi-V 2 Vi-V 2 

The total refracting power F will have the same sign as 
that of the lens which has the greater r-value; for example, 
the combination will act like a convex lens provided the 
v-value of the positive element exceeds that of the nega- 
tive element. 

Thus, being given the values of F, v x and v 2} we can em- 
ploy the above relations to determine the required values 
of F\ and F 2 . Moreover, if Ki denotes the algebraic differ- 
ence of the curvatures of the two faces of the first lens, and, 
similarly, if K 2 denotes the corresponding magnitude for 
the second lens, then 



ni-1 n 2 -l 

where n h n 2 denote the indices of refraction of the two kinds 



504 



Mirrors, Prisms and Lenses 



[§174 



of glass for some standard wave-length, as already stated, 
which is usually light corresponding to the Fraunhofer 
D-line. Thus, while the magnitudes denoted by Ki and K 2 
may be computed, the actual curvatures or radii of the lens- 
surfaces remain indeterminate; so that there are still two 




Fig. 220. — Dollond's 
telescope objective. 



Fig. 221. — Fraun- 
hofer's telescope 
objective, No. 1. 



Fig. 222. — Fraun- 
hofer's telescope 
objective, No. 2. 



Fig. 223. — Herschel's 
telescope objective. 




Fig. 224.— Barlow's 
telescope objective. 




Fig. 225. — Gauss's 
telescope objective. 



other conditions which may be imposed on an achromatic 
combination of this kind. For example, in some cases it 
may be conventient to cement the two components together, 
and then one of the conditions will be that the curvatures of 
the two surfaces in contact shall be equal. Usually, how- 
ever, a more important requirement will be the abolition of 
two of the so-called spherical errors due to the fact that the 



Ch. XIV] Problems 505 

rays are not paraxial, so that the image will be sharp and 
distinct, especially at the center. 

Some historic types of achromatic object-glasses of a tel- 
escope are illustrated in the accompanying diagrams. Dol- 
lond's achromatic doublet (Fig. 220) consisted of a double 
convex crown glass lens combined with a double concave flint 
glass lens; whereas Fraunhofer's constructions show a com- 
bination of a double convex and a plano-concave lens (Fig. 
221) and of a double convex and a meniscus lens (Fig. 222). 
J. Herschel's form (1821) is shown in Fig. 223, Barlow's 
(1827) in Fig. 224; and, finally, the Gauss type made by 
Steinheil in 1860 is exhibited in Fig. 225. The newer va- 
rieties of Jena glass make it possible to construct an achro- 
matic objective of two lenses which is far superior in achro- 
matism to any of the older types above mentioned. 

PROBLEMS 

1. Find the values of the reciprocals of the dispersive 
powers (§ 165) of alcohol and water, using data given in 
table in § 163. Ans. Alcohol, 60.5; water, 55.7. 

2. The indices of refraction of rock salt for the Fraun- 
hofer lines C, D and F are 1.5404, 1.5441 and 1.5531, re- 
spectively. Calculate the value of the reciprocal of the dis- 
persive power. Ans. 42.84. 

3. White light is emitted from a luminous point on the 
axis of a thin lens. If the yellow rays are brought to a focus 
at a point whose distance from the lens is denoted by u', 
show that the distance between the foci of the red and blue 
rays is approximately equal to 2 F.u'/v, where F denotes the 
refracting power of the lens for yellow light and v denotes 
the reciprocal of the dispersive power of the lens-medium. 

4. A lens is made of borate flint glass for which ^ = 55.2. 
The focal length of the lens for sodium light is 30 inches. 
Find the distance between the red and blue images of the 
sun formed by the lens. Ans. 0.54 in. 



506 Mirrors, Prisms and Lenses [Ch. XIV 

5. A crown glass prism of refracting angle 20° is to be 
combined with a flint glass prism so that the combination 
will be achromatic for the Fraunhofer lines C and F. The 
indices of refraction are as follows : 

n c n-D n F 

Crown 1.526 849 1.529 587 1.536 052 
Flint 1.629 681 1.635 036 1.648 260 

Using the approximate formulae for thin prisms, show that 
the refracting angle of the flint prism will be 9° 54' 11", and 
that the deviation of the rays corresponding to the D-line 
will be 4° 18' 7". 

6. A direct vision prism combination is to be made with 
the same kinds of glass as in the preceding problem; so that 
rays corresponding to the D-line are to emerge without de- 
viation. If the refracting angle of the crown glass prism is 
20°, show that the refracting angle of the flint glass prism 
will be 16° 40' 48", and that the angular dispersion between 
C and F will be 9' 33". 

7. An Amici direct vision prism (§ 171) is to be made of 
crown glass and flint glass whose indices of refraction for 
the D-line are 1.5159 and 1.9626, respectively. If the re- 
fracting angles of the two equal crown glass prisms are each 
equal to 45°, show that the refracting angle of the middle 
flint glass prism will be 98° 7.4'. 

8. A Kessler prism (§ 172) in the form of a rhombus is 
made of glass of index n D = 1.6138. Find the angles of the 
prism. Ans. 35° 5' and 144° 55'. 

9. A thin lens is made of crown glass for which z>i = 60.2. 
Another thin lens is made of flint glass for which *> 2 = 36.2. 
When the two lenses are placed in contact they form an 
achromatic combination of focal length 10 cm. Find the 
focal length of each lens. Ans. /i =3.99 cm. ; / 2 = —6.63 cm. 

10. An achromatic doublet is to be made of two thin 
lenses cemented together, and the focal length of the com- 
bination for the D-line is to be 25 cm. The first lens is a 
symmetric convex lens of barium silicate glass and the other 



Ch. XIV] Problems 507 

lens is a concave lens of sodium lead glass. The indices of 
refraction are: 

^D Up — Tie 

Barium silicate 1.6112 0.01747 

Sodium lead 1.5205 0.01956 

Find the radii of the surfaces on the supposition that the rays 
corresponding to the lines C and F are united. 

Ans. The radii of the first and last surfaces are +14.60 
and —22.65 cm., respectively. 

11. A symmetric double convex lens is made of rock salt 
for which n c = 1.5404 and n F = 1.5531. Find the thickness 
of the lens if the focal lengths for the colors C and F are equal. 

Ans. d = 3.4363. r, where r denotes the radius of the first 
surface of the lens. 

12. Two thin lenses of the same kind of glass, one convex 
of focal length 9 inches, the other concave of focal length 
4 inches, are separated by an interval of 20 inches. A small 
white object is placed 36 inches in front of the convex lens. 
Show that the various colored images are all formed at the 
same place. 

13. Two thin lenses of the same kind of glass, one convex 
and the other concave, and both of focal length 4 inches, are 
adjusted on the same axis until the colored images of a white 
object placed 12 inches in front of the convex lens are formed 
at the same place. Show that the interval between the lenses 
must be twelve inches. 

14. A lens-system surrounded by air is composed of m 
spherical refracting surfaces. Assuming that the total thick- 
ness of the system is negligible, show that the condition of 
achromatism is 

k=m 

2 (# k _i-i4) Sn k = 0, 

k=2 

where R k denotes the curvature of the kth surface and 8n k 
denotes the dispersion of the medium included between the 
(k — l)th and kih surfaces for light of the two colors to be 
compensated. 



CHAPTER XV 

RAYS OF FINITE SLOPE. SPHERICAL ABERRATION, 
ASTIGMATISM OF OBLIQUE BUNDLES, ETC. 

175. Introduction. — The theory of the symmetrical op- 
tical instrument, as it has been developed in the preceding 
chapters, is based on the assumption that the rays concerned 
in the formation of the image are entirely confined to the 
so-called paraxial rays (§ 63) whose paths throughout the 
system are contained within an exceedingly narrow cylindri- 
cal region of space immediately surrounding the axis. With 
this fundamental restriction it was found that there was 
perfect collinear correspondence between object-space and 
image-space; so that a train of spherical waves emanating 
from an object-point was transformed by the optical system 
into another train of spherical waves accurately converging 
to or diverging from a corresponding center called the image- 
point; and so that, in general, a plane object at right angles 
to the axis was reproduced point by point by a similar plane 
image. As a matter of fact, these ideal conditions are never 
realized in any actual optical system except in the case of a 
plane mirror or combination of plane mirrors. Moreover, 
according to the wave-theory of light, a mere homocentric 
convergence of the rays is not sufficient for obtaining a point- 
image of a point-source ; for this theory lays particular stress 
on the further essential requirement that the effective por- 
tion of the wave-surface which contributes to the produc- 
tion of the image shall be relatively large in comparison with 
the radius of the surface, if the light-effect is to be concen- 
trated as nearly as possible at a single point and not spread 
over some considerable area in the vicinity of the point. This 
condition implies, therefore, that the aperture of the bundle 

508 



§ 176] Young's Construction 509 

of effective rays must not be below a certain finite limit, in 
other words we are compelled by a practical necessity, wholly 
aside from the principles at the basis of geometrical optics, to 
employ more or less wide-angle bundles of rays. Moreover, 
if a wide-angle bundle of rays is a requirement of a distinct, 
clear-cut image, it is also equally essential for a bright image. 
Thus, on both theoretical and practical grounds, it is found 
necessary to extend the limits of the effective rays beyond 
the paraxial region. 

Instead, therefore, of the ideal case of collinear correspond- 
ence of object-space and image-space, the theory of optical 
instruments is complicated by numerous practical and, for 
the most part irreconcilable difficulties, due chiefly to the 
so-called aberrations or failure of the rays to arrive at the 
places where they might be expected according to the 
simple theory of collineation or point-to-point corre- 
spondence (punctual imagery). In the preceding chapter 
brief reference was made to the chromatic aberrations arising 
from the differences in the color of the light; but now we 
have to deal with the monochromatic aberrations of rays of 
light of one definite wave-length which are caused by the pe- 
culiarities of the curved surfaces at which the rays are re- 
flected and refracted. These surfaces are nearly always 
spherical in form, and hence the aberrations of this latter 
kind are usually called spherical aberrations. A complete 
treatment of this intricate subject lies wholly outside the 
scope of this volume. In the present chapter it must suffice 
to point out the general nature of some of the more important 
of the so-called spherical errors. First, however, we must 
see how to trace the path of a single ray through a centered 
system of spherical surfaces before we are in a position to 
study a bundle of rays. 

176. Construction of a Ray Refracted at a Spherical 
Surface. — In § 34 a method was explained for constructing 
the path of a ray refracted from one medium into another, 
which is always applicable to a refracting surface of any form. 



510 



Mirrors, Prisms and Lenses 



[§176 



The following elegant and useful construction of the path of 
a ray refracted at a spherical surface was published in 1807 
by Thomas Young (1773-1829.) 




Fig. 226. — Construction of ray refracted at convex spherical surface (n'>ri). 

In the accompanying diagrams (Figs. 226 to 229) the 
center of the spherical refracting surface ZZ is designated 
by C. The point R is any point on the path of the incident 
ray lying in the first medium of refractive index n. The point 




Fig. 227. — Construction of ray refracted at concave spherical surface (w'>n). 

where the ray meets the spherical refracting surface is marked 
B. The plane of the paper which contains the incident ray 
RB and the incidence-normal BC is the plane of incidence. 



176] 



Young's Construction 



511 



The index of refraction of the second medium is denoted by 
n' and the radius of the spherical refracting surface by r. 
Around C as center and with radii equal to n'.r/n and n.rjn' 




Fig. 228. — Construction of ray refracted at convex spherical surface (n'<n). 

describe, in the plane of incidence, the circular arcs k and k', 
respectively; and let S designate the point where the straight 
line RB, produced if necessary, meets the arc k. Draw the 
straight line CS ntersecting the arc k' in the point S'. Then 




Fig. 229. — Construction of ray refracted at concave spherical surface (n'<n). 

the straight line BT drawn from B through S' will represent 
the path of the refracted ray. In making this construction, 
care must be taken to select for the point S that one of the 



512 Mirrors, Prisms and Lenses [§ 177 

two points in which the straight line RB cuts the circle k 
which will make the segments BS and BS' both fall on the 
same side of the incidence-normal, since the angles of in- 
cidence and refraction are described always in the same 
sense, both clockwise or both counter-clockwise. 

The proof of the construction is simple. Since the radius 
r = BC is a mean proportional between the radii SC=n'.r/n 
and S'C=n.r/n / , that is, since 

CS :CB = CB :CS'-n' :n, 
the triangles CBS and CBS' are similar, and hence Z CBS = 
Z BS'C. In the triangle CBS : 

sinZCBS : sinZBSC = CS : CB=n' : n. 
By the law of refraction: n.sma=n'.sma', where a = 
ZCBS. Consequently, ZBSC = ZCBS'= a', so that the 
straight line BS' is the path of the refracted ray. 

This construction can be employed to trace the path of 
a ray graphically from one surface to the next through a 
centered system of spherical refracting surfaces. 

177. The Aplanatic Points of a Spherical Refracting Sur- 
face. — Incidentally, in connection with the preceding con- 
struction, attention is directed to the singular character of 
all pairs of points such as S, S' determined by the intersec- 
tions of the two concentric auxiliary spherical surfaces with 
any straight line drawn from their common center C. To 
every incident ray directed towards the point S there will 
correspond a refracted ray which will pass ("really" or 
" virtually") through the other point S'; so that in this 
special case we obtain a homocentric bundle of refracted 
rays from a homocentric bundle of incident rays, for all 
values of the aperture-angle of the bundle. Thus, S' is a 
point-image of the object-point S. The distances of S and 
S' from the center C are connected by the invariant-relation: 
w'.CS'=w.CS. 

That pair of these points which lies on the optical axis is 
especially distinguished and called the pair of aplanatic 



§178] 



Spherical Aberration 



513 



points of the spherical refracting surface; they are designated 
by J, J' (Fig. 230). Thus, we have: 

CJ :AC=AC :CJ'=n' :n, 
or 

CJ.CJ' = r 2 , n.CJ=rc'.CJ'. 
The aplanatic points, therefore, he always on the same side 
z 







Fig. 230. — Aplanatic points of spherical refracting surface. 

of the center C so that whereas the rays must pass "really" 
through one of them, they will pass " virtually'' through 
the other. In geometrical language the pdints J, J' are said 
to be harmonically separated (§ 67) by the extremities of 
the axial diameter of the refracting sphere. 

178. Spherical Aberration Along the Axis. — However, in 
general, a homocentric bundle of rays incident on a spheri- 
cal refracting surface will not be homocentric after refrac- 
tion. The diagram (Fig. 231) represents the case of a merid- 
ian section of a bundle of incident rays which are all parallel 
to the axis of a convex spherical refracting surface for which 
n f >n. It will be seen that, whereas the paraxial rays after 
refraction meet on the axis at the second focal point F', the 
outermost or edge rays cross the axis at a point L' between 
the vertex A and the focal point F'; and the intermediate 
rays cross the axis at points lying between F' and I/. The 
segment F'L' is the measure of the spherical aberration along 



514 



Mirrors, Prisms and Lenses 



[§17S 



the axis or the axial aberration of the edge ray of a direct 
cylindrical bundle of incident rays. (By a " direct" bundle 
of rays is meant a bundle of rays emanating from a point on 
the axis.) In the figure this segment is negative, that is, meas- 
ured in the sense opposite to that of the incident light; and 




Fig. 231. — Spherical aberration. 

this effect is usually described by saying that a convex spheri- 
cal refracting surface at which light is refracted from air to 
glass is spherically under-corrected; whereas, under the same 
circumstances, a concave spherical refracting surface will be 
found to be spherically over-corrected, that is, the segment 
F'L' in this case will be positive. In fact, the points of in- 
tersection of pairs of consecutive rays lying in the plane of 
a meridian section of a spherical refracting surface form a 
curved line lying symmetrically above and below the axis, 
if the bundle of incident rays is symmetric with respect to 
the axis; and this plane curve is the so-called caustic curve 
of the meridian rays. The two branches on opposite sides 
of the axis unite in a double point or cusp at the point on 
the axis where the paraxial rays intersect, so that the axis 
is tangent to both branches at this point, which in the figure 



§ 179] Spherical Zones 515 

is the point F'. The system is said to be spherically over- 
corrected or under-corrected according as the cusp is turned 
towards the incident light (<) or away from it (>), respec- 
tively; on the supposition that the incident rays are parallel 
to the axis. Each refracted ray in the meridian plane touches 
the caustic curve, and hence this curve is said to be the geo- 
metrical envelope of the meridian section of the bundle of 
refracted rays. 

