Google
This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other maiginalia present in the original volume will appear in this file - a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing tliis resource, we liave taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, non-commercial purposes.
+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for in forming people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liabili^ can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at|http: //books .google .com/I
Phys ^7^0 , ^0
HARVARD COLLEGE
LIBRARY
FROM THE
FARRAR FUND
The hetgukut of Mrt. Eliaa Farrar in
memory cfher husband, John Farrar,
HoUie PnfesMor of MathemaHes,
Aetronomy and Natural Fhiioeophy,
t807'18S6
®
THE
PRINCIPLES AND METHODS
OF
GEOMETRICAL OPTICS
ESPECIALLY AS APPLIED TO THE
THEORY OF OPTICAL INSTRUMENTS
BY
JAMES P. C. SOUTHALL
Professor of Physics in the Alabama Polytechnic Institute
Neto Yortt
THE MACMILLAN COMPANY
LONDON: MACMILLAN & CO., LTD.
IQIO
lku3 U
lSd,lo
'^^' — ^ Q.
^a^La/L,
Copyright, 1910
Bv JAMES P. C. SOUTHALL
PRtss or
The New Era printin« company
Lancaster. Pa.
TO
HENRY C. LOME, Esq.
WHOSE KIND ENCOURAGEMENT AND EFFECTUAL AID
WILL ALWAYS BE REMEMBERED
THIS VOLUME IS GRATEFULLY INSCRIBED BY
THE AUTHOR
PREFACE.
From time to time, almost like a voice crying in the wilderness,
some one is heard to lament the apathy with which Geometrical Optics
is regarded in this country and in England;^ although it is sufficient
merely to call the roll of such names as Barrow, Newton, Cotes,
Smith, Blair, Young, Airy, Hamilton, Herschel, Rayleigh, etc.,
in order to be reminded that this domain of science was once at any
rate within the sphere of British influence. At present, however, it
can hardly be gainsaid that the great province of applied optics is
almost exclusively German territory; so that not only is it a fact that
nearly all of the extraordinary developments of modern times in both
the theory and construction of optical instruments are of German
origin, but it is equally true also that until at least quite recently*
there was actually no treatise on Optics in the English language where
the student could find, for example, hardly so much as a reference to
the remarkable theories of Petzval, Seidel and Abbe — to mention
only such names as are inseparably associated with the theory of optical
imagery. Partly with the object of supplying this deficiency, and
partly also in the hope (if I may venture to express it) of rekindling
among the English-speaking nations interest in a study not only
abundantly worthy for its own sake and undeservedly neglected, but
still capable, under good cultivation, of yielding results of far-reaching
^Referring to Czapski's Theory of Optical Instruments (the first edition of which was
published in 1893) and to the volume on Optics in the ninth (1895) edition of Mueller-
Pouillet's Physics, Professor Silvanus P. Thompson, in the preface of his valuable
translation of Dr. O. Lummer's Contributions to Photographic Optics (London. 1900).
writes as follows:
"Both these works are in German, and most unfortunately no translation of either has
appeared — most unfortunately, for there is no English work in optics that is at all com-
parable to either of these. I say so deliberately, in spite of the admirable article by Lord
Rayleigh on 'Optics' in the Encyclopadia Britannica, in spite of the existence of those
excellent treatises. Heath's Geometrical Optics, and Preston's Theory of Light, No
doubt such books as Heath's Geometrical Optics and Parkinson's Optics are good in
their way. They serve admirably to get up the subject for the Tripos; but they are far
too academic, and too remote from the actual modern applications. In fact, the science
of the best optical instrument-makers is far ahead of the science of the text-books. The
article of Sir John Herschel 'On Light' in the Encyclopedia Metropolitana of 1840
marks the culminating point of English writers on optics."
«H. Dennis Taylor's A System of Applied Optics (London, 1906) is a most valuable
work by the inventor of the celebrated "Cooke" lenses for photography.
vi Preface.
importance in nearly every field of scientific research, I have prepared
the following work, wherein my endeavour has been to lay before the
reader a connected exposition of the principles and methods of Geo-
metrical Optics, especially such as are applicable to the theory of
optical instruments; and although I am regretfully aware of many
shortcomings in the execution of this task, I cling to the hope that
they will be perhaps not so apparent to many of my critics as they are
to myself.
I have not hesitated to use, especially in connection with the geo-
metrical theory of optical imagery in Chapters V and VII, the elegant
and direct methods of the modern geometry, but these applications
are always so simple and elementary that it is hardly to be feared that
any readers will be deterred thereby.
In the theory of optical imagery developed by Gauss with such rare
analytical skill, it is assumed that both the aperture and the field of
view of the optical system of centered spherical surfaces is exceedingly
small, so that all the rays concerned in the production of the image
are comprised within a narrow cylindrical region immediately sur-
rounding the optical axis. In the design of telescopes with objectives
of considerable diameter, the necessity of taking account of the so-
called spherical aberration due to the increase of the aperture was first
recognized; which led to the well known investigations on this subject
of EuLER, Bessel, Airy, Gauss, Seidel and others. With the de-
velopment of the microscope and the birth and growth of photography,
new requirements had to be filled in order to portray parts of the
object which were not situated on the optical axis, so as to correct,
if possible, the aberrations due not only to increase of the aperture
but also to increase of the field of view. This difficult problem, under-
taken first by Petzval with only partial success, was investigated by
Seidel, professor of mathematics in the University of Munich, in a
series of papers contributed to the Astronomische Nachrichten in the
year 1855; wherein, by an extension of Gauss's methods so as to
include in the series-developments the terms of the next higher order,
elegant and entirely general formulae are derived in a comparatively
simple way, which enable one to perceive almost at a glance how the
faults in an image formed by a centered system of spherical refracting
surfaces are due partly to the size of the aperture and partly also to
the extent of the field of view. These methods and theories are treated
at length in Chapter XII.
Prism-Spectra and the Chromatic Aberrations of Dioptric Systems
are the subjects that are included under the head of "Colour-Phe-
nomena" in Chapter XIII.
Preface. vu
One of the most important divisions is Chapter XIV, wherein the
reader will find a fairly complete treatment of Abbe's theory of the
limiting of the ray-bundles by means of perforated diaphragms or
"stops," which has so much to do with the practical efficiency of an
actual optical instrument.
Without entering more fully into the contents of the various chap-
ters, it may be stated that the work as a whole is designed as a general
introduction to the special theory of optical instruments (telescope,
microscope, photographic objective, etc., including also the eye itself).
To discuss properly and fully each of these types would require a
separate and extensive volume, which I may be induced to undertake
at some future time as a sequel to the present work.
A complete system of notation which is free from objection is dif-
ficult to devise; and, in spite of the pains I have bestowed on the
matter and the importance which I have attached to it, I do not
doubt that fault will be found not only with the plan which I have
adopted but with many of the characters which I have introduced.
My object has been to make the work convenient as a book of reference,
so that the meaning of a symbol and of the marks that distinguish it
would be immediately obvious as far as possible; but in order to aid
the reader still further in this respect, the principal uses of the letters
in both the diagrams and the formulae are quite fully explained in an
appendix at the end of the volume. In some instances the same letter
or sign has been employed deliberately in two or more totally different
senses, but only where there seemed to be no chance of confusion, and
because also I have tried carefully to avoid resorting to strange and
uncouth symbols which often make a mathematical work appear to
be far more difficult and uncanny than it really is.
The original sources from which I have borrowed have been given,
as far as possible, either in the text or in the foot-notes. I am espe-
cially aware of how much I have derived in one way or another from
Dr. CzAPSKi's epoch-making book. Die Theorie der optischen Instru-
mente nach Abbe, and from Die Theorie der optischen Instrumente,
Bd. I (Berlin, 1904) edited by Dr. M. von Rohr under the auspices
of CzAPSKi himself and in collaboration with the staff of optical
engineers connected with the world-famous establishment of Carl
Zeiss in Jena. This latter work — which is, in fact, the offspring of
the former, and in whose praise one might well exclaim, **0 matre
pulchra filia pulchrior!*' — is a vast treasury of optical theory amassed
by experts in the various branches of Geometrical Optics which will
remain for many years to come the standard book of reference on this
subject.
viii • Preface,
I gladly take this opportunity of expressing my thanks to
Professor Charles Hancock, of the University of Virginia, who
made the drawings of the diagrams, and to my colleague Professor
A. H. Wilson and my assistant Mr. C. D. Killibrew who have
helped me with the proof-reading. I esteem it a pri\41ege to be per-
mitted to dedicate the work to Henry C. Lome, Esq., of Rochester,
N. Y.
James P. C. Southall.
AuBVRN. Ala..
December i. 1909.
CONTENTS.
CHAPTER I.
Page
Metiiods and Fundamental Laws of Geometrical Optics,
Arts, i-io, §§ 1-34 1-32
Art. I. The Theories of Light, §§1,2 1,2
Art. 2. The Scope and Plan of Geometrical Optics, §§ 3, 4 2, 3
Art. 3. The Rectilinear Propagation of Light, §§ 5-8 3-8
§ 6. HuYGENS's Construction of the Wave-Front 4
§ 7. Fresnel's Extension of Huygens's Method 7
Art. 4. Rays of Light, §§9, 10 8-10
§ 9. Principle of the Mutual Independence of the Rays of Light. 8
Art. 5. The Behaviour of Light at the Surface of Separation of two
Isotropic Media, §§ 11-13 ia-12
Art. 6. The Laws of Reflexion and Refraction, §§ 14-22 13-20
§ 15. The Laws of Reflexion and Refraction 13
§ 18. Principle of the Reversibility of the Light-Path 15
§ 19. The Laws of Reflexion and Refraction as derived by tb^
Wave-Theory (Huygens's Construction) 16
§ 20. HuYGENs's Construction of Reflected Wave 18
§ 21. HuYGENS's Construction of Refracted Wave 18
Art. 7. Absolute Index of Refraction of an Optical Medium, §§ 23-26 20-22
§ 24. Absolute Index of Refraction 20
§ 26. Reflexion Considered as a Special Case of Refraction 22
Art. 8. The Case of Total Reflexion, § 27 22-25
Art. 9. Geometrical Constructions, etc., §§ 28-30 25-28
§ 28. Construction of the Reflected Ray 25
§ 29. Construction of the Refracted Ray 26
§ 30. The Deviation of the Refracted Ray 27
Art. 10. Certain Theorems Concerning the case of so-called Oblique
Refraction (or Reflexion), §§ 31-34 28-32
CHAPTER II.
Characteristic Properties of Rays of Light, Arts. 11-15,
§§ 35-49 33-50
Art. II. The Principle of Least Time (Law of Perm at), §§ 35-38 33-36
§38. The Optical Length of a Ray; and the Principle of the
Shortest Route 35
ix
X G>ntents.
Page
Art. 12. Hamilton's Characteristic Function, §§ 39-41 36-39
Art. 13. The Law of Malus, §§ 42, 43 39f 40
Art. 14. Optical Im^es, §§44, 45 40-42
Art. 15. Character of an Infinitely Narrow Bundle of Optical Rays,
§§ 46-49 42-50
§ 46. Caustic Surfaces 42
CHAPTER HI.
Reflexkm and Refraction of Lic^t-Rays at a Plane
Smfacei Arts. 16-20, §§ 50-64 51-73
Art. 16. The Plane Mirror, §§ 50, 51 51-55
Art. 17. Trigonometric Formulae for Calculating the Path of a Ray
Refracted at a Plane Surface. Imagery in the case of
Refraction of Paraxial Rays at a Plane Surface, §§ 52, 53. . 55-59
§ 53. Refraction of Paraxial Rays at a Plane Surface 57
Art. 18. Caustic Surface in the case of a Homocentric Bundle of
Rays Refracted at a Plane Surface, §§ 54, 55 59-64
Art. 19. Astigmatic Refraction of an Infinitely Narrow Bundle of
Rays at a Plane Surface, §§ 56-62 64-70
§ 57. The Meridian Rays 65
§ 58. The Sagittal Rays 65
§ 59. Position of the Primary Image-Point y, and Convergence-
Ratio of Meridian Rays 67, 68
§ 60. Position of the Secondary Image-Point 3', and Convergence-
Ratio of Sagittal Rays 68, 69
§ 61. Astigmatic Difference 69
§62. Refraction at a Plane Surface of an Infinitely Narrow
Astigmatic Bundle of Incident Rays 69
Art. 20. Refraction of Infinitely Narrow Bundle of Rays at a Plane:
Geometrical Relations between Object-Points and Image-
Points, §§ 63, 64 70-73
§ 64. Construction of the I. Image-Point 71
CHAPTER IV.
Refraction tfarough a Prism or Prism-System,
Arts. 21-31, §§ 65-107 74-133
Art. 21. Geometrical Construction of the Path of a Ray Refracted
through a Prism in a Principal Section of the Prism,
§§ 65-69 ^ 74-80
Art. 22. Analytical Investigation of the Path of a Ray Refracted
through a Prism in a Principal Section, §§ 70-73 80-87
§ 71. Analytical Investigation of the case of Minimum Deviation 81
Contents. xi
Pagb
§ 72. Other Special Cases 83
Art. 23. Path of a Ray Refracted across a Slab with Parallel Faces,
§§ 74» 75 88-90
Art. 24. Refraction, through a Prism, of an Infinitely Narrow,
Homocentric Bundle of Incident Rays, whose Chief Ray
lies in a Principal Section of the Prism, §§ 76-81 90-97
§76. Construction of the I. and II. Image-Points corresponding
to a Homocentric Object- Point 90
§ 77. Formulae for Calculation of the Positions on the Chief
Emergent Ray of the I. and II. Image-Points 92
§ 78 and § 79. Convergence-Ratios of the Meridian and Sagittal
Rays 93, 94
§ 80. The Astigmatic Difference 94
§ 81. Magnitude of the Astigmatic Difference in Certain Special
Cases : 95
Art. 25. Homocentric Refraction, through a Prism, of Narrow, Homo-
centric Bundle of Incident Rays, with its Chief Ray lying
in a Principal Section of the Prism, §§ 82-85 97"I05
§ 83. Analytical Method 98
§ 84. Geometrical Investigation (according to Burmester) 99
Art. 26. Apparent Size of Image of Illuminated Slit as seen through
a Prism, § 86 105, 106
Art. 27. Astigmatic Refraction of Infinitely Narrow, Homocentric
Bundle of Incident Rays across a Slab with Plane Parallel
Faces, §§ 87-90 106-111
§ 87. Construction of the I. and II. Image-Points 106
§ 88. Formulae for the Determination of the Positions of the I.
and II. Image-Points 107
§ 89. Astigmatic Difference in case of a Slab 108
Art. 28. Path of a Ray Refracted through a system of Prisms, in the
case when the Refracting Eldges of the Prisms are all
Parallel, and the Ray lies in a Principal Section Common
to all the Prisms, §§ 91-94 111-115
§ 92. Construction of the Path of the Ray 112
§ 93. Formulae for the Trigonometrical Calculation of the Path of
the Ray through the System of Prisms 113
§ 94. Condition that the Total Deviation shall be a Minimum ... 114
Art. 29. Refraction, through a System of Prisms, of an Infinitely
Narrow, Homocentric Bundle of Incident Rays: the Chief
Ray thereof lying in a Principal Section Common to all the
Prisms, §§ 95-99 115-123
§ 95. Geometrical Construction of the I. and II. Image-Points. . . 115
§96. Formulae for Calculation of the Positions on the Chief
Emergent Ray of the I. and II. Image-Points 117
xii G>ntents.
Pack
§ 97. The Convergence-Ratios of the Meridian and Sagittal Rays 120
§ 98. Formula for the Astigmatic Difference 121
§ 99. Homocentric Refraction through a System of Prisms 122
Art. 30. Path of a Ray Refracted Obliquely through a Prism, §§ 100-
103 123-128
§ 100. Construction of the Path of the Ray 123
§ loi. Formulae for Calculating the Path of a Ray Refracted
through a Prism obliquely 124
§ 102. Deviation (D) of Ray Obliquely Refracted through a
Prism 125
Art. 31. Homocentric Refraction through a Prism of an Infinitely
Narrow, Homocentric Bundle of Obliquely Incident Rays,
§§ 104-107 128-133
CHAPTER V.
Reflexion and Refraction of Paraxial Rays at a
Spherical Surface, Arts. 32-38, §§ 108-134 134-173
Art. 32. Introduction. Definitions, Notations, etc., §§ 108, 109. . . . 134-136
§ 109.* Paraxial Ray 136
I. Reflexion of Paraxial Rays at a Spherical
Mirror, Arts. 33, 34, §§ iia-117 I37-I47
Art. 33. Conjugate Axial Points in the case of Reflexion of Paraxial
Rays at a Spherical Mirror, §§ i ia-112 137-142
§ 112. Focal Point and Focal Length of a Spherical Mirror 140
Art. 34. Extra-Axial Conjugate Points and the Lateral Magnification
in the case of the Reflexion of Paraxial Rays at a Spherical
Mirror, §§ 113-117 142-147
§ 113. Graphical Method of Showing the Imagery by Paraxial
Rays 142
§ 116. The Lateral Magnification 145
II. Refraction of Paraxial Rays at a Spherical
Surface, Arts. 35-37, §§ 1 18-129 147-162
Art. 35. Conjugate Axial Points in the case of the Refraction of
Paraxial Rays at a Spherical Surface, §§ 1 18-120 147-152
§ 119. Construction of the Image-Point M' conjugate to the Axial
Object-Point M 149
§ 120. The Focal Points Fand E' of a Spherical Refracting Surface 150
Art. 36. Refraction of Paraxial Rays at a Spherical Surface. Extra-
Axial Conjugate Points. Conjugate Planes. The Focal
Planes and the Focal Lengths, §§ 121-124 153-158
§ 122. The Construction of the Image-Point Q' Corresponding to
Contents. xiii
Pagb
the Extra-Axial Object-Point Q 153
§ 123. The Focal Planes of a Spherical Refracting Surface 154
§ 124. The Focal Lengths/ and e' of a Spherical Refracting Surface 155
Art. 37. The Image-Equations in the case of the Refraction of Par-
axial Rays at a Spherical Surface, §§ 125-129 158-162
§ 125. The Abscissa-Equation in Terms of the Constants «, «'
and r 158
§ 126. The so-called Zero-Invariant 159
§ 127. The Lateral Magnification 160
§ 128. The Image-Equations in Terms of the Focal Lengths/, «'. . 161
III. Supplement: Containing Certain Simple Appli-
cations OF THE Methods of Projective
Geometry, Art. 38, §§ 130-134 162-173
Art. 38. Central Collineation of Two Plane-Fields, §§ 130-134 162^173
§ 131. Projective Relation of Two Collinear Plane-Fields 163
§ 132. Geometrical Constructions 165
§ 133. The Invariant in the Case of Central Collineation 168
§ 134. The Characteristic Equation of Central Collineation 170
CHAPTER VI.
Refraction of Paraxial Rays through a Thin Lens or tfarough
a System of Thin Lenses, Arts. 39-42, §§135-152 174-197
Art. 39. Refraction of Paraxial Rays through a Centered System of
Spherical Surfaces, §§ 135-139 174-179
§ 135. Centered System of Spherical Surfaces 174
§ 138. The Lateral Magnification Y 178
§ 139. The Principal Points of a Centered System of Spherical
Surfaces 178
Art- 40. Types of Lenses. Optical Centre of Lens, §§ 140-142 179-182
§ 142. Optical Centre of Lens 181
Art. 41. Formulae for the Refraction of Paraxial Rays through an
Infinitely Thin Lens, §§ 143-149 182-191
§ 144. Conjugate Axial Points in the case of the Refraction of Par-
axial Rays through an Infinitely Thin Lens 183
§ 145. The Focal Points of an Infinitely Thin Lens 184
§ 146. The Focal Lengths /and «' of an Infinitely Thin Lens 186
§ 147. Lateral Magnification of an Infinitely Thin Lens 187
§ 148. Construction of the Image Formed by the Refraction of
Paraxial Rays through an Infinitely Thin Lens 187
§ 149. Refraction of Paraxial Rays through a Combination of
Infinitely Thin Lenses 190
Art. 42, CoTEs's Formula for the "Apparent Distance" of an Object
viewed through any number of Thin Lenses, §§ 150-152. . 19* -197
xiv Contents.
CHAPTER VII.
The Geometrical Theory of Optical Imageiyt
Arts. 43^52A§ 153-187 i98r-262
I. Introduction, Art. 43, §§153-156 198-201
Art. 43. Abbe's Theory of Optical Imagery, §§ 153-156 198-201
II. The Theory of Collineation, with Special Refer-
ence TO ITS Applications to Geometrical
Optics, Arts. 44-47, §§ 157-171 201-217
Art. 44. Two Collinear Plane-Fields, §§ 157-161 201^206
§ 157. Definitions 201
§ 158. Projective Relation of Two Collinear Plane-Fields 202
§ 159. The so-called "Flucht** Points of Conjugate Rays 203
§ 160. The so-called "Flucht** Lines (or Focal Lines) of Conjugate
Planes 204
§ 161. Affinity of Two Plane-Fields 206
Art. 45. Two Collinear Space-Systems, §§ 162-165 206-210
§ 164. The so-called "Flucht" Planes, or Focal Planes, of Two
Collinear Space-Systems 208
§ 165. Affinity- Relation between Object-Space and Image-Space. 209
Art. 46. Geometrical Characteristics of Object-Space and Image-
Space, §§ 166-168 210-212
§ 166. Conjugate Planes 210
§ 167. The Focal Points and the Principal Axes of the Object-
Space and the Image-Space 211
§ 168. Axes of Co-ordinates 212
Art. 47. Metric Relations, §§ 169-171 213-217
§ 169. Relations between Conjugate Abscissae 213
§ 170. The Lateral Magnifications 214
§ 171. The Image-Equations 216
III. Collinear Optical Systems, Arts. 48-52,
§§ 172-187 218-262
Art. 48. Characteristics of Optical Imagery, §§ 172-176 218-229
§ 172. Signs of the Image-Constants a, b and c 218
§ 174. Symmetry around the Principal Axes 221
§ 175. The Different Types of Optical Imagery 223
Art. 49. The Focal Lengths, Magnification-Ratios, Cardinal Points,
etc., §§ 177-182 229-242
§ 177. Analytical Investigation of the Relation between a Pair of
Conjugate Rays 229
§ 178. The Focal Lengths / and e^ 232
G)ntents. xv
Pagb
§ 179. The Magniiication-Ratios and their Relations to one an-
other 234
§ 180. The Cardinal Points of an Optical System 236
§ 181. The Image-Equations referred to a Pair of Conjugate Axial
Points 239
§ 182. Geometrical Constructions of Conjugate Points of an Op-
tical System 241
Art. 50. Telescopic Imagery, §§ 183, 184 243-245
§ 183. The Image-Equations in the Case of Telescopic Imagery. . 243
§ 184. Characteristics of Telescopic Imagery 244
Art. 51. Combination of Two Optical Systems, §§185, 186 245-255
§ 185. The Problem in General 245
§ 186. Special Cases of the Combination of Two Optical Systems 251
Art. 52. General Formulae for the Determination of the Focal Points
and Focal Lengths of a Compound Optical System, § 187. . 255-262
CHAPTER VIII.
Ideal Imageiy by Paraxial Rays. Lenses and Lens-
Systems, Arts. 53-58, §§ 188-204 263-286
Art. 53. Introduction, §§ 188, 189 263, 264
Art. 54. The Focal Lengths of a Centered System of Spherical Sur-
faces, §§ 190-193 264-267
§ 193. Ratio of the Focal Lengths/ and e' 266
Art. 55. Several Important Formulae for the case of the Refraction of
Paraxial Rays through a Centered System of Spherical
Surfaces, §§ 194-196 267-273
§ 194. Robert Smith's Law 267
§ I95« Formulae of L. Seidel 269
Lenses and Lens-Systems, Arts. 56-58, §§ 197-204. . 273-286
Art. 56. Thick Lenses, §§ I97-I99 273-283
§ 199. Character of the Different Forms of Lenses 276
Art. 57. Thin Lenses, §§ 200, 201 283-284
§ 201. Infinitely Thin Lenses 284
Art. 58. Lens-Systems, §§ 202-204 284-286
CHAPTER IX.
Eiact Methods of Tmdng the Path of a Ray Refracted
at a Spherical Surface, Arts. 59-65, §§ 205-220 287-315
Art. 59. Introduction, § 205 287, 288
Art. 60. Geometrical Method of Investigating the Path of a Ray
Refracted at a Spherical Surface, §§ 206-208 288-294
{ 206. Construction of the Refracted Ray 288
{ 207. "Aplanatic" Pair of Points of a Spherical Refracting Surface 290
{ 208. Spherical Aberration 292
xvi Contents.
Pack
Trigonometric Computation of the Path of a Ray of Finite Inclina-
tion TO THE Axis, Refracted at a Single Spherical
Surface, Arts. 61-65, §§ 209-220 294-315
Case I. When the Path of the Ray Lies in a Principal
Section of the Spherical Refracting Surface,
Arts. 61-63, §§ 209-212 294-304
Art. 61. The Ray-Parameters, and the Relations between them,
§§ 209, 210 294-297
Art. 62. Trigonometric Computation of the Path of the Refracted
Ray, § 211 298-302
Art. 63. Formulae for Finding the Point of Intersection and the
Inclination to each other of a Pair of Refracted Rays lying
in the Plane of a Principal Section of the Spherical Refract-
ing Surface, § 212 302-304
Case II. When the Path of the Ray Does not Lie in
A Principal Section of the Spherical Refracting
Surface, Arts. 64, 65, §§ 213-220 3<H-3i5
Art. 64. Parameters of Oblique Ray, §§ 213-215 304-310
§ 214. Method of A. Kerber 305
§ 215. Method of L. Seidel 307
Art. 65. Trigonometric Computation of Path of Ray Refracted Ob-
liquely at a Spherical Surface, §§ 216-220 310-315
§ 216. The Refraction-Formulae of A. Kerber 310
§ 219. The Refraction-Formulae of L. Seidel 313
CHAPTER X.
Trigonometric Formulae for Calculating the Path of
a Ray through a Centered System of Spherical
Refracting Surfaces, Arts. 66-69, §§ 221-229 316-330
Case L When the Ray Lies in the Plane of a
Principal Section, Arts. 66, 67, §§ 221-224 316-321
Art. 66. Calculation-Scheme for the Path of a Ray lying in the Plane
of a Principal Section of a Centered System of Spherical
Refracting Surfaces, §§ 221-223 316-318
Art. 67. Numerical Illustration, § 224 318-321
Case IL When the Path of the Ray Does not Lie
IN THE Plane of a Principal Section of the
Centered System of Spherical Refracting
Surfaces, Arts. 68, 69, §§ 225-229 322-330
Art. 68. Trigonometric Formulae of A. Kerber for Calculating the
Path of an Oblique Ray through a Centered System of
Spherical Refracting Surfaces, §§ 225, 226 322-325
G>ntents. xvii
Pagb
§ 226. The Initial Values 323
Art. 69. Trigonometric Formulae of L. Seidel for Calculating the
Path of an Oblique Ray through a Centered System of
Spherical Refracting Surfaces, §§ 227-229 325-330
§ 228. Sbidel's "Control" Formulae 328
§ 229. The Initial Values 329
CHAPTER XL
General Case of the Refraction of an Infinitely Narrow Bundle
of Rajs tiirough an Optical System. Astigmatism,
Arts. 70-78, §§ 230-251 331-366
Art. 70. General Characteristics of a Narrow Bundle of Rays Re-
fracted at a Spherical Surface, §§ 230-232 331-336
§ 230. Meridian and Sagittal Rays 331
§ 231. Different Degrees of Convergence of the Meridian and
Sagittal Rays 333
§ 232. The Image-Lines 334
Art. 71. The Meridian Rays, §§ 233-237 336-343
§ 233. Relation between Object-Point S and the I. Image-Point S' 336
§ 234. Centre of Perspective K 339
§ 235. The Focal Points / and i' of the Meridian Rays ^ 340
§ 236. Formula for Calculating the Position of the I. Image-Point
5' corresponding to an Object-Point 5 on a given incident
chief ray u 342
§ 237. Convergence-Ratio of Meridian Rays 342
Art. 72. The Sagittal Rays, §§ 238-241 343-345
§ 238. Relation between the Object-Point 3 and the II. Image-
Point 5' ._. 343
§ 239. The Focal Points 7, /' of the Sagittal Rays 343
§240. Formula for Calculating the Position of the II. Image-
Point y corresponding to an Object-Point 5 on a given
chief incident ray u 344
{ 241. Convergence-Ratio of the Sagittal Rays 345
Art. 73. The Astigmatic Difference, and the Measure of the Astigma-
tism, § 242 345-347
Art. 74. Historical Note, concerning Astigmatism, § 243 347, 348
Art. 75. Inquiry as to the Nature and Position of the Image of an
Extended Object formed by Narrow Astigmatic Bundles
of Rays, § 244 349*351
Art. 76. Collinear Relations in the case of the Refraction of a Narrow
Bundle of Rays at a Spherical Surface, §§ 245, 246 35i"356
{ 245. The Principal Axes of the Two Pairs of Collinear Plane
Systems 351
xviii Contents.
Pack
§ 246. The Focal Lengths 354
Art. 77. Refraction of Narrow Bundle of Rays through a Centered
System of Spherical Refracting Surfaces, §§ 247, 248 356-360
§ 247. Formulae for Calculating the Astigmatism of the Bundle
of Emergent Rays 356
§ 248. Collinear Relations 358
Art. 78. Special Cases, §§ 249-251 360-366
§ 249. The Special Case of the Refraction of a Narrow Bundle of
Rays at a Plane Surface 360
§ 250, Reflexion at a Spherical Mirror Treated as a Special Case
of Refraction at a Spherical Surface 361
§ 251. Astigmatism of an Infinitely Thin Lens 363
CHAPTER XIL
The Theory of Spherical Aberratioiis, Arts. 79-104,
§§ 252-326 367-473
L Introduction, Arts. 79, 80, §§ 252-259 367-376
Art. 79. Practical Images, § 252 367-369
Art. 80. The so-called Seidel Imagery, §§ 253-259 369-376
§ 254. Order of the Image, according to Petzval 370
§ 255. Parameters of Object-Ray and Image-Ray, according to
L. Seidel 371
§ 256. The Correction-Terms or Aberrations of the 3rd Order. . . 373
§ 257. Planes of the Pupils of the Optical System 374
§ 258. Chief Ray of Bundle 375
§ 259. Relative Importance of the Terms of the Series-Develop-
ments of the Aberrations of the 3rd Order 375
II. The Spherical Aberration in the Case When
THE Object-Point Lies on the Optical Axis,
Arts. 81-85, §§ 260-275 376-400
Art. 81. Character of a Bundle of Refracted Rays Emanating Origi-
nally from a Point on the Optical Axis, §§ 260-262 376-380
§ 260. Longitudinal Aberration, or Aberration along the Optical
Axis 376
§ 261. Least Circle of Aberration 378
§ 262, The so-called Lateral Aberration 379
Art. 82. Development of the Formula for the Spherical Aberration of
a Direct Bundle of Rays, §§ 263-266 380-386
§ 266. Abbe*s Measure of the "Indistinctness** of the Image 385
Art. 83. Spherical Aberration of Direct Bundle of Rays in Special
Cases, §§ 267-272 386-394
Contents. xix
Page
§ 267. Case of a Single Spherical Refracting Surface 386
§ 268. Case of an Infinitely Thin Lens 387
§ 271. Case of a System of Two or More Thin Lenses 392
Art, 84. Numerical Illustration of Method of Using Formulae for
Calculation of Spherical Aberration, § 273 394-397
Art. 85. Concerning the Terms of the Higher Orders in the Series-
Development of the Longitudinal Aberration, §§ 274, 275. . 397-400
§ 275. The Aberration Curve 398
III. The Sine-Condition. (Optical System of Wide
Aperture and Small Field of Vision.),
Arts. 86-90, §§ 276-284 400-415
Art. 86. Derivation and Meaning of the Sine-Condition, §§ 276-278. 400-407
§ 278. Other Proofs of the Sine-Law 405
Art. 87. Aplanatism, § 279 407, 408
§ 279. Aplanatic Points 407
Art. 88. The Sine-Condition in the Focal Planes, § 280 408, 409
Art 89. Only One Pair of Aplanatic Points Possible, § 281 409-412
Art. 90. Development of the Formula for the Sine-Condition on the
Assumption that the Slope- Angles are Comparatively Small,
§§ 282-284 412-415
IV. Orthoscopy. Condition that the Image Shall Be
Free From Distortion, Arts. 91-94, §§ 285-294 — 415-429
Art, 91. Distortion of the Image of an Extended Object Formed by
Narrow Bundles of Rays, §§ 285-287 415-418
§ 286. Image-Points regarded as lying on the Chief Rays 416
§ 287. Measure of the Distortion 417
Art, 92. Conditions of Orthoscopy, §§ 288-291 418-422
§ 288. General Case: When the Centres of the Pupils are Affected
with Aberrations 418
§ 289. Case when the Pupil-Centres are without Aberration 420
§ 290. The Two Typical Kinds of Distortion 421
§ 291. Distortion when the Pupil-Centres are the Pair of Aplanatic
Points of the System 421
Art. 93. Development of the Approximate Formula for the Distortion
Aberration in case the Slope-Angles of the Chief Rays are
Small, § 292 422-427
Art. 94. The Distortion-Aberration in Special Cases, §§ 293, 294 427-429
§ 293. Case of a Single Spherical Refracting Surface 427
§ 294. Case of an Infinitely Thin Lens. 428
XX Contents.
Pack
V. Astigmatism and Curvature of the Image,
Arts. 95-98» §§ 295-306 429-444
Art. 95. The Primary and Secondary Image-Surfaces, § 295 429, 430
Art. 96. The Aberration-Lines, in a Plane Perpendicular to the Axis,
of the Meridian and S^ittal Rays, §§ 296-298 430-434
§ 297. Case when the Slope- Angles of the Chief Rays are Small . . 432
Art. 97. Development of the Formulae for the Curvatures i/R', i/!R',
§§ 299-304 434-441
§ 299. The Invariants of Astigmatic Refraction 434
§ 300. Developments of 1/5 and 1/5 in a series of powers of 4> 435
§ 301. The Expressions for the Co-efficients B, B and 5', 15'. . . . 437
§ 303. Curvature of the Stigmatic Image 439
§ 304. Formulae for the Magnitudes of the Aberration-Lines 440
Art. 98. Special Cases, §§ 305, 306 442-444
§ 305. Case of a Single Spherical Refracting Surface 442
§ 306. Case of an Infinitely Thin Lens 443
VI. Aberrations in the Case of Imagery by Bundles of
Rays of Finite Slopes and of Small, Finite
Apertures, Arts. 99-101, §§ 307-315 444-456
Art. 99. Coma, §§ 307, 308 444-448
§ 307. The Coma Aberrations in General 444
§ 308. The Lack of Symmetry of a Pencil of Meridian Rays of
Finite Aperture 447
Art. 100. Formulae for the Comatic Aberration-Lines, §§ 309-313. . . . 448-455
§ 309. Invariant-Method of Abbe 448
Art. loi. Special Cases, §§ 314, 315 455, 456
§ 314. Case of Single Spherical Refracting Surface 455
§ 315. Case of Infinitely Thin Lens 456
VI I. Seidel's Theory of the Spherical Aberrations of
THE Third Order, Arts. 102-104, §§ 31^326 456-473
Art. 102. Development of Seidel*s Formulae for the y- and 2- Aberra-
tions, §§ 316-322 456-468
§ 316. GAUSsian Parameters of Incident and Refracted Rays. . . . 456
§ 317. Approximate Values of the GAUSsian Parameters, and the
Correction-Terms of the 3rd Order 458
§ 318. Relations between the Ray-Parameters of Gauss and
Seidel 459
§ 322. Conditions of the Abolition of the Spherical Aberrations of
the 3rd Order 467
Art. 103. Elimination of the Magnitudes Denoted by h, u, § 323. . . 468-470
Art. 104. Remarks on Seidel's Formulae; and References to Other
General Methods, §§ 324-326 470-473
Contents. xxi
Page
CHAPTER XI I L
Colour-Phenomena, Arts. 105-1 13, §§327-359 474-531
I. Dispersion and Prism-Spectra, Arts. 105-107,
§§ 327-342 474-503
Art. 105. Introductory and Historical, §§ 327-330 474-484
§ 327. Relation between the Refractive Index and the Wave-
Length 474
§ 328. Newton's Prism-Experiments and the Fraunhofer Lines
of the Solar Spectrum 475
§ 329. The Jena Glass 478
§ 330. Combinations of Thin Prisms 483
Art. 106. The Dispersion of a System of Prisms, §§ 331-335 484-492
§ 332. Dispersion of a Single Prism in Air 487
§ 333. The Dispersion of a Train of Prisms composed alternately
of glass and air 488
§ 334. Achromatic Prism-Systems 489
§ 335- Direct-Vision Prism-System 491
Art- 107. Purity of the Spectrum. Resolving Power of Prism-Sys-
tem, §§ 336-342 492-503
§ 336. Measure of the Purity of the Spectrum 492
§ 337- Purity of Spectrum in case of a Single Prism 494
§ 338. Diffraction-Image of the Slit 495
§ 339. Ideal Purity of the Spectrum 497
§ 340. Resolving Power of Prism-System 498
II. The Chromatic Aberrations, Arts. I08-113,
343-359 503-531
Art. 108. The Different Kinds of Achromatism, § 343 503-505
Art. 109. The Chromatic Variations of the Position and Size of the
Image, in Terms of the Focal Lengths and Focal Distances
of the Optical System, §§ 344, 345 505-508
Art. no. Formulae Adapted to the Numerical Calculation of the
Chromatic Variations of the Position and Magnifications
of the Image of a given Object in a Centered System of
Spherical Refracting Surfaces, §§ 346-348 508-513
§ 346. Chromatic Longitudinal Aberration 508
§ 347. Differential Formulae for the Chromatic Variations 510
Art. III. Chromatic Variations in Special Cases, §§ 349-353 513-522
§ 349* Optical System consisting of a Single Lens, surrounded on
both sides by air 513
§ 350. Infinitely Thin Lens 516
i 351. Chromatic Aberration of a System of Infinitely Thin Lenses 517
xxii Contents.
Pack
§ 352. Two Infinitely Thin Lenses in Contact 519
§ 353* System of Two Infinitely Thin Lenses Separated by a
Finite Interval (d) 520
Art. 112. The Secondary Spectrum, §§ 354, 355 523-526
Art. 113. Chromatic Variations of the Spherical Aberrations, §§ 356-
359 526-531
CHAPTER XIV.
The Aperture and the Field of View. Brightness of Optical
Images, Arts, i 14-123, §§ 360-396 532-582
Art 114. The Pupils, §§ 360-364 532-540
§ 360. Effect of Stops 532
§ 361. The Aperture-Stop 533
§ 363. The Aperture-Angle 538
§ 364. The Numerical Aperture 538
Art. 115. The Chief Rays and the Ray-Procedure, §§ 365-367 540-544
§ 365. Chief Ray as Representative of Bundle of Rays 540
§ 366. Optical Measuring Instruments 541
Art. 1 16. Magnifying Power, §§ 368, 369 544-549
§ 368. The Objective Magnifying Power 544
§ 369. The Subjective Magnifying Power 545
Art. 117. The Field of View, §§ 370, 371 549-551
§ 370. Entrance- Port and Exit-Port 549
Art. 1 18. Projection-Systems with Infinitely Narrow Aperture (® =0),
§§ 372-374 551-554
§ 372. Focus-Plane and Screen-Plane 551
§ 373- Perspective-Elongation 553
§ 374- Correct Distance of Viewing a Photograph 554
Art. 119. Optical Systems with Finite Aperture, §§ 375-381 555-563
§ 375* Projected Object and Projected Image in the case of Pro-
jection-Systems of Finite Aperture 555
§ 377* Focus-Depth of Projection-System of Finite Aperture 557
§ 379. Lack of Detail in the Image due to the Focus-Depth 560
§ 380. Focus-Depth of Optical Systems of Finite Aperture used in
Conjunction with the Eye 560
§ 381. Accommodation-Depth 561
Art. 120. The Field of View in the case of Projection-Systems of
Finite Aperture, §§ 382-386 563-571
§ 382. Case of a Single Entrance-Port 563
§ 383. Case of Two Entrance-Ports 568
Contents. xxiii
Page
Intensity of Illumination and Brightness,
Arts. 121-123, §§ 387-396 571-582
Art. 121. Fundamental Laws of Radiation, §§ 387-389 571-575
§ 387. Radiation of Point-Source 571
§ 388. Radiation of Luminous Surface-Element 573
§ 389. Equivalent Light-Source 574
Art. 122. Intensity of Radiation of Optical Images, §§ 390-393 575-579
5390. Optical System of Infinitely Narrow Aperture (Paraxial
Rays) 575
§ 391. Optical System of Finite Aperture 576
§ 393- The Illumination in the Image-Space 578
Art, 123. Brightness of Optical Images, §§ 394-396 579*582
§ 394. Brightness of a Luminous Object 579
§ 396. Brightness of a Point-Source 581
APPENDIX.
Ezplanatioiis of Letters, Symbols, Etc 583-612
I. Designations of Points in the Diagrams 583-593
II. Designations of Lines 593, 594
III. Designations of Surfaces 594-59^
IV. Symbols of Linear Magnitudes 596-604
V. Symbols of Angular Magnitudes 604-608
VI. Symbols of Non-Geometrical Magnitudes (Constants,
Co-efficients, Functions, Etc.) 608-612
Index 613-626
GEOMETRICAL OPTICS.
CHAPTER I.
METHODS AND FUNDAMENTAL LAWS OF GEOMETRICAL OPTICS.
ART. 1. THE THEORIES OF LIGHT.
1. According to the Corpuscular or Emission Theory of Light,
nudntained and developed by Newton, the sensation of Light is due
to the impact on the retina of very minute particles, or corpuscles^
projected from a luminous body with enormous speeds and proceed-
ing in straight lines. Thus, in Newton's famous work on Optics,*
published in 1704, he asks: **Are not rays of light very small bodies
emitted from shining substances? For such bodies will pass through
uniform mediums in right lines without bending into the shadow,
which is the nature of the rays of light." Opposed to this view was
the Undulatory Theory of Light, which, notwithstanding the specu-
lations that have been found in the writings of earlier philosophers,
such as Leonardi da Vinci, Galileo and others, must beyond doubt
be attributed to Huygens as its author, whose work, entitled TraiU
^ la lumiire (Leyden, 1690), was based on the assumption that the
phenomena of light were dependent on an hypothetical Ether ^ or very
subtle, imponderable and exceedingly elastic medium, which pot only
pervaded all space but penetrated freely all material bodies solid,
liquid and gaseous. According to this theory, the something that was
emitted from a luminous body was not matter at all but a kind of
Wave-Motion which was propagated through the all-pervading ether
with a finite speed which is different according to the different cir-
cumstances in which the ether through which the disturbance advanced
is conditioned by the presence of ordinary gross matter. This remark-
able and ingenious theory encountered at first great difficulties, and
even Huygens himself was not able to give satisfactory explanations
of some of the most familiar phenomena of light. In the end, however,
it was destined to triumph, and, in the hands of such advocates
as Young and above all of Fresnel (who, in order to account for the
^ I. Nkwton: Opiicks: or a treatise of the reflexions^ inflexions, and colours of light (Lon-
don, 1704); Me Book ill., Qu. 29.
2 1
2 Geometrical Optics, Chapter I. [ § 4.
Polarisation of Light, was led to assume that the ether-vibrations were
transversal), the Wave-Theory won its way to the front rank of science,
where it remains to-day more firmly established than ever.
The Electromagnetic Theory of Light, which is a development from
the Wave-Theory, is a monument to the genius and mathematical
insight of Maxwell, but the experimental basis of this theory is to
be found in the investigations of Hertz, who showed that electrical
energy also was propagated by means of ether- waves which, under
certain circumstances, obeyed the laws of Reflexion and Refraction
and travelled with the speed of light.
2. But, independent of any of the theories as to the real nature
of light, there are certain well-ascertained facts about the mode of
propagation of light which may themselves be made the basis of a
certain science of light, and which — provided we are careful to con-
fine our investigations along these lines within justifiable limits — will
lead us often by the easiest route to a true knowledge, at any rate, of
the behaviour and effects of light. Moreover, these cardinal facts,
which we may call the fundamental characteristics of the mode of
propagation of light, are so few and so simple and suffice to explain
such a large class of important phenomena, especially those phenomena
on which the design and construction of optical instruments chiefly
depend, that the advantage of this method has been long recognized.
ART. 2. THE SCOPE AND PLAN OF GEOMETRICAL OPTICS.
3. The fundamental characteristics of the mode of propagation of
light may be enumerated under three heads as follows:
(i) The Law of the Rectilinear Propagation of Light, from which
we derive the ideas of "rays" of light; (2) the assumption that the
parts of a beam of light are mutually independent; so that, for ex-
ample, the effect produced at any point is to be attributed only to the
action of the so-called "rays" which pass through that point; and,
finally, (3) The Laws of Reflexion and Refraction of Light.
These laws, inasmuch as they are concerned essentially only with
the direction of the propagation of light, are purely geometrical; and,
hence, the science which is based upon them, and which seeks, by their
means, to explain the phenomena of light, either as they occur in na-
ture or as they are produced by the agency of optical instruments, is
called Geometrical Optics.
4. But while it is the peculiar office of Geometrical Optics to give
as far as possible explanations of such phenomena of light as depend
simply on changes in the directions in which the light is propagated, it
j S.] Fundamental Laws of Geometrical Optics. 3
do^ not pretend to be able to explain all such phenomena; and,
esp>^ally, it excludes as outside of its province all cases in which the
li^ht is propagated in anisotropic or crystalline media.
^4oreover, also, although the fundamental laws above-mentioned are
sufficient in themselves to construct a very complete and satisfactory
syst:em of explanation of a large class of optical phenomena, it must
not: be supposed that Geometrical Optics is willing to dispense entirely
or even partially with the more accurate ideas and conceptions which
are to be derived only by the consideration of the real and essential
nature of light. If such were to be our procedure, we should often go
astray, and, indeed, we know by experience that when Geometrical
Optics has ignored or even lost sight of the notions of the Wave-Theory
of Light, and pushed too far the geometrical consequences of the fun-
damental theorems on which it is based, erroneous results have been
obtained. On the contrary, the wave-phenomena of interference and
the like must be kept throughout constantly in view even when they
are not paraded to the front, and every result should be subjected to
the test of the methods of Physical Optics. Viewed in this way,
Geometrical Optics is -not to be regarded as a mere mathematical
disdpline — as is sometimes said by way of reproach — but it takes its
rank as a useful and important branch of Physics.