If the entire figure is revolved around the optical axis the 
arc ZZ will generate a zone of the spherical refracting sur- 
face containing the vertex A; and each incident ray pro- 
ceeding parallel to the axis will generate a cylindrical sur- 
face, and all the refracted rays corresponding to the incident 
rays which lie on the surface of one of these cylinders will 
intersect in one point lying on the axis between F' and L'. 
The revolution of the caustic curve will generate a caustic 
surface, which will be the enveloping surface of the bundle of 
refracted rays (see § 187.) 

The caustic curve terminates at the point H' where the 
edge ray intersects the next consecutive ray in the meridian 
section. If a plane screen erected at right angles to the axis 
so as to catch the light transmitted by the bundle of refracted 
rays is placed initially in the transversal plane that passes 
through the extreme point H' and then gradually shifted 
parallel to the axis towards the second focal plane, there will 
appear on the screen at first a circular patch of light sur- 
rounded on its outer edge by a brighter ring, which will grad-. 
ually contract as the screen approaches L'. Between L' 
and F' there will be seen at the center of the circular patch 
of light an increasingly bright spot. For a certain position 
G' where the distance of the screen from F' is about three- 
fourths of the length of F'L' the cross-section of the bundle 
of refracted rays will have its narrowest contraction. This 
section is sometimes called the least circle of aberration. 

179. Spherical Zones. — Since, in general, it is not possible 
to abolish the spherical aberration of a single spherical re- 



516 Mirrors, Prisms and Lenses [§ 180 

fracting surface, the only means available is to try to ac- 
complish this result by distributing the duty of refracting 
the rays over a series of surfaces whose curvatures and dis- 
tances apart are so nicely adjusted with respect to each other 
that when the rays finally emerge they will all unite in one 
focus on the axis. Thus, for example, if the incident rays 
are supposed to be parallel to the axis of the system, and if 
the system has been designed so as to be spherically corrected 



Fig. 232. — Graphical representation of the spherical zones of a lens. 

for the edge ray which meets the first surface at the distance 
h from the axis, it is conceivable that all the intermediate 
rays of incidence-heights z (where h > z > 0) might perchance 
emerge from the system along paths which all likewise passed 
through the focal point F'; but practically this never hap- 
pens. If the edge ray intersects the axis at F', an intermedi- 
ate ray of incidence-height z will cross the axis at some other 
point I/, and the segment F'L' is called the spherical aberra- 
tion of the zone of radius z or simply the spherical zone z. The 
spherical zones of a lens may be exhibited graphically by 
plotting a curve whose abscissae are the values of F'L' and 
whose ordinates are the corresponding values of z, as repre- 
sented in Fig. 232. 

180. Trigonometrical Calculation of a Ray Refracted at 
a Spherical Surface. — The diagram (Fig. 233) represents a 
meridian section ZZ of a spherical refracting surface of radius 
r (=AC) separating two media of indices of refraction n, n' . 
A ray RB incident on the surface at B at an angle <x = 
Z NBR = Z CBL crosses the axis at L at a slope-angle 6 = 
ZALB. If the central angle is denoted by </> = ZBCA, 



180] 



Calculation of Refracted Ray 



517 



and if the abscissa of the point L with respect to the center C 
is denoted by c, that is, if c = CL, then in the triangle CBL, 
we have the relations: 

a=d + 4>, c.sin# = — r.sina. 
The path of the corresponding refracted ray is shown by the 
straight line BT which crosses the axis at the point L'; and 