ART. 3. THE RECTUIlfEAR PROPAGATION OF UGHT.
5. In an isotropic medium light travels in straight lines, is the state-
ment of a fact, which, if not absolutely and unexceptionally true,
certainly cannot be far from the truth; and, indeed, until compara-
tively recent times this statement had never been called in question,
yhe fact is confidently assumed not merely in the ordinary affairs of
*«€ but in the most exact measurements both in Geodesy and in Astron-
<>niy, and, so far as these sciences are concerned, its validity has never
l^^ doubted. In order to view a star through a long narrow tube,
the axis of the tube must be pointed so that it coincides with the
straight line which joins the (real or apparent) position of the star
^th the eye of the observer. In aiming a rifle or in any of the proc-
esses that we call "sighting" the method is based with certainty upon
this commonest fact of experience. The most conclusive proof that
a hne is straight consists in showing that it is the path which light
pursues. The greatest difficulty that Huygens encountered in his
wave-theory of light was to explain its apparent rectilinear propaga-
tion. It was from this law that the idea of a *Vay of light" originated.
Nevertheless, the law is only approximately true, as has been well
4 Geometrical Optics, Chapter I. I § 6.
ascertained now for more than a century. For when we proceed to
subject it to as rigid a test as possible, and try, by means of screens
with very narrow openings, to separate from a beam of light the so-
called "rays'* themselves, we discover that these latter have in reality
no physical existence; and that the narrower we succeed in making
the opening, the less do we realize the idea conveyed by the term
"ray**. When the light arrives at the narrow opening, it does not
merely pass through it without changing its direction, but it spreads
out laterally as well, utterly misbehaving itself so far as the law of
rectilinear propagation is concerned. Thus, although the straight
line joining a point-source of light with an eye may pierce an inter-
posed screen at an opaque part of the screen, a narrow slit in another
part of the screen may enable the eye to perceive the source. When
an opaque object is interposed between a point-source of light and a
screen, the shadow on the screen will be found to correspond less and
less with the geometrical shadow in proportion as the dimensions of
the opaque body are made smaller and smaller, and, in fact, the very
places where, on the hypothesis of the rectilinear propagation of light,
we should expect shadows often prove to be places of quite contrary
effects, and vice versa. The fact is, light is propagated not by "rays"
but by waves, and the rectilinear propagation of light is practically true
in general because the wave-lengths of light are so minute. But when
we have to do with narrow apertures and obstacles whose dimensions
are comparable with those of the wave-lengths, we have the so-called
Diffraction-effects which are treated at great length in works on Physi-
cal Optics and which can only be alluded to here.
6. However, in order to arrive at a clear comprehension of the
matter, let us consider briefly the explanation afforded by the wave-
theory of the mode of propagation of light in an isotropic medium.
We may begin by giving Huygens's Construction of the Wave-Front|
which enables us to see how Huygens himself tried to explain the
assumed rectilinear propagation of light.
Let 0 (Fig. i) bea point-source* of light, or a luminous point, from
which as a centre or origin ether-waves proceed with equal speeds in
all directions. At the end of a certain time the disturbances will have
arrived at all the points which He on a spherical surface a described
around the centre 0, which is the locus of all the points in the iso-
^ An actual " point-source " of light by. itself cannot be physically realized. A '* lumi-
nous point " is an infinitely small bit of luminous surface. Nevertheless, exactly as in
Mechanics we are accustomed to speak of " particles of matter", and similarly in all
branches of Theoretical Physics, we may make use in Optics of this convenient and useful
conception, whether it be actually realizable or not.
I&l
Fundamental Laws of Geometrical Optics.
F THB WiVB-
tropic medium that are in this particular initial phase of vibration,
and which is the Wave-Front at this instant. According to Huygens,
every point P in the wave-front,
from the instant that the disturb-
ance reaches it, will become a new
source or centre of disturbance,
from which secondary waves will be
propagated in all directions. More-
over, Huygens assumed that these
secondary waves, originating at all I
the points affected by the principal
wave, interfere with each other in
such fashion that their resultant
sensible effects are produced only
at the points of the surface which
envelops at any given instant all
the secondary wave-fronts, and that
this enveloping surface is, therefore,
the wave-front at that instant.
Obviously, in an isotropic medium, such as is here supposed, this sur-
face will be a sphere described around 0
as centre.
Accordingly, if waves diverge from a
luminous point 0 (Fig. 2), and if an
opaque plane screen H/with an opening
AB is interposed in front of the advanc-
ing waves, the wave-front at any time /
may be constructed as follows: Consider
all the points, such as M, which lie in
the plane of the screen at the place
where the aperture is made. As soon
as the disturbance arrives at one of these
points, it will become a new centre of
disturbance, from which will divei^e,
therefore, secondary spherical waves. In
general, the radii of these secondary
waves will be different. Thus, in the
diagram, as here drawn, the point des-
ignated by A is nearer the source 0
than the point designated by M, so that
the disturbance must arrive at A first, and hence the secondary
FlO. 1.
Hmonra'a ConsrRucnoit op tbs
Wavs-Pboitt. Sidierical waves dl-
wsinff fnHn Uie poiiit«nrce O and
pwiny thiQQxh an opening AB In
Ibe opaque acncti ff/. The arc I>C
If a Mctkia of Ilw portion of the ipber-
ieal wave-fnjct » whkh containa the
pointa berond the acncD which have
Geometrical Optics, Chapter I.
IS6.
wave proceeding from A will have had time to travel farther than
the secondary wave originating at M. If we put OM^ x, and if we
denote the radius of the secondary spherical wave around M at the
time / by r, then d = x + r will denote the distance from 0 to which
the disturbance is propagated in the time /, which shows that as x in*
creases, r decreases; that is, the greater is the distance of the jwint M
from the source 0, the smaller will be the radius of the secondary wave-
surface around this point M. Thus, the enveloping surface is seen to
be the portion of a spherical surface of radius d around 0 as centre:
it is that part of this spherical suriace which is comprised within the
cone which has 0 for its vertex and the opening AB oi the screen for
its base. The wave proceeds, therefore, from 0 into the space on the
other side of the screen, but on this side of the screen the wave-sur-
face is limited by the rays drawn from 0 to the points in the edge of
the opening. According to Huygens's view, the disturbance is propa-
gated within this cone just as though the screen were not interposed
at all, whereas points on the far side of
the screen but outside this cone of rays
are not affected at all. This mode of ex-
planation leads to the theory of the recti-
linear propagation of light.
If the luminous point 0 (Fig. 3) is at such
a distance from the screen that the dimen-
sions of the opening A B may be regarded
as vanishingly small in comparison there-
with, we shall have a cylindrical bundle of
rays, and the wave-fronts will be plane in-
stead of spherical.'
The most obvious objection to Huy-
gens's construction is. What right has he
to assume that the points of sensible effects
are the points on the surface which envelops
the secondary waves? And why is the light not propagated backwards
* The single points of a lumtnojs heavenly body are to be regarded as at an Infinite
distance !n comparison with the dimensions of our apparatus, so that the wave-fronts oT
the disturbances emitted from such points are plane. But the rays wliich come from dif-
ferent points of a celestial body cannot be regarded as parallel unless the parallax of the
■tar Is sufficiently small. This angle has a right considerable magnitude in the caaet of
both the sun and the moon, so that the divergence of the rays which come from opposite
ends of the diameters of these bodies may amount to more than half a degree. For most
experiments in Optics this divergence is negligible, and a tieam of sunlight may be regarded
as consisting of parallel rays. We may obtain bundles of parallel rays from tcrrettrial
•ouTces of light by means of lenses, etc.
HtrtoBsfa
THE Wavb-Pront.
procecdiDE throush
§ 7.] Fundamental Laws of Geometrical Optics. 7
as well as forwards? Moreover, if the opening in the screen is very
narrow, this construction does not correspond at all with the observed
facts.
7. FresnePs Extension of Huygens's Method. In place of Huy-
GENs's arbitrary assumption that the places
where there are sensible effects are to be found
only on the surface which envelops the ele-
mentary waves, Fresnel insisted that these
secondary waves, encountering each other,
must therefore be regarded as interfering with
each other, and thus he conceived that the ^ _
. . PRBSNBL'S Method. The
disturbance at any point P (Fig. 4) must be eflfcctatthepointpof adu-
due to the superposition of the component turbanceorijfinatiniratoisto
^ ^ iif *** attributed almost entirely
disturbances propagated to P from all the to the disturbance that is prop-
points of the wave-surface <r. According to aa»tedaionff the straight line
T- ^i_ r f i_^ /r ^ ^ i_ 0/>; provided the wave-lengths
Fresnel, therefore, light-effects are to be are very smau.
found, not on the enveloping surface, but at
all points where the secondary waves combine to reinforce each other.
On investigation — ^which we do not attempt to show here — ^it ap-
pears that the disturbances which arrive at P along all the straight
lines joining P with points on the wave-front in great measure neu-
tralize each other, and the result is (assuming that the wave-length
is small) that the actual effect at P may be considered as due wholly to the
action of a very small element of the wave-front situated at the point A
where the straight line joining 0 with P intersects the wave-front <r-
(This point A is called the "pole" of the wave with respect to the
point P; it is the point of the wave-front that is nearest to P, so that
the disturbance from this point arrives at P before the disturbance
from any other point of the wave-front.) Hence, if between 0 and P
we interpose a small opaque screen which exactly shuts off from P
the effect due to the small ''zone*' around A, there will be darkness at
P; moreover, what is true of this point P is true also of any point
which, like P, is situated on the straight line OP. On the other hand
if a plane screen is placed tangent to the wave-surface at A , with a
small circular opening in it the centre of which is at -4, so that the
point P is screened from the entire wave-surface except the very small
"effective zone" immediately around A, the effect at P, as also at all
points along the straight line i4P, is found to be precisely the same
as though the screen had not been interposed. It is thus that the
idea of Huygens as developed by Fresnel leads, as we see, to the
theory of the approximate rectilinear propagation of light — that is.
8 Geometrical Optics, Chapter I. [ § 9.
light does in fact behave very nearly as if it were propagated in
straight lines.
8. Although, therefore, this fundamental law has always to be
stated with certain reservations, and, as a matter of fact, is never
strictly true, yet even when it is regarded from the standpoint of the
wave-theory, the law of the rectilinear propagation of light loses very
little of its meaning. On the contrary, in agreement with experience,
that theory shows that in the cases which ordinarily occur, especially
in those cases where we have to do with beams of light of finite di-
mensions, the effects at any rate are for all practical purposes the
same as if these beams of light were composed of separate rays, each
independent of the others, along which the light is propagated in
straight lines. But, however useful and generally safe this simple and
convenient rule may be, it must be borne in mind that it is inexact
and we must be prepared, therefore, to meet here and there excep-
tional cases where the rule is plainly inadmissible. It is only in this
way that the methods of Geometrical Optics can be approved.
ART. 4. RAYS OF UGHT.
9. A self-luminous point is said to emit "rays of light" in all direc-
tions. In an isotropic medium (§io) the ray-paths are straight lines
proceeding from the centre of the expanding spherical wave-surface;
and whether the medium is isotropic or not, the direction of the ray-
path at any point is to be considered as being always along the normal
to the wave-surface that goes through that point (see §42). What
are called '*rays of light" in Geometrical Optics are in fact those short-
est paths, optically speaking (§38), along which the ether-disturbances
are propagated. Employed in this sense, the word "ray" is a purely
geometrical idea. However, there is a certain sense in which we can
attach a physical meaning also to these so-called "light-rays". For,
as a rule, it is approximately true that the ether-disturbance at any
point of the path of a ray of light is due to disturbances which have
occurred successively at all points along the ray that are nearer to the
source than the point in question; so that, according to this view,
the effect at any point P is to be considered as in no degree arising
from disturbances at other points which do not He on a ray passing
through P. This is, in fact, the Principle of the Mutual Independence
of the Rays of Light, which is also one of the fundamental laws of
Geometrical Optics, and which assumes that each ray in a beam of
light is somehow separate and distinct from its fellows, and has, there-
fore, a certain physical existence. Thus, for example, if we have a
§ 10.] Fundamental Laws of Geometrical Optics. 9
wide-angle cone of rays incident on a screen and producing there a
comparatively large light-spot, and if we interpose an opaque object
so as to intercept a considerable fraction of the rays before they
reach the screen, a corresponding portion of the light on the screen
will vanish; and, hence, it can be inferred that we may suppress some
of the rays in a beam without altering, apparently, the effect pro-
duced by the remaining rays.
Here also, however, when this principle is examined from the stand-
point of the wave-theory, we find that it, too, has to be stated with
reservations. According to Fresnel, the disturbance at the point P
(Fig. 4) is to be considered as the resultant of an infinite number of
partial disturbances propagated to P from all points situated on the
wave-front <r; so that in a certain sense P may be considered as being
at the vertex, or "storm-centre", of a cone of rays which are by no
means independent of each other. Every point, such as P, which lies
ahead of the advancing wave-front is in similar circumstances. But,
as has been stated (§7), the resultant effect at the point P is due in
the main to the disturbance that is propagated along the central ray
of the cone of rays that converge to P\ and, thus, the law of the
Mutual Independence of Rays, if it is true at all, can only be said to
be true of these central rays of all such cones of rays as are here meant.
In point of fact, the resultant effect at the point P is to be ascribed
not merely to the disturbance propagated along this central ray from
the pole A of the wave, but to a zone of the wave-surface of very
small, but finite, dimensions, with its vertex at A. And the moment
we attempt to isolate physically the ray -4 P by screening P from the
effects of this zone, the effect at P vanishes entirely and the ray ceases
to exist.
10. It is best, therefore, without any reference to its physical mean-
ing, to define a ray of light as a line or path along which the ether-
disturbance is propagated. An optical medium is any space, whether
filled or not with ponderable matter, which may be traversed by rays
of light. In Geometrical Optics, where we have to do only with iso-
tropic media, the rays of light are straight lines (Art. 3). At a sur-
face of separation of two media the direction of the ray will usually
be changed abruptly, either when the ray passes from one medium into
the next or is bent away at the surface of a body; so that under such
drcumstances the ray-path will consist of a series of straight line-
segments. If, for example, Bj^ designates the point where the ray
meets the ith surface, then the straight line-segment Bj^^Bj^ will
represent the path of the ray in the jfeth medium : and here it may be
10 Geometrical Optics, Chapter I. [ § 12.
ivuiiUki\l that, so long as we are speaking of this portion of the ray-
^Mih, «uvv jKxint Plying on the straight line determined by Bj^_, and
H^ ib u» W a>n5jidered as situated in the jfeth medium, even though
i\w :>ul^iance of which the medium is composed does not extend out
io ihv* iKuut P. If the point P is situated on the straight line B^_iB^
Uiwwu iheae two incidence-points, we say that the ray in this medium
^wi:>ac'6 rc%illy through the point P; otherwise, we say that the ray
^iv^ vutuaUy through the point P.
AMI* S. IBM UHAVIOUR OF LIGHT AT THE SURFACE OF SEPARATIOH
OF TWO ISOTROPIC MEDIA.
U. lu order to have clear ideas of certain matters mentioned in
I lie ^xrciv\iiug articles, it will be necessary to know how the rays of
li^ia aic aftccted when they arrive at the boundary-surface separating
l\\v> adjixiuiug optical media. At such a surface the ''incident^ light
^tib ii ib called) will, in general, be di\'ided into two portions, which are
pi v>^ vacated from the places where the light falls on the surface in
uliiupily changed directions:
V I M Uiu pcxrtion of the light is turned back or ''reflected^' at the sur-
(av t , aud pursues its progress in the first medium along new ray-paths
vwvcpi uuilcT s^XHTial cimditions).
v-i^ I he lunaining portion, crossing the surface and entering the
L.vxoiul uu'iUuiu, u>akes its way, in general, in this new region; this
lo lUv :>o valk-'d *Vc?/rat/c\i** light.
I J. 1 U»v\c\cr, here alsi^ a closer study of these phenomena reveals
k\\< lav I ih.a luiihcr of the alx)ve statements is an entirely accurate
\ . iipiu.u lhu:>. it will l>e found that even that part of the light
\\ltuU lo aul iv» W ivtKvted and which ultimately returns into the
iu I uu vUmu liavl vuv:isc'd ihe boundar>'-surface and penetrated a little
\» »^ ioi • k\w .iAMUil uwdium. This is the explanation of the colour of
• » '. I V I vv u l».\ u iK\ lal light : the incident light falling on the body
ui I ,.. yu lining u» a bU^hl extent Wow the surface is there, according
« . •. I Uw.i \ v.i Svlw u\c AKsi^rption" (into which we cannot enter
'. • ' . .Mv.l ».i vsiuuu v^f iis cxmstituent parts, and only the remain-
' i.M iiK iviiwud The depth of penetration depends on the
.» « iUv lyyx. lusdu M\\\ \\\ a very great degree on the character
» ^ > i» iM^u. .uaav\'. Thus, for example, if the second medium
.( > M t ' ^> i
\\Ul iu\\vl\e the knowledge of whether the glass
U\h .vaiv sa iu the form of a fine powder; andif the
' x^ KnI, vUx ^vx \i ^^viviiUm would be as to the surface, whether
\> '' ..\'v I' U-Ussl \^ uv^v, eic\
§ 13.) Fundamental Laws of Geometrical Optics. 11
When a beam of sunlight is admitted through an opening in a shut-
ter into an otherwise dark room, and is allowed to fall, for example,
on a metallic surface, the reflected light itself consists of two portions,
viz., one part (in this case the greater part) which leaves the metallic
surface in a perfectly definite direction, and which is said therefore
to be regularly reflected, and another part which leaves the surface in
countless different directions, and which is said to be "scattered".
This scattering or "diffusion" of the reflected light is due to the in-
equalities or rugosities of the surface; it may be greatly diminished
by cleaning and polishing the surface. If the reflecting surface is geo-
metrically regular and physically smooth, the reflected light will be
nearly all regularly reflected. And even in those cases where the light
is irregularly reflected or diffused, as, for example, when a beam of
sunlight is reflected from a ground-glass surface, it would be more
correct to attribute the irregularity not so much to the behaviour of
the rays of light as to the peculiarity of the surface itself. Perhaps,
if we knew precisely the arrangement and orientation of the elements
of such a reflecting surface, we should discover that the reflexion was
quite regular after all. However, the actual dimensions of these
rugosities of the surface will also affect the phenomenon, inasmuch as
when these dimensions are sufficiently small, the assumptions which
lie at the foundation of Geometrical Optics will cease to be valid.
It is in consequence of this fact, that the light which is incident on
a rough surface is subjected to different experiences at the different
places in the surface, that these irregularities are made visible to us
as themselves sources of rays of light; whereas if the reflecting sur-
face were perfectly smooth, so that the rays were regularly reflected
all according to the same law, we should not be able to see the surface
at all, we should see merely the images of objects from which the rays
had come — objects which were either self-luminous or else illuminated
by diffusely reflected light. Moreover, in order to view the images,
the eye would have to be placed somewhere along these special routes
of the reflected rays; otherwise, none of these rays would enter the
eye and nothing would be visible by the reflected light. Most objects
are seen by diffusely reflected light, and no matter where the eye is
situated, it will intercept some of the rays that are scattered from the
surface of the body.
13. In large measure the above observations concerning the por-
tion of the light that is reflected apply also to the other portion that
is refracted. If the surface of separation of the two media is smooth,
the directions of the refracted rays will, in general, depend only on the
12 Geometrical Optics, Chapter I. [ § 13.
directions of the incident rays according to the so-called Law of Re-
fraction; and in this case the light is said to be regularly refracted.
But if the boundary-surface is rough, the rays will be diffusely re-
fracted in all directions (* 'irregular refraction").
The light which enters the second medium may be modified in vari-
ous ways. A greater or less portion of it, depending on the character
and peculiarity of the medium, will be absorbed; that is, the ether
loses some of its energy and ordinary matter gains it. Invariably, a
fraction of the light-energy will be transformed into heat, possibly
also into chemical and electrical forms of energy. If the medium is
perfectly transparent, the rays of light traverse it without being ab-
sorbed at all; whereas if the medium absorbs all the light-rays, it is
said to be perfectly opaque. No medium is absolutely transparent
on the one hand or absolutely opaque on the other. A perfectly trans-
parent body would be quite invisible, although we may easily be made
aware of the presence of such a body by the distortion of the images
of bodies viewed through it. As a rule, the absorptive power of a
medium will depend on the colour (or wave-length) of the light. Thus,
a piece of green glass will allow only certain kinds of light to pass
through it, and therefore when the rays of the sun fall on it, it will
absorb some of these rays and be transparent to others, and the trans-
mitted light falling on the retina of the eye, will produce a sensation
which we describe vaguely as green light. An interesting phenomenon
occurs called Fluorescence, whereby the colour of the light undergoes
a change in the second medium.
Again, there are some media which, although they cannot be called
transparent, nevertheless permit light to pass through them in a more
or less irregular and imperfect fashion; for example, such substances
as porcelain, milk, blood, moist atmospheric air, which contain sus-
pended or imbedded in them particles of matter of a different optical
quality from that of the surrounding mass. The light undergoes in-
ternal diffused reflexion at these particles. Objects viewed through
such media can be discerned, perhaps, but always more or less indis-
tinctly. These so-called **cloudy media" are said, therefore, to be
translucent, but not transparent.
It is usually assumed in Geometrical Optics that the media are not
only homogeneous, but perfectly transparent; and also that the sur-
faces of separation between pairs of adjoining media are perfectly
smooth.
§15.]
Fundamental Laws of Geometrical Optics.
13
ART. 6. THE LAWS OF REFLEXION AND REFRACTION.
14. Let MM (Fig- S) be the trace in the plane of the diagram of
the smooth reflecting or refracting surface separating two transparent
isotropic media. Let PB represent the rectilinear path of a ray of
light in the first medium (a). The ray PB is called the incident ray^
the point B where this ray meets the boundary-surface between the
two media (a) and (ft) is called the incidence-pointy the normal N N'
to the surface at the point B is called the incidence-normal ^ and the
plane PB N determined by the incident ray and the incidence-normal
(which is here the plane of the paper) is called the plane of incidence.
In general, to an incident ray PB there will correspond two rays,
viz., a reflected ray BR, which re-
mains in the first medium (a) and
a refracted ray BQ, which shows
the path taken by the light in the
second medium (6). The acute
angles at the incidence-point B be-
tween the incidence normal NB N'
and the rays PB, BR and BQ
are called the angles of incidence, re-
flexion and refraction, respectively.
Each of these angles is defined as
ike acute angle through which the
incidence-normal has to be turned in
order to bring it into coincidence with
the straight line which shows the path
of the ray in question. Thus, in the
diagram the angles of incidence, re-
flexion and refraction are ^ NBP,
Z NBRsLtid ^ iV'^C, respectively;
where the order in which the let-
ters are written indicates the sense
of rotation. These angles are to be reckoned as positive or negative
according as the sense of rotation is counter-clockwise or clockwise.
15. The Laws of Reflexion and Refraction, as determined by ex-
periment, may now be set forth in the following statements:
(i) Both the reflected and the refracted rays lie in the plane of inci-
dence,
(2) The reflected ray in the first medium and the refracted ray in the
second medium lie on the opposite side of the normal from the incident
ray in the first medium. Or if we prolong the refracted ray backwards
Pio. 5.
Laws op Reflexion and Refraction.
M/A is a section in the plane of incidence (plane
of paper) of the surface separating the first
medium (a) from the second medium (b).
The point B is the point of incidence, and
NN* is the normal to the surface at this point.
PB, BR and BQ are the incident, reflected and
refracted rays, respectively.
- ~ :he straight line
"- the same >irle c..
".r.vcted rays He .;i
^-.> 'A incidence an«.
::.:ence and retJexior
: 'r:cx:cft ere ecia::.
- • .
9* ,"
crc IK a iot:-
>;•' *-
i::c'-.V ;:/;/; or
•ioi: :ed by a. a\
L-: Taction mav he
-hi.'h :-.;r light of a
"> . r. :he nature of
• • • H,
:r the index
The order
icr in which
• • 1
- •■ - « >.«.•• lit \1
. . . • • 1
— ... -• .t w.. . •Ll
• » *
« ^m
»« • ft •
-ore
— "-■•• •« •■» ]
> .;..u
■••■»-
§ J. ^.1 Fundamental Laws of Geometrical Optics. 15
ut:^?d to EucxiD (3CX) B. C). On the other hand, the law of refraction
is much more modern. Claudius PxoLEMiEUs, the great astrono-
itt^r, who flourished during the reigns of the Antonines, published a
tr^stise on optics ('Oirroc^ irpayiuirtia) in which he describes a number
of experiments whereby he measured the angles of incidence and re-
fraction, without, however, discovering the law. The next experi-
ments along this line of which we have any record are those of Alhazen
who died in Cairo in 103 8 ; he repeated the experiments of PxoLEMiEUS,
but: added nothing to the previous knowledge of the matter. Kepler
alscD made experiments, but was equally unsuccessful. The real dis-
co^vrerer of the law was Willebrord Snell, of Leyden, who announced
it some time prior to 1626. It was first published by Descartes*
ia 1637; who seems undoubtedly to have obtained it from Snell,
although he failed to mention his name in connection with it.
18. In the case of Reflexion, it is obvious that the directions of
the incident and reflected rays may be reversed, so that if PBR (Fig. 5)
represents the path pursued by the light in going from P to i?, under-
going reflexion at the incidence-point -B, then RBP will represent the
path which the light takes in going from i? to P under the same cir-
ciimstances, that is, via the incidence-point B. Experiment shows
that the same rule holds good also for the ray refracted at ^ ; so that
if -PjBQ is the route followed by a ray in going from a point P in the
medium (a) to a point Q in the medium (6), undergoing refraction at
the incidence-point B, the same route will be pursued in the reverse
sense QBP by a ray whose direction in the medium (6) is from Q
towards 5. And, hence, since
sin a sin a!
^e have obviously, the relation:
^06*^60 = I- (2)
This general law of optics, known as the Principle of the Reversi-
bility of the Light-Path, may be stated as follows:
If a ray of light, undergoing any number of reflexions and refrac-
tions, pursues a certain route from one point A to another point A'^
and if at A' it is incident normally on a mirror so that it is reflected
' Kwst Du Perron Descartes: Discours de la mithode pour bien conduire sa raison
dckercherlaveritidans Us scitnus; plus la Dioptrique, Us MHloreset la Gfametrie (Leyden,
1637).
air -tKC «
;fii
TUiT aini: Toute in ihe miera
« ..
bytlu
•Tc veiling ii
A-
.V- KrrtArTET. TTArF-FKCWrT!: X2C TEX C^S
■ -- > E section i= the plane of incaia:
•- sirwiiuir -.c, froffi -Jit second snedirc ►
V ^»' ^ the paj^r of the xaddenL rrfie«e
. i...
'v-^ its plane, so that ih<
■- v.x^jrular to the plane wa\-e
X <.v.: h soometric surface sepa-
V =<-.:r,Npic optical medium (b).
- ■^^'•'■'-- "i«liuni will, in general,
• ^^' A: the same time also a
).l Fundamental Laws of Geometrical Optics. 17
ve will be reflected back from the boundary-surface into the first
dium, which likewise may be changed both in form and in direc-
n. But the speed of propagation of the reflected wave will be the
ne as that of the incident wave; whereas the speed of propagation
the refracted wave in the new medium (6) will be different from
t of the incident wave in the medium (a).
"or the sake of simplicity, let us suppose that the two media (a)
I (6) are separated by a plane surface. We proceed to give HUT-
vs's Construction of the Reflected and Refracted Wave-Fronts for
5 case. In the diagram (Fig. 6) mm represents the trace in the plane
the paper of the plane surface separating the media (a) and (6);
\ AB \s the trace in the same plane of a portion of the advancing
ident plane wave; so that the incident rays in the plane of the
»er will be represented by straight lines perpendicular to A 5, such
BC and DE. At the instant when we begin to reckon time the
dent wave-front is supposed to be in the position shown by AB^
hence at this moment the disturbance will have just arrived at
point A of the plane surface mm- From this moment, therefore,
Drding to Huygens's idea, this disturbed point A is itself to be
irded as a centre of disturbance, and from it as centre elementary
lispherical waves are propagated not only into the second medium
but also back into the first medium (a). Exactly the same con-
Dn will be true at this instant (/ = o) of every point in the plane
ace situated on the straight line perpendicular to the plane of the
er at the point A. The envelope of each of these two sets of equal
^spherical surfaces will be a semi-cylinder, whose axis is the straight
just mentioned. A little later the disturbance which was ini-
y at Z> will reach the point E in the line nn\ and if v^ denotes the
•d with which the disturbance is propagated in the medium (a), the
nent when it arrives at E will be / = DE/v^, Beginning from
moment the two sets of semi-cylindrical surfaces which have for
r common axis the straight line perpendicular at E to the plane
he paper will begin to be formed. And, thus, at successively later
later instants, the disturbance will arrive in turn at all the points
iM which lie between A and C; until, finally, at the time / = BC/v^
disturbance reaches the extreme point C Meanwhile, around all
straight lines perpendicular to the plane of the diagram at the
nts on MM which lie between A and C two sets of co-axial semi-
ndrical elementary wave-surfaces have been forming, one set being
pagated back into the first medium (a) and the other set being
pagated forward into the second medium (b). The nearer one of
.1
18 Geometrical Optics, Chapter I. [ § 21.
these points between A and C is to the point C, the smaller will be the
radius of the corresponding semi-cylinder.
20. Let us consider, first, the Reflected Wave. At the moment t —
B C/i\, when the point C begins to be disturbed, the semi-cylindrical
wave 5| whose axis passes through A will have expanded in the first
medium until its radius is equal to BC. At this same instant the
semi-cylindrical wave 5, whose axis is determined by the point E
will have been expanding into the first medium during the time B C/v^
— DE/v^, so that the disturbance will have been propagated a dis-
tance BC -- DE^ JC^ which is therefore the radius of this cylindrical
surface.
According to Huygens's Principle, the surface which at any instant
is tangent to all the elementary semi-cylindrical reflected waves will
be the required reflected wave-front at that instant. We shall show
that this reflected wave-front is a plane surface which at the moment
when the disturbance reaches C contains this point; or, what amounts
to the same thing, we shall show that if the line CG in the plane of
the diagram touches at G the semi-circle in which the plane cuts the
semi-cylinder 5|, it will be the common tangent of all such semi-
circles; for example, it ^nll be tangent to the semi-drcle 5, around
an>' jioint E as centre. From C draw CG tangent to S^ at G and CF
tangent to 5, at F. Draw AG and EF. The triangles CGA and
ABC are congruent, since the angles at B and G are both right angles
and AG = BC. Hence, j^GCA =^ Z BAC. Similarly, from the
congruence of the triangles CFE and CEJ, it follows that Z FCE =
/ J EC. And since Z 5^4 C = Z JEC, we have Z GCA = Z FCE;
anil, nuisequently, the tangent-lines CG and CF coincide. Hence,
llie trace in the plane of the paper of the reflected wave-front is the
sliiiiglu line CFG, This reflected plane wave will be propagated
imwauU, parallel with itself, in the direction shown in the diagram
liy i\\c rertei ted rays AG, EF, etc. It is evident from the construc-
tion tluit the ray incident at A, the normal A N to the reflecting sur-
laii: at .1 tiUil the corresponding reflected ray AG are all situated in
llu- biiuu! plane, viz., here the plane of the paper which is the plane
III iiuiilciuu fi>r the ray in question. It only remains therefore to
^Imw ilia I the angles of incidence and reflexion are equal. This is
i.Li\itm:i almi from the congruence of the triangles CGA and CBA.
«il. TUtt Refracted Wave. If the velocity of propagation of the
\Na\t ia tlif. bi'ioiul medium (6) is denoted similarly by r^, it is plain
lint ul ilii^ ntoiuent / = 5 C/i'„ when the disturbance reaches the
I'^'iiu I ', the siciiaulary disturbance which proceeds from A as centre
§ 22.] Fundamental Laws of Geometrical Optics. 19
will have been propagated into the medium (6) to a distance AH =
vj = Vf,' BC/v^; and, similarly, the disturbance at any intermediate
point, as E, between A and C, will have been propagated in the second
medium to a distance EK — {BC — DE)vJv^= EJ-vJv^. Thus,
the radii of the elementary semi-cylindrical refracted waves S[ and
5,, whose axes are perpendicular to the plane of the paper at A and
£, are BC-vJv^Sind EJ-vJv„, respectively. The refracted wave-
front at any instant will be the surface which is tangent to all these
elementary cylindrical surfaces at this instant. Exactly the same
method as we used in finding the reflected wave-front can be employed
here; and we shall find that at the instant when the disturbance
reaches C the refracted wave-front is the plane containing the point
C which is perpendicular to the plane of the paper and tangent to
the elementary wave S[ at H.
Snell's law of refraction may be deduced at once by observing
that in the figure AG ^ AC-sin a, where a = Z -4 i5 C is equal to the
angle of incidence of the parallel incident rays, and i4 fl'=i4 C'sin a',
where a! ^ Z. ACH is equal to the angle of refraction of the parallel
refracted rays; and, consequently:
sin a AG V. , .
—. — 7 = -7-fr = - = constant. (3)
sin a AH Vf, ^•''
22. In the figure the case is represented where the disturbance is
propagated faster in the first medium (a) than in the second medium
(&), that is, v^ is greater than i;^. In this case the angle of refraction
a' is less than the angle of incidence a, and hence the refracted rays
are bent towards the normal, as, for example, when light is refracted
from air into glass. According to the Wave-Theory of Light, there-
fore, the velocity of propagation in the optically denser of the two
media is less than it is in the other medium. Now the NEWTONian or
Emission Theory of Light leads to precisely the opposite conclusion.
The two theories are here in direct conflict with each other, and ex-
periment has decided in favor of the Wave Theory. Arago, in 1838,
suggested the method of measuring the speed of propagation of light
which was afterwards (1865) successfully employed by Foucault.
Foucault's experiments demonstrated that light travelled faster, for
example, in air than in water. These experiments were subsequently
repeated by Michelson, with an improved form of apparatus, and
MiCHELsON found that the speed of light in air was 1.33 times as
great as that in water, which agrees with the value of the relative
20 Geometrical Optics, Chapter I. [ § 24.
index of refraction of air and water. The same experimenter found
that the speed in air was 1.77 times the speed in carbon bisulphide,
whereas the value of n for these two substances is about 1.63, so that
in this case the agreement was not so close.
ART. 7. ABSOLUTE INDEX OF REFRACTION OF AN OPTICAL MSDIITM.
23. According to the Wave-Theory, therefore, the relative index
of refraction of two media (a) and (6) is equal to the ratio of the speeds
of propagation of light in the two media. And, hence, if we know the
indices of refraction of a medium (c) with respect to each of two media
(a) and (6), we can easily compute the value of the relative index of
refraction of the two media (a) and (6) with respect to each other.
For, according to formulae (i) and (3), we shall obtain:
and therefore:
K ^i^ ^k
^k ^C ^C
m — — 2F'
which, according to (2), may be written also:
For example, suppose that the substances designated by the letters a, h
and c are water, glass and air, respectively, and that we know the
values of the relative indices of air and water and of air and glass, viz.,
n^ = 4/3 and n^ = 3/2; then the value of the relative index of refrac-
tion from water to glass will be n^ = (3/2): (4/3) = 9/8.
Generally, it may be shown that if the letters a, fr, c, . . . f , j, h are
employed to designate a number of optical media, then:
««6 • »ftc • »«i • • «<f * »/» = »•»• (4)
And, in particular, if the last medium {k) is identical with the first
medium (a), the continued product of the relative indices of refraction
will be equal to unity; formula (2) states this law for the case where
there are only two media (a) and (6).
24. The fact that n„^ = «e» • ^ca suggests the idea of employing
some standard optical medium (c) with respect to which the indices of
refraction of all other media could be expressed. The medium that
{25.] Fundamental Laws of Geometrical Optics. 21
is selected for this purpose is that of empty space or vacuum, and the
index of refraction of a medium with respect to empty space is called,
therefore, the absolute index of refraction of the medium, or, simply,
the refractive index of the medium. Accordingly, the absolute index
of refraction of empty space is itself equal to unity, and if w^, w^ denote
the absolute indices of two media (a) and (6), then evidently:
^ot = -• (S)
The absolute indices of refraction of all known transparent media
are greater than unity. However, Kundt^ determined, in i888, the
'idices of refraction of a number of metallic substances, using very
"^n prisms of the materials which he subjected to investigation ; and
^^ values of n which he obtained in the case of silver, gold and copper
^^re all less than unity: which implies that light travels faster in each
^^ these metals than it does in vacuo. See also more recent experi-
ments with such substances as these, especially those of Drude and
^i>JOR in 1903.
The index of refraction of air, at 0° C. and under a pressure of
^^ cm. of mercury for light corresponding to the Fraunhofer D-line
^^^ been found to be equal to 1.000293; it is usually taken as equal
^^ unity.
25. With every isotropic optical medium there is associated, there-
^*^, a certain numerical constant n; and thus when a ray of light is
'^^r^cted from a medium of index n into another medium of index n',
^^^ trigonometric formula of the law of refraction may be written in
^^^ following symmetrical form:
« • sin a = n' • sin a'; (6)
^^hich may be stated by saying:
-4/ every refraction of a ray of light from one medium to another, the
P^o<iuct of the refractive index of the medium and the sine of the acute
^^ZU between the ray and the incidence-normal remains unchanged.
This product
Jf = w • sin a (7)
^ sometimes called the Optical Invariant.
* A. Kundt: Ueber die Brechungsexponenten der Metalle: Ann, der Phys, (3), zzxiv.
22 Gcocnetrical Opdcs. Chapter I. [ § 27.
26. Reflexion considered as a Special Case of Refmctioii. Whereas
the angles of incidence and refraction ha^-e like signs alwa>'5, on the
contrar\' the signs of the angles of incidence and reflexion are alwajrs
opposite. In order, therefore, that formula (6) may be apfdicable also
to the case of reflexion as well as to that of refracticm, the values of
fi and n' in the former case must be such that a' ^ ~ a is a solution
of the equation in question; and the condition that we shall have
this solution is e\'identlv:
Ji' = — II, or n'/n = — i.
Thus, it will not be necessar\' to in\^estigate separately and independ-
ently each problem of reflexion; for so soon as we have discovered in
any special case the relation between the incident ray and the corre-
sponding refracted ray, we ha\'e merely to impose the condition
w' = — H
in order to ascertain directly the relation which under the same cir-
cumstances exists between the indd^it ray and the corresponding
reflected ray. This procedure, which will be frequently emplo>-ed in
the folloiK-ing pages, will be found to be exceedingly omvenient and
serv-iceable, besides sa\"ing much needless labour.
Here* also, we take occasion to say that hereafter whenever we
speak of the "direction of a straight line" — that is, ike pasiiiue direc-
tion of the line — ^^'e shall mean alwa\*s the direction from a point on
the lint in the medium of the incident rays toxairds the point where the
line meets the rejtectins, or refrci<tin^ surface. If the straight line is itself
the path of an incident or refracted ray of light, the po6iti\-e direction
as thus defined will U^ the direction alon^ the line in xckick the tight goes;
but if the straight line is the path of a rejected ray^ the po6iti\-e direc-
tion in this case i^assumiui: that there is only one reflecting surface)
will be opposite to that which the lii:ht actually follows. It will be
well to Ixwr this in mind, esixvially in derix-inj: reflexion-formulae from
the cvrresjx^nding rvfraction-fonnuLv by the method abo^-e mentioned.
^See §170: see alsi> §251.^
ART. S. THE CASE OP TOTAL REFLEXIOH.
27. The formula
sm a «■ . sm a
enables us to calculate the maiiuituvle of the an^ile of refraction a\
{ 27.] Fundamental Laws of Geometrical Optics. 23
so soon as we know the values of the indices n, n' of the two media
and the magnitude of the angle of incidence a; and thus we can de-
termine the direction of the refracted ray corresponding to a given
incident ray. However, the solution of the above equation is not al-
ways possible, for if the magnitudes denoted by the symbols », n' and
a are such that the expression on the right-hand side of the equation
turns out to have a value greater than unity, evidently there will be
no angle a' that can satisfy the equation, and hence in such a case
there will be no refracted ray corresponding to the given incident ray.
In order to make this matter clear, let us distinguish here two cases
as follows:
(i) The case when n' > n; as, for example, when the light is re-
fracted from air to water {v! \n = 4/3). In this case the second
medium is said to be more highly refracting, or '^optically denser",
than the first medium. The angle of incidence a will be greater than
the angle of refraction a', so that a ray, entering the second medium
from the first, will be heni towards the incidence-normal. Under these
circumstances, the value of the expression on the right-hand side of
the above equation will be always less than unity, so that there is
^ways a certain angle a' whose sine has this value. Provided the second
ffiedium is optically denser than the firsts to every incident ray there wUl
ohoays be a corresponding refracted ray.
(t) The case when n' <n; as, for example, when the light is re-
fracted from water to air {n'/n = 3/4) ; in which case the first medium
is the optically denser of the two. The angle of incidence a now will
be less than the angle of refraction a', so that the refracted ray
^U be bent away from the incidence-normal. When n is greater
than n', the expression on the right-hand side of the above equation
roay be less than, equal to or greater than unity, depending on the
^ue of the incidence-angle a. For a certain limiting value a == A
of the angle of incidence, we shall have n-sina/n' = i, and hence
«' * 90®. In this case, therefore, the refracted ray corresponding to
^ incident ray which meets the refracting surface at an angle of in-
cidence A such that
sin il = - = n^, (8)
n
^11 He in the tangent-plane to the refracting surface at the point of
incidence. If the two media (a) and (b) are separated by a plane
surface, the refracted ray in this limiting case will proceed along the
surface, or, as we say, just "graze" the surface. This angle A be-
24
Geometrical Optics, Chapter 1.
IS 27.
b A
//^
l////^\^ ^
S
Total Iittbui&i. RBn-mziox.
tween the incidence-normal and the direction of the ray in the denser
of the two media is called the critical angle for the two media (a) and
(6). In formula (8) n denotes always the refractive index of the den-
ser of the two media (n^ < i); so that, for example, if the two media
are air and water, the water
corresponds to medium (a)
and the air to medium (fr),
and hence we have n^
= 3/4, for which we find
A = 48° 27' 40'. For air
and glass, n^ = 2/3 and
A = 42" 37'.
In this case {n>n'),ii the
incident ray meets the re-
fracting surface at an angle
of incidence a greater than
the critical angle A, the
expression « ■ sin afn' will be greater than unity, which means that
there will be no real value belonging to the angle a', and hence to such
an incident ray there will be no corresponding refracted ray. The
ether-disturbance propagated in the denser medium in such a direc-
tion as this will not cross the boundary-surface between the two media,
but will be totally reflected there. Consider, for example, the diagram
(Fig. 7), where the point designated by 5 represents a point-source
of light supposed to be situated in a medium (a) which is optically
denser than the medium (6) from which it is separated by a plane
refracting surface, the trace of which in the plane of the paper is the
straight line nii. Rays are emitted from 5 in all directions, but only
those rays are refracted into the rarer medium {ti) that are comprised
within the conical surface whose vertex is at S, whose axis is the per-
pendicular SA let fall from S on n)t, and whose semi-angle is A ASB
=■ L A— sin"' H^. The ray SB is refracted along the plane refract-
ing surface in the direction S)i, as shown by the arrow-head; whereas
a ray SR which has an angle of incidence a greater than the critical
angle A is not refracted at all.
Another way of regarding this diagram is to suppose that an eye
were placed at the point S, and that the rays were being refracted
from the medium (6) into the denser medium (o) ; so that in this case
the directions of the arrow-heads on the rays in the figure should all
be reversed. All the rays entering the eye at S will be comprised
within the cone generated by revolving the right triangle SAB around
§28.]
Fundamental Laws of Geometrical Optics.
25
SA as axis. For example, suppose that the media (a) and (6) are
water and air, respectively, so that mm represents, therefore, the hori-
zontal free surface of tranquil water, and suppose that S marks the
position below the water of the eye of an observer. An object situ-
ated on the horizon (determined by the water-surface) would be made
visible by means of the ray BS, and the eye under water would lo-
cate the object as being in the air in the direction SB. A ray coming
from a star and falling on the surface of the water between A and B
might enter the eye at 5, but the apparent zenith-distance of the star
would always be less than its actual zenith-distance, except when the
star was actually at the zenith-point of the celestial sphere.
The phenomenon of total reflexion of light at the boundary-surface
between water and air is beautifully exhibited in the luminous foun-
tains and cascades that in recent years have been spectacular features
at expositions and places of amusement.
Incidentally, it may be remarked here that the ratio of the inten-
sity of the reflected light to that of the refracted light increases stead-
ily with increase of the angle of incidence, from the least value of this
angle when the rays are
normally incident to its
greatest value when the
rays are totally reflected.
The rays that are totally
reflected from the inside of
one of the faces of an equi-
lateral triangular gla^
prism placed in the sun-
light are seen at a glance
to be brighter than the
rays reflected at the outside
face of the prism.
ART. 9. GBOMBTRICAL COH-
STRUCnOHS, ETC.
Pig. 8.
Construction of Rbflrcted and Rbfractbd Rats.
The straiffht line tt U the trace ia the plane of the paper
of the tansrent-plane at the incidence-point B to the
reflecting or refracting surface. BP* =» n' • BPln ;
PA=^AP"\ PB, BR and BQ represent the paths of the
incident, reflected and refracted rays, respectively.
28. Construction of the
Reflected Ray. In the dia-
gram (Fig. 8) the straight
line PB represents the path
of an incident ray meeting a reflecting surface at the incidence-point
B^ and NN' represents the normal to this surface at 5; so that, if a
denotes the angle of incidence, Z NBP = a. The straight line per-
26 Geometrical Optics, Chapter I. [ § 29.
pendicular to NN' at the point B is the trace in the plane of inci-
dence of the tangent-plane tt to the reflecting surface at B. In order
to construct the corresponding reflected ray, we draw from any point
P of the incident ray the straight line PA perpendicular to tt at A,
and prolong this perpendicular to P" until AP" = PA, and from P"
draw the straight line P'^BR; then BR will represent the path of
the corresponding reflected ray. The proof of the construction is
obvious from the figure, since we have:
ZNBR = ZPP'R = ZBPP"^ ZPBN = - a;
according to the law of reflexion.