*f 








~~~eT"~- 




A 






C^ 


^N» 


L'^- 




L 


Tl 


\ n» 












- T 



Fig. 233. — Diagram for trigonometrical calculation of refracted ray. 

if we put a' = ZN'BL', 0' = ZAL'B and c'=CL', we ob- 
tain a similar pair of formulae from the triangle CBL', namely: 

a'=d'-\-(j>, c'.sin#'= — r.sina'. 

Accordingly, being given the constants denoted by n, n' and 
r, and the parameters (c, 8) of the incident ray, we can find 
the parameters (c', 8') of the refracted ray by means of the 
following system of equations: 

c . a 

sin a — — - sin 8, 
r 



sin a = —sin a, 
n 



c' = 



8'= 8+ a' -a, 



sin a 
sin 8' 



It is easy to see that if we have given two incident rays 
which both cross the axis at the same point L, so that the 
abscissa c has the same value for both rays while the slope- 
angles 8 are different, different values of c' will, in general, 
be obtained for the abscissae of the points where the two cor- 



518 Mirrors, Prisms and Lenses [§ 180 

responding refracted rays cross the axis. This is the analyti- 
cal statement of the fact of spherical aberration (§ 178). 

The formulae for calculating the path of a ray reflected at 
a spherical mirror may be derived immediately by putting 
n'=—n (§ 75) in the preceding system of equations. Thus 
we find: 

sina = — sin 0, a'= — a, 0'= 0— 2a, c' =r . - - _ . 

r sm(0-2a) 

Incidentally, a number of other useful relations may be 
obtained from Fig. 233. For example, if the distances of the 
points L and L' where the ray crosses the axis before and 
after refraction measured from the incidence-point B are 
denoted by I and V, respectively, that is, if l = BL, Z' = BL', 
where I and V are to be reckoned positive or negative ac- 
cording as these lengths are measured in the same direction 
as the light traverses the ray or in the opposite direction, 
respectively; then 

Z'.sin0' = Z.sin0; 
and, since by the law of refraction, 

n'.c'.sin 0'=ft.c.sin 0, 
we obtain the useful invariant relation : 

n'.c' _n.c 

1 T' 

Moreover, by projecting the two sides c and I of the triangle 
CBL on the third side r, the following formula is obtained : 

r = Lcosa— c.cos0, 
which may be written : 

c_ r /cosa_l\ 
I coscj) \ r If 
Similarly, in the triangle CBL' : 

c'_ r /cosa/_l\ 

Multiplying the first of these equations by n and the second 
by n' and equating the resulting expressions, we find : 



,/cosa' 1\ /cos a 1\ 



§ 181] Path of Ray through Centered System 519 

which may also be written : 

n' n n'. cos a'— ft.cosa ^ , N 
V~T r =0(say); 

or finally: 

L' = L+D, 

where L = n/l, L r = n'll f . 

If the ray is a paraxial ray, we may put cos a = cos o! = 1 
(§ 63) ; and now if we write u, v! in place of I, V , respectively, 
the formula above will reduce to the abscissa-equation for 
the refraction of a paraxial ray at a spherical surface (§ 78). 
Moreover, if in the last formula we put ft' = ft (§ 75), we 
find the corresponding relation for the reflection of a ray at 
a spherical mirror, namely: 

1 1 _ 2cos a 
1 + P~~r~' 
181. Path of Ray through a Centered System of Spher- 
ical Refracting Surfaces. Numerical Calculation. — Using 
the same system of notation as in § 118, we may write the 
formula for the refraction of a paraxial ray at the kth sur- 
face of a centered system of spherical refracting surfaces, 
as follows : 

W = U k +F k , 
where 

U k = n k /u k , U k ' =w k +i/%', and F k = (n k +i-n k )/r k ; 
ft k =A k M k , V = A k M k +i, 7k = A k C k . 
And if d k = A k A k .f i, then also: 

i/^ k +i = i/t/'k-4K+i. 

According to the relations given in § 180, we have the 

following system of formulae for the refraction at the A;th 

surface of a ray whose slope-angles before and after refraction 

have the finite values 6 k = Z A k L k B k and k +i = Z A k L k+ iB k , 

respectively : 

c k . n . . ft k 

sin a k = — — .sin k , sin a k = — — .sin a k , 

sin a k 



#k + i= #k+ k - a k , c k =-r k . 



sin k + 



520 Mirrors, Prisms and Lenses [§ 181 

where c k = C k L k and c k ' = C k L k+ i. Moreover, if we put 

a k = C k C k + i = d k +r k _}_i — r k , 
then 

c k _j_i = c k — a k . 

In order to exhibit the methods of calculations by means of 
these formulae, a comparatively simple numerical illustra- 
tion is appended. The actual example here chosen is one 
given by Dr. Max Lange in his paper entitled " Vereinfachte 
Formeln fiir die trigonometrische Durchrechnung optischer 
Systeme" (Leipzig, 1909), pages 24, foil. The optical sys- 
tem is a two-lens object-glass of a telescope for which the 
data were published by Dr. R. Steinheil in the Zeitschrift 
fiir Instrumentenkunde, xvii (1897), p. 389, as follows: 

Indices of refraction (for D-line) : 

ni = n3 = Ub= l (air); n 2 = 1.614 400 (flint); n 4 =1.518 564 
(crown) . 

Thicknesses: 

di = 2; d 2 = 0.01; d 3 = 5. 

Radii: 

ri=+ 420; r 2 = +181.995; r 3 = +178.710; r 4 = — 40 133.8. 
The incident rays are parallel to the axis, so that 

1 = O, ui = ci= oo (C/i = 0). 
The calculation is divided into two parts, namely: (1) the 
calculation of the paraxial ray, and (2) the trigonometric 
calculation of the edge ray which meets the first surface of 
the object-glass at the height hi =33 above the axis. When 
C\= oo , we find sinai = /ii/ri, which, according to the above 
data, gives lg sin a\ = 1.5185139. This is the starting point 
of the calculation of the edge ray. 

Each vertical column contains the calculation for one 
spherical refracting surface. The sign written after a log- 
arithm indicates the sign of the number to which the loga- 
rithm belongs. Generally the calculations do not have to be 
performed to the degree of accuracy to which they are carried 
here. 



181] 



Scheme of Numerical Calculation 



521 



1. PARAXIAL RAY 





k=l 


k=2 


k=3 


k=4 


lg(ttk+l-ftk) 
clgrk 


9.7884512+ 
7.3767507 + 


9.7884512- 
7.7399406 + 


9.7148D24 + 

7.7478511 + 


9.7148024- 
5.3964898- 


IgFk 


7.1652019 + 


7.5283918- 


7.4626535 + 


5.1112922+ 


Fk 
Uk 


+0.00146286 
0.00000000 


-0.00337592 
+0.00146552 


+0.00290171 
-0.00191036 


+0.00001292 
+0.00099459 


Uk' 


+0.00146286 


-0.00191040 


+0.00099135 


+0.00100751 


clg Uk' 


2.8347972+ 


2.7188755- 


3.0037761 + 


2.9967506 + 


Igdk 

clg nk +i 


0.3010300+ 

9.7919889 + 


8.0000000 + 
0.0000000 + 


0.6989700+ 
9.8185669+ 




lg(dk/nk+i) 


0.0930189 + 


8.0000000 + 


0.5175369 + 




-dk/nk+i 
1/Uk' 


- 1.2389 
+683.5924 


- 0.0100 

- 523.4504 


3.2926 
+ 1008.7327 


« 


1/C/k+i 


+682.3535 


- 523.4604 


+ 1005.4401 




lg Uk+i 


7.1659906+ 


7.2811162- 


6.9976438 + 





lg 1*4' = clg U A ' = 2.9967506 + ; < = A 4 F , = +992.546 

clg (Ui , .U 2 , .U 3 , .Ua , ) = 1.5541994— 
lg (U 2 .U 3 .Ud = 1. 4447506— 



lg/= 2.9989500 + 
/= +997.585 



2. EDGE RAY 





k=l 


k=2 


A:=3 


k=4 


— flk-l 
Ck'-l 


8.8952646 + 
9.7919889 + 


+236.0050 

+682.2850 


+ 3.2750 
-685.6727 


+40307.51 
+ 1353.49 


ck 


+918.2900 


-682.3977 


+41661.00 


IgCk 

lg sin#k 
clgrk 


2.9629799 + 
8.4765370- 
7.7399405+ 


2.8340376- 
8.8114112+ 

7.7478511 + 


4.6197297 + 
8.3325613- 

5.3964898- 


lg sinak 
lg nkfnk +i 


9.1794574+ 
0.2080111 + 


9.3932999+ 
9.8185669+ 


8.3487808- 
0.1814331 + 


lg sinak' 
lg^k 

clg sin0k +i 


8.6872535 + 
2.6232493+ 
1.5234630- 


9.3874685 + 
2.2600595+ 

1.1885888+ 


9.2118668+ 
2.2521489 + 
1.6674387- 


8.5302139 - 
4.6035102- 
1.4803948- 


lgck' 


2.8339658+ 


2.8361168- 


3.1314544+ 


4.61411^9 + 



522 Mirrors, Prisms and Lenses [§ 182 

_ ai = - 4° 30' 23.24" c 4 ' = +41126.23 
ai' = + 2° 47' 22.69" r 4 = -40133.80 
1 = O, ai'-ai= 2 =- 1° 43' 0.55"A 4 L 5 = + 992.43 
-a 2 = - 8° 41' 40.45" -i* 4 '= - 992.55 
- 10° 24' 41.00" F'L 5 = - 0.12 
a 2 ' = + 14° T 31.28" 
03 = + 3 42' 50.28" 
- a3= -14° 19' 13.26" 

-10° 36' 22.98" 
a 3 '= + 9° 22' 26.69" 

4 =- 1° 13' 56.29" 
-a 4 = + 1° 16' 45.13" 

+ 0° 2' 48.84" 
a 4 '= - 1° 56' 33.95" 

O = - 1° 53' 45.11" 

Thus, we see that this object-glass has a slight spherical 
aberration of —0.12, that is, it is a little under-corrected 
(§ 178). 

182. The Sine-Condition or Condition of Aplanatism. — 
Suppose that for a certain object-point M (Fig. 234) on the 
axis of a symmetrical optical instrument the spherical aber- 
ration has been abolished for all the zones of the system, so 
that rays proceeding from this point will all be accurately 
focused at the conjugate image-point M'. On a straight 
line perpendicular to the axis at M take a point Q very close 
to M; and let ?/' = M'Q' denote the size of the image of the 
object 2/ = MQ which is produced by the central zone, that is, 
by the paraxial rays. Now even though .the system is spher- 
ically corrected with respect to the pair of axial points M, M', 
it by no means follows that rays emanating from Q will all 
meet again in Q'. In order that this shall be the case, the 
magnification-ratio must be equal to y'/y for all the zones 
of the system. Draw the object-ray MBi and the corre- 
sponding image-ray B 2 M'; if the slopes of these rays are 



§182] 



Sine-Condition 



523 



denoted by 6 and 0', it may be shown that for the zone 
corresponding to the incidence-point Bi the magnification- 
ratio is equal to n.sin d/n'.s'n 6'; and if this is equal to y'jy, 
then the image formed by rays belonging to this zone will 
be of the same size as the image y' made by the paraxial rays. 




Fig. 234. — Sine-condition. 

Thus, in order that with the employment of wide-angle 
bundles of rays a symmetrical optical instrument may pro- 
duce a sharp image of a little plane element perpendicular 
to the axis of the instrument, not only must the system be 
spherically corrected for the pair of conjugate axial points 
M, M', but it must also satisfy the so-called Sine-Condition, 
namely, 

n.sin 6 _y f _ 
n'. sin d' y 
This celebrated principle was clearly formulated by Abbe 
in 1873, but it had already been recognized by Seidel, and 
it may be deduced from a general law of radiant energy which 
was first given by Clausius (1864). The proof of it must be 
omitted here. It may be stated in words as follows: The 
necessary and sufficient condition that all the zones of a 
spherically corrected system shall produce images of equal 
size at the point M' conjugate to the axial point M is that, 
for all rays proceeding from M, the ratio of the sines of the 
slope-angles 6, 6' of each pair of corresponding incident 
and emergent rays shall be constant ; that is, 

sin d n' wm , , 

- — tt. = — . y = constant, 
sin n 



524 Mirrors, Prisms and Lenses [§ 182 

The sine-condition 

n.y. sin 6 = n'.y f . sin 0' 
is essentially different from the Smith-Helmholtz law for 
paraxial rays (see §88 and §118), namely, n.?/.tan0 = 
n'.i/'.tan 6 f , although when the angles 6, 6' are small, both 
conditions may be expressed by the equation n.y. d = n f .y r . 6 r . 

If the optical system is spherically corrected for the pair 
of axial points M, M', and if at the same time the sine-condi- 
tion is satisfied, the points M, M' are called the aplanatic 
pair of points of the system. It may be demonstrated that 
no optical system can have more than one pair of such apla- 
natic points. In the case of a single spherical refracting sur- 
face the two points J, J' (§ 177) whose distances from the 
center C are such that 

CJ.CJ' = r 2 , n.CJ=n'.CJ', 
are a pair of aplanatic points as above defined; for they are 
free from spherical aberration and if they are joined by 
straight lines BJ, BJ' with any point B on the spherical re- 
fracting surface, and if we put 6 = Z CJB, d' = Z CJ'B, we 
have sin 0/sin d' = nfn' = constant. This property of the 
points J, J' of a refracting sphere has been ingeniously util- 
ized in the construction of the objective of the compound 
microscope. 

If in Fig. 234 we put Z = BiM, then smd=—h/l, where h 
denotes the height of the point Bi above the axis. Hence, 
the sine-condition may be written : 



or since (§ 124) 



where /, f denote the focal lengths of the system and x de- 
notes the abscissa of M with respect to the primary focal 
point F (x = FM), we obtain also: 

h _lf 
sin 6' x 



I. 


h 

sin0' 


n 


-y; 


V 


=/ = 


n 


f 


X 


~~n' 


' x 1 



§ 183] Caustic Surfaces 525 

Suppose now that the object-point M is infinitely distant so 
that x = l— oo ; then for a ray parallel to the axis meeting the 
first surface at the height h, we shall have: 

sin 0' J ' 
Thus, if the aplanatic points are the infinitely distant point 
of the axis and the second focal point F', and if around F' as 
center we describe a sphere of radius equal to /', the parallel 
object-rays will meet their corresponding image-rays on the 
surface of this sphere; whereas in the case of collinear imagery 
with paraxial rays the points of intersection of the incident 
and emergent rays under the same circumstances will all lie 
in the secondary principal plane (§ 119), which touches the 
sphere above mentioned at the point where the axis crosses it. 
If therefore we put h/sin 6' = e, the sine-condition for an 
infinitely distant object is e+f=0. For example, in the 
case of the telescope objective calculated in § 181: 

lg hi = 1.5185139 + 
clg sin 05=1 .4803948- 

lge =2.9989087- e= -997.490 
/ = +997.5 85 
e+f= + 0.095 
Accordingly, the sine-condition is very nearly satisfied in the 
case of this object-glass. 

183. Caustic Surfaces. — The characteristic geometrical 
property of a bundle of light-rays emanating originally from 
a point-source is expressed in a law announced by Maltjs in 
1808 (§ 39), which may be stated in terms of the undulatory 
theory of light as follows : The rays of light are always normal 
to the wave-surfaces. In fact, what is meant by a wave-sur- 
face is any surface which cuts the rays orthogonally. In 
general, the curvatures of the normal sections at any point of 
a curved surface will vary from one azimuth to another; but, 
according to Euler's theorem (§ 111), the normal sections of 
greatest and least curvature, called the principal sections of 



526 Mirrors, Prisms and Lenses [§ 184 

the surface at the point in question, are always at right 
angles to each other. It is well known that the normal drawn 
to any point of a curved surface will not meet the normal at 
a consecutive point taken arbitrarily. But if the consecutive 
point is taken in the direction of either of the principal sec- 
tions, the two consecutive normals will intersect. Thus, 
along each normal to a curved surface there are two points 
called the principal centers of curvature (§ 111), where con- 
secutive normals lying in the two principal sections intersect. 

Accordingly, if we regard a bundle of rays of light as a sys- 
tem of normals to the wave-surface, we may say that each 
ray determines two principal sections of the bundle, and that, 
in general, there will be two points on the ray, the so-called 
image-points (cf. § 113), where contiguous rays in each of the 
two principal sections intersect the ray in question. The as- 
semblage of these pairs of image-points on all the rays of a 
wide-angle bundle of rays emanating originally from a 
point-source form a surface of two sheets called the caustic 
surface (cf. § 42). Each ray of the bundle is tangent to both 
sheets of the caustic surface. In the special case when the 
bundle of rays is symmetrical about an axis, one sheet of 
the caustic surface will be a surface of revolution, whereas 
the other sheet will be a portion of the axis of symmetry (see 
§ 178). 

184. Meridian and Sagittal Sections of a Narrow Bundle 
of Rays before and after Refraction at a Spherical Surface 
— The apertures of the bundles of effective rays which are 
transmitted through a symmetrical optical instrument are 
all limited by the position and dimensions of the aperture- 
stop (§ 134). For the present it will be assumed that the 
diameter of the stop is very small. Each point of the object 
lying in the field of view is the source of a narrow bundle of 
rays which contains one ray, called the chief ray (§ 140), 
which in traversing the medium where the stop is placed, 
passes through the center of the stop. Accordingly, the chief 
ray will he in the meridian plane determined by the object- 



§184] 



Astigmatism of Oblique Bundle 



527 



point where the bundle of rays originates. The path of this 
chief ray may be traced geometrically by Young's construc- 
tion (§ 176) or it may be calculated trigonometrically by 
means of the system of formula? given in § 181. We have 
now to investigate the positions on this chief ray of the two 
image-points produced by the intersections of this ray with 
the rays immediately adjacent to it lying in the two prin- 
cipal sections of the bundle as determined by its chief ray 
(§ 183). Whenever a narrow bundle of rays has two such 
image-points, it is said to be a tigmatic. Practically, this is 
always the case if the chief ray is incident on a refracting 
surface at an angle a which is not vanishingly small. Under 
such conditions the bundle of refracted rays will be astig- 
matic, and we have the case which some writers call " astig- 
matism by incidence" but which is better described as the 
astigmatism of an oblique bundle of rays, as distinguished from 




Fig. 235. — Meridian section of narrow bundle of rays refracted at spherical 

surface. 



the astigmatism produced by direct (normal) incidence on 
an astigmatic refracting surface or surface of double curva- 
ture (Chapter IX). 

In the diagrams (Figs. 235, 236) , which show the meridian 
section ZZ of a spherical refracting surface whose center is at 
C and vertex at A (Fig. 235), the point designated by P (or Q) 
represents an object-point which is the source of a narrow 



528 



Mirrors, Prisms and Lenses 



184 



homocentric bundle of rays whose chief ray PB (or QB) is 
incident on the surface at the point B at the angle of inci- 
dence a. This ray crosses the axis at the point marked L 
in Fig. 235 and the corresponding refracted ray crosses the 