29. Construction of the Refracted Ray. Let n, n' denote the ab-
solute indices of refraction of the two isotropic media separated by
a smooth refracting surface, and let B (Fig. 8) designate the point
where the given incident ray PB meets this surface. With the inci-
dence-point B as centre, and with any radius r = BP describe in the
plane of incidence the arc of a circle cutting the incident ray in a point
P; and in the same plane describe also the arc of a concentric circle
of radius equal to n'r/n. Through P draw a straight line perpendicular
at A to the plane tt which is tangent to the refracting surface at the
incidence-point B ; and let the straight line A P, produced if necessary,
meet the circumference of the latter circle in a point P' lying on the
same side of the tangent-plane as the point P. Through the point
B draw the straight line P'BQ. Then BQ will represent the path
of the corresponding refracted ray. For
sinZ APB BP' ^ n'
sin ZAP'B" BP "" n '
and, since Z A PB = Z NPB = a, it follows from the law of refrac-
tion that L AP'B ^ Z N'BQ = a\ where a' denotes the angle of
refraction.
The diagram, as drawn, exhibits the case when the ray is refracted
into a denser medium {n' > n) ; but the construction given above is
equally applicable to the other case also.
Assuming that n' > n, we see from Fig. 8 that when the angle of
incidence Z NBP = 90°, the incident ray PB will be tangent to the
refracting surface at the incidence-point B, and then BA == BP, so
that AP' will be tangent to the construction-circle of radius BP.
In this case we shall have:
a'^ Z PP'B = sin-' ^~ = sin"' -, = ^4,
P'B n
830.1
Fundamental Laws of Geometrical Optics.
27
where A denotes the magnitude of the so-called critical angle of the
two media (§ 27).
30. The Deviation of the Refracted Ray. The angle between the
directions of the incident and refracted rays is called the angle of devi-
ation, and will be denoted here by the symbol €. Thus, if P JB (Fig. 9)
represents the path of a ray incident on a refracting surface at the
point B, and if P'B (constructed as explained
in § 29) shows the direction of the correspond-
ing refracted ray, then Z P'BP = c; that is,
c denotes 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 incident ray. Ifa= Z NBP = /LA PB,
«' = Z NBP' = AAP'B denote the angles
of incidence and refraction, the angle of devi-
ation is defined by the following relation:
t^ a — a
(9)
Fio. 9.
Deviation of thb Rs-
FRACTBO Rat. The straisrht
lines PB, P'B show the direct-
ions of the incident and re-
fracted rays.
IP*PB^*, <
a — a'
The diagram is drawn for the case when
n' > n, for which the sign of the angle € is
positive. By merely interchanging the letters
P, P' in the figure, we obtain the case when
n' < n, for which the angle denoted by € is negative.
It is apparent from the figure that the intercept P'P included be-
tween the circumferences of the two construction-circles, which re-
mains always parallel to the normal BN, increases in length as the
angle of incidence a increases; and, since the other two sides BP and
BP' oi the triangle BPP' have constant lengths, it follows that the
deviation of the refracted ray increases with increase of the angle of in-
cidence. This is true both for n' > n and for »'< n.
Differentiating equation (6), we obtain (after eliminating n, n'):
da! tan a'
da tan a
(10)
and, since from the figure tan a' /tan a = AP/AP\ we have there-
fore the following relations:
da'idaidt = APiAP'iPP';
GO that in the triangle PP'B the side PP' opposite the angle € is
28 Geometrical Optics, Chapter I. I § 32.
divided externally at A into segments which are inversely propor-
tional to the corresponding variations of the angles at P and P'.
Moreover, since
d€ _PP' I
da" AP'" i + AP/PP''
and since as the angle a increases, not only does A P decrease but PP'
increases by an equal amount, it follows that de/da increases with
increase of a. Hence,
The greater the angle of incidence, the greater will be the corresponding
rate of increase of the angle of deviation.
This characteristic property of refraction is true both for n'/n
greater than unity and for n'/n less than unity. In the case of re-
flexion, the law will be different, for the deviation of the reflected
ray decreases in proportion as the angle of incidence increases.
ART. 10. CERTAIN THEOREMS CONCERirnf O THE CASE OF SO-CALLBD
OBLIQUE REFRACTION (OR REFLEXION).
31. The plane of incidence containing the normal to the refracting
(or reflecting) surface at the point of incidence is a normal section of
the surface at that point; and, whenever feasible, it will be conven-
ient to select this plane as the plane of the diagram. In the following
pages, however, we shall often have occasion to investigate the path of
a ray which is incident in succession on a series of refracting (or re-
flecting) surfaces; in which case the plane of incidence with respect to
one such surface will, in general, make with the plane of incidence with
respect to the next following surface an angle different from zero.
Accordingly, in our diagrams it may happen that the normal section
of the refracting (or reflecting) surface which lies in the plane of the
paper may not coincide with the normal section which contains the
ray incident on that surface and its corresponding refracted (or re-
flected) ray. In such a case as this the ray is said to be **obliquely"
incident on the surface; and in this connection the following theorems
will be found useful.
We remark that it will be necessary to treat here only the problem
of refraction; as the corresponding theorems relating to reflexion,
which may easily be proved independently also, may be derived im-
mediately by merely putting n' = — », according to the general prin-
ciple explained in § 26.
32. In the diagram (Fig. 10) the straight line NN' is the normal
at the point B to the refracting surface, so that the plane of the dia-
§32.1
Fundamental Laws of Geometrical Optics.
29
gram is therefore the plane of a normal section of the surface with
respect to the point B; and the straight line tt is the line of inter-
section of the plane of the
paper with the plane tan-
gent to the refracting sur-
face at B. Let RB rep-
resent the path of a ray
incident on the surface at
the point B, and from any
point R of this ray draw
R N perpendicular at N to
the normal NN'. The
plane of incidence RB N is
also the plane of a normal
section of the surface; but
we shall suppose here that
RB is "obliquely" incident
on the refracting surface,
so that the plane of inci-
dence is not the same as the
plane of the paper. The
corresponding refracted ray
BR' will lie in the plane of incidence. On this refracted ray take a
point R\ such that
RBiBR' ^n:n',
and from R' draw R' N' perpendicular to NN' at N'. Draw also
RPj R'P' perpendicular to the plane of the paper. Then the two
planes RPN, R'P' N' will be parallel to the tangent-plane at B. By
the law of refraction:
n ' sin a = n' ' sin a',
where Z NBR ^ a, Z N'BR' = a'. By the construction:
Pio. 10.
Oblique Refraction. J?B, BR' represent the paths of
the incident and refracted rays. Z NBR = a, Z N'BR*
= a', Z RBP=' ri, Z R'BP' = V. Z NBP^- y. Z N'BP*
and, therefore:
RN--RBsma, KN' = RB • sin a'\
RN = N'R\
Since R N and N'R\ lying both in the plane of incidence, are equal
and parallel, PN and N'P\ which are the projections oi RN and
N'R' in the plane of the paper, are also equal and parallel; so that the
triangles NPR and N'P'R' are congruent, and RP = P'R'.
30 Geometrical Optics, Chapter I.
If the symbols ly, r;' are employed to denote the angles made I
incident and refracted rays i?B, BR' with their projections PB
in the plane of the normal section which is the plane of the ]
so that
Z RBP = 1,, Z R'BP' = n\
then, since
RP--RB sin n, R'P' = R'B - sin 17',
we have:
RB'sinfi = BR''sinri';
and, hence:
wsiniy = n' sin 17'.
Accordingly, we have the following result:
The sines of the angles which the incident and refrctcted rays
with the plane of any normal section of the refracting surface at the
of incidence have the same ratio as the sines of the angles of incident
refrcLCtion themselves.
33. Moreover, since, by construction (Fig. 10),
and
we have:
Putting
so that
we find:
n'RB^n-BR,
PB^RB' cos 17, BP' = BR' • cos 17',
n'PB cos 17 = n-BP' • cos 17'.
Z NBP = 7, ^ N'BP' = 7',
PB • sin 7 = BP' ' sin 7',
n 'COS t; • sin 7 = »' • cos 17' • sin 7';
a result which may be stated as follows:
The projections of the incident and refracted rays on a plane of <
maJ section of the refracting surface at the point of incidence ar
subject to a law of refraction, the absolute indices of refraction n
and n' 'COS rj' being dependent on the angles rj and ri' made by tki
dent and refracted rays with the plane of the normal section.
If we put
ny^^n^ cos t;, «,' = n' • cos 17',
and bear in mind that we have also the relation:
nsin77 = n'sinT7',
J3il
Fundamental Laws of Geometrical Optics.
31
we can derive easily the formula:
n, " n
in the form given by Cornu/
34. The following is a convenient method of constructing a draw-
ing representing the path of a ray obliquely refracted at the surface
of separation of two isotropic optical media.
Let the plane of the paper (Fig. ii) be designated as the ^3^-plane
and let the tangent-plane to the refracting surface at the incidence-
point B be designated as the 3^-plane, which is represented as making
Fio. 11.
COlfSTRUCTION OF OBLIQUELY RBFRACTBD RAT.
^ acute angle with the plane of the paper. From B, in a plane ocz
P^ndicular to the plane of the paper, draw B N normal at B to
tile tangent-plane yz, and draw BM normal at M to the plane of the
P^per. Suppose, for example, that the real length of BM is twice its
length as shown in the figure. Let P designate the position of the
point where the given incident ray meets the plane of the paper, so
tiiat BP shows the direction of the incident ray lying in the plane of
wcidence BPN. If the triangle BPN is revolved around PN sls
^ until it comes into the plane of the paper, the point B will arrive
^^ a point C on the straight line drawn from M perpendicular to NP,
A. CoRNU: De la refaction k travers un prisme suivant une loi quelconque: Ann,
^•^vim. (2). i. (1873). 237. See also E. Reusch: DieLehre von der Brechung u.Far-
****»rtreuung des Lichts an ebenen Flaechen und in Prismen, die in mehr synthetischer
Form dargestellt: Pogg. Ann, cxvii. (1862), 247 ; and A. Bravais : Notice sur lea
^•riifliei qui sont situ^ a la m6me hauteur que le soleil: Journ. ic, polyt., xviii., cah. 30
^**4S). 79; and Mdmoire sur les halos: Joum. ic. polyt., xviii.. cah. 31 (1847). 27.
I i
I ;
32 Geometrical Optics, Chapter I. [ §
and Z NCP = a. Hence, with C as centre and with radius eq
to n' ' CPjn describe in the plane of the paper the arc of a ciir
meeting in a point R the straight line drawn through P parallel
CN\ evidently, as in §29, Z NCR = a'. Therefore, the straight la
BQ joining the incidence-point B with the point Q where CR m
NP will represent in the diagram the direction of the refracted
CHAPTER II.
CHARACTERISTIC PROPERTIES OF RAYS OF LIGHT.
ART. 11. THE PRinCIPLE OF LEAST TIME (LAW OF FERMAT).
. Fermat* ( 1 608-1 665), arguing from an assumed law of the
3my of nature that light must be propagated from one point to
lier in the shortest time, was able to deduce the law of refraction
e case of a ray refracted from one isotropic medium to another
« a plane boundary-surface; or, conversely, that the time required
le light to be transmitted from any point P on the incident ray
ly point Q on the corresponding refracted ray is less than it would
ong any other route between the points P and Q. A correspond-
aw in regard to light reflected at a plane mirror dates back to
0 of Alexandria (150 B. C).
the boundary-surface separating the two media is curved, the time
n by the light to be transmitted from P to Q along the actual
may not, however, be always a minimum; on the contrary, in cer-
cases it may be a maximum. A simple illustration is given by
iVm. Rowan Hamilton,* who instances the fact that "if an eye
aced in the interior, but not at the centre, of a reflecting hollow
re, it may see itself reflected in two opposite points, of which one
ed is the nearest to it, but the other on the contrary is the farthest;
lat of the two different paths of light, corresponding to these two
>site points, the one indeed is the shortest, but the other is the
est of any."
►. A characteristic property of a ray of light may be stated quite
orally as follows:
a ray of light, undergoing any number of reflexions and refractions,
^ts two points P and Q, the time taken by the light to be transmitted
'i P to Q along the CLCtual path of the ray is either a minimum or a
irnum.
will be entirely sufficient if we prove the truth of this statement
-ly for the case of a single refraction; as it can then be extended
ediately to the case where the ray suffers any number of reflexions
refractions.
^ Fbrmat: LiUera ad P. Mbrsbnum contra Dioptricam CarUsianam (Paris, 1667}.
V. R. Haicilton: On a General Method of expressing the Paths of Light and of
lanets: Dublin UniversUy Review, October, 1833.
33
34
Geometrical Optics, Chapter II.
I§3^
In the diagram (Fig. 12) the point designated by P represents th<
starting point in the first medium and the point designated by (
represents the terminal point in the second medium; and i^i is thi
trace of the refracting surface in a plane containing the two points Jf
and Q (which is represented here as th<
plane of the paper). The problem t(
be solved is, What must be the positioi
on the refracting surface of the incidence
point Bf in order that the time taker
by the light to be transmitted from /
to Q, viz.,
PB . BQ
where v and v' denote the speeds ol
propagation of light in the first anc
second medium, respectively, shall b(
either a minimum or a maximum? Let
us suppose that this point B is also situ
ated in the plane of the paper, whid
will be therefore the plane of inddena
of the ray PB, Evidently, this critical
position of the inddence-point B will
be such that an infinitdy small variation from this position would
not alter the time taken by the light in going from P to Q. It will
suffice to consider a variation of the position of B in the plane of in-
ddence; accordingly, let us designate by B^ the position of a point
on the normal section /x/li of the refracting surface infinitely dose tc
the critical point of incidence B. If the light travdled from P to C
along the route PBiQ^ the time taken would be:
Pio. 12.
PERlfAT'S I«AW OF I,BA8T TiMB.
PB, BO represent paths of incident
and refracted rays. /*BiO is another
hypothetical route from the point P
in the first medium to the point Q in
the second medium, which differs infi-
nitesimally from the actual nuttPBO.
and, consequently, the condition which has to be imposed is that
PB - PB, ^BQ - B,Q
H — ; — -^ = O.
V V
From B and B, draw BR, and B^R perpendicular to B^Q and PE
at Ri and at R, respectively; then
PB - PBi = RB and B,Q - BQ^ B^R^;
§ 38.] Characteristic Properties of Rays of Light. 35
audi hence, the condition above becomes:
RB B,R.
V-^ = o.
V V
Draw NBN' normal to the curve mm at the point B, and put
£ HBP = a, Z N'BQ = a'; then
RB = BBi • sin a, B^R^ = BB^ • sin a'.
The condition may be written, therefore:
sing _ V
"• 7 "" ~/ »
sm a V
which will be recognized as the law of refraction (§ 2i). But the actual
path of the ray from P to Q is according to this law. Consequently,
the time t along this path will be either a minimum or a maximum.
Whether in any given special case the time is a minimum or a maxi-
mum, can be determined only by investigating the form of the re-
fracting (or reflecting) surface.
37. This result, as was stated above, can be immediately extended
to the case where the ray is compelled in its progress from P to Q
to traverse any number of media or to bend away at certain surfaces
of separation between two bodies, that is, where the ray is constrained
to undergo a certain prescribed series both of refractions and of re-
fedons. If we denote by t^ the time occupied by the light between
two successive adventures of this kind, the analytical expression of
the so-called Principle of Least Time may be written in the following
form:
«(S0 = o; (13)
that is, the time taken by the light to be transmitted, under certain
P'^scribed conditions, from one point P to another point Q along the
actual path of the ray differs from the time which would be taken
along any other hypothetical route, which is infinitely near to the
actual route, by an infinitesimal of an order higher than the first order.
38. The Optical Length of a Ray; and the Principle of the Short-
^ Route. The sum of the products of the length of the path of a
^y in each medium by the refractive index of that medium is called
the Optical Length, sometimes also the reduced length, of the ray. Thus
tf '1, /„ etc., denote the actual lengths of the ray-path in the media
36 Geometrical Optics, Chapter II.
whose indices of refraction are denoted by «i, n„ etc., resp<
the optical length of the ray is:
where n^^, /^ denote the values of the magnitudes n, / for
medium. When the ray is reflected at a body i, we must ]
n^ = — «i_i, according to the rule given in § 26; so that the d
given above applies to reflexions as well as to refractions.
Since n^ = V/vj^, where V and v^^ denote the speeds of pro]
of light in vacuo and in the ith medium of the series, resp
and l^ = Vffi^, we have njj^ = F/j^; whence we see that the
length of the ray (=FS^J is equal to the distance that ligJ
travel in vacuo in the same length of time as it takes to go
actual path. This explains the use of the term * 'reduced
We also see that equation (13) is equivalent to the follow
6(XnJf) = o;
whence is derived the so-called Principle of the Shortest Roui
may be stated as follows:
When light is transmitted from one point P to another point {
going during its progress any prescribed series of reflexions an
Uons, the optical length measured along the actucU path of the
minimum or a maodmum.
▲RT. 12. HAMILTOirS CHARACTERISTIC FUNCTION.
39. The statement at the end of the last article recalls I
TUis*s celebrated ''Principle of Least (or Stationary) Action
wards developed by Euler and other great mathematicians;
we define the vague term "action" in the case of a ray of
mean the optical length of the ray. The function
is the so-called Characteristic Function, the idea of which '
introduced into mathematical optics by Sir W. R. Hamili
which reduces the solution of all problems, in theory at leas
common process.^
* Professor P. G. Tait in his book on Light (Edinburgh, 1889) says (Art. 189)
TON was in possession of the germs of this grand theory some years before 1824,
first communicated to the Rojral Irish Academy in that year, and published ii
instalments some years later." Hamilton's papers on this subject publishec
title ** Theory of systems of rays " are to be found in the TransactUms cf ike \
Academy, xv. (1828), 69-174; xvi. (1830), 3-62; and 93-126; and zvii. (1837
i40-l Characteristic Properties of Rays of Light. 37
In the application of this method the co-ordinates of the two ter-
minal points P{a, 6, c) and Q{a\ b\ c') which are connected by the ray
are to be regarded as known, and, therefore, invariable. The equa-
tions of the reflecting and refracting surfaces must likewise be given.
But the co-ordinates of the points where the ray meets these surfaces
are the variables in the problem. The equation of the kxh. surface
may be written:
and ance the co-ordinates jc^, y^, Zj^ of the point where the ray meets
this surface must satisfy this equation, we may regard 2^ as a known
function of x^, y^. The actual length of the ray-path between the
(i-i)th and the ftth surfaces will be:
Since IL = o, we must have:
dL dL
-- = o, J- = o,
where s^ is to be considered as the dependent variable, so that:
dL _dL^^dL^dz^ dL _dL dL dzj^
dxj, dxj, dZf, dxj, ' dyj, dy^ dz^ dy^ *
M. In order to illustrate the use of the method in a simple case,
«t us suppose that there is only one refracting surface separating two
°*edia of refractive indices n and »'; then:
^ = r = Via' - x,y + {V - y,y + {c' - z,)\
and
2nd, according to the equations above, we derive here:
,^ (x.-a) + (..-c)g; (a'-»,) + (c'-..)g-;
dy, " * 'I "' V °-
38 Geometrical Optics, Chapter 11. [ § 41.
If the incidence-point is taken as the origin of co-ordinates, then
Xi = yi = ail = o. Moreover, if the incidence-normal is taken as the
2-axis, then also dzjdxi = dz^jdy^ = o. Introducing these simpli-
fying values, we find:
na n'a' nb n'b'
If, further, we take the plane of incidence for the yz-plane, we must
put a = o; whence it follows from the first of the two equations above
that a' = o also; and hence the point Q, and therefore also the re-
fracted ray, must lie in the plane of incidence, in accordance with a
fundamental law of refraction. Finally, if a, a' denote the angles of
incidence and refraction, it is evident that:
b . V . ,
J = — sm a, -y = sin a ,
and hence the second equation above is equivalent to the other fun-
damental law of refraction:
n • sin a = n' • sin a'.
Thus we see how this process leads to the ordinary laws of refraction.
41. If the characteristic function of a system is known, it is pos-
sible in theory to deduce from it all the optical properties of the sys-
tem. In some comparatively simple cases this process enables us to
get results with almost magical facility. It must be admitted, how-
ever, that the method, so fascinating on account of its generality, is
difficult in its applications, involving as it does the theories of the
higher analytical geometry and demanding mathematical knowledge
and skill of the highest order. In addition to Hamilton, a number
of other investigators, among whom may be mentioned espedally
Maxwell^ and Thiesen* and Bruns,^ have developed in one way or
^ J. C. Maxwell: A dynamical theory of the electromagnetic field. Proc, Roy. Soc.,
xiii. (1864). 531-536; Phil. Trans., civ. (1865), 459-512; PML Mag,, (4) nix. (1865).
z 52-1 5 7. Also, On the application of Hamilton's characteristic function to the theoiy
of an optical instrument symmetrical about an axis: Proc. of London Maih, Soc, vL
(1874-5). 1 1 7-122; and On Hamilton's characteristic function for a narrow beam oC
light; Proc. London Math. Soc, vi. (1874-5), 182-190.
*M. Thiesen: Bcitraege zur Dioptrik: Berl. Ber., 1890, 799-8i3- Also, Ucber voU-
kommene Diopter, Wied. Ann. der Phys. (2), xlv. (1892), 82i-'3. Sec also, Ueber die
Construction von Dioptem mit gegebenen Eigenschaften, Wibd. Ann, der Phys, (a),
xlv., 823-4.
' H. Bruns: Das Eikonal: Saechs. Ber, d. Wiss., xxi. (1895), 321-436. See also F.
Klein: Ueber das BRUNSche Eikonal; and, also, Raeumliche KoUineation bel
Instrumenten: Zft. f. Math. u. Phys., xlvi. (1901).
§ 42.] Characteristic Properties of Rays of Light. 39
another the theory of the characteristic function in optics. But the;
greatest difficulty is encountered in turning the theory to account,
and, so far as the practical optician is concerned, the HAMiLTONian
method has not been found to smooth his way.
ART. 13. THE LAW OF MALUS.
42. The wave-front at any instant due to a disturbance emanat-
ing from a point-source is the surface which contains 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 will be
concentric spheres described around the point-source as centre; but
if the wave-front arrives at a reflecting or refracting surface m» at
which the directions of the so-called rays of light are changed, the form
of the wave-surface thereafter will, in general, be spherical no longer;
and even in those cases when the refracted (or reflected) wave-front
is spherical, the centre (except under certain very special circum-
stances) will not coincide with the centre of the incident wave-surfaces.
The function 2n/ (§38) has the same value for all actual ray-paths
between one position of the wave-surface and another position of it;
so that knowing the form of the wave-front at any instant and the
paths of the rays, we may construct the wave-front at any succeeding
instant by laying off equal optical lengths along the path of each ray.
It follows that the ray is always normal to the wave-surface. For,
suppose that the straight line PB represents the path of a ray inci-
dent at the point B on a surface fi separating two media, and that the
straight line BQ represents the path of the corresponding refracted
(or reflected) ray; and let <r designate the wave-surface whereon the
point Q lies. From the incidence-point B draw any straight line BR
meeting the wave-surface <r in the point designated by R. Then, by
the minimum property of the light-path, the route PBQ is less than
the route PBR^ because the natural route from P to i? is not via
the incidence-point B; and hence the straight line BQ must be shorter
than the straight line BR, and therefore BQ is the shortest line that
can be drawn from the incidence-point B to the wave-surface <r. It
follows that BQ meets the wave-surface <r normally. The same rea-
soning will be applicable also in the case of every other refraction or
reflexion, so that we may state generally:
The light-rays meet the wave-surface normally, and, conversely, the
system of surfaces which intersect at right angles the rays emanating
originally from a point-centre is a system of wave-surfaces.
40 Geometrical Optics, Chapter II. [ § 44.
43. The fact which has just been proved is equivalent to the law
enunciated by Malus/ in i8o8, which may be stated as follows:
An orthotomic system of rays remains orthotomic, no matter what r«-
fractions {or reflexions) the rays may undergo in traversing a series of
isotropic media. (An orthotomic system of rays is one for which a
surface can be constructed which will cut all the rays at right angles.)
A proof of this law which does not contain any reference to the
ideas of the Wave-Theory is given by Heath* as follows:
Let A BCD E (Fig. 13) and A^B^C^DiE^ be two infinitely near ray-
paths, and suppose that they cross nor-
"»- ^' mally at A and Ai a certain surface <r. On
each ray of the system, reckoning from the
points Af Ai, etc., where the rays cross the
i surface <r, let a series of points E, Ej, etc.,
^ be determined such that the optical lengths
from A to £, from Ai to Ej, etc., are all
^A ' "^^ / equal. We propose to show that the surface
^■^^A/^ ^^"^^^^ a' which contains the terminal points £, £,,
_ „ etc., of these rays will cut the rays at right
i^w OF malcs. angles.
In order to prove this, we draw the
straight lines AiB and DE^ as shown in the figure. The optical
length Xnl measured along the infinitely near hypothetical route
AiBCDE^ is, by Fermat's Law, equal to 2n/ along AiBiCiDiEi
or along ABCDE, Hence, subtracting from each the part BCD
which is common to the routes ABCDE and A^BCDE^^ we have:
n-i4B + n'-D£ = n-i4iB + »'D£i,
where n, n' denote the refractive indices of the first and last medium,
respectively. But since AB \s normal to the surface <r, ultimately
A^B = AB; and, hence, ultimately also DE = DEi; that is, DE
must be normal to the surface a'. In the same way we can show
that any other ray D^E^ will likewise be normal to o"'.
ART. 14. OPTICAL IMAGES.
44. In case we do not wish to utilize all the rays emitted from a
luminous body, we may interpose a screen with a suitable opening
in it, whereby some of the rays are intercepted, while others, called
' E. L. Malus: Optique: Joum. de VEcoU Polyt., vii. (1808), 1-44; 84-129.
• R. S. Heath: A Treatise on Geometrical Optics (Cambridge, 1887). Art. 87.
§ 45.] Characteristic Properties of Rays of Light. 41
the "effective rays", are permitted to pass through the opening. Thus,
each separate point of a luminous body is to be regarded as the vertex
of a cone or bundle of rays. In every bundle of rays there is always a
certain central or representative ray, usually coinciding with the axis
of the cone, or distinguished in some special way, called the chief ray^
of the bundle. A pencil of rays is obtained from a bundle of rays by
passing a plane through the axis or chief ray of a bundle. This use
of this term is convenient and is also in accordance with the usage
of some writers on geometry.
An optical system is a combination of isotropic media arranged in
a certain sequence so that they are traversed by the effective rays
all in the same order. In this case the effective rays emitted by a
luminous point P are those rays coming from P which succeed finally
in passing through the system from one end to the other without being
intercepted at any point on the way. In general, through any point
P', within the region reached by the bundle of emergent rays which
had their origin at the luminous point P, one ray, and one ray only,
will pass, since the optical route between P and P\ for a given dis-
podtion of the optical media, will usually be uniquely determined.
However, within this region there may be found a number of points
F where two or more rays intersect; and under certain circumstances
it may indeed happen that all of the effective rays emanating from
the point P will, after traversing the optical system, meet again in
one point P'; and in this exceptional case the point P' is said to be
the optical image of the point P, and the two points P and P', object-
point and image-point, are called conjugate points or conjugate foci.
'f the rays actually pass through P', the image is said to be real;
'^'hereas if it is necessary to produce backwards the actual portions of
tte rays in order to make them intersect in P', the image is said to
^ virtual. Thus, in the case of a perfect imager all of the **emer-
^^t rays" corresponding to the rays of a given bundle of ''incident
'^ys" proceeding from the object-point P will intersect in the image-
point P".
^5. In order, therefore, to have an image in the sense above de-
™^, the optical system must transform a train of spherical waves
^^th the object-point P as centre into another train of spherical waves
^th the image-point P' as centre. The optical lengths along all the
tay-paths between P and P' will be equal, so that the disturbances
The term " chief ray " is a happy rendering of the German Hauptstrahl which has
^ introduced into English Optics by Professor Sil vanus P. Thompson in his transla-
^ of Dr. O. LuMMXR's Photographic Optics (London, 1900).
42 Geometrical Optics, Chapter II. [ § 46.
arrive at P' along all these different routes all in the same phase, and
hence conspire to produce at P' a maximum effect. According to
the notions of Geometrical Optics, there will be no light-effects what-
ever at points which lie outside of the cone of rays which meet in P';
but when the matter is investigated by the surer methods of Physical
Optics, we discover that this conclusion is not justified, and that there
are light-effects at points which are not comprised within this geo-
metric cone. In fact, instead of a single image-point P', we find that
we have around P' a so-called diffraction-pattern. But the wider the
cone of rays that meet in P', the more nearly will the distribution
of light around P' approach as its limit the ideal image-point of Geo-
metrical Optics; and this is the only meaning which Physical Optics
can attach to the idea of an image-point.
ART. 15. CHARACTER OF AN HfFIinTELT NARROW BUNDLE OF OPTICAL RATS.
46. Caustic Surfaces. According to the Law of Malus, the direc-
tion of the ray-path at any point P is along the normal to the wave-
surface which passes through P. In the special case when the wave-
surface is spherical, the normals all meet in one point at the centre of
the sphere; but if the wave-surface has any other form, a pair of nor-
mals drawn to the surface at two different points will, in general,
not intersect at all. The curved line which is traced on the surface
by a plane containing the normal to the surface at the point P is
called a normal section of the surface at this point. The curvatures
of these lines at the point P will generally be different for different
normal sections; and Euler has shown that at each point P of a curved
surface the normal sections of maximum and minimum curvature are
at right angles to each other; and, accordingly, the two normal sections
thus distinguished are called the Principal Sections of the surface at
the point P.
An investigation of the theory of the curvature of surfaces shows
that the normals at consecutive points of a curved surface wUl intersect
each other, provided those points are taken along the curves of greatest
and least curvatures; but that, in general, the normals at consecutive
points do not intersect.
Applying these results from the theory of curved surfaces, let us
designate by the symbol u the chief ray of an infinitely narrow bundle
of rays, and let P designate the position of the point on the wave-
surface (T where the chief ray u crosses this surface. Only those rays
of the elementary bundle which cross the wave-surface <r at the points
lying in the principal sections of the surface through the point P will
§ 46.1 Characteristic Properties of Rays of Light. 43
meet the chief ray u; so that this ray u is to be regarded also as the
chief ray of each of two infinitely narrow pencils of rays lying in two
perpendicular planes: the vertices of these two pencils of rays being
the centres of greatest and least curvature of the surface with respect
to the point P. The other rays of the infinitely narrow bundle which
do not lie in the planes of the principal sections will generally not meet
the chief ray u at all. Thus, on each ray u determined by a point
P of the wave-surface, there are to be found two points, the centres
of greatest and least curvature with respect to the point P, which are
the vertices of two narrow pencils of consecutive rays of which u is the
chief ray.
A line of curvature is a curve traced on a surface such that the nor-
mals at any two consecutive points of the curve intersect each other.
Therefore, through every ordinary point of the surface two such lines
of curvature will pass intersecting each other at right angles. The
totality of each of these two systems of lines of curvature completely
covers the entire surface. The locus of the points of intersection of
rajrs belonging to points which lie along a line of curvature will be
the evolute of that line of curvature; and in optics this evolute, which
is also the envelop of the rays crossing the wave-surface at points
lying along the line of curvature, is called a caustic curve. The total-
ity of the caustic curves corresponding to one system of lines of cur-
vature of the curved surface will constitute a caustic surface. Thus,
there will be two caustic surfaces, one for each of the two systems of
the lines of curvature of the wave-surface; these caustic surfaces being
indeed the lod of the two centres of principal curvature of the wave-
surface. Each ray is evidently a common tangent of the two caustic
surfaces.
In the special case when the wave-surface is a surface of revolu-
tion, so that the orthotomic system of rays is therefore symmetrical
with respect to the axis of revolution, it is easy to obtain a clear idea
of the caustic surfaces. For here one system of lines of curvature
are the meridian curves of the surface, and consequently the caustic
surface corresponding thereto is generated by the revolution about the
axis of symmetry of the evolute of the meridian curve. And the
other system of lines of curvature are circles with their centres ranged
along the axis of symmetry, and, since the rays which cross the wave-
surface at points lying in the circumference of one of these circles will
all lie in the surface of a right circular cone whose vertex is on the
axis of revolution, the caustic surface corresponding to this system of
lines of curvature reduces to a segment of the axis of revolution itself.
44 Geometrical Optics, Chapter II. [ § 47.
In Chapter VI. of Heath's Geometrical Optics (Cambridge, 1887)
the reader who wishes to pursue this subject will find an extensive
investigation of the forms and properties of caustic lines and surfaces
in a number of interesting special cases. Wood's Physical Optics
(New York and London, 1905), wherein the caustic surfaces are
studied especially from the standpoint of the Wave-Theory, and
experimentally rather than mathematically, contains also much on this
subject that is both novel and suggestive. However, so far as the
theory and design of optical instruments is concerned, it will hardly
repay us here to attempt to investigate these surfaces in detail;
although in the next chapter, by way of illustration, we shall study
briefly the caustic in the case of the refraction of a spherical wave
at a plane surface (§54).
47. The main thing that it concerns us to know at present is that
a narrow bundle of optical rays, originally homocentric (or monoceniric,
as it is sometimes called, that is, emanating all from one and the same
point or **focus"), is, in general, transformed by reflexion or refraction
at a surface of any form into a non-homocentric or astigmatic bundle
of rays, all the rays of which, at least to a first approximation, inter-
sect two infinitely short image-lines, the so-called image-lines of the
bundle. We proceed to explain how this occurs, according to the
theory of Sturm,* the originator of the theory of astigmatism.
Let P (Fig. 14) designate the position of a point on the wave-sur-
face <r, and let the ray u determined by the point P be represented
in the diagram by the straight line P35. This ray coincides with the
normal to the surface at the point P, and the points designated by
5, 3 are, in the case shown in the diagram, the centres of greatest and
least curvature, respectively, with respect to the point P. With S
as centre, and with radius equal to SP, describe in the plane of the
paper the infinitely small arc APB of a circle, making AP ^ PB:
' J. C. Sturm: M^moire sur Toptique: Liouville's Joum, de Maih,, iii. (1838), 357-384-
Also, M6moire sur la thdorie de la vision: CompUs rend., xx (1845), 554^560; 761-767;
1 238-1 257. This latter paper was translated and published in Pogg. Ann,, Izv. (1845).
See also the following writers on this subject:
E. E. Kummer: AUgemeine Theorie der gradlinigen Strahlensjrsteme: Crbllss Joam,,
Ivii. (i860), 189-230. Modelle der allgemeinen, unendlich duennen. gtadlinigen Scrab-
lenbuendel: Berl. Akad. Ber., i860. 469-474. Ueber die algebraiachen StimhleiiBytteiiie.
in's Besondcre ueber die der ersten und der zweiten Ordnung: Berl, Akad, Mcmatsha,
1865. 288-293. Berl, Akad. Ahh., 1866, No. i, 1-120.
H. Helmholtz: Handbuch der physiologischen Optik, ii. Thl. (i86o), 246.
A. F. MoEBius: Geometrische Entwickelung der Eigenschaften unendlich dnenner
Strahlenbuendcl: Sitzungsber. d. Saechs. Akad. Math.-phys. CL, ziv. (1862), I~x6.
F. Lippich: Ueber Brechung und Reflexion unendlich duenner Strahtenssratenie an
Kugelflaechen: Denkschr. d. Wien. Akad., math.-phys. Cl„ xxxviii. (i878), 163-192.
C. Neumann: Sitzungsber. d. Saechs. Akad., math.-phys. Cl„ 1879, 42.
§47.]
Characteristic Properties of Rays of Light.
45
and through the point S draw in the plane of the paper the straight
lin^ dSd' perpendicular to the ray u. Let the figure thus obtained
be rotated about dd' as axis through an infinitely small arc above and
below the plane of the paper, so that the arc APB will thereby gen-
Fio. 14.
CoHsnruTioM of IimifiTBLT Narrow. Astigmatic Bundle op Optical Rats. Aa'p'b'Bbpa
b the element of Uie wave-surface <r around the i>oint P. PSS represents the normal to the surface
^ the point/*, and S, 5 designate the positions on this normal of the centres of principal curvature
^th respect to the point P. The chief ray u of the astifirmatic bundle of rays is represented by the
^fxvaal PsS ; S and .?are the primary and secondary imanre-points. respectively ; the infinitely short
lioocc'uid diT are the primary and secondary imaire-lines. respectively, cc' is perpendicular to
tbe chief Fay u at the i>oint Sand lies in a plane perpendicular to the plane of the paper: dd' is
l>cnieiidicalar to the chief ray u at the point Sand lies in the plane of the paper.
«^te the element of surface apbBVp'a'A, which, to a first approx-
iniation, may be regarded as an element d<r of the wave-surface <r in
the immediate vicinity of the point P. We shall investigate, there-
fore, the infinitely narrow bundle of rays which cross the wave-surface
^t points lying within this surface-element d<r, of which the ray u
P">ceeding from P is the chief ray.
Inuring the rotation around dd' as axis the point P traces the in-
finitdy short arc pp\ which is an element of one of the lines of curva-
^ which pass through P, the arc APB being an element of the other
'uie of curvature at this point. The point S traces the infinitely short
^ cc' parallel to pp'\ the arcs cc' and pp' being both perpendicular
to the plane of the paper. The rays proceeding from the points of
the wave-surface which lie along pp' constitute a narrow pencil of
[^ys lying in a plane perpendicular to the plane of the paper and hav-
"^ its vertex at the point 3. And, similarly, the rays proceeding
46 Geometrical Optics, Chapter II. [ § 47.
from the points of the wave-surface which lie along the arc APB con-
stitute a narrow pencil of rays lying in the plane of the paper and hav-
ing its vertex at the point S. The chief ray u is common to both of
these pencils.
The entire bundle of rays may be regarded as composed of a sheaf of
pencils of rays in either of two ways, as follows:
First, the entire bundle of rays may be regarded as arising from the
rotation about cc' as axis of the pencil of rays pSp', so that the element
of arc pp' generates the element of surface da.
Again, the entire bundle of rays may be generated by rotating the
pencil of rays ASB about dd' as axis through infinitely small arcs
above and below the plane of the paper.
The centres of curvature 5, 5, both situated on the chief ray «, are
(as has been stated) the so-called image-points of the infinitely narrow
astigmatic bundle of rays. The point 5 which is the vertex of the
pencil of rays lying in the plane of the paper (the meridian or primary
principal section of the bundle) is called the primary image-point; and
the point 5 which is the vertex of the pencil of rays lying in the plane
perpendicular to the plane of the paper (the sagittal or secondary prin-
cipal section of the bundle) is called the secondary image-point. In
the figure as here shown, the point designated by S is the centre of
the greatest curvature of the surface with respect to the point P, and
the point designated by S is the centre of least curvature; but this
will depend entirely on the form of the surface at P.
The two infinitely short lines cc' and dd\ through both of which
all the rays of the bundle pass, and which, regarded as straight lines,
lie in planes at right angles to each other, and which, moreover,
are both perpendicular co the chief ray u of the bundle, are the
two image-lines of the narrow astigmatic bundle of rays. The primary
image-line cSc' lies in the primary principal section, and the secondary
image-line dSd' lies in the secondary principal section.
Thus, according to Sturm's Theory, the general characteristics
of an infinitely narrow bundle of optical rays may be enumerated as
follows:
(i) The direction of propagation of the disturbance at the point P
is along the ray u which is normal to the wave-surface <r at P, As a
first approximation, the element dc of the wave-surface at this point
may be regarded as bounded by arcs which are parallel to the arcs
of greatest and least curvature of the surface at the point P-
(2) All rays of the bundle which cross the wave-surface at points
lying along the arc APB (Fig. 14) intersect in the primary image-
§ 48.] Characteristic Properties of Rays of Light. 47
point S; whereas the rays which cross the wave-surface at points
lying along the arc pPp^ intersect in the secondary image-point 5. The
rays which cross the element da of the wave-surface at points lying
along any arc drawn parallel to the arc APB will all meet (as a first
approximation) in a point of the primary image-line cc\ and such rays
will also cross the plane of the pencil ASB (the primary principal
plane) at points which (to the same degree of approximation) will
lie in the secondary image-line dd\ Similarly, rays which cross the
element dc of the wave-surface at points which lie along any arc
parallel to the arc pPp' will (as a first approximation, also) meet in
a point of the secondary image-line dd\ and will cross the plane of
the pencil pSp' (the secondary principal plane) at points which lie
in the primary image-line cc'.
(3) If through the normal u to the wave-surface at the point P we
pass a plane making with the plane of the paper an angle between
0° and 90®, and consider the system of rays which cross the surface-
element da at points lying along the arc traced by this plane, we find
that in general these rays will not intersect each other at all; for
ordinarily this system of rays will not lie in the plane of a normal
section of the wave-surface.
The image-line at S (or at 3) contains the vertices of all those pen-
cils of rays which have their planes perpendicular to the plane of prin-
cipal curvature for which 3 (or 5) is the centre. If we are given
the chief ray u and the two image-lines, we can construct the entire
bundle of rays by joining each point of one image-line with all the
points of the other image-line.
48. With the passage of time, the element da of the wave-front
advances in the direction of the wave-normal «, each point of da
travelling along the normal belonging to it. Approaching the image-
^ne at 5, the element shrinks in dimensions, collapsing finally at 5
into the image-line cc\ Thereafter, the surface element begins to open
^^ again, and, later, it begins again to contract until it collapses at
•^into the image-line dd'; after which the wave expands again in a
^Jt of wedge-shaped opening. In any position of the element da
v^ng between the two image-points S and 3, the principal curvatures
^I necessarily be of opposite signs, so that while the element will
** expanding along one dimension, it will be contracting along the
pther. At some place, therefore, between the primary and secondary
*^ge-points a plane perpendicular to the chief ray will cut the bundle
0' rays in a section whose contour will have a form similar to that of
^ element da of the wave-surface. This is the so-called place of
48 Geometrical Optics, Chapter II. [ § 4
least confusion. For example, if the element dc is in the form of
circle, the sections of the bundle of rays made by planes at right angl
to the axis of the bundle or chief ray will generally be elliptical
form, and at the place of least confusion the two axes of the elliptic
section will be equal, so that we have here a ''circle of least conf
sion". Between the wave-surface and the place of the circle of lea
confusion the major axes of the elliptical sections will be parallel
the primary image-line cc' ; whereas the major axes of the other elli
tical sections beyond the circle of least confusion will be parallel
the secondary image-line dd\
49. In the above discussion it has been assumed that the lines
curvature at the different points of the element d<r of the wave-surfa
are parallel to the lines of curvature at the point P; which is trii
however, only in case we neglect magnitudes of the second order
smallness. Hence, the results which are given above as to the co
stitution of an infinitely narrow bundle of optical rays, known
Sturm's Theory, are valid only to that degree of approximation. Ta
ing account of magnitudes of the second order of smallness, we stn
find that, instead of image-lines going through the image-points
the bundles of rays, we have bits of image-surfaces, which, howevc
in special cases may collapse into image-lines having any indinatioi
to the chief ray.^
In the case when the magnitudes of the second order of sma!
ness are neglected, the question has been raised, also, especially I
Matthiessen,* as to the part of Sturm's Proposition which asser
that the two image-lines are perpendicular to the chief ray u. If, fc
example, the wave-surface is a surface of revolution, and if we draw tm
infinitely near normals in the plane of a meridian curve, and rota
the meridian plane through a small angle about the axis of revoli
tion, we obtain for the secondary image-line the piece of the axis inte
cepted by the two normals, which may not, and generally will no
be perpendicular to the chief normal u. The other image-line he
' See, for example, Ludwig Matthibssbn: Ueber die Form der unendlich dttenni
astigmatischen Strahlenbuendel und ueber die KuMMER'schen Modelle: SitMun^
der malh.'phys. CI. der koenigl. bayer. Akad. der Wiss. zu Muenchen, ziii. (1883). 35-51.
'L. Matthiessen: Neue Untersuchungen ueber die Lage der Brennlinien iinendlii
duenner copulirtcr Strahlenbuendel gegen einander und gegen einen Hauptstnthl: Ai
Ifa/A..iv(i884), 177-192. Also, published in the " Supplement " of Zs, f. Maik. u. Phy.
xxix. (1884), 86. See also, by same author, Untersuchungen ueber die Constitutloa u
endlich duenner astigmatischer Strahlenbuendel nach ihrer Brechung in einer kmmiiM
Oberflaeche: Zft. f. Math. u. Phys., xxxiii. (i888), 167-183.
In connection with this question, see especially also:
S. CzAPSKi : Zur Frage nach der Richtung der Brennlinien in unendlich duenaan o
tischen Buescheln: Wied. Ann., xlii. (1891), 332-337.
.] Characteristic Properties of Rays of Light. 49
vill be the infinitely small arc of a circle described about the axis by
iie point of intersection of the two normals.
In order to make the matter clear, consider the diagram (Fig. 15),
where P designates a point on the wave-surface, and where the straight
line P55, drawn normal to the surface at the point P, represents the
Pio. 15.
CORSllIU'liON OP iNFIinTBLY NARROW, ASTIOICATIC BUNDLB OF OPTICAI. RATS.
chief ray u of the bundle. The arcs A PB and pPp' are elements of
the lines of curvature of the surface which go through P and which
Ue In two planes at right angles to each other. The rays which cross
the wave-surface at points lying in the arc A PB meet at the centre
of curvature S of this arc ; and, similarly, the rays which cross the wave-
surface at points lying in the arc pPp' meet at the centre of curvature
•S. Neglecting magnitudes of the second order of smallness, we may
consider the element of the wave-surface around the point P as a
^rvilinear rectangle, with its sides parallel to the arc APB and pPp'.
^€ two principal planes pp'^ and A BS will be tangent to the two
Rustic surfaces in the primary image-line cc' and in the secondary
"^e-line dd\ respectively. Obviously, as Matthiessen contends,
^cse image-lines may not be, and, indeed, generally, will not be, per-
I*odicular to the chief ray u. In case the curvatures of the wave-
^ace are symmetrical on both sides of pp' or of A B, or if the cur-
^ture at the point P is constant, the image-lines will be perpendicu-
^ to the chief ray. Thus, for example, if a bundle of rays is re-
fracted at a surface of revolution whose axis lies in the same plane as
"^c chief ray of the bundle, there will be symmetry on both sides of
^ arc pp\ and, hence, in such a case as this the primary image-
j"^cc' will be perpendicular to the chief ray u at 5, but the secondary
"**age-line dd' will not meet w at 5 at right angles. If the vertex of
^ homocentric bundle of incident rays lies on the axis of revolu-
^^n, the secondary image-line will be an element of the axis of revo-
*^tion containing the point 5.