Fig. 236. — Sagittal section of narrow bundle of rays refracted at spherical 

surface. 

axis at I/. One of the principal sections of the bundle of in- 
cident rays will be the meridian section (§§ 112, 113) made by 
the plane containing the optical axis and the vertex P (or Q) 
of the bundle, that is the plane of the paper; whereas the 
other principal section, called the sagittal section (Fig. 236), 
is made by a plane which intersects the meridian plane 
at right angles along the chief ray of the bundle. The point 
G (Fig. 235) is a point on the spherical refracting surface in 
the meridian section, taken exceedingly close to the point B. 
Likewise, the point D (Fig. 236) lies on the spherical refract- 
ing surface very near to B ; but it is contained in the sagittal 
section and is represented in the diagram as lying slightly 
above the plane of the paper. The ray PG (Fig. 235) after 
refraction meets the chief refracted ray at the image-point 
P' of the narrow pencil of refracted meridian rays. Similarly, 
the ray QD (Fig. 236) after refraction will meet the chief 
refracted ray at the image-point Q' where the straight line 
QC intersects this ray, as will be immediately obvious by 



§ 185] Sagittal Section of Narrow Bundle 529 

supposing that the triangle QBQ' is revolved around the 
central line QQ' as axis through a small angle out from the 
plane of the paper. Thus, whereas the meridian section of 
the bundle of refracted rays is contained in the same plane as 
the meridian section of the bundle of incident rays, the sagit- 
tal sections are in two different planes BDQ and BDQ' (Fig. 
236) which intersect each other in a straight line perpendic- 
ular to the meridian plane at the point B, that is, in the line 
BD, which, since the point D is infinitely near to B, may be 
regarded as a straight line. 

185. Formula for Locating the Position of the Image- 
Point Q' of a Pencil of Sagittal Rays Refracted at a Spher- 
ical Surface. — As was explained (§ 184), the image-point Q' 
(Fig. 236) in the sagittal section corresponding to the object- 
point Q is at the point of intersection of the straight line QC 
with the chief ray of the bundle of refracted rays. This con- 
struction suggests at once a method of obtaining an analyt- 
ical relation connecting the points Q and Q'; for if the straight 
line QQ' is regarded for the time being as the axis of the spher- 
ical refracting surface, and if we put g = BQ, q' = BQ' (where 
the distances denoted by q, q' are to be reckoned positive or 
negative according as they are measured from the incidence- 
point B in the same direction as the light takes along the 
chief ray or in the opposite direction, respectively), we have 
merely to write q, q r in place of the symbols I, V in the formula 
derived in § 180 in order to obtain the desired relation, 
namely, 

q' q 

where the function denoted here by D is a constant for a 
given chief ray and is defined by the following expression: 
n _w'.cosa/— n.cosa_n.sin(a — a') 
r r.sin a' 

Thus having ascertained the path of the chief ray, and know- 
ing the position of the object-point Q, that is, being given 



530 Mirrors, Prisms and Lenses [§ 186 

the value of q, we may calculate the value of q' by means of 
the above formula and thus locate the position of the image- 
point Q' of the sagittal section of the bundle of refracted rays. 
186. Position of the Image-Point P' of a Pencil of Me- 
ridian Rays Refracted at a Spherical Surface. — The angles 
of incidence and refraction of the chief ray are denoted by 

a, a', respectively. Moreover, let 6, d f (Fig. 235) de- 
note the angles which the chief ray makes with the axis of 
the spherical refracting surface before and after refraction, 
respectively; and also let the central angle BCA be denoted 
by 4>. Then for a contiguous ray in the meridian section 
which is incident at the point G very close to the point B, 
these angles may be denoted by a+da, a'+da'; d+dd, 

B'+dd'; and <j)+d<j), where da, da', etc., denote the 
little increments in the magnitudes of the angles a, a', etc., 
in passing from the chief ray to an adjacent ray in the merid- 
ian section. Now since for the rays PB and PG these angles 
are connected by the formulae (§ 180); 

a= d+4>, a+da= d+dd+ct>+d<j), 
we obtain by subtraction: 

da = dd+d<f). 
Around P as center and with radius equal to PB describe the 
small arc BU which subtends ZBPG = <i0; so that we may 
write: 

, n arc BU 

a u = , 

V 

where p = BP denotes the distance of the object-point P from 
the incidence-point B, being reckoned positive or negative 
exactly in the same way as q in § 185. Now the sides of the 
little curvilinear triangle BGU may be considered as straight 
to the degree of approximation with which we are concerned 
at present, and since the sides of the angle GBU are per- 
pendicular to the sides of the angle of incidence a, so that 
Z GBU = a, we obtain : 

arc BU = arc GB.cosa. 



§ 186] Meridian Section of Narrow Bundle 531 

Combining this relation with the one above, we have there- 
fore : 

in arc GB. cos a 

dd = — . 

V 
Moreover, since Z GCB = dcf>, 

, , arc GB 
d<f> = ——-; 

and, therefore, by adding this equation to the last and taking 
account of the relation above, we find : 

da- g-2!^. aw OB. (1) 

Similarly, for the corresponding refracted rays BP' and 
GP' which intersect at the image-point P', for which BP' = 
p', we can derive the analogous relation : 

da'=g-^).awGB. (2) 

Now according to the law of refraction, 

ft.sina = ft'.sina', n.sin(a+da) = n' '.sin( a' +d a') , 
and if in the expansions of sin(a+<ia) and sm(a'+da') 
we write da and da' in place of sinda and sinda' and put 
cosd a = cosd a' = 1, as is permissible on account of the small- 
ness of these angles, we may derive the following relation 
between d a and d a' : 

nxosa.da = n'.cosa'.da'. (3) 

Hence, multiplying equation (1) by n.cosa and equation (2) 
by n'.cos a', and equating the two expressions thus obtained, 
according to equation (3), we find, after removing the com- 
mon factor, arc GB, the following formula connecting the 
ray-intercepts p and p' : 

, , /cos a' \\ /cos a 1\ 

n.cosa ( — — — - ) =n.cosa( — -) : 

V p r! \ p rl 

which may also be written thus: 

n'. cos 2 a/_ft.cos 2 a _ ~ 

— — P u ' 

where the symbol D has the same meaning as before in § 185. 
If we introduce Abbe's differential notation and use the 



532 



Mirrors, Prisms and Lenses 



186 



operator A placed in front of a symbol to denote the differ- 
ence in the value of the magnitude denoted by the symbol 
before and after refraction, that is, for example, if £±z = z' — z; 
then we may write the two formulae for p and q in the 
following abbreviated form: 

\ n _ A^-cos 2 a_ n 
q~ p 

The position of the image-point P' of the meridian section 
of a narrow bundle of rays refracted at a spherical surface 
may also be quickly ascertained by a simple geometrical 




Fig. 237. — Construction of center of perspective (K) with respect to a given 
ray refracted at a spherical surface. 

construction which depends on finding a point K called the 
center of perspective, which bears precisely the same relation 
to the pair of points P, P' as the center C of the spherical 
surface bears to the pair of points Q, Q' (§ 184) ; that is, just 
as the straight line QQ' must pass through C, so also the 
straight line PP' must pass through K. The existence of 
this point K was first recognized by Thomas Young (1801). 
In the diagram (Fig. 237) the chief incident ray is represented 
by the straight fine RB and the chief refracted ray, con- 
structed by the method given in § 175, is represented by the 
straight fine BT. From the center C draw CY and CY' 
perpendicular to RB and BT at Y and Y', respectively. The 



§ 187] Astigmatic Difference 533 

point K will be found to lie at the point of intersection of the 
straight lines YY' and SS'; and hence if P designates the 
position of an object-point lying anywhere on the chief in- 
cident ray, the corresponding image-point P' in the meridian 
section will lie at the point where the straight line PK meets 
the chief refracted ray. This beautiful construction is ex- 
ceedingly useful in graphical methods of investigating the 
imagery in the meridian section along a particular ray. The 
proof of the construction is not at all difficult, but it cannot 
be conveniently given here. 

187. Measure of the Astigmatism of a Narrow Bundle of 
Rays. — We have seen that, in general, a narrow homocentric 
bundle of rays falling obliquely on a spherical refracting 
surface is transformed into an astigmatic bundle of refracted 
rays, so that corresponding to a given object-point P (or Q) 
there will be two so-called image-points P' and Q' lying on 
the refracted chief ray at the points of intersection of the 
rays of the meridian and sagittal sections, respectively. The 
interval between these image-points, that is, the segment 
P'Q' = q' — p' is called the astigmatic difference. However, it 
is more convenient to measure the astigmatism by the dif- 
ference between the reciprocals of the linear magnitudes p' 
and q'. If, for example, according to the system of notation 
introduced in § 106, we put 

n/p=P, rt/p'=P', n/q = Q, n'jq'^Q', 
the formulae derived in §§ 185, 186 may be written as follows: 

Q'-Q = P'.cos 2 a'-P.cos 2 a = D; 
where, on the assumption that the meter is taken as the unit 
of length, the magnitudes denoted by the capital letters will 
all be expressed in terms of the dioptry. The astigmatism 
of the bundle of refracted rays is measured by (P f — Q r ) . If 
the bundle of incident rays is homocentric (Q = P), the as- 
tigmatism of the bundle of refracted rays will be : 
P'-Q'= P'.sin 2 a' - P.sin 2 a. 

Accordingly, we see that the astigmatism of a bundle of 
rays refracted at a spherical surface will vanish provided 



534 Mirrors, Prisms and Lenses [§ 188 

Q = P and P'.sin 2 a' — P.sin 2 a = 0; which may happen in 
two ways, as follows: 

(1) If a' = a = 0, that is, if the chief ray of the narrow 
bundle meets the refracting surface normally, as, for ex- 
ample, when it is directed along the axis, then no matter 
where the object-point may lie, the two image-points will 
coincide. In fact, in case of the axial ray we may put Q = 
P=U, Q' = P f =U f , D = F, so that the formulae for the me- 
ridian and sagittal sections both reduce in this case to the 
fundamental equation for the refraction of paraxial rays at 
a spherical surface, namely, U' = U+F. 

(2) But for any value of a, we shall have P' — Q' = 0, 
that is, P'.sin 2 a' = P.sin 2 a, provided P7n' 2 = P/n 2 or 
n'.p' = n.p. In this case the points designated by P, P' 
(or Q, Q') are identical with the points S, S' in Figs. 226 to 
229. If the vertex of the homocentric bundle of incident rays 
lies at any point S on the surface of the sphere described 
around C as center with radius equal to n'.rjn, the bundle 
of refracted rays will likewise be homocentric with its ver- 
tex at the corresponding point S' on the surface of the con- 
centric sphere of radius n.r/n' (see § 177). 

188. Image-Lines (or Focal Lines) of a Narrow Astig- 
matic Bundle of Rays. — In all the preceding discussion of 
the properties of an astigmatic bundle of rays, it cannot have 
escaped notice that only such rays have been considered as 
are contained in the two principal sections of the bundle. If 
there were no other rays to be taken into account besides 
these, we might say that to each point of the object P (or Q) 
there corresponded two image-points P' and Q'. But this is 
by no means a complete or even approximately complete 
statement of the image-phenomenon in this case; for, indeed, 
the rays which he in neither of the two principal sections do, 
as a matter of fact, constitute by far the greater portion of 
the total number of rays of the bundle. According to the 
theorem of Sturm (1803-1855), the constitution of a narrow 
bundle of rays is exhibited in the accompanying diagram 



188] 



Sturm's Conoid 



535 



(Fig. 238) called Sturm's conoid (§ 113). All the rays of the 
bundle pass through two very short focal lines or image-lines 
XX and YY which are both perpendicular to the chief ray. 
The image-line XX which goes through the point of intersec- 
tion P' of the meridian rays lies in the plane of the sagittal 
section; and, similarly, the image-line YY which goes through 
the point of intersection Q' of the sagittal rays lies in the 




Fig. 238. — Sturm's conoid. 

plane of the meridian section. Strictly speaking, this theo- 
rem can be regarded as representing the actual facts only on 
the assumption that the bundle of rays is infinitely thin ; and 
on this assumption the entire bundle may be conceived as 
generated by a slight rotation either of the meridian section 
around the image-line YY as axis, whereby the point P' will 
trace the image-line XX, or of the sagittal section around 
the image-line XX as axis, whereby the point Q' will trace 
the image-line YY. Thus, according to Sturm's theorem, 
with an object-point P (or Q) lying on the chief ray of an 
infinitely narrow bundle of incident rays there are associated 
two exceedingly tiny image-lines lying in the principal sec- 
tions of the bundle of refracted rays at right angles to the 
chief ray. Not only as to the orientation of the image-lines 
of Sturm, but as to their practical, nay, even as to their 
mathematical existence, there has been much controversy, 
but we cannot enter into this discussion here. In spite of 
its limitations and admittedly imperfect representation, 
Sturm's conoid remains a very useful preliminary mode of 
conception of the character of a narrow astigmatic bundle 



536 



Mirrors, Prisms and Lenses 



[§189 



of rays. The only proper way of arriving at a more accurate 
knowledge of the constitution of a bundle of light-rays is by 
the aid of the powerful methods of the infinitesimal geom- 
etry. Mathematical investigations of this kind have been 
pursued with great skill by Gullstrand whose writings con- 
tained in a series of published papers and treatises dating 
from about 1890 have extended the domain of theoretical 
optics far beyond the narrow limits imposed upon it by 
Gauss and the earlier writers on this subject. 

189. The Astigmatic Image-Surfaces. — Thus, the effect 
of astigmatism is that the rays of a narrow oblique bundle, 
instead of being brought to a focus at a single point, pass 
through two small focal lines at right angles to the path of 



ccv 



oo 




oc- 



cc- 



Fig. 239. — Astigmatic image-surfaces. 



the chief ray in the image-space. If the chief rays proceeding 
from the various object-points lying in a meridian plane of a 
symmetrical optical instrument are constructed, and if along 
each of these rays the positions of the image-points P', Q' of 
the pencils of meridian and sagittal rays are determined, the 
loci of these points will be two curved lines, both symmetri- 
cal with respect to the axis, which touch each other at their 
common vertex on the axis. In the diagram (Fig. 239) the 
object is supposed to be infinitely distant, as, for example, 
in the case of a landscape photographic lens. The contin- 



§ 189] Astigmatic Image-Surfaces 537 

uous curved line represents the locus of the points of inter- 
section of the sagittal rays, whereas the dotted curve repre- 
sents the locus of the points of intersection of the meridian 
rays. These curved lines are the traces in the meridian plane 
of the two astigmatic image surfaces which are generated by 
revolving the figure around the axis of symmetry. The two 
image-surfaces which correspond to a definite transversal 
plane in the object-space, and which are the loci of the most 
sharply defined images of object-points lying in this plane, are 
not to be confused with the two sheets of the caustic surface 
of a wide-angle bundle of rays emanating from a single point 
of an object (§ 183). The focal lines of the narrow pencils of 
meridian rays lie on one of these surfaces and the focal lines 
of the narrow pencils of sagittal rays lie on the other surface. 
The positions and forms of the image-surfaces will depend 
essentially on the place of the stop; for it is evident that if 
the stop is shifted to a different place, the chief ray of each 
bundle (§§ 140, 184) will be a different ray, and the points 
P' and Q' will all occupy entirely different positions. If a 
curved screen could be exactly adjusted to fit one of the 
image-surfaces, a fairly sharp image of the object might be 
focused on it, but not only would the image be curved in- 
stead of flat, but there would also be a certain astigmatic 
deformation due to the fact that each point of the object 
would be reproduced not by a point but by a little focal line, 
as has been explained. Between the two image-points P' and 
Q' on each chief ray there lies a certain approximately circu- 
lar cross-section of the narrow astigmatic bundle known 
(§ 113) as the "circle of least confusion," and the locus of 
the centers of these circles will lie on a third surface interme- 
diate between the other two, which is sometimes taken as a 
kind of average or compromise image-surface. 

There can be no doubt that astigmatism of oblique bundles 
is responsible for serious defects in the image produced by 
an optical instrument, and much pains has been bestowed on 
trying to remedy this fault as far as possible. Fortunately, 



538 



Mirrors, Prisms and Lenses 



189 



-5 




+2 



the possibility of abolishing astigmatism of this kind, that 
is, of making the two image-surfaces coincide in a single 
surface, is afforded by the fact that the astigmatic difference 
(§ 187) is opposite in sign according as the refracting surface 
is convergent or divergent. For example, Fig. 240 shows 

graphically the opposite 
effects of a convergent 
and a divergent spheri- 
cal refracting surface 
under otherwise equal 
conditions. The two 
curves on the left-hand 
side relate to the con- 
vergent system, and the 