5
\
50 Geometrical Optics, Chapter II. [ § 49«
Concerning the matter here under discussion, Czapski's argument
(in his paper referred to above) is substantially as follows:
All the rays of the bundle may be regarded as intersecting both of
the image-lines cc\ dd', provided we neglect infinitesimals of the sec-
ond order. But with this same proviso, we may consider as image-
line any section of the bundle of rays made by a plane passing through
either 5 or 5. The form of this section will resemble more or less
a figure 8. The axes of the two lemniscate-like sections at S and at
3 will be at right angles to each other, and these axes may themselves
be regarded as the image-lines. Therefore, taking the sections normal
to the chief ray, we have, according to this view, a perfect right to
say that the iitiage-lines are perpendicular to the chief ray; it being
merely a question as to what is meant by an image-line.
Thus, the image-lines of the bundle of rays may be defined in two
ways, and the only question is as to which is to be preferred. Accord-
ing to the first definition,* the image-lines are lines traced on the caustic
surfaces, and as such are distinguished, therefore, by the following
properties: (i) E^ch point of them is the "focal" point or meeting-
place of elementary wave-trains (or rays) of equal optical lengths;
so that assuming fhat these waves have a common origin, they will
re-inforce each other at this convergence-point on the image-line, and
hence at this point there will be a maximum light-effect. (This latter
item is not mentioned by Matthiessen, but Czapski directs atten-
tion to it as being a matter not to be overlooked in this discussion.)
And (2), as Matthiessen very particularly remarks, in these lines
the section of the bundle is a minimum, in some cases indeed an infini^
tesimal of order higher than that of any other section.
However, from a practical point of view the Sturm Image-Line^
perpendicular to the chief ray of the bundle possess also certain ad-^
vantages; by their very definition, they have the distinguishing prop-*
erty of being the places in the bundle where the element of wave-sur-
face is smallest. The relative merits of the two modes of defining th^
image-lines are discussed very thoroughly by Czapski in the pape^
referred to (of which the above is a digest and partial translation) 9
and the conclusion which he reaches is that he can see no advantage
in abandoning the "classical" image-lines of Sturm. See also § 232.
' See L. Matthiessen: Berlin- Eversbusch, Zs, f, vergU Augenheilk,, vi. (z889), p. 104*
CHAPTER III.
REFLEXION AND REFRACTION OF LIGHT-RAYS AT A PLANE SURFACE.
ART. 16. THE PLANE MIRROR.
50. In the diagram (Fig. i6) the plane reflecting surface or mirror
is supposed to be perpendicular to the plane of the paper; the straight
line AB showing its trace in this plane. The reflected ray BQ cor-
Pio. 16.
Path of a Rat Rbflbctbd
Plank Mirror.
AT A
HOMOCBNTRIC BXTNDLB OP RATS
FLBCTED AT A PLANB MIRROR.
'^sponding to an incident ray LB will lie in the plane of incidence
(^di is here the plane of the paper), and the path of the reflected
'^y will be such that, if it is produced backwards to meet at L', the
*Wght line drawn from L perpendicular to the plane of the mirror
*t the point -4, we shall have VA = AL (see § 28). Moreover, since
^ position of the point L' is independent of the position on the plane
"^^r of the incidence-point 5, all incident rays which go through
^ point L will be reflected along paths which, prolonged backwards,
^ meet in the point L' (Fig. 17); so that to a homocentric bundle of
'«^ifen/ rays reflected at a plane mirror there corresponds also a homo-
^^^ bundle of reflected rays.
Tl^e points designated by L and L', which are the vertices of these
51
52 Geometrical Optics, Chapter III.
two corresponding homocentric bundles of rays, are a pair of <
gate points with respect to the plane mirror (§ 44). In this ca
point V is a virtual image of the object-point L. But if the t
of incidence rays converge towards a * Virtual object-point" L sit
behind the mirror, we shall have then a real image at a point
front of the mirror. Thus, the object-point L may be situatec
where in infinite space, and there will be always a corresponding i
point L'. It may be remarked here that the plane mirror is th<
optical system which, without any restrictions whatever as 1
angular apertures of the bundles of rays concerned in the fom
of the image, satisfies perfectly the geometrical condition of coi
correspondence, viz., that to every object-point there shall corre
one, and only one, image-point.
The straight line LU is bisected at right angles by the pU
the mirror; and hence if we put
AL = V, AV = v\
that is, if the symbols v, v' denote the abscissae,* with respect 1
point A as origin, of the points L, L', respectively, we may wri
so-called abscissa-equation for the case of reflexion at a plane ti
as follows:
t/ = — w;
whereby, knowing the position of the object-point L, we can asc
the position of the corresponding image-point L\
* The word abscissa will be employed throughout this book (unless otherwise ;
ally stated) to describe the position of the point where a ray crosses the optical .
of a refracting or reflecting surface with respect to the vertex A of the surface ai
This optical axis (which will be particularly defined hi a following chapter) Is i
here with the straight line drawn from the luminous point perpendicular to the
Thus, for example, in Fig. 16, the abscissa of the object-point L is AL, which is a
be reckoned in the sense in which the letters are written, so that if p = A L. then — s
So far as our immediate purposes in this chapter are concerned, it is entirel]
terial which direction along the axis we take as the positive direction; the opposite
ion will, of course, have to be reckoned as negative. Subsequently, we shall «
as a rule, it will be convenient to reckon the positive direction along any ray of lig
direction which the light pursues along that ray; and we may. therefore, use this
here (cf. J 26). Generally, in all our diagrams the incident light will be represi
travelling from left to right.
In this place we take occasion, also, to say expressly that, if A, B. C, Z> . .
designate the positions of a number of points ranged all along a straight line, in ai
whatever, we have always the following relations:
AB-hBA = o. or AB = — BA;
AB-{- BC-\-CAr=o;
AB -f BC -f CD -I- • • • -\-JK = AK\ etc.. etc. (See App
§50.)
Reflexion and Refraction of Light-Rays.
53
Ai XL
M
PlO. 18.
ixaob of bztbndbd omrct in a plans
Mirror.
If, instead of a single luminous point L, we have an extended object
consisting of an aggregation or system of luminous points, to each
point of the object there will correspond one image-point, and the
imaage of such an object will be
formed by the system of image-
points. Thus, if L|, L,, etc.
(Fig. 1 8), designate the positions
of the points of the object, the
portions of the corresponding
image-points L|, L^, etc., will
be determined by the fact that
the plane of the mirror must bi-
sect at right angles the straight
lines joining each pair of conju-
gate points. It is obvious that
the image in this case will have exactly the same dimensions as the
object. In general, however, the two will not be congruent; that is,
Mnage and object, although similar and equal, cannot be superposed,
because, being symmetrically situated with respect to the mirror,
their corresponding parts face opposite ways; so that, for example,
the situation of the object with respect to right and left is reversed
in the image. The object and image will be
congruent only in case the object is a plane
figure, as shown in the diagram.
The extent of the portion of the mirror that
is actually utilized will depend on the magni-
tude and position of the object whose image
is to be viewed and also on the position of the
eye of the observer. Thus, for example, in
order that a man standing erect in front of a
vertical plane mirror may be able to view his
image from head to foot, the height of the
mirror must be at least half the height of the
man, and then the lower edge of the mirror
must be placed at a level half-way between
the levels of the eyes and feet; as may easily
be verified.
Wherever the eye is placed in front of a plane mirror, the image
^ an object will appear always at the same place and of the same
^^niensions. The more inclined towards the mirror is the cone of rays
"^t enter the eye, the greater will be the piece of the mirror utilized
A
Fig. 19.
^ two bundles of reflected
^ htfiag cqnal an^rular
J^^WM intercept nneqiud
"•*■<>* the plane mirror.
I
54 Geometrical Optics, Chapter III. I § 51.
by these rays. For example (Fig. 19), the cone of rays which enters
the eye at £1 intercepts on the mirror a shorter piece of the mirror
than will be intercepted when the eye is placed at £j in the figure.
The nearer the object and the eye are to the mirror, and the farther
they are from one another, the greater will be the piece of the mirror
that will be utilized in viewing the image. If the surface of the mirror
is not accurately plane, any irregularities in it will be made apparent
by viewing the image at very oblique incidence; for in this case each
element of the mirror that is used will produce an image, and the
resulting image will be more or less blurred or indistinct. In this
way it is possible to test with a high degree of accuracy whether a sur-
face is truly plane or not. The method has been employed to show
the curvature, due to the spherical form of the earth's surface, of the
free surface of tranquil mercury.
51. A number of the most important uses of the plane mirror de-
pend on the fact that when the mirror is turned through any angle about
an axis perpendicular to the plane of incidence, the reflected ray will
be turned through an angle just twice as great. This follows imme-
diately from the law of reflexion. For if the plane of the mirror is
turned through any angle d, the normal to the mirror will be turned
through an equal angle, and hence the angle between a given incident
ray and the normal at the point of incidence will be changed by the
amount 0, and therefore the angle between the incident and the re-
fleeted rays will have been increased (or diminished) by 2$, It was
PoGGENDORFF who first suggested the method of measuring small
angles which depends on this principle, and which has been extensively
employed for this purpose in a great variety of scientific instrumettts.
such as the reflexion-lever, the mirror galvanometer, Gauss's mag"
netometer, etc. Essentially the same idea is employed in the goni-
ometer in measuring the angles of crystals and prisms.
In this connection, we may mention here also the case of two pla^^
mirrors at which the rays are reflected back and forth alternately*
The incident rays emanating from a luminous point placed anywhe^
in the dihedral angle between the planes of the two mirrors whi^
fall on mirror No. i will give rise to one series of images, while th^
incident rays which fall on mirror No. 2 will give rise to a second seri^
of images. The images of both series will evidently all be rang^
on the circumference of a circle whose centre is at the point of the lio^
of intersection of the planes of the mirrors determined by a plafl^
through the luminous point perpendicular to this line, and whose
radius is equal to the length of the straight line joining the centre with
j
§ 52.] Reflexion and Refraction of Light-Rays. 55
the luminous point. The last image of each of the two series will be
the first image of that series which is so situated as to be behind both
mirrors, and which lies, therefore, in the equal dihedral angle formed
by continuing the planes of the mirrors backwards beyond their com-
mon line of intersection. The total number of images in any case
will depend on the angle included between the planes of the two
mirrors, and also on the position of the object-point with respect to
the mirrors. If 0 denotes the angle between the two plane mirrors,
and if the angular distances of the object-point from the two mirrors
are denoted by w and <p, so that d = cj + (p, the total number of
images may be shown to be as follows:
(i) If the angle 0 is contained a whole number of times, say t, in
i8o**, so that i8o/^ = t, the number of images in this case will be
2t — I, no matter what may be the values of the angles denoted by
(2) But if the angle 0 is contained in 1 80® a whole number of times
i with a remainder € < ^, so that 180/^ = * + t/0, there are four
cases here to be distinguished as follows:
(a) If € > 0/2, the number of images in this case = 21 + 2;
(b) If € = 0/2, the number of images in this case = 2t + i ;
(c) If € < 0/2, but > w, the number of images = 2t + i ;
and
((Q If € < ^ and also < co, the number of images = 21.
See Heath's Geometrical Optics (Cambridge, 1887), Art. 32.
This theory explains Sir David Brewster's Kaleidoscope, in which
multiple images are formed by two plane mirrors inclined to each
other. When the mirrors are parallel and facing each other (^ = o),
the number of images will be infinite.
Another theorem of inclined mirrors given in Heath's Geometrical
Optics, Art. 14, which is applied in the instrument known as the Sextant,
is as follows:
When a ray of light is reflected an even number of times (2%) in
succession at two plane mirrors (the reflexions occurring in a plane
it right angles to the planes of the mirrors), the total deviation is
^ual to 21 times the angle of inclination of the mirrors.
ART. 17. TRIGOHOMBTRIC PORMULiB FOR CALCULATIK G THE PATH OP A
RAT REFRACTED AT A PLANE SURFACE. IMAGERY IN THE CASE OF
REFRACTION OF PARAXIAL RATS AT A PLANE SURFACE.
52, Let L (Fig. 20) designate the position of a point on a ray
incident at the point 3 on a plane refracting surface which separates
I
56
Geometrical Optics, Chapter III.
(§5
two isotropic optical media of absolute indices of refraction n and n
The straight line LA or x which, going through the point L, mee
the surface normally at the point A is called the aocis of the refractir
plane with respect to the point L. Tl:
^ magnitudes
AL ^v, L ALB = a,
which determine completely the positic
of the incident ray, may be called tl
ray-cO'OrdinaUs, Similarly, if L' desij
nates the position of the point where tl
refracted ray, produced backwards fro
the incidence-point B, crosses the ax
X, then
AL' = r', Z ALB = a'
Rat of I^xoht Refractbd at a
Plane.
lAUB'^**.
will be the ray-co-ordinates of the refracted ray L'B. The proble
is, being given the incident ray (v, a), to determine the refracted ra
From the figure we derive immediately:
V tan a!
moreover, by the law of refraction:
«
n-sin a = n'-sin a';
whence are obtained the following formulae:
»' = -
V l^w' — n*sin*a
1
n
cos a
sin a'
n
= —, sin a.
n
(I'
These equations enable us to find the magnitudes v\ a' and to detc
mine, therefore, the refracted ray.
For a given value of v, we see that the value of v' will depend <
the angle of incidence a. Only those rays emanating from the poi
L which meet the plane refracting surface at equal angles of inciden
(and which lie, therefore, on the surface of a right circular cone ge
erated by the revolution of the straight line LB about LA as axi
151]
Reflexion and Refraction of Light-Rays.
57
...
ffW^^
J(/!.."J^
^
^^
'«t^
^
^^\^
will, after refraction, all intersect at a point V on the axis x. So
that if L is a luminous point emitting rays in all directions, an eye
placed in the second medium («') will, in general, not see a distinct,
but only a blurred and distorted, image of the object-point at L; as
Till be more fully explained in the section which treats of the caustic
by refraction at a plane surface (Art. i8).
53. RefractioQ of Paraxial Rays at a Plane Surface. In one
special case, however, the imagery produced by refraction at a plane
surface is ideal. Let MA (Fig.
ii) be the axis of the plane re-
fracting surface tt with res[>ect
to tlie object-point M; and let
us suppose that all the points of
the refractir^ plane are screened
from M except those points
which are infinitely near to the
poiac A where the axis meets
the surface; so that of all the
lays proceeding from M only
tiwse whose paths lie very close
to the axis can meet the refract-
><msur[ace. We shall have thus
M infinitely narrow bundle of
^raiio/ incident rays (enor-
nousiy exaggerated in the dia-
pam) whose chief ray coincide
ing with the axis of the refracting
pW meets this plane normally.
The angles of incidence and re-
traction of the chief ray are both
•flual to zero; whereas in the
<3se of all the other rays these
*^les will both be infinitely
•"all. If we suppose that the
''isles a, a' are so small that we
■>|3V neglect the second and
■"Eher powers of these angles,
the angle « disappears entirely from the first of equations (i6) ; and if
^Ik absdssfe with respect to the point A of the conjugate axial points
^,W are denoted by «, «', respectively, that is, if here we put
AM = «, AM' = u'.
REnucnon of Pakaiui. RAvi at a Pi.&itB.
In thcw dlasranu the Inddent nyi are mipiMMd
to meet Uie RcfracCini Plane almiMt nonuallv.
Tbe angular aiwrturu of tbe conea of raya an in
realltr lnfioltely ■mall, alUiouEli Uicy are here
momunuly maffnlfitd. Paraiia] RSI's dlveislDE
from a point M arc rctimcted at the Plime Surface
■a thoueh they came from M'.
AIU'^K, AM' — u'.
58
Geometrical Optics, Chapter III.
[§53.
we have evidentiy the following relation:
u = —u\
n
(17)
which is the so-called abscissa-equation for the refraction of paraxial
rays at a plane surface. Provided we know the position on the axis
of the object-point M , this equation enables us to determine the ix)si-
tion of the corresponding image-point M\ Thus, to a homocentric
bundle of incident paraxial rays refracted at a plane surface there
corresponds also a homocentric bundle of refracted rays.
Within the infinitely narrow cylindrical region immediately around
the axis of the refracting plane, we have, therefore, a point-to-point
correspondence of object and im^^.
According to (17), since «, u' have
the same signs, the points Af , M^ lie
always on the same side of the re-
fracting plane, that is, the point M'
is a virtual image of the point M.
If the object is an infinitely short
line MQ (Fig. 22) perpendicular to
the axis at M, obviously, the image
of the point Q will be a point Q*
lying on the straight line drawn
through Q perpendicular to the re-
fracting plane and at the same dis-
tance from this plane as the axial
image-point M\ Consequently, the
image of the infinitely short object-line MQ at right angles to the axis
is an equal and parallel line M'Q', The ratio y jy, where MQ = y^
M'Q' = y' is called the Lateral Magnification or the Linear Magnifir
cation, and will be denoted here by the symbol F. Thus, in the case
of the imagery produced by the refraction of paraxial rays at a plane
surface, we have:
IBIAGERY IN THB CASB OF RBFRACTION
OF Paraxial Rats at a Plane. The im-
aire of the infinitely small object-line MQ
parallel to the Refracting Plane ia an equal
imasre-line Af*Q' havinff the same direction
BsMQ.
AAf=u, aat^m', AfQ='y, Ar(y=y.
F= - = + i.
y
(18)
The two equations (17) and (18) show that the image is always vir-
tual and erect and of the same size as the object, provided the latter
is a line at right angles to the axis. If ^AMQ is not a right angle,
the image-line will not be parallel to the object-line nor of the same
length as the object-line. We have here, in fact, a special case of
i
{S4.I
Reflexion and Refraction of Light-Rays.
59
collinear correspondence, known as Central Collineaiion, the refracting
plane being itself the plane of collineation and the centre of collinea-
tion being the infinitely distant point of a straight line perpendicular
to the refracting plane. It is the relation that in geometry is called
affinity.
ART. 18. CAUSTIC SURFACE IH THE CASE OF A HOMOCENTRIC BUNDLE
OF RATS REFRACTED AT A PLANE SURFACE.
54. In general, as we saw (§ 52), to a homocentric bundle of rays
incident on a plane refracting surface there corresponds a system of
refracted rays which is not homocentric. It will be an instructive
exercise to investigate in this comparatively simple case the form of
the caustic surface (§ 46), especially as this example will afford a very
good illustration of the
general principles ex-
plained in Art. 15 of the
preceding chapter.
Let the vertex of the
homocentric bundle of
incident rays be desig-
nated by S (Fig. 23), and
let the straight line
marked m show the trace
in the plane of the paper
of the refracting plane.
Since everything is sym-
metrical with respect to
the normal SA^ drawn
from 5 to the plane re-
fracting surface, it will
be sufficient to investi-
gate the form of the re-
fracted wave-surface in
the plane of the paper.
Let the straight line SB
drawn in the plane of the
paper and meeting the
refracting plane in the point B represent the path of an incident ray,
and let U designate the point where the corresponding refracted ray,
produced backwards, intersects the straight line SA . In the case which
-we shall consider here the first medium (n) is supposed to be optically
Pio. 23.
BPHBRICAL WAVB DrVBRGINO FROM A POINT S AXU RE-
FRACTED AT A Plane into an optically rarer medium
(n' < It). SB is ray incident on refractinsr plane at the
point B, and BP\a the corresponding refracted ray.
60 Geometrical Optics, Chapter III. [§ .S-4.
denser than the second medium (n'), as, for example, when the r
are refracted from water into air; hence, n > «', where «, n' den
the absolute indices of refraction of tiie two media. In this ca
therefore, the point V will lie between 5 and -4, as shown in the figu
Produce the normal SA into the second medium to a point Q
that AQ ^ SA, and pass a circle through the points 5, B and Q, a
produce the refracted ray backwards to meet the circumference
this circle in the point designated in the figure by K. The angle
is evidently bisected by the straight line KB, and we have:
LSKB = LBKQ = LBSQ = a,
since these inscribed angles stand on equal arcs of the circle. TXr^^e
two angles at V are equal to the angle of refraction a' and to tMr^^e
supplement of this angle; hence, in the triangle SL' K we have:
SL'iKS = sin a:sin a';
and, similarly, in the triangle QKV\
VQ\KQ = sin a:sin a';
so that, by the law of refraction:
SL'iKS -- L'QiKQ^ n'ln,
or
{SV + L'Q):(KS + KQ) = n'ln;
that is,
KS + KQ= -, SQ = constant.
n
Thus, we see that the locus of the point K is an ellipse with its f(
at the points 5 and Q. Moreover, the refracted ray, which bisects:---^^
the angle SKQ, is normal to the ellipse at K. The ellipse is, therefore
an orthotomic curve for the system of refracted rays which lie in th<
plane of the paper.
The meridian section of the refracted wave-front at any moment
may be found by measuring off equal distances from the points ol
this ellipse along each refracted ray; that is, the refracted wave-fronts-^^"
are parallel curves to this orthotomic ellipse (Fig. 24). These curves
will not be themselves ellipses, since the parallel to a conic is, in gen-
eral, a curve of the eighth degree.^ But the parallel to a conic has the
same evolute as the conic itself; so that the caustic curve which is the
* See Salmon's Conic Sections, 6th edition. Art. 372. Ex. 3.
{ 55.] Reflexion and Refraction of Light-Rays. 61
evdate of the wave-line will in the present case be the evolute of an
ellifMe.
If here we put AS = c (Fig. 23), and if the centre A of the ellipse
is taken as origin of a system of rectangular axes {SA, AB being the
Waut
FtaiU
f
Paint
Fio. 2*.
^^*i?»ik: CnvB and Wavb-Lihb m thb case of Refbactioh of CnctlLAK Waves at a
^^*A«OHt Uke. Roys refracted from wBWr IdIo air.
/'Actions of the positive axes of x, y, respectively) , the Cartesian equa-
**«»n of the eUipse will be:
" n — n n
^**I the rationalized equation of the evolute of this ellipse is:
{«V + (n' - n'*)y' ~ n'*^]' + 27tfn'n'\n' - n'Vy* = o.
^ »is is, therefore, the equation of the caustic curve in the case here
'^'^^dered. The caustic here is a "virtual" caustic.
It has been assumed above that the first medium was optically den-
"**■ than the second. In the opposite case, viz., n < n', the ortho-
w>tnic curve for the system of meridian refracted rays proves to be a
"yperbola with the same foci as the ellipse above, so that the caustic
^^"r^ for this case will be the evolute of the hyperbola.
55. The equation of the caustic by refraction at a straight line
■"ay also be deduced directly, as follows:
62
Geometrical Optics, Chapter III.
[§5S.
Taking, as above, the point A (Fig. 25), which is the foot of the
perpendicular let fall from the object-point S on the refracting straight
line, as the origin of a system of rectangular
axes, where SA and AB are the positive
directions of the axes x and y, respectively,
and putting:
AS = c, ZASB = a, ZHS'B = a\
ZBSG = da, ZBS'G = da\
we obtain immediately the following relations:
da
AB = — ctan a; BG = — c
Pio. 25.
Used in Deriving Equa-
tion OP Caustic by Refrac-
tion AT a Straight I«ine.
The plane of Uie paper is the
plane of incidence of the inci-
dent ray SB, to which corre-
sponds the refracted ray whose
direction is along the straight
line S'B. (7 is a point in the
plane of incidence on the re-
fracting straight line and in-
finitely near to the incidence-
point ^. .S(7. 5* (? incident and
refracted rays.
COS* a
„ ^. GB- cos a' cos a'-da ^
BS' = = — = — ^•
Since
we have:
da' cos^ a 'da'
n-sin a = n'-sina',
_ . n cos a
da = — ; da.
n cos a
Hence, eliminating a', we obtain:
c(n — «*-sin*a)
S5' =
nn' cos' a
Now
X =J?5' = S5'cosa'
c
= —;» {n'' - {n* - n^) tan* a}l
fin
Again, since
AB= -C'tsLiia, BH^BS'sina, and AH^AB + BH,
we find after several reductions:
AH = y-= -—^(n^-n'^ tan' a.
n'
Eliminating tan a from these expressions for x and y, we obtain the
Cartesian equation of the locus of the primary image-point Sf (S 47)
corresponding to the object-point 5, as follows:
fe')'-p'^^)'-^
Reflexion and Refraction of Light-Rays.
63
dng rationalized, gives precisely the same equation as is given
id of § 54.
Lustic turns its convex side towards the refracting straight
touches it at a point designated by V in Fig. 26 whose dis-
m the point A = n'clVn — w^, and which is therefore the
point of incidence on the positive side of the y-axis. Of
lere is also another point of tangency at an equal distance
m the negative side of the y-axis. Thus, for example, if the
oint 5 is in water, and if the rays emerge from water into air
\h)y we shall find AV = 1.14-45. Putting y = o in the
of the caustic curve, we find the cusp of the caustic at
f on the normal to the refracting surface at -4, such that
'- AS/fif and hence (§53) this point M' is the image-point
ial rays of the object-point S,
diagram is revolved around SA as axis, the caustic curve
rate the caustic surface of the
rays corresponding to the
trie bundle of incident rays
g in all directions from the
point S, There are always
itic surfaces, but in case the
I surface is a surface of revolu-
: of the caustic surfaces col-
to a piece of the axis of sym-
[lich in this case is the segment
§46).
*ye were placed at the point
(Fig. 26), and if at a point 5
e surface of still water there
ated a radiant point, the pri-
age of the radiant point 5
; located at the point of tan-
of the tangent to the caustic
awn from the point E. If
s placed vertically above the
)oint 5, the image will be seen
fiat is, at a depth one-fourth
0 the surface of the water
object-point actually is. We
jfore how it is that an object under water viewed by an
le air above will, in general, appear not only to be raised
FiO. 26.
Caustic by Refraction at a Plans
Surface for case when h > n\ Dia-
flrram is drawn for the case when the
first medium is water and the second
medium is air. The horizontal surface
of the still water is the refracting plane,
the rays beinsr refracted from below
upwards.
The refracted ray BE, correspondinir
to the incident ray SB, when produced
backwards is tangent to the caustic
curve at the point marked 5* and meets
the normal SA at the point marked J'.
These points S', S' are the positions of
the I. and II. Iraai;e-Points of the astiir-
matic bundle of refracted rays that enter
the eye at E,
64
Geometrical Optics, Chapter III.
1!
towards the surface, but also to be displaced sideways more and n
towards the observer, the more obliquely he regards the object,
viously, incident rays meeting the water-surface at points beyond
extreme point V will be totally reflected (§ 27).'
AST. 19. ASTIGMATIC EEFKACTIOR OF AR IKFIHITBLT HABKOW BUITDLl
RATS AT A PLAITE SURFACE.
56. In the diagram (Fig. 27) the refracting plane designated t
is supposed to be perpendicular to the plane of the [laper. The p(
S in the first medium (n) is the vertex of an infinitely narrow hoi
centric bundle of incident rays, whose chief ray, viz., the ray S£
AiriOMATIC BUHDLB
NA>KOir HOMOCHHTUC
tlys. Sit Ok Object-Poi
D ItAyaDUB TO RBPKACTIOH AT A PL&HII OF AN IKfini
,B OF INCIDBHI Ravs. k. h' ir Hie Chief inddent and re(n
Is I. Imase-Point: .S* UII. Imue-Point
», meets the refracting plane at the incidence-point B. The seel
of this bundle of rays made by the refracting plane will be a si
closed curve GJGJ; which will be elliptical in case the cone of indd
*See In connection nlth this section L, Matthibssbn: Das aKlgmatbcbe'BlM
1, ebenen Gcundes eioes Wasseibassins: Ann. dtr Phyt. (1901), 347-3S>
§ 58.] Reflexion and Refraction of Light-Rays. 65
rays has a circular cross-section. The chief refracted ray u\ corre-
sponding to the chief incident ray w, will, if produced backwards from
5, meet in the point designated by S' the straight line SA which is
normal to the refracting plane at the point A, In general, the other
rays of the bundle of refracted rays will not intersect the chief ray
tt', but will pass from one side of it to the other both above and below
the plane of the paper. The refracted rays will constitute an astigmatic
bundle of rays (§ 47), on whose chief ray v! the two image-points will
lie. In order to ascertain the positions of these image-points, we must
investigate those rays of the astigmatic bundle which meet the chief
niyii'.
57. The Meridian Rays. The three points 5, B and A determine
the plane of incidence of the chief ray u\ in the diagram this is the
plane of the paper. This is likewise the plane of incidence of all the
rays of the bundle which meet the refracting plane at points lying
along the diameter GG of the closed curve GJGJ, The rays, there-
fore, which are refracted at the points G, G will necessarily meet the
refracted chief ray w'; and, since we assume that the bundle of rays
is infinitely narrow, the rays refracted at the points G, G will meet the
chief refracted ray u' in one and the same point 5', provided we neg-
kct infinitesimal magnitudes of the second order; and the same thing
^ be true of all the rays refracted at points lying along the line-
dement GG. The plane of incidence of the chief ray u which contains
^ pencil of rays is one of the principal planes of curvature (§ 46) of
tlte refracted wave-surface at the incidence-point By and these rays
^ the so-called meridian rays of the bundle. The meridian rays of
^ refracted bundle of rays all intersect at the I. Image-Point 5'.
In this statement it is assumed that we neglect magnitudes of the
*cond order of smallness, and hence the convergence of the rays at
•^ is said to be a "convergence of the first order" only.
Moreover, to a pencil of incident rays proceeding from the radiant
point 5 and meeting the refracting plane at points lying along a chord
of the curve GJGJ which is parallel to the diameter GG there corre-
sponds a pencil of refracted rays lying in the same plane as the pencil
^ incident rays (the plane determined by the chord and the radiant
Pomt) whose vertex will be a point infinitely close to the point 5',
^ve or below it, lying in the I. Image-Line at S' which is perpen-
^lar to the plane of incidence of the chief ray u {% 47).
58. The Siqpttal Rays. Let us next consider the rays of the in-
finitely narrow bundle which meet the refracting plane at points lying
along a diameter JJ oi the curve G/G/ which is at right angles to the
66 Geometrical Optics, Chapter III.
diameter GG. The rays of this pencil which are incident at th
points J, J of the diameter // are symmetrical with respect
normal SA, so that after refraction they will intersect the
refracted ray «' in the point 5' where u' meets 5^ . This can ix
clearer, if necessary, by imagining that the right triangle S
rotated through an infinitely small angle above and below the pi
the paper around SA as axis, so that the point B traces the lii
ment //, and the chief incident ray u coincides in succession w
the rays of the pencil SJJ. It is obvious that all the rays i
pencil will, after refraction, intersect the chief refracted ray «'
II. Image-Point 5'. The plane 5*// which is the plane of this
of refracted rays and which is perpendicular to the plane of inc
of the chief incident ray » is the other principal plane of curvat
the refracted wave-surface at the incidence-point B. This
determines the sagittal section of the bundle of refracted rays, a)
pencil of rays S'JJ contains the sagittal rays after refraction;
refracted rays correspond to the incident sagittal rays belong
the pencil SJJ. The rays of the sagittal section of the bun
refracted rays intersect in 3^ not merely approximately, but es
because in the sagittal section there is symmetry with respect
plane of incidence of the chief ray u, so that rays from the n
point S which make equal angles with the plane SAB on op
sides of this plane will, after refraction, all pass through S'; b
at the II, Image-Point the convergence is of the second order,
To a pencil of incident rays which meet the refracting pU
points lying along a chord of the closed curve GJGJ which is p
to the diameter // there corresponds a pencil of refracted rays
meet all at one point of the II. Image-Line. This latter lies
plane of incidence of the chief ray u, and, according to Sturh, i
pendicular at 5' to the chief refracted ray «'. However, we ma
consider as 11. Image-Line of the astigmatic bundle of refracted
not the line-element in the plane of incidence of the chief ray u t
perpendicular to the refracted chief ray u' at the point y, but tl
ment mn of the normal to the refracting plane at the point A wl
intercepted on this normal by the two extreme rays of the me
pencil of refracted rays. Through this bit of the normal all thi
of the bundle of refracted rays must pass, as may easily be se
rotating the plane of the pa[}er around SA as axis through a
angle above and below this plane. In the course of this rotatio
rays of the meridian section will trace out all the other rays i
bundle, but the element mn of the normal SA will remain unch
in magnitude and in position. As to this matter, see § 49.
§59.1
Reflexion and Refraction of Light-Rays.
67
We proceed now to determine the positions of the two image-points
S* and 3' of the astigmatic bundle of refracted rays.
59. Position of the Primary Image-Point S\ In the diagram (Fig.
28) the straight line A B shows the trace in the plane of the paper of the
refracting plane. The straight
line SB represents the chief
ray tf of an infinitely narrow
homocentric bundle of incident
rays emanating from the ob-
ject-point 5- The plane of the
paper is the plane of incidence
of the chief ray «. Infinitely
near to the incidence-point B
of the chief ray and in the
plane of the paper let us take
the point G, so that SG repre-
sents a secondary ray of the
pendl of meridian incident rays.
The I. Image-Point S' will be
at the point of intersection of
the refracted rays correspond-
ing to the incident rays SB
and SG, Let a, a' denote the angles of incidence and refraction of the
chief ray; so that, referring to the figure, we may write:
I NBS = a, Z NBS' = a', Z BSG = da, Z BS'G = da'.
Fig. 28.
RBFRAcnoK OP Narrow Bundle op Rays at
A Plane. Figure for the determination of the posi-
tions of the I. and II. Image-Points y and S* on the
chief refracted ray m' corresponding to the object-
point S on the chief incident ray «.
INSS-'-fi, lNBS' = a'. I BSG ^ da,
lBS'G=-da', BS=s, BS' = s', BS'^V,
Then m the triangles BSG, BS'G we have :
BG da BG da'
^1 therefore:
And smce
we obtain finally:
^if we put here:
SB cos a' S'B cos a"
BS' __ cosg^ da
BS cos a da' '
nsina = n'-sina',
BS' ^n'cos^a'
BS ncos'a *
55 = 5, BS' = s',
68 Geometrical Optics, Chapter III. [ § <X).
the formula above may be written:
- w'cos'a' . .
s' = 2 — 5. (19)
If the chief incident ray u is given and the position on it of the
radiant point 5, this formula enables us to determine the position of
the corresponding I. I mage- Point 5' on the chief refracted ray u\
The convergence-ratio or angular magnification of the rays of the
meridian section is the ratio da! {da of the angular apertures of the
pencils of incident and refracted rays in the meridian section. If this
ratio is denoted by the symbol Z^ (where the subscript indicates the
chief ray of the pencil), we have evidently:
„ da' wcosa . .
Z^= T-=-?^ 7- (20)
da w cosa
60. Position of the Secondary Image-Point S\ In order to deter-
mine the position of the II. Image-Point S\ which is at the point
of intersection of the straight line drawn through the homocentric
object-point 5 perpendicular to the refracting plane with the chief
refracted ray u' of the astigmatic bundle of refracted rays, we have
from the triangle SB3':
BS sin Z 55'5 sin a'
and, therefore:
SS' sin Z BSS' sin a
BS' n'
If we put
BS n'
BS = 5, BS' = s',
we shall have the following equation:
5' = - 5. (ai)
n
Thus, if we know the position of the homocentric object-point 5 on the
chief incident ray u, this formula enables us to locate the poaitkm of
the II. Image-Point on the corresponding chief refracted ray «'.
All incident rays lying on the surface of the cone generated by the
revolution of the ray SB around the normal 5^4 as axis will after re-
fraction at the plane refracting surface lie on the surface of a coae
generated by the revolution of ^'B around the same axis; as is
immediately from the formula just obtained.
\ 62.] Reflexion and Refraction of Light-Rays. 69
If SJ is a ray of the sagittal section of the homocentric bundle of
incident rays which meets the plane refracting surface at a point /
in&nitely near to the incidence-point B of the chief incident ray w,
"SJ will show the direction of the corresponding ray of the sagittal
secdon of the astigmatic bundle of refracted rays; and the ratio of
the angles BlS'J and BSJ is the convergence-ratio or angular magnifi-
cation of these corresponding pencils of sagittal rays. If here we put:
Z BSJ = dX, Z BS'J = d\\
and if the symbol Z^ denotes the convergence-ratio of the pencils of
inddent and refracted sagittal rays with the chief incident ray u, we
have evidently: _
61. The Astigmatic Difference of the bundle of refracted rays is
the piece of the chief refracted ray u' comprised between the II. and
I. Image-Points of the astigmatic bundle of rays; that is,
3'5' = S'B + BS' = 5' - y.
In the case of an infinitely narrow homocentric bundle of incident
'^ys refracted at a plane, we obtain from formulae (19) and (21) :
y _ cos* a
^ for the astigmatic difference of the bundle of refracted rays:
3'y =
"" I 2 I ) • (24)
n \cos a / ^
f
^ astigmatic difference vanishes only in case a = a' = o; that is,
^*n the chief incident ray u is normal to the refracting plane; which
* the case of paraxial rays (§ 53).
6* Refraction at a Plane Surface of an Infinitely Narrow Astig-
. iBitic Bundle of Incident Rays. If the bundle of incident rays is
^pnatic, and if we designate by S and 5 (Fig. 29) the vertices of the
pencils of incident meridian and /sagittal rays, respectively, the bun-
^ of refracted rays will, in general, be astigmatic also, and the I. and
IJ. Image-Points 5' and 5', lying on the chief refracted ray «', will
torespond to the points 5 and 5, respectively, lying on the chief in-
CJdent ray u. We may call the point S the I. Object-Point and the
70
Geometrical Optics, Chapter III.
point 5 the II. Object-Point. The pencil of meridian inddenl
emanating from the I. Object-Point 5 and lying in the plane ol
dence of the chief incident ray u will be transformed by refractio
Fio. 29.
ASTXoiCATic BuiTDLB OP ItrciDBiTT Ray8 Rbfractbd AT A Planb. The chicf njs of t]
matic bundles of inddent and refracted rays are designated by u and u*, S, ^designate 1
lions on M of the I. and II. Object-Points. To .S on m corresponds the I. Imase-Point 5* oc
to.? on M corresponds the II. Image-Point 1^ on u\ In thediagram the plane of the paper o
with the plane of the meridian rays.
a pencil of meridian refracted rays lying in the same plane wi
vertex at the I. Ima^-Point S'. Hence, putting
BS = s, BS' = s\
we have, according to formula (19):
, n' cos* a'
s = r~ s.
Similarly, putting
we have by formula (21)
n cos a
5 = — 5.
n
The bundle of incident rays wilt have been rendered asdgma
consequence, for example, of previous refractions.
ART. 20. REFRACTION OF INFINITELY NARROW BUNDLE OF RATS
PLANE: GEOMETRICAL RELATIONS BETWEEN OBJECT-
POINTS AND IMAGE-POINTS.
63. If on a given incident ray u (Fig. 30) we take a range of c
points P, 0, i?, 5, • • • , whereto on the refracted ray u' correspoi
range of I. Image-Points P', Q\ P', 5', • • • and the range of II. I
Points T\ Q\ H.'j 5', • • • , then, according to formula (19), we must
BP' BQ' BR' _ BS' ^
BP " BQ" BR" BS^ ""'
which means that the straight lines PP', QQ^ RR\ SS\ • • • , joini
S 64.] Reflexion and Refraction of Light-Rays. 71
object-points on the incident ray u with their corresponding I. Image-
Points on the refracted ray u' are a system of parallel straight lines;
and, hence, the point-ranges P, Q,R, • • • and P', Q\ J?', • • • are similar
ro-^ges of points. And, since the straight lines PT\ QQ\ RR\ • • •
Pio. 30.
ntAcnoN OF Narrow Bim dlb of Rays at a Plans. The nmse of Object-Points P,Q,"'
^TtLmmm oa the chief incident ray u is similar to the ran^e of I. Imase-Points P',Qf ,'" and also to the
of n. Image-Points ^. ^. • • • lying on the chief refracted ray u\
w^hich connect the Object-Points on the incident ray u with their cor-
responding II. Image-Points on the refracted ray u' are all perpendicu-
lar to the refracting plane, and therefore parallel to each other, it
follows that the point-ranges P, Q, i?, • • • and P', ^', 3J', • • • are also
similar ranges of points.
Conjugate to any object-point X, lying in the plane of incidence
of the incident ray u, there will be on the refracted ray x\ which cor-
responds to an incident ray x parallel to u and going through the
obiect-point X, the I. Image-Point X' and the II. Image-Point X';
and the range of Object-Points lying along the incident ray x is simi-
lar to the ranges of I. and II. Image-Points lying along the refracted
r^y x'. Thus, the plane system 17 of the Object-Points X, • • • , which
lie in the plane of incidence of the incident ray «, is in affinity with the
pbne-system ri' of the I. Image-Points X\ • • • and also with the plane-
system V of the II. Image-Points X\ • • • : hence, also, the plane-
systems fi' and "rj' are in affinity with each other. The straight line in
which the plane refracting surface meets the plane of incidence is the
afiniiy-axis^ for all three of these ''affin" plane-systems.
64. Constnsction of the I. Image-Point. Let u (Fig. 31) be an
inddent ray meeting the plane refracting surface m at the point B,
1 The afi^ty-Azis of two plane " a fin *' systems is the straight line common to the
two syttems which corresponds with itself point by point. Obviously, any pair of corre-
^poodiiig straight lines of the two systems will meet in the affinity-axis.
72
Geometrical Optics, Chapter III.
and let u' be the corresponding refracted ray. Corresponding i
Object- Point 5 on tt we find the II. Image- Point 5' on u' at the
of intersection with u' of the perpendicular SA drawn from 5 t
refracting plane. Draw SXy S' Y perpendicular to the incid
normal BN at X, F, respectively, and from X draw XP perpen
lar to u at P, and from Y draw YP' perpendicular to u' at P'.
BP = 55- cos* a, BP' = jB5'- cos* a';
and since
we have:
BS' = - 55,
n
BP'
BP
n' ' cos* a'
wcos a
Consequently, according to formula (19), the points P^P' are c
sponding points of the '*affin** systems ly, 17'; so that if P is an 01
Point of the chief incident ray u, P' will be the I. Image-Point
on the chief ref racte
tt'. The I. Image-]
S' corresponding tc
Object-Point 5 of
chief incident ray
found by drav
through 5 a straigh
parallel to PP' w
by its intersection
the chief refracted r
will determine th<
quired point S'. T
essentially the cons
tion given by Rei:
Another constru
of the two corresi
ing points of the **<
systems 17, 17' is as fol
Through S' dn
straight line perpendicular to the refracted ray u' and meetini
straight line BA in the point U\ draw the straight line US me
the incidence-normal -BiV^ in the point Z; and from Z let fall a
pendicular ZQ on the incident ray at the point Q. Then the
5^ (or Q') is the I. Image-Point corresponding to the Object-Poi
*B. Rbusch: Reflexion und Brechung des Lichts an sphaerischen Flaechec
Voratiwetxung endlicher Einfallswinkel: Pogg. Ann., czxx. (1867), 497^51 7*
*
*
^'.-
^^'
PlO. 31.
RBniACTXON OP NARROW BUNDLE OP RAYS AT A PLANS.
Construction of I. Imaffe-PoinL
§64.] Reflexion and Refraction of Light-Ra}^. 73
on the chief incident ray u; as we proceed to show. Let the straight
line joining U and IS' meet the incidence-normal B N in the point N;
then
^_BN _ BU_BZ_BZ _BQ
BP' " AS'~ AU" AS" BX" BP'
and, therefore: _
BS' _ BP\
BQ" BP'
and, hence, ^5' (or Q(y) is parallel to PP'. Therefore, 'S' (or Q')
is the point of iy' which corresponds to the point Q of r^}
'For other methods of construction of the I. Image-Point see F. Kbsslbr: Beitraege
wr paphiachen Dioptrik: ZfL /. Math, u. Phys,, xxix. (1884), 65-74.
See also the construction of the I. Image Point in the case of refraction at a plane
coniidered aa a special case of refraction at a sphere, as given in § 249.
Sace a plane surface may be regarded as a spherical surface with its centre at an in-
finite distaDce. obviously, all the problems treated in this chapter can be considered as
*pscial cases of the problem of refraction at a spherical surface, as will be seen hereafter.
Aaading to this view, this entire chapter might be regarded as superfluous.
CHAPTER IV.
REFRACTION THROUGH A PRISM OR PRISM^YSTEM.
ART. 21. GEOMETRICAL CONSTRUCTION OF THE PATH OF A RAT
REFRACTED THROUGH A PRISM IN A PRINCIPAL
SECTION OF THE PRISM.
65. In optics the term Prisin is applied to a portion of a trans-^
parent, isotropic substance included between two non-parallel plan^
refracting surfaces called the faces or sides of the prism. These
distinguished as the first and second faces of the prism in the ordeC
in which the light-rays arrive at them. The straight line in whi
the two plane faces meet is called the edge of the prism, and the dih
dral angle between the two faces is called the refracting angle. Thii
angle, which will be denoted by the symbol /3, may be defined mo
precisely as the angle through which the first face of the prism has to
turned, around the prism-edge as axis, in order to bring this face i
coincidence with the second face. A principal section of the prism is
made by any plane perpendicular to the edge of the prism. At first
we shall consider only such rays as lie in a principal section of the
prism or infinitely narrow bundles of rays whose chief rays lie in a prin-
cipal section.
In the general case of the problem of refraction through a single
prism we have to do with as many as three optical media, viz.: the
medium of the incident rays or the first medium, the medium of which
the prism-substance is composed and the medium of the emergent rays.
The absolute indices of refraction of these media will be denoted by
n^, n[ and n'n in the order named. In most cases the third medium is
identical with the first, as, for example, in the case of a glass prisia
surrounded by air; and unless the contrary is expressly stated, wfe
shall assume that this is the case. Thus, we shall have ni ^ n^; anA^
the symbol
«, n,
n — — /
will be employed to denote the relative index of refraction of the la^ —
dium of the prism-substance with respect to the surrounding mediuxicB..
66. The following construction of the path of a ray
I Refraction Through a Prism or Prism-System. 75
:)Ugh a prism in a principal section was published by Reusch^ in
ii; the same construction was published by Radau* in the follow-
gyear.
In the diagrams (Figs. 32 and 33) the plane of the paper represents
. pnndpal section of the prism, and the point V in this plane shows
q:
FlO. 32.
CoscBTEucnoir op thb Path of a Rat Refracted through a Prism xm a Principal
8«noi». Cue when m' = i»i and m' > hi.
where the prism-edge meets the plane of the principal section. The
two plane faces of the prism, designated by Mu M2 ^^^ shown therefore
by two straight lines meeting in the point V. The straight line L^B^
(or tt,) represents the path of the given incident ray meeting the first
face of the prism at the incidence-point B^; and the problem is to con-
struct the remainder of the ray-path both within the prism and after
emergence from the prism. The method is in fact