Fig. 240. — Astigmatism of convergent two Curves On the right- 
spherical refracting surface (plotted , , •-, -, , , ,, . 
on the left) and astigmatism of diverg- nana - siae relate tO tUe 
ent spherical refracting surface (plotted divergent System J and 
on the right). ,1 , , -, 

we see that not only are 
the curvatures opposite in the two cases, but the relative 
positions of the curves are different. It will not be difficult 
to understand that it may be possible, by suitable choice of 
the radii of the refracting surfaces and of their distances 
apart and also of the position of the stop, to design a system 
which will be free from astigmatism at any rate for a certain 
zone of the lens; so that, although we may not be able to 
make the two astigmatic image-curves coincide absolutely 
throughout their entire extent, we may contrive so that the 
two curves are nowhere very far apart, while at one point, 
corresponding to the corrected zone, they actually intersect 
each other. 

190. Curvature of the Image — Now let us suppose that 
the astigmatism of oblique bundles has been completely 
abolished for a certain angular extent of the field of view, so 
that at last there is strict point-to-point correspondence by 
means of narrow bundles of rays between object and image. 
The two image-surfaces have thus been merged into one, and 



190] 



Curvature of Image 



539 



over this surface, within the assigned limits, the definition 
of the image is clear-cut and distinct. There still remains, 
however, another trouble due to the fact that the image is 
curved and not flat; consequently, if the image is received 
on a plane focusing screen, only those parts of the stigmatic 
image which lie in the plane of the screen will be in focus 
(Fig. 241), whereas the. rest of the image on the screen will 
be blurred. 
Now this error of the curvature of the image cannot be 



Stigmatic 

Surtax 



Focus 
Scrc«U 




Fig. 241. — Curvature of stigmatic image. 

overcome by employing methods similar to those above de- 
scribed for the abolition of astigmatism. For the correction 
of the latter error the particular kinds of glass of which the 
lenses were made were not essential; whereas with unsuitable 
kinds of glass there is no choice of the radii, thicknesses, etc., 
which will yield an image which is at the same time stig- 
matic and flat. This fact was well known to Petzval (1807- 
1891). Petzval's formula (published in 1843) for the abo- 
lition of the curvature of a stigmatic image produced by a 
system of infinitely thin lenses in contact with each other is 

F 



540 Mirrors, Prisms and Lenses [§ 191 

where Fj denotes the refracting power and n x denotes the 
index of refraction of the ith lens of the system. This 
formula is equivalent also to the following statement: The 
curvature of the stigmatic image of an infinitely distant 
object in a system of lenses whose total thickness is negli- 
gible is equal to 

- 2 (refracting powers of all lenses of index n) \ 
n J 

The general principal of this equation was discovered by Airy 
and was given by Coddington in his treatise published in 
1829. Seidel pointed out that the two faults of astigmatism 
and curvature could not both be corrected at the same time 
unless some of the convex lenses of the system were made of 
more highly refracting glass than the con- 
cave lenses. Now with the varieties of 
pf glass which were available before the 
production of the modern Jena glass, 
this requirement was directly opposed to 
the condition of achromatism, and as the 
latter error was considered more serious 
than the curvature-error, the earlier lens- 



Ff designers made no attempt to obtain a 

Fig. 242. — stigmatic g a ^ stigmatic image. But with the new 

image in trans- . . _ - _ .. . ., . 

versai focal plane kinds of glass now at our disposal, It IS 
for a given zone possible to design the optical system so 
that not only is the astigmatism corrected 
for a certain zone, as explained in § 189, but the point of in- 
tersection of the two image-lines lies in the same transversal 
plane as the axial point where the two image-lines touch 
each other (Fig. 242). Accordingly, we may say that for 
this zone the image is both flat and stigmatic. The construc- 
tion of modern photographic lenses which are practically 
free from these spherical errors is an almost unsurpassed 
triumph of human ingenuity. 

191. Coma. — Astigmatism implies that the bundles of 
rays concerned in producing the image are very narrow, and 



§ 191] Symmetry in Sagittal Section 541 

this means that the diameter of the stop is very small. But 
the validity of the assumptions which are at the foundation 
of geometrical optics begins to be caUed in question in the 
case of narrow bundles of rays, as was pointed out in § 175; 
so that we must be careful here not to push our conclusions 
too far. As a matter of fact, in various optical instruments 
and particularly in some modern types of photographic 
lenses, the diameter of the stop is by no means small and the 




Fig. 243. — Symmetrical character of sagittal section. 

field of view is extensive. The spherical aberrations which 
are encountered in an optical system of this kind are of an 
exceedingly complicated nature which cannot be described 
here in detail. 

A bundle of rays of finite aperture emanating from a point 
outside the optical axis will show aberrations of a general 
character similar to the aberrations along the axis of a direct 
bundle of rays (§ 178). But the effects in the two principal 
sections of the bundle will be very different from each other; 
because, whereas the rays in the sagittal section, being sym- 
metrically situated on opposite sides of the meridian plane, 
are therefore symmetrical with respect to the chief ray, as 
represented in Fig. 243, there will, in general, be a complete 
absence of symmetry in the meridian section (Fig. 244) . The 
image (if indeed we may continue to use this term) of an 
extra-axial object-point under such circumstances will be 
at best an element of one or other of the two sheets of the 
caustic surface. Usually, however, what is called the image 



542 



Mirrors, Prisms and Lenses 



[§191 



is the light-effect as obtained on a focusing screen placed at 
right angles to the axis at the place where the central parts of 
the object are best delineated. The appearance on the screen 
may be described as a kind of balloon-shaped flare of light, 




Fig. 244. 



-Unsymmetrical character of meridian section, 
coma. 



giving rise to 



with a bright nucleus growing fainter as it expands in some 
cases towards, in other cases away from, the axis. This de- 
fect of the image is known to practical opticians as side-flare 
or coma (from the Greek word meaning "hair" from which 
the word "comet" is likewise indirectly derived). The def- 
inition in the outer parts of the field of the object-glass of a 
telescope depends on the removal of this error; and this ap- 
plies also to the case of a wide-angle photographic lens. The 
only way to obtain a really clear and accurate conception of 
this important spherical aberration is to study the forms of 
the two sheets of the caustic surface. Generally speaking, 
we may say that the convergence of wide-angle bundles of 
rays will be better in the case of an optical system which has 
been corrected for astigmatism, but even then there will be 
lack of symmetry in all the sections of a bundle of rays ex- 
cept in the sagittal section. If the slope of the chief ray is 
comparatively slight, although not negligible, the condition 
of a sharp focus is equivalent to Abbe's sine-condition (§ 182). 
But for greater inclinations of the chief rays, it will generally 
be necessary to resort to the exact methods of trigonometri- 



192] 



Distortion 



543 



cal calculation of the ray-paths in order to determine the 
nature and degree of the convergence. 

192. Distortion; Condition of Orthoscopy. — Let us assume 
that the system has been corrected for both astigmatism and 
curvature of the image, in the sense explained in § 190; so 
that by means of narrow bundles of rays a flat stigmatic 
image is obtained of a plane object placed at right angles to 
the axis. The next question will be to inquire whether the 
image is a faithful reproduction of the object or whether it is 
distorted. If the image in the " screen-plane " (§ 134) is 
geometrically similar to the object-relief projected from the 
center of the entrance-pupil on the " focus-plane " (§ 141), 
then we may say that the optical system is orthoscopic or 
free from distortion. 

The dissimilarity which may exist between an object and 
its image is a fault of an essentially different kind from those 
which have been previously considered, and there is no in- 




Focus - Piano Screen- Plane 

Fig. 245. — Condition of orthoscopy (freedom from distortion) . 

timate connection between this defect and the others. Here 
we are not concerned so much with the quality and defini- 
tion of the image on the screen as with the positions of the 
points where the chief rays cross the screen-plane. The po- 
sitions of these representative points will not be altered by 
reducing the stop-opening (§§ 141, 142); and accordingly the 
image in the screen-plane is to be regarded merely as a cen- 
tral projection on this plane along the chief rays proceeding 
from the center of the exit-pupil. 



544 



Mirrors, Prisms and Lenses 



[§192 



In the diagram (Fig. 245) the centers of the entrance-pupil 
and exit-pupil of the optical system are designated by and 
0'. The straight lines PO, P'O' represent the path of a chief 
ray which crosses the focus-plane in the object-space at the 
point P and the screen-plane in the image-space at the point 
P'. If ?/ = MP, s/' = M'P' denote the distances of P, P' from 
the axis, then the condition that the image in the screen- 
plane shall be similar to the projected object in the focus- 




t 



a 



Fig. 246. 



-Object (a) reproduced by image (b) barrel-shaped distortion 
or by image (c) cushion-shaped distortion. 



plane, that is, the condition of orthoscopy (freedom from 
distortion) is that the ratio y'/y shall have a constant value 
for all values of y within the limits of the field of view. If, 
on the contrary, this is not the case, and if the ratio y'/y is 
variable for different values of y, then the image will be dis- 
torted ; and this distortion will be one of two kinds according 
as the ratio y'/y increases or decreases with increase of y. 
For example, if the object is in the form of a square, as shown 
in Fig. 246, a, then, on the supposition that y'/y decreases 
as y increases the image of the diagonal will be shortened 
relatively more than the image of a side of the square, and 
the square will be reproduced by a curvilinear figure with 
convex sides as shown in Fig. 246, b; this is the case of barrel- 
shay ed distortion, as it is called. On the other hand, if the 
ratio y'/y increases in proportion as the object-point is taken 
farther and farther from the axis, we have the opposite type 
known as cushion-shaped distortion (Fig. 246, c). 

If in Fig. 245 we put OM = z, 0'M' = z', ZMOP = w, 



§ 193] Airy's Tangent-Condition 545 

Z M'O'P' = a/, the condition of orthoscopy may be expressed 

as follows : 

y' z'.tano/ 

— = = constant: 

y 2. tana; 

and if we assume, as has been tacitly assumed in the pre- 
ceding discussion, that the chief rays all pass through the 
pupil-centers O, O', so that the abscissae denoted by z, z' have 
the same values for all distances of the object-point P from 
the axis, then we derive at once Airy's tangent-condition of 
orthoscopy, namely, tana/ : tan co = constant. But although 
a chief ray must pass through the center of the aperture- 
stop (§ 140), it will not pass through the centers of the pupils 
unless the latter are free from spherical aberration. The 
constancy of the tangent-ratio by itself is not a sufficient 
condition for orthoscopy; in addition, the spherical aberra- 
tion must be abolished with respect to the centers of the 
pupils. 

If the optical system is symmetrical with respect to an in- 
terior aperture-stop, the tangent-condition will be immedi- 
ately satisfied, because on account of the symmetry of the 
two halves of the system, every chief ray will issue in exactly 
the same direction as it had on entering, and therefore 
tan0: tan0' = l. Accordingly, if a "symmetric doublet" 
of this kind is spherically corrected with respect to the center 
of the aperture-stop, it will give an image which will be free 
from distortion. 

193. SeidePs Theory of the Five Aberrations. — In the 
theory of optical imagery which was developed according to 
general laws first by Gauss (§ 119) in his famous Dioptrische 
Untersuchungen published in 1841, the fundamental assump- 
tion is that the effective rays are all comprised within a nar- 
row cylindrical region of space immediately surrounding the 
optical axis; this region being more explicitly defined by the 
condition that a paraxial ray is one for which the angle of 
incidence (a) and the slope-angle ( 6 ), in the case of each 
refraction or reflection, are both relatively so minute that the 



546 Mirrors, Prisms and Lenses [§ 193 

powers of these angles higher than the first can be neglected 
(§ 63). Evidently, therefore, Gauss's theory is applicable 
only to optical systems of exceedingly small aperture and 
limited extent of field of view. But with the development 
of modern optical instruments and especially with the in- 
crease of both aperture and field demanded for certain types 
of photographic lenses, it became necessary to take account 
of rays which lie far beyond the narrow confines of the central 
or paraxial rays. Long prior to the time of Gauss important 
contributions to the theory of spherical aberrations had been 
made in connection with certain more or less special problems ; 
but the first successful attempt to extend Gauss's theory in 
a general way by taking account of the terms of higher orders 
of smallness was made by Seidel (1821-1896) in a re- 
markable series of papers published between the years 1852 
and 1856 in the Astronomische Nachrichten. Seidel's 
method consisted in tracing the path of the ray through the 
centered system of spherical refracting surfaces and in de- 
veloping the trigonometrical expressions in series of ascend- 
ing powers which were finally simplified by neglecting all 
terms above the third order. If the ray-parameters are re- 
garded as magnitudes of the first order of smallness, it is 
easy to show that on account of the symmetry around the 
optical axis these series-developments can contain only terms 
of the odd orders of smallness; so that in Seidel's theory 
the terms neglected are of the fifth and higher orders. It is 
impossible to describe here in detail the elegant mathemati- 
cal treatment by which Seidel was enabled to arrive at 
his final results; suffice it to say, that he obtained a sys- 
tem of formulae from which it was possible to ascertain the 
influence both of the aperture and the field of view on the 
perfection of the image. In Seidel's formulae the aber- 
rations of the ray, that is, its deviations from the path pre- 
scribed by Gauss's theory, are expressed by five different 
sums, denoted by Si, $2, S3, S i} and S 5 , which depend only on 
the constants of the optical system and the position of the 



§ 193] Seidel's Five Sums 547 

object-point, and which are, in fact, the coefficients of the 
various terms in the equations. The condition that there 
shall be no aberration demands that all of these five sums 
shall vanish simultaneously, that is, 

oi = 02 = 03 = 04 = 05 = 0. 

If, on the other hand, these conditions are not satisfied, the 
image yielded by the lens-system will not be faultless; and 
therefore it will not be without interest to inquire more par- 
ticularly into the separate influence of each of these five ex- 
pressions which occur in Seidel's formulae. 

Thus, for example, if the optical system is designed so that 
Si = 0, then there will be no spherical aberration at the center 
of the field (§ 178) for the given position of the axial object- 
point. And if not only $1 = but also $2 = 0, then there 
will be no coma (§ 191). The condition S 2 = means also 
that Abbe's sine-condition (§ 182) will also be satisfied, so 
that the image of the parts of the object in the immediate 
vicinity of the axis is sharply defined. 

But even when we have 0*1 = 0*2 = 0, the optical system 
will, in general, still be affected by astigmatism of oblique 
rays (§ 184), so that an object-point lying at some little dis- 
tance from the axis will not be reproduced by an image-point 
but at best by two short focal lines at different distances 
from the lens-system and directed approximately at right 
angles to each other. Moreover if the distance of the object- 
point from the axis is varied, the positions of these two focal 
lines will vary also both with respect to their distance from 
the lens-system and with respect to their mutual distance 
apart. In other words, when both Si and $2 vanish, then, in 
general, there is no unique image of a transversal object- 
plane, but this latter may be said to be reproduced by two 
so-called image-surfaces (§ 189) which are surfaces of revolu- 
tion around the optical axis and which unite and touch each 
other at the point where the axis crosses them. The ex- 
pressions for the curvatures of these surfaces at this common 
point of tangency are given by Seidel's sums £3 and 



548 



Mirrors, Prisms and Lenses 



[§193 



St; so that if also $3— #4 = 0, the two image surfaces will 
coalesce and now the image of the plane object will be sharply 
denned, that is, stigmatic, although it will usually still be 
curved. But if also S 3 = Si = 0, the image will be both plane 
and stigmatic. However, it may still show unequal magnif- 
ications toward the margin, which means that there is dis- 
tortion (§ 192). This last error will be abolished provided 
aS 5 = 0; and now the image may be said to be ideal inasmuch 
as it is flat and sharply defined not only in the center but 
out .toward the edges and is at the same time a faithful re- 
production of the plane object. 

To attempt to derive Seidel's actual formula? or even 
to discuss the equations would be entirely beyond the scope 




Fig. 247. — Diagram representing the (i — l)th and ith lenses of a system of 
infinitely thin lenses. 

of this volume. But it may be convenient to insert here 
without proof the expressions of Seidel's five sums for 
the comparatively simple case of an optical system considered 
as composed of a series of m infinitely thin lenses each sur- 
rounded by air. 

Let Ai (Fig. 247) designate the point where the optical axis 
crosses the ith lens of the system, the symbol i being employed 
to denote any integer from 1 to m; and let us consider two 
paraxial rays which traverse the optical system, one of which 
emanating from the axial object-point Mi (AiMi=Wi) and 
meeting the first lens at a point Bi such that A]Bi = /ii, 
crosses the axis after passing through the (i— l)th lens at a 
point Mi (A i M i = w i ) and meets the ith lens at a point B A 



§ 193] System of Thin Lenses 549 

such that AiBi = h Y whereas the other ray, which emanates 
from an extra-axial object-point and which in the object- 
space passes through the center Oi of the entrance-pupil 
(§ 139) of the system (AiOi = si) and meets the first lens at 
a point Gi such that gfi = AiGi, crosses the axis after passing 
through the (i— l)th lens at a point Oi AiOi = Si) and meets 
the ith lens at a point Gj such that AiGi = g^ Then if we put 

Ui = l/ui, £i=l/si, 
it may easily be shown that 

9i(S i +F i )=g i+ i.S i+1 ; 

where F { denotes the refracting power of the ith lens. Now 
if ft; denotes the index of refraction of the ith lens and if R[ 
denotes the curvature of the first surface of this lens; and if, 
further, for the sake of brevity, the symbols A i} B- lf C l} D i} 
and E{ are introduced to denote the following functions of 
n u F h R v U\ and Si, namely: 

Ai = n J+? Fi R\- (4(^+0^ 2^+1 1 

m { m m-l J 

m m-l \m-l' 

m [ m m-l J 

+ n j±i Fi m+ 2 ^±i F iUi s i+ 2 -^ m 

m m m - 1 

+ J* F % + (JH-Yf\; 

m-l \m-l/ 

Ci = 3(« i± 2) m _ (6^+1) ^^ 3(2^+1) F \ FiRi 

+ i Fi m+ 2{3n ' + 2 W+ 3 -^ f& 

m m Wi 

m-l m-l \m-i/ > 



550 Mirrors, Prisms and Lenses [§ 193 



Di = '2-LZ Fi R\- t^L^ (Ui+Sd +^~ F { FiRi+^FiUi 

m { rii ?2i-l J ?ii 

ni ?2i- 1 



+ ^? Si+ (3_)V?; 

m-l Vm-l/ 



Wi ?2i— 1 rzi-1 \ni-l/ 

then Seidel's formulae for the spherical errors of a sys- 
tem of m infinitely thin lenses may be expressed as follows: 

i==m /7>-\ 4 i=m /h\ s n- 

S 3=I(|i.^)c i; ^(fi.^A; 

i=i V/ii fifi/ i=i \/ii 0i/ 

i=ihi\gi/ 

The greatest practical value of these formulae is to guide 
the optician to a correct basis for the design of his instrument 
and to supply him, so to speak, with a starting point for a 
trigonometrical calculation of the optical system which he 
aims to achieve. But the reader who wishes to pursue this 
subject further will find it necessary to consult the more ad- 
vanced treatises on applied optics. 



Ch. XV] Problems 551 

PROBLEMS 

1. If L, L' designate the points where a ray crosses the 
axis of a spherical refracting surface before and after refrac- 
tion, respectively, and if C designates the center of the sur- 
face, show that 

0+0' 

, , cos 

n n n —n 2 



a -|- a' 
cos- 



2 

where c = CL, c' = CL', a, a' denote the angles of incidence 
and refraction, 6, 6 r denote the slope-angles of the ray 
before and after refraction, r denotes the radius of the sur- 
face, and n, n' denote the indices of refraction. Also, show 
that 

a' +6' 



c +r 



sin 0cos- 



c+r . Ql a + 
sin o cos- 



2 

2. A ray parallel to the axis meets the first surface of a 
glass lens (index 1.5) at a height of 5 cm. above the axis, and 
after emerging from the lens crosses the axis at a point I/. 
The thickness of the lens is 1 cm. Determine the aberration 
F'L', where F' designates the position of the second focal 
point, for each of the following cases: (a) First surface of 
lens is plane and radius of curved surface is 50 cm.; (6) 
Second surface of lens is plane and radius of curved surface 
is 50 cm.; and (c) Lens is symmetric, radius of each surface 
being 100 cm. 

Ans. (a)/=±100cm.,F'L'=q=1.13 cm.; (6)/==*= 100cm., 
W=^f0.29 cm.; (c) /==»= 100.17 cm., F'L'==f0.42 cm.; 
where in each case the upper signs apply to positive lens and 
the lower signs apply to negative lens. 

3. An incident ray crosses the axis of a lens at an angle 6\ 
and meets the first surface at a point Bi, the angle of inci- 
dence being en; the slope of the refracted ray BiB 2j which 



552 Mirrors, Prisms and Lenses [Ch. XV 

meets the second surface at the point B 2 is 2 , and the angle 
of incidence at this surface is a 2 . If the radii of the surfaces 
are denoted by ri and r 2 , show that 

r 2 .sin(ci2— #2)— ri.sin(ai— 00 

-D1-D2 = : — 5 . 

sin C/ 2 

4. The chief ray of a narrow bundle of parallel rays is in- 
cident on a spherical mirror of radius 32 cm. at an angle of 
60°. Find the distance between the two image-points P' 
and Q' of the bundle of reflected rays. Ans. 24 cm. 

5. The chief ray of a narrow bundle of parallel rays is in- 
cident on a spherical mirror of radius r at a point B, the angle 
of incidence being 60°. Determine the positions of the image- 
points P' and Q'. Ans. BP' = r/4, BQ' = r. 

6. A narrow bundle of parallel rays in air is refracted at 
a spherical surface of radius r into a medium whose index of 
refraction is -y/s. If the angle of incidence is 60°, find the 
positions of the image-points P' and Q'. 

Ans. p' = 3rV3/4, g'=r\/3. 

7. A narrow bundle of parallel rays is incident on a spheri- 
cal refracting surface at an angle of 60°. If the meridian rays 
are converged to a focus at a point P' lying on the surface of 
the sphere, show that the angle of refraction of the chief ray 
is equal to the complement of the critical angle of the two 
media. 

8. The radius of each of the two surfaces of an infinitely 
thin double convex lens is 8 inches, and the index of refrac- 
tion is equal to \/S. The chief ray of a narrow bundle of 
parallel rays inclined to the axis at an angle of 60° passes 
through the optical center of the lens. Find the positions of 
the foci of the meridian and sagittal rays. 

Ans. The focal point of the meridian rays is 1 inch and 
that of the sagittal rays is 4 inches from the optical center. 

9. If in Young's construction of a ray refracted at a spher- 
ical surface (§ 176) a semi-circle is described on the incidence- 
radius BC as diameter intersecting the incident and refracted 
rays in the points Y, Y', respectively, show that the straight 



Ch. XV] Problems 553 

line YY' is perpendicular to the straight line CS. The point 
K where the straight lines YY' and CS meet is the center of 
perspective of the range of object-points lying on the inci- 
dent ray and the corresponding range of meridian image- 
points lying on the refracted ray (see § 186). Show that 
nTr _ n.r.sin 2 a 

0K w—> 

and that 

tanZBKC = tana+tana'. 

10. If the chief ray of a narrow homocentric bundle of 
rays is incident on a plane refracting surface at a point B, 
and if a, a' denote the angles of incidence and refraction, 
show that 

BP' = Hl^Jl . BP, BQ' = - . BQ, 
n cos 2 a n 

where P (or Q) designates the position of the vertex of the 

incident rays and P' and Q' designate the positions of the 

image-points of the meridian and sagittal rays, respectively. 

11. In the preceding problem show that the straight line 
QQ' is perpendicular to the plane refracting surface. 

12. The position of the image-point P' of a pencil of me- 
ridian rays refracted at a plane surface may be constructed 
as follows: Through the object-point P (or Q) draw PQ' per- 
pendicular to the refracting plane and meeting the chief re- 
fracted ray in Q'; and from P and Q' draw PX and Q'Y per- 
pendicular to the incidence-normal at X and Y, respectively. 
Draw XG perpendicular to the chief incident ray at G and 
YG' perpendicular to the corresponding refracted ray at G'. 
Then the straight line PP' drawn parallel to GG' will inter- 
sect the chief refracted ray in the required point P'. Using 
the result of No. 10 above, show that this construction is 
correct. 

13. The chief ray RB of a narrow pencil of sagittal rays 
meets a spherical refracting surface at the point B and is re- 
fracted in the direction BT. Through the center C draw CV 



554 Mirrors, Prisms and Lenses [Ch. XV 

parallel to BT meeting BR in V and CV parallel to BR meet- 
ing BT in V. If Q, Q' designate the positions of the points 
of intersection of the sagittal rays before and after refraction, 
respectively, and if BQ = g, BQ,' = q', show that 

BV BV , 

+— =1, 

Q 2 
and that 

VQ.V'Q' = VB.V'B. 

(Compare this last result with the Newtonian formula for 
refraction of paraxial rays at a spherical surface, viz., x.x' = 

14. The chief ray RB of a narrow pencil of meridian rays 
meets a spherical refracting surface at the point B, and is re- 
fracted in the direction BT. Through the center of perspec- 
tive K (see § 186; see also problem No. 9 above) draw KU par- 
allel to BT meeting BR in U and KU' parallel to BR meeting 
BT in U\ If the positions of the points of intersection of the 
meridian rays before and after refraction are designated by P 
and P', respectively, and if BP = p, BP' = p', show that 

BU BU' , 

V V 

and that 

UP.U'P' = UB.U'B. 

(Compare this result with that of the preceding problem.) 

15. If J, J' designate the positions of the aplanatic points 
of a spherical refracting surface, and if 6, 6' denote the 
slopes of the incident and refracted rays BJ, BJ', respec- 
tively, show that 

sin 8 _ n' 
s!nT'~n ' Vi 
where y denotes the magnification-ratio for paraxial rays. 

16. A. Steinheil's so-called "periscope" photographic 
lens is composed of two equal simple meniscus lenses, both 
of crown glass, separated from each other with a small stop 
midway between. The data of the system, as given in Von 



Ch. XV] Problems 555 

Rohr's Theorie und Geschichte des photographischen Objektivs 
(Berlin, 1899), p. 288, are as follows: 

Indices: ni=nz=ns = l; ^2=^4= 1.5233 

Radii: r\ = — r 4 = +17.5 mm.; r2= — r 3 = +20.8 mm. 

Thicknesses: di = d z — +1.3 mm.; d 2 = 12.6 mm. 

Distance of center of stop from second vertex of first lens 
= +6.3 mm. ; diameter of stop = 2.38 mm. ; diameter of each 
lens = 11.32 mm. 

Employing the above data, determine (1) the position and 
size of the entrance-pupil, (2) the angular extent of the field, 
(3) the position of the second focal point F'; and (4) the 
point where an edge-ray directed towards a point in the 
circumference of the entrance-pupil and parallel to the axis 
crosses the axis after emerging from the system. 

Ans. (1) Distance of center of entrance-pupil from second 
vertex of first lens is +6.45 mm.; diameter of entrance- 
pupil is 2.53 mm. (2) The angular extent of the field is 
nearly 90° . (3) Distance of F' from last surface is A4F' = 
+90.946 mm. (4) The edge-ray crosses the axis at a dis- 
tance A 4 L 4 = +90.432 nun. 

17. The abscissae of the points M k , M k+ i where a par- 
axial ray crosses the axis of a centered system of m spherical 
refracting surfaces before and after refraction at the &th sur- 
face are denoted by u k = A k M k , w k ' = A k M k+ i. If the ray 
proceeds in the first medium of index n\ in a direction par- 
allel to the axis, it may be shown (cf. problems Nos. 16 and 
17, end of Chapter X) that the primary focal length of the 
system is given by the formula 

f _ U2- Us • • . U m (TT _ n \ 

f -w.u t '...u m " (Ul -° h 

where U k = n k /u k , U k ' = n k+ i ju k . Having calculated the 
path of the paraxial ray in the preceding problem, em- 
ploy the above formula to determine the focal length of 
Steinheil's "periscope." Ans. /= +98.696 mm. 

18. The path of a chief ray which in traversing the air- 
space between the two lenses of Steinheil's " periscope " 



556 Mirrors, Prisms and Lenses [Ch. XV 

(see No. 16) goes through the center of the stop will be sym- 
metrical with respect to the two parts of the optical system, 
so that for such a ray we must have : 

Ca=— ci, c 4 =— ci, C3=—c 2 , c 3 =— c 2 '; 
a 4 = a/, a/=ai, ct3= a-2, a 3 /= a 2 ; 
0$= 6 1 } 6i= Si. 
Show that if 3 = — 30° for a ray which goes through the 
stop-center, the ray must have been directed initially at a 
slope-angle di= — 28° 2' 54 .43" towards a point Li on the 
axis whose distance from the second vertex of the first lens is 
A 2 Li = +6.563 mm. 

19. The astigmatism of a narrow bundle of rays refracted 
through a centered system of spherical surfaces may be com- 
puted logarithmically by means of the following recurrent 
formulae : 

w k .sin(a k — a k ') 



D t = 



r k .sm a k 



■/i k = r k .sin(a k — k ), t k = 



*k +1 



w k+ i.sin0 k +i ' 
Sagittal Section 

Qk =Qk+D k , Q k +1 = 



l-fc.Qk' ' 



Meridian Section 

p ,_ P k .eos 2 a k +D k P k ' m 

cos 2 ak 1-ikA 

where the symbols a, a', 0, n and r have their usual mean- 
ings and where P, P' and Q, Q' and D are the magnitudes de- 
fined in §§ 186 and 184. The calculations according to these 
formulae will be considerably simplified in the case of a chief 
ray which traverses a system like Steinheil's " periscope" 
(see No. 16) which is symmetric with respect to the stop- 
center. For example, for this particular system we can write 
for a chief ray: 



Ch. XV] Problems 557 

Di = D\, D 3 = D 2 , h±= — hi, hz = —h 2 , h = ti; 

0.1— Oi = — (oi— 0.4), o 2 — o 2 ' = — (a.3 — a 3 ')> 

ai— B\= a/— 6 b — a«- # 4 , a 2 — 6 2 = 0,3— Q\= Oz — Oz. 

Apply the above formulae to the optical system of problem 
No. 16 to calculate the astigmatic difference (§ 186) of a nar- 
row bundle of emergent rays whose chief ray is the ray whose 
path was determined in problem No. 18; assuming that the 
bundle of incident rays was cylindrical, that is, Pi = Qi = 0. 

Ans. p 4 '— 5/= +4.849 mm. 

20. Using Seidel's formulae as given in § 193, show that 
the condition that an infinitely thin lens surrounded by air, 
and provided with a rear stop, shall yield a punctual or 
stigmatic image of a plane object placed in the primary focal 
plane of the lens, is as follows: 

\n-l I n(n-l) 2 {n-iy n{n-l) 

where F denotes the refracting power of the lens, F 2 denotes 
the refracting power of the second surface, 1/S denotes the 
distance of the stop, and n denotes the index of refraction 
of the lens. If the stop is a rear stop at a distance of 30 mm. 
from the lens, and if n = 1.52, show that the maximum value 
of the refracting power of a convex lens which will give a 
punctual image of a plane object placed in the primary focal 
plane is F = + 14.87 dptr. 

21. Using Seidel's formulae as given in § 193, show 
that the condition that an infinitely thin lens surrounded by 
air, and provided with a rear stop, shall give a punctual or 
stigmatic image of an infinitely distant object, is: 



fcV+ c ) 



n n— 1 n 



where n denotes the index of refraction of the lens, F denotes 
its refracting power, Ri denotes the curvature of the first 
surface of the lens, and C = S—F, the magnitude S being 
equal to the reciprocal of the stop-distance. 



INDEX 

The numbers refer to the pages 



Abbe, E.: Porro prism system, 50; refract ometer, 128; definition of 
focal length, 344; pupils, 401; magnifying power, 454; v-value of 
optical medium, 480; optical glass, 482, 489; sine-condition, 523, 
542, 547; differential notation, 531. 

Aberration, Chromatic: see Chromatic Aberration, Achromatism, etc. 

Aberration, Least circle of, 515. 

Aberration, Spherical: see Spherical Aberration. 

Aberrations, Chromatic and monochromatic, 509; Seidel's five sums, 
545-550, 557. 

Abney's formula for diameter of aperture of pinhole camera, 5, 26. 

Abscissa formula for plane refracting surface, 97, 191, 269; spherical 
mirror, 154, 155, 191, 276, 285; spherical refracting surface, 191, 
193, 200, 274, 285; infinitely thin lens, 228, 229, 279, 285; centered 
system, 332, 519. See also Image Equations. 

Absorption of light, 2. 

Accommodation of eye, 433-439; amplitude, 437-439; range, 438; 
diminishes with age, 435, 436; effected by changes in crystalline 
lens, 434; refracting power of eye in accommodation, 436, 437. 

Achromatic combinations: Prisms, 480, 481, 491-493; lenses, 480, 481, 
499-505. 

Achromatic system, 488. 

Achromatic telescope, 480, 481, 505. 

Achromatism, 480, 481, 487 and foil.; optical and actinic or photo- 
graphic, 489-491. 

Airy, Sir G. B.: Cylindrical lens, 315; tangent-condition of orthoscopy, 
545; curvature of image, 540. 

Ametropia, 439 and foil.; axial, curvature and indicial ametropia, 442. 

Ametropic eye, 440 and foil.; distance of correction-glass, 445, 446. 

Amici, G. B.: Direct vision prism system, 495, 497, 506. 

Amplitude of accommodation, 437-439. 

Anastigmatic (or stigmatic) lenses, 314. 

Angle, Central, 152, 516. 

559 



560 * Index 

Angle, Critical: see Critical Angle, Total Reflection. 

Angle, Slope, 151, 334, 516. 

Angle, Visual: see Visual Angle, Apparent Size. 

Angle of deviation, in case of inclined mirrors, 43; in case of refraction, 
78; in prism, 50, 51, 125; in lens, 293. See also Prism, Thin prism, 
Prism-dioptry, Prismatic power of lens. 

Angles of incidence, reflection and refraction, 30, 31, 65. 

Angles, Measurement of, by mirror and scale, 56. 

Angstrom unit of wave-length, 10; see also 'i enth-meter. 

Angular magnification (or convergence-ratio), 351. 

Anterior chamber of eye, 425. 

Anterior and posterior poles of eye, 431, 432. 

Aperture-angle, 404. 

Aphakia, 213, 442. 

Aplanatic points of optical system, 524; of spherical refracting surface 
(J, J'), 512, 513, 554. 

Aplanatism, 524. See Sine-Condition. 

Apochromatism, 489. 

Apparent place and direction of point-source, 15-18. 

Apparent place of object viewed through plate of glass, 102, 103, 105, 
106. 

Apparent size, 20-22, 446 and foil.; in optical instrument, 449 and foil. 

Aqueous humor, 213, 371, 425. 

Astigmatic bundle of rays, 25, 310-314, 526-538, 552 and foil.; image- 
lines, 100, 312, 313, 534-536, 547; image-points, 312, 526, 527, 
529-534; principal sections, 311, 528. See also Meridian rays, 
Sagittal rays, Image-points, Image-lines, Sturm's conoid, Astig- 
matism. 

Astigmatic difference, 533. 

Astigmatic image-surfaces, 536-538, 547. 

Astigmatic lenses, Chap. IX, 300 and foil.; 314. 

Astigmatism by incidence, 527. 

Astigmatism, Measure of, 533. 

Astigmatism of oblique bundles of rays, 527, 547. 

Astigmatism, Sturm's theory, 313, 534. 

Astronomical telescope, 411, 456; field of view, 411, 412; magnifying 
power, 454-460. 

Axial ametropia, 442; static refraction and length of eye-ball, 442, 443. 

Axial (or depth) magnification 351. 

Axis of collineation, 243. 

Axis of lens, 217; spherical refracting surface, 149. See also Optical 
axis. 

Axis, Visual, 433. 



Index 561 

B 

Back focus of lens, 365. 

Badal's optometer, 422, 423. 

Barlow's achromatic object-glass, 504, 505. 

Barrel-shaped distortion, 544. 

Bending of lens, 284. 

Blind spot of eye, 430, 431. 

Blur-circles, 414-417, 419. 

Brewster, Sir D.: Kaleidoscope, 47. 

Bundle of rays, Character of, 24, 25, 508, 509, 525; "direct," 514; 

homocentric (or monocentric), 25, limitation by means of stops, 

397-399. See also Astigmatic bundle of rays. 
Bunsen burner, 66, 473. 
Burnett, S. M.: Prism-dioptry, 135. 



Calculation of path of ray: refracted at spherical surface, 516-519; 
reflected at spherical mirror, 518; refracted through prism, 124, 
125; refracted through centered system, 332, 519-522; numerical 
example in case of paraxial and edge rays, 520-522. 

Camera: see Pinhole camera. 

Cardinal points of optical system, 334-339. 

Cataract: see 'Aphakia. 

Caustic curve, 514. 

Caustic surface, in general, 526; by refraction at plane surface, 98, 99; 
by refraction at spherical surface, 515. 

Center: Of collineation, 243; of curvature, 260, 526; of perspective (K), 
532, 554; of rotation of eye, 432, 434, 448, 452. 

Centered system of spherical refracting surfaces: Optical axis, 329; 
construction of paraxial ray, 330, 331; calculation of path of parax- 
ial ray, 332; conjugate axial points (M, M'), 346, 347; extra-axial 
conjugate points (Q, Q'), 339-342; lateral magnification, 333, 349; 
Smith-Helmholtz formula, 334; focal planes, 333-335; focal 
points, 332-335; ray of finite slope, 519-522. 

Centers of perspective of object-space and image-space, 416, 417. 

Centrad, 134, 294. 

Central angle (<p), 152, 516. 

Central collineation, 242-247. • 

Central ray, 243. 

Chief rays, 24, 413, 420, 526. 

Choroid, 425. 

i 



562 Index 

Chromatic aberration, 487-489, 509. 

Ciliary body, 427; mechanism of accommodation, 434. 

Circle of aberration, Least, 515. 

Circle of curvature, 260. 

Circle of least confusion, 314, 537. 

Circles of diffusion: see Blur-circles. 

Clausius, R.: Sine-condition, 523. 

Coddington, H. : Curvature of image, 540. 

Collineation : Central, 242-247; center of, 243; axis of, 243; invariant 
of, 246. 

Collinear correspondence, 242, 508. See also Punctual imagery. 

Color and frequency of vibration, 472-476; and wave-length, 475. 

Color of a body, 2. 

Colors of spectrum, 466. 

Coma, 542, 547. 

Combination of three optical systems, 374-376. 

Combination of two lenses, 366-370; achromatic, 499 and foil. 

Combination of two optical systems, 356-362; focal lengths, 359; focal 
and principal points, 358, 361; refracting power, 361. 

Complete quadrilateral, 162. 

Compound optical systems: Chap. XI, 356, foil. 

Concave: Lens, 221; surface, 150. 

Concentric lens, 221, 232, 387, 388. 

Cones and rods of retina, 428, 429. 

Conjugate planes, 172, 194, 236. 

Conjugate points on axis (M, M'): Centered system of spherical re- 
fracting surfaces, 346, 347; infinitely thin lens, 227-229, 232; 
plane refracting surface, 97; plate with parallel faces, 105; spherical 
mirror, 154, 164; spherical refracting surface, 181, 183. 

Conjugate points off axis (Q, Q') : Centered system of spherical refract- 
ing surfaces, 339-342; infinitely thin lens, 234-236; spherical 
mirror, 171-175; spherical refracting surface, 193-196. 

Conoid, Sturm's, 313, 314, 535. 

Convergence-ratio: see Angular Magnification. 

Convergent and divergent optical systems, 186, 339, 340. 

Convergent lens, 221. 

Convex: Lens, 221; surface, 150. 

Cornea of human eye, 425; optical constants, 371, 372, 401; vertex, 
431. 

Correction-glass: Refracting power and vertex-refraction, 443-446; 
distance from eye measured by keratometer, 421, 422; second 
focal point of glass at far point of eye, 445. 

Crew, H.: ''dioptric," 287. 



Index 563 

Critical angle of refraction, 80. 

Cross-cylindrical lens, 315, 317, 319, 320, 325. 

CrystaUine lens of human eye, 213, 371, 372, 373, 378, 381, 395, 428; 

optical constants, 371-373, 395, 434; changes in accommodation, 

395, 434; "total index," 436. See also Aphakia. 
Culmann, P.: Smith-Helmholtz formula, 202. 
Curvature of arc: total, 258; mean, 259; center of, 260; circle of, 260; 

radius of, 260; sign of, 260; measure, 260-264. 
Curvature of image, 538-540, 547, 548. 
Curvature of normal sections of surface, 300-303; principal sections, 

302, 303, 525. 
Curvature, Unit of, dioptry, 286-288. 
Curvature ametropia, 442. 
Curvature-method in geometrical optics, 282. 
Cushion-shaped distortion, 544. 
Cylindrical lenses, 217, 310, 314-317; types, 315-317; combinations, 

318-326; transposition, 318-320. 
Cylindrical surface, 265, 305-308, 310-313; refracting power, 307, 308, 



Dennett: Centrad, 134. 

Depth-magnification, 351. 

Descartes, R: Law of refraction, 67. 

Deviation of ray : See Angle of deviation, Minimum deviation. 

Deviation without dispersion, 481, 491-493. 

Diamond, 70, 479. 

Diaphragms or stops for cutting out rays, 397-399. 

Diffraction-effects, 14. 

Dioptry, 286-288; "dioptrie," "dioptre," "diopter," etc., 286," 287; 

millidoptry, Hectodioptry, and Kilodioptry, 287. 
" Direct' ' bundle of rays, 514. 
"Direct vision," 448. 
Direct vision prism-systems, 493-499. 
Direction of ray or straight line: See Positive direction. 
Direction of source from observer's eye, 15-18. 
Dispersion, Chromatic: Chap. XIV, 465 and foil.; anomalous, 477; 

irrationality of, 477-479; partial, 479, 483; relative, 479, 483. 
Dispersion without deviation, 481, 493-499. 
Dispersive power (or strength), 479-481; dispersive strength of lens, 

503. 
Distinct vision, Distance of, 452, 453. 
Distortion, 543-545. 



564 Index 

Divergent lens, 221; divergent and convergent optical systems, 339, 
340. 

Dollond, J.: Achromatic object-glass, 481, 482, 504, 505. 

Donders's "reduced eye," 214; astigmatism of eye corrected by cylin- 
drical glasses, 316; loss of accommodation with increasing age, 
435, 436. 

Double concave lens, 219. 

Double convex lens, 217. 

Double ratio (or cross ratio), 156-164. 

Drysdale, C. V., 287. 

Dutch telescope, 456; field of view, 412, 413; "eye-ring," 413, 458; 
magnifying power, 455-460. 

Dynamic refraction of eye, 438. 

E 

Effective rays, 23. 

Emergent rays, 24. 

Emmetropia and ametropia, 439-443. 

Emmetropic eye, 440. 

Entrance-port, 406-409, 410, 413. 

Entrance-pupil, 43, 179, 400 and foil., 543; two or more entrance-pupils, 
405, 406; entrance-pupil of eye, 401, 448. 

Ether, Light transmitted through, 10, 472-476. 

Euler, L.: Theory of curved surfaces, 303, 306, 525; achromatism,481. 

Exit-port, 409, 410, 413. 

Exit-pupil, 400-405, 411-413, 415, 417, 419, 420, 448, 543. 

Eye: Accommodation, 433-439; anterior chamber, 425; aqueous hu- 
mor, 371, 425; bacillary layer of rods and cones, 428; "black of the 
eye," 401; blind spot, 430; center of rotation, 432, 434, 448, 452; 
change of refracting power in accommodation, 436, 437; choroid, 
425; ciliary body, 427; cornea, 371, 372,401,425; cornea- vertex, 431; 
crystalline lens, 371-373, 428; decrease of power of accommodation 
with age, 435, 436 ; description of human eye, 425-43 1 ; entrance-pupil, 
401, 448; expressions for refraction of eye, 439; far point and near 
point, 434, 435; field of fixation, 432, 435; focal lengths, 343, 374, 
389, 432; focal lengths in case of maximum accommodation, 437; 
focal points, 374, 389, 423, 432; fovea centralis, 429, 432, 433, 446; 
iris, 401, 425; line of fixation, 432; motor muscles, 431, 432; nodal 
points, 422, 432; optical axis, 431; optic nerve, 430; point of fixa- 
tion, 432; positions of cardinal points in state of maximum accom- 
modation, 437; posterior pole, 432, 438; principal points, 374, 432; 
pupil, 23, 401, 409-413, 421, 425; refracting power, 374, 432; 



Index 565 

retina, 428; static and dynamic refraction, 438 and foil.; suspen- 
sory ligament (zonule of Zinn), 428, 434; variation of principal 
points in accommodation, 437; visual axis, 433; visual purple, 430; 
white of the eye, 425, yellow spot {macula lutea), 428. 

Eye: see also Schematic eye, Ametropic eye, Emmetropic eye, Hyper- 
metropic eye, Myopic eye, "Reduced eye." 

Eye-axis, Length of, 438, 440-443, 448. 

Eye-ring of telescope, 413, 458, 459. 

Eye-glasses: See Correction-glass, Astigmatic lenses, Cylindrical lenses, 
Ophthalmic prisms, etc. 



Faraday, M.: Optical glass, 482. 

Far point, 434, 438, 440, 442; far point sphere, 434; senile recession, 
436; coincides with second focal point of correction-glass, 445; 
in case of schematic eye, 461. 

Far point distance, 437, 444. 

Far-sighted eye, 435. See Hypermetropia. 

Fermat, P.: Principle of least time, 86. 

Field of fixation of eye, 432, 435. 

Field of view, 18, 19, 406-409, 448; of plane mirror, 40-43; of spherical 
mirror, 176-179; of infinitely thin lens, 247-249, 409-411; of 
Dutch telescope, 412, 413; of astronomical telescope, 411, 412; 
"ragged edge," 412. 

Field-stop, 19, 178, 249, 406, 410. 

" Fish-eye camera," 81. 

Fixation: field of, 432, 435; line of, 432; point of, 432. 

Flat image, 539, 540, 548. 

Fluorite, 479, 485. 

Focal lengths of schematic eye, 343, 374, 389, 432 ; in case of maximum 
accommodation, 437. 

Focal lengths of spherical mirror, 167; of spherical refracting surface, 
191, 192, 193, 199, 281; of infinitely thin lens, 229, 240-242; of 
compound system, 359; of combination of two lenses, 367; of thick 
lens, 363; of optical system in general, 342-344. 

Focal planes of spherical refracting surface, 197-199; of infinitely thin 
lens, 232; of optical system, 334, 335, 341; of centered system of 
spherical refracting surfaces, 333. 

Focal point angle, 447; as measure of size of retinal image, 449. 

Focal points of spherical mirror, 166, 189; of spherical refracting sur- 
face, 186-189; of infinitely thin lens, 229-232; of centered system 
of spherical refracting surfaces, 332, 333; of optical system, 334, 



566 Index 

335; of compound system, 358, 361; of thick lens, 363; of com- 
bination of two lenses, 367. 

Focal points of schematic eye, 374, 389, 423, 432. 

Focus plane, 400, 402-404, 406-408, 414-417, 543. 

Fovea centralis, 429, 432, 433, 446. 

Fraunhofer, J.: 145, 479, 493, 494, 506; dark lines of solar spectrum, 
470, 472, 475, 476, 477; measurement of index of refraction, 129; 
notation of dark lines, 472 ; production of optical glass, 482 ; achro- 
matic object-glass, 504, 505. 

Frequency of vibration and color, 472-476; connection with wave- 
length, 475. 

Fresnel, A. J.: Principle of interference, 14; use of cylindrical lens, 315. 



Galileo: Telescope and astronomical discoveries, 456, 462, 463, 464. 

Gauss, K. F.: Reduced distance, 279, 280; theory of optical imagery, 
334, 536, 545, 546; principal points, 335; achromatic object-glass, 
504, 505. 

Glass, Optical: see Optical Glass. 

Gleichen, A.: Lehrbuch der geometrischen Optik, 352. 

Goerz, P.: "Double anastigmat" photographic lens, 352. 

Graphical methods: Paraxial ray diagrams, 168-171; path of paraxial 
ray through centered system, 331; Young's construction, 509-511. 

Gregory, J., achromatism, 480. 

Grimsehl, E., Lehrbuch der Physik, 363. 

Gullstrand, A.: Reduced distance, 280; schematic eye, 343, 370, 371, 
374, 381, 382, 389, 395, 432, 436, 442, 443, 461; formulae for com- 
pound systems, 260, 361; schematic eye in state of maximum 
accommodation, 395, 436, 461; writings, 536. 

H 

Hadley's sextant, 58-60. 

Hall, C. M.: Achromatic telescope, 481. 

Harcourt, W. V.: Optical glass, 482. 

Harmonic range of points, 161-164. 

Heliostat, 54, 55. 

Helmholtz, H. Von: Ophthalmometer, 103; Smith-Helmholtz 

equation, 201, 202, 214, 215, 334, 338, 342, 459, 524; Handbuch der 

physiologischen Optik, 371. 
Hero of Alexandria, 87. 
Herschel, Sir J. F. W.: Achromatic object-glass of telescope, 504, 505. 



Index 567 

Homocentric bundle of rays, 25. 

Houstoun, R. A.: Newton and colors of spectrum, 466, 469. 

Huygens, C: Construction of wave-front in general, 10-13, 123; in 
case of reflection at plane mirror, 33-37, 61; in case of refraction 
at plane surface, 70-72; Huygens 's ocular, 396, 501, 502. 

Hypermetropia, 441, 443, 445. 

Hypermetropic eye, 441; correction glass, 445. 



Image, 5, 17, 18, 25; ideal, 25, 506, 548; real and virtual, 17, 18. 

Image, Rectification of, by successive reflections, 50, 51. 

Image, Size of retinal, 448, 449. 

Images in inclined mirrors, 43-51. 

Image-equations of optical system: Referred to focal points, 345; 
referred to principal points, 345-347; referred to pair of conjugate 
points in general, 347, 348; referred to nodal points, 348; in terms 
of refracting power and reduced "vergences," 348. 

Image-equations of spherical refracting surface, 200, 201. 

Image-lines of narrow astigmatic bundle of rays, 100, 312, 313, 534- 
536, 547. 

Image-lines of narrow astigmatic bundle of rays refracted at plane 
surface, 100. 

Image-point, 25. 

Image-points of narrow astigmatic bundle of rays, 312, 526, 527, 529- 
534. 

Image-rays, 24. 

Image-space and object-space, 242, 243. 

Image-surfaces, Astigmatic, 536-538, 547. 

Incidence: Angle of, 30; height, 151; normal, 30; plane of, 30. 

Incident rays, 24, 30. 

Inclined mirrors, 43-51. 

Index of refraction: Absolute, 74; limiting value of, 70; relative, 66; 
measurement of, 106, 107, 128, 129; function of wave-length, 
476, 477. 

Indicial ametropia, 442. 

"Indirect vision," 446. 

Infinitely distant plane of space, 197, 434. 

Infinitely distant point of straight line, 158. 

Infinitely thin lens, Paraxial Rays: 217-257, 276-279, 285; abscissa- 
formula, 226-229, 285; character of imagery, 237-240; conjugate 
axial points, 227-229, 232-234; construction of image, 236; extra- 
axial conjugate points, 234-236; field of view, 247-249, 409-411; 



568 Index 

focal lengths, 229, 240-242; focal planes, 232; focal points, 229- 
232; lateral magnification, 236, 237; principal planes, 239; pris- 
matic power, 291-295; refracting power, 283, 284. 

Infinitely thin lens, Central Collineation, 246. 

Infinitely thin lens, Conventional representation, 226. 

Infinitely thin lens, Refraction of spherical wave through, 276-279. 

Infinitely thin lens-system, 289-291; formulae for spherical aberrations, 
548-550. See also Achromatic combinations. 

Invariant: Of refraction, 76; of central collineation, 246; in case of 
refraction of paraxial rays at spherical surface, 191. 

Iris of eye, 401, 425. 

Isotropic medium, 3, 4. 



Jack son, Professor: New optical glass, 484. 
Jansen, Z.: Reputed inventor of telescope, 456. 
Jena glass, 482-485, 540. 



K 

Kaleidoscope, 47. 

Kepler, J.: Astronomical telescope, 455, 456, sagitta, 202. 

Keratometer, 421, 422. 

Kessler, F. : Direct vision prism, 497, 498, 499, 506. 

Klingenstierna, S.: Achromatic combination of prisms, 481. 

Kohlrausch, F. : Measurement of index of refraction, 128. 



Lagrange, J. 1^.: Smith-Helmholtz formula, 202. 

Landolt, E.: Physiological Optics, 287. 

Lange, M.: Calculation-system, 520. 

Lateral magnification: Centered system, 333, 349; infinitely thin lens, 

236, 237; spherical mirror, 176; spherical refracting surface, 196. 
Law: Of independence of rays of light, 15; of rectilinear propagation, 

3, 4; of reflection, 31; of refraction, 66; of Malus, 89-91, 525. 
Least circle of aberration, 515. 
Least confusion, Circle of, 314. 
Least deviation: see Prism. 
Least time, Principle of, 86-89. 



Index 569 

Lens: see Astigmatic lens, Cylindrical lens, Infinitely thin lens, Thick 
lens, Toric lens, etc. 

Lens: Axis, 217; bending of, 284; concentric, 221, 232, 387, 388; concave 
and convex, 222; convergent or positive and divergent or negative, 
223; definition, 217; dispersive strength, 503; double convex and 
double concave, 217, 219; meniscus, 219, 226, 385, 386, 387; of 
zero curvature, 221, 386; optical center, 223-226; plano-convex 
and plano-concave, 219; refracting power, 283, 363; symmetric, 
217, 385, 388; thickness, 219. 

Lens, Crystalline: see Crystalline lens. 

Lens-gauge, 263-265, 288, 289. 

Lenses, Forms of, 217-223. 

Lens-system: see Combination of two lenses. 

Lens-system, Thin: 289-291; achromatic combination, 502-505. 

Light: Rectilinear propagation, see Chap. I; wave-theory, 9, 10, 472 
and foil.; velocity, 10, 72, 75, 474, 475. 

Line of fixation, 432. 

Lippershey, F.: Reputed inventor of telescope, 456. 

Listing, J. B.: " Reduced eye," 214; nodal points, 337. 

Luminous bodies, 1. 

Luminous point, Direction and location, 15-18. 

M 

Macula lutea or yellow spot, 428. 

Magnification: see Angular magnification, Axial magnification, Lateral 

magnification, Magnification-ratios, Magnifying power. 
Magnification-ratios, 349-351. 
Magnifying power, 199, 344, 452 and foil.; Abbe's definition, 454; 

absolute, 454; individual, 454. 
Magnifying power of magnifying glass, 453; of microscope, 454; of 

telescope, 455-460. 
Malus, E. L.: Law, 89-91, 525. 
Medium: see Optical medium. 
Meniscus lens, 219, 226, 385, 386, 387. 

Meridian rays, 311. See Meridian section of narrow bundle of rays. 
Meridian section of narrow bundle of rays, 311, 528, 530-533, 535, 552, 

553, 554, 556; lack of symmetry in, 541. 
Meridian section of surface of revolution, 305. 
Michelson, A. A.: Velocity of light, 474. 
Minimum deviation of prism: see Prism. 
Mirror: see Plane mirror, Spherical mirror, "Thick mirror," (l Thin 

mirror," Inclined mirrors, etc. 



570 Index 

Mirror and scale for angular measurement, 56-58. 

Moebius, A. F.: Principal points, 335. 

Monocentric bundle of rays, 25. 

Monochromatic aberrations, 509. See Spherical aberration. 

Monochromatic light, 66, 467, 473-477. 

Monoyer, F.: "dioptrie," 286. 

Moser, C: Nodal points, 337. 

Muscles, Motor, of eye, 431, 432. 

Myopia, 441, 443, 445. 

Myopic eye, 441; correction-glass, 445. 



N 

Near point, 434, 435, 438; near point sphere, 434, 435; near point re- 
cedes from eye with increase of age, 435, 436; in case of schematic 
eye, 436, 461. 

Near point distance, 437. 

Near-sighted eye, 435. See Myopic eye. 

Negative lens, 223. 

Negative principal points, 338. 

Neutralization of lenses, 291. 

Newton, Sir I.: 11; prism experiments and dispersion, 66, 465, 466, 
467, 469, 470, 480, 481. 

Newtonian formula (x.x f =ff), 168, 201, 237, 345, 554. 

Nodal planes, 337. 

Nodal points, 337, 338; construction, 340; relation between nodal 
points and principal points, 341, 343; image-equations referred to 
348; of lens, 226, 363. 

Nodal points of eye, 422, 432. 

Normal sections of curved surface, 300-305, 525, 526; cylindrical 
surface, 306. 



Object-point, 25. 
Object-rays, 24. 

Object-space and image-space, 242, 243. 
Obliquely crossed cylinders, 320-326. 
Oculars of Huygens and Ramsden, 502. 
Opaque bodies, 2. 

Ophthalmic lenses: See Astigmatic lenses, Cylindrical lenses, Correction' 
glass, Toric lenses, etc. 



Index 571 

Ophthalmic prism: Base-apex line, 135; combination of two ophthal- 
mic prisms, 138-142; deviation, 133; power, 134; rotary prism, 141. 

Ophthalmometer, 103. 

Optic nerve, 430. 

Optical achromatism, 489, 490. 

Optical axis, axis of symmetry, 23; of centered system, 329; of lens, 217. 

Optical axis of eye, 431. 

Optical center of lens, 223-226. 

Optical disk for verifying law of reflection, 32; refraction, 67, 68; total 
reflection, 83, 84. 

Optical glass, 481 and foil.; process of manufacture, 485-487. 

Optical image, 5, 17, 18, 25. See also linage. 

Optical instrument, 23. 

Optical invariant of refraction, 76. 

Optical length, 89-91, 278, 279. 

Optical medium, 3; media of different refractivities, 70. 

Optical system, 23. 

Optometer of Badal, 422, 423. 

Origin of coordinates, 149. See also Image-equations. 

Orthoscopy, Conditions of, 543-545. 



Paraxial ray, Definition, 152. 

Paraxial rays, Diagrams showing imagery by means of, 168-171. 

Paraxial rays: Centered system, 329-334, 519-521; infinitely thin lens, 
217-257, 276-279, 285; plane refracting surface, 96-98, 191, 265- 
269; plate with parallel faces, 105-107; spherical mirror, 153-179, 
189, 274-276, 285; spherical refracting surface, 179-202, 269-274, 
285, 519, 534; thin lens-system, 289-291. 

Paraxial ray, Calculation of, 519-521. 

Pencil of rays, 24. 

Pendlebury, C. : Lenses and systems of lenses, 280. 

Penumbra, 7. 

Period of vibration, 473. 

Perspective in art, 22. 

Perspective, Center of, 159; so-called center (K), 532; pupil-centers as 
centers of perspective, 416, 417. 

Perspective elongation of image, 419, 420. 

Perspective ranges of points, 159-161. 

Perspective reproduction in screen-plane, 417. 

Petzval, J.: Curvature of image, 539. 

Photograph, Correct distance of viewing, 417-419. 



572 Index 

Pinhole camera, 5. See also "Fish-eye" camera. 

Plane image, Conditions of, 538-540, 548. 

Plane mirror: Conjugate points, 38; reflection of plane and spherical 

waves at, 33-37; image of extended object in, 37-40; uses of, 52; 

rotation of, 32, 56; field of view, 40-43; punctual imagery, 508; 

reflecting power, 380, 381. See also Inclined Mirrors, Mirror and 

scale, "Thick mirror," Sextant, Heliostat, etc. 
Plane Mirrors, Inclined, 43-51; rectangular combinations for rectifying 

image, 50, 51. 
Plane refracting surface: Caustic surface, 98, 99; narrow astigmatic 

bundle of rays, 98-100, 553; paraxial rays, 96-98, 191, 265-269; 

plane wave, 70-72; principle of least time, 87-89. 
Plane wave, 13; reflection at plane mirror, 33-35; refraction at plane 

surface, 70-72; refraction through prism, 123, 124; mechanical 

illustration, 72, 73. 
Plano-convex and plano-concave lenses, 219, 225. 
Piano-cylindrical lenses, 315-317. 
Plate (or slab) with plane parallel faces: Path of ray through, 101-103; 

refraction of paraxial rays, 105-107; apparent position of object 

viewed through plate at right angles to line of sight, 102, 103, and 

inclined to line of sight, 105-107; multiple images by reflection and 

refraction, 107-110; parallel plate micrometer, 103. 
Point of fixation, 432. 

Point-source of light, 1; apparent place and direction, 15-18. 
Porro, I.: Prism-system for rectification of image, 50, 51. 
Porta 's pinhole camera, 5. 
Porte lumiere, 53. 

Ports: See Entrance-port, Exit-port. 
Positive and negative directions along a straight line, 104; positive 

direction along the axis, 149, 219. 
Positive lens, 223. 
Posterior pole of eye, 432, 438. 
Power of lens or prism: See Prism, Prismatic power of lens, Reflecting 

power, Refracting power. 
Power of accommodation: See Accommodation. 
Prentice, C. F.: Crossed cylinders, 321; diagrams, 308, 309, 310; power 

of ophthalmic prism, 135. 
Presbyopia, 435. 

Principal planes, 335; of a thin lens, 239; of a spherical refracting sur- 
face, 196, 335. 
Principal point angle, 447; as measure of size of retinal image, 

448. 
Principal points, 334, 335; relation to nodal points, 341, 343; image 



Index 573 

equations referred to, 345-347; of combination of two lenses, 367, 
369, 370; of compound system, 361; of compound system of three 
members, 375; of infinitely thin lens, 239; of "thick mirror," 377- 
379, 383; of thick lens, 363. 

Principal points of eye, 374, 432; of eye in state of maximum accom- 
modation, 437; as points of reference, 437. 

Principal section of prism, 113. 

Principal sections: Of curved surfaces, 302, 525; of surface of revolution, 
305; of cylindrical surface, 306; of toric surface, 309; of toric lenses, 
310; of a bundle of rays, 304, 311-314, 528, 535. 

Prism, 85, 86, 113 and foil.; base-apex line, 134; edge, 113; refracting 
angle, 113, and its measurement, 55; principal section, 113. See 
also Thin prism, Ophthalmic prism. 

Prism, Dispersion by, 465 and foil. 

Prism, Path of ray through a: Calculation, 124, 125, and construction 
of, 113-116; deviation, 116; deviation away from edge, 122; "graz- 
ing" incidence and emergence, 117, 118; limiting incident ray, 118; 
minimum deviation, 119-122, 128-133, normal emergence, 129; 
symmetrical ray, 119-122, 129-133. 

Prism, Refraction of plane wave through, 123, 124. 

Prism-dioptry, 135, 294. 

Prism-system: Achromatic combination of two thin prisms, 491-493; 
direct vision prism combinations, 493 and foil.; direct vision 
prism of Amici, 495-497, and of Kessler, 497-499. 

Prismatic power of infinitely thin lens, 291-295. 

Problems, 25-27, 60-63, 92-94, 110-112, 142-148, 203-216, 249-257, 
295-299, 326-328, 351-355, 384-396, 423-424, 461-464, 505-507, 
551-557. 

Projected image and object, 415, 416. 

Pulfrich, C: Refractometer, 128. 

Punctual imagery, 313, 314, 397, 508, 509; in plane mirror, 508. 

Punctum ccecum (blind spot), 430. 

Punctum proximum (near point), 434, 435. 

Punctum remotum (far point), 434. 

Pupil of eye, 23, 401, 409-413, 421, 425. 

Pupils of optical system: See Entrance-pupil, Exit-pupil. 

Purity of spectrum, 469-471. 

Purkinje images by reflection in the eye, 378; calculation of equiv- 
alent optical system, 381, 382. 

Q 

Quartz, 485. 



574 Index 



Radius: Of curvature, 260; of spherical reflecting or refracting surface, 
150. 

Ramsden circle, 458. 

Ramsden ocular, 463, 502. 

Range of accommodation, 438. 

Rays, Chief: see Chief rays. 

Rays of finite slope, Chap. XV, 508, foil. 

Rays of light, 9; mutual independence, 15; meet wave-surface nor- 
mally, 13, 14, 89-91. See also Bundle of rays, Effective rays, Emer- 
gent rays, Image rays, Incident rays, Obiect-rays, Pencil of rays, etc. 

Ray-coordinates (or ray-parameters), 95, 517. 

"Real and "virtual," 17; images, 17, 18. 

Rectangular combinations of plane mirrors, 50, 51. 

Rectilinear propagation of light, 3-5. 

Reduced abscissa, and "vergence," 284-286, 348. 

Reduced distance, 279-281; reduced distance (c) between two optical 
systems, 360. 

"Reduced eye," 214, 437. 

Reduced focal lengths, 281; focal point "vergences," 284-286. 

Reflecting power of mirror, 283; plane mirror, 380, 381; "thick mirror " 
379. 

Reflecting surface, Quality of, 29, 30. 

Reflection, Angle of, 31, and laws of, 31. 

Reflection, Regular and irregular (diffuse), 28-30. 

Reflection as special case of refraction, 182, 183, 189. 

Reflection and refraction, Generalization of laws of, 86-89. 

Refracted ray, Construction of, 76-78; deviation, 78. See also Plane 
refracting surface, Spherical refracting surface, etc. 

Refracting angle of prism, 113; measurement of, 55. 

Refracting power, 281-284; in normal section of refracting surface, 303; 
of spherical refracting surface, 282, 300; of compound system of 
two members, 361, and of three members, 375; of thick lens, 363; of 
thin lens, 283, 284; of thin lens-system, 290; of combination of two 
lenses, 367. 

Refracting power of correction-glass, 444. 

Refracting power of schematic eye, 374, 432; in state of maximum 
accommodation, 437, 438, 439. 

Refraction of eye, 438, 439; dynamic, 438, and static refraction, 438. 

Refraction of light, 64, 65; angle of, 65; laws of, 66, and experimental 
basis, 67-69; mechanical illustration of, 72, 73. See also Index of 
Refraction, Total Reflection, etc. 



Index 575 

Resolving power of eye, 21, 22. 

Resultant prism equivalent to two thin prisms, 138-142. 

Retina, 428. 

Retinal image, Size of, 448, 449. 

Reversibility of light-path, 69. 

Rotary prism, 141. 

S 

Sagitta of arc, 262. 

Sagittal rays, 311. See Sagittal section of narrow bundle of rays. 

Sagittal section of narrow bundle of rays, 311-314, 528-530; symmetry 

in, 541. 
Scheiner, C: Astronomical and terrestrial telescopes, 456. 
Schematic eye: Far point, 461; focal lengths, 343, 374, 389, 432; focal 

points, 374, 389, 423, 432; length of eye-axis, 432, 442, 443; near 

point, 436, 461; optical constants, 370-374, 389, 432, 436, 437, 443, 

461; in state of maximum accommodation, 395, 436, 437, 461. 
Schott, O.: Optical glass, 482, 489. 
Sclerotic coat or sclera, 425. 
Screen-plane, 400, 402, 414-417, 419, 543. 
Searle, G. F. C: "Thick mirror," 376, 377. 
Secondary spectrum, 488. 
Segments of straight line, 104, 105. 
Seidel, L. Von : Theory of the five spherical aberrations, 545, 546, 547, 

548, 550, 557; curvature of image, 540; sine-condition, 523. 
Self-conjugate point, 243. 
Self-conjugate ray, 243. 
Sextant, 58-60. 
Shadows, 6-9. 

Sine-condition, 522-525, 547. 
Slab with plane parallel faces: See Plate. 
Slope of ray, 151, 334, 516. 
Smith, R.: Smith-Helmholtz formula, 201, 202, 214, 215, 334, 383, 

312, 459, 524. 
Snell (or Snellius), W.: Law of refraction, 67, 72. 
Spectrum, 466 and foil.; purity of, 469-471. 
Spectrum, Solar, 466 and foil.; Newton's experiments, 465 and foil. 

Wollaston's experiments, 469, 470; Fraunhofer's experiments, 

472; dark lines, 472. 
Spherical aberration, Chap. XV, 509, 513 and foil.; along the axis, 

513-516, 518, 522, 547. 
"Spherical lens," 217. 



576 Index 

Spherical Mirror, Ray reflected at, 518, 519. 

Spherical mirror, Paraxial Rays: 153-179, 189, 274-276, 285; abscissa 
formula, 154, 285; construction of conjugate axial points, 164-166 
focal points, 166, 189; focal length, 167; Newtonian formula, 168 
extra-axial conjugate points, 171-173; construction of image, 173 
imagery, 174, 175; lateral magnification, 176; field of view, 176- 
179; reflecting power, 283; spherical wave reflected at spherical 
mirror, 274-276. See also "Thick Mirror." 

Spherical over- and under-correction, 514, 515. 

Spherical refracting (or reflecting) surface: Axis, 149; convex and 
concave, 150; convergent and divergent, 186; magnifying power, 
199; radius, 150; vertex, 149. 

Spherical refracting surface: Aplanatic points, 512, 513, 524; calcula- 
tion of refracted ray, 516-519; construction of refracted ray, 509- 
512; formulae for refracted ray, 517-519. 

Spherical refracting surface, Astigmatism of oblique bundle of rays, 
526-534, 553, 554, 556. 

Spherical refracting surface, Paraxial rays: 179-202, 269-274, 285, 519, 
534; abscissa formula, 191, 193, 285; conjugate axial points, 
179-186, 191, 192; conjugate planes, 193, 194; construction of 
image, 194-196; construction of refracted ray, 199, 200; extra- 
axial conjugate points, 193-196; focal lengths, 191-193, 199; 
focal planes, 197-199; focal points, 186-189; image-equations, 200, 
201; lateral magnification, 196; refracting power, -179-202; re- 
fraction of spherical wave, 269-276. 

Spherical wave reflected at plane mirror, 35-37; at spherical mirror, 
27^-276. 

Spherical wave refracted at plane surface, 265-269; at spherical surface, 
269-274; through infinitely thin lens, 276-279. 

Spherical zones, 515, 516. 

Sphero-cylindrical lens, 217, 315, 317. 

Spherometer, 263. 

Static refraction of eye, 438, 440, 441, 442, 443; connection with length 
of eye-ball in case of axial ametropia, 442, 443; relation with re- 
fracting power or vertex refraction of correction-glass, 444-447. 

Steinheil, A.: Data of "periscope" photographic lens, 554, 555, 556; 
achromatic object-glass, 505. 

Steinheil, R.: Calculation of object-glass of telescope, 520. 

Stigmatic (or anastigmatic) lenses, 314. 

Stokes, Sir G. G.: Optical glass, 482. 

Stop, Effect of, 398, 399; front, rear or interior stop, 398. See also 
Aperture-stop, Field-stop, etc. 

Sturm, J. C. F.: Conoid, 310, 313, 534, 535. 



Index 577 

Surface of revolution, 305; meridian section, 305; principal sections, 

305. 
Surfaces, Theory of curved, 300-303, 525, 526; normal sections, 300- 

303, 525, 526; principal sections, 302, 525. 
Suspensory ligament, 428, 434. 
Symmetric lens, 217, 385, 388. 
Symmetric points, 339. 



Tangent-condition of orthoscopy, 545. 

Telecentric optical system, 420-423. 

Telescope: see Astronomical telescope, Dutch (or Galilean) telescope, 

Terrestrial telescope. 
Telescope: Eye-ring or Ramsden circle, 413, 458, 459; magnifying 

power, 445-460; invention, 456, 457; object-glass and ocular, 455; 

simple schematic telescope, 455. 
Telescopic imagery, 359 
Telescopic system, 359. 
Tenth-meter, 10, 475. 
Terrestrial telescope, 457. 
Thick lens, 362-366; focal points, nodal points, principal points, and 

refracting power, 363; vertex refraction, 365, 366. 
"Thick mirror," 376-384, 392, 393; principal points, 377-379, 383; 

reflecting power, 379. 
Thin lens: see Infinitely thin lens, Infinitely thin lens-system. 
"Thin mirror," 377. 
Thin prism: combination of two thin prisms, 138-142; deviatitn, 133, 

134 and power, 134-138. See also Ophthalmic prism. 
Thin prisms, Achromatic combination of, 491-493; and direct-vision 

combination of, 493-495. 
Thompson, S. P., 38, 135; axial (or depth) magnification, 351; obliquely 

crossed cylindrical lenses, 321; symmetric points of optical system, 

338. 
Toepler, A.: Negative principal points optical system, 338. 
Toric lens, 310, 314, 316, 317. 
Toric surface, 265, 305, 306, 308-310, 320. 
Total reflection, 79-86; experimental illustrations, 83-89. See also 

Prism. 
Total reflection prism, 85, 86, 125, 127. 
Translucent body, 3. 
Transparent body, 2. 

Transposing of cylindrical lenses, 318-320. 
Tscherning, M.: Physiological Optics, 287. 



578 Index 

U 

Umbra, 7. 

Undulatory theory of light: see Wave Theory. 

Unit planes and unit points of optical system, 335. 



Velocity of light in different media, 72-75, 475; varies with color, 474; 

in vacuo, 10, 75, 474, 476. 
Verant, 418. 

Vertex of spherical surface, 149; of cornea, 431. 
Vertex refraction of lens, 365, 366; of correction-glass, 445, 446. 
"Vertex-depth" of concave surface of meniscus lens, 298. 
Vertices of lens, 219. 
Vibration frequency and color, 472 and foil.; and wave-length, 473 

and foil. 
"Virtual" and "real," 17; images, 17, 18. 
Virtual image, 17, 18; in case of plane mirror, 38. 
Virtual object in case of plane mirror, 38. 
Vision, "Direct," 448; and "indirect," 446. 
Vision, Distance of distinct, 452, 453. 
Visual angle, 20, 446 and foil.; principal point angle, 447, 448; focal 

point angle, 447, 449. 
Visual axis, 433. 
Visual purple, 430. 
Vitreous humor, 213, 371, 428. 
Von Rohr, M.: Abbreviation "dptr.," 287; verant, 418; Theorie und 

Geschichte d. photograph. Objektivs, 555. 

W 

Wave-front, Plane, 13, and spherical, 11. See also Plane wave, Spherical 
wave, Huygens, Malus. 

Wave-length, in vacuo, 5, 475; wave-length and frequency, 475; wave- 
length and index of refraction, 476, 477; wave-length and color, 
474-477. 

Wave-surface, Rays normal to, 13, 14, 89-91, 525. 

Wave-theory of light, 9, 10, 472 and foil., 508. 

Wollaston, W. H.: Dark lines of solar spectrum, 472; dispersion ex- 
periments, 469. 

Wood, R. W.: "Fish-eye" camera, 81; velocity of light of different 
colors, 474. 



Index 579 



Yellow spot (or macula lutea), 428. 

Young, T.: center of perspective (K), 532; construction of ray re- 
fracted at spherical surface, 509, 510, 511, 527, 552; principle of 
interference, 14. 

Z 

Zinn's zonule (or suspensory ligament), 428. 



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QC385.S7 



ASTRO 



. 3 5002 00188 2526 

Southall, James Powell Cocke 

Mirrors, prisms and lenses; a text-book 





QC 
385 

57 






AUTHOR 

Southall 






96184 


TITLE 

M-i T»T»ors . 


■orisms 


and 


lenses 



Astronomy Library 

QC 

385 

S7 96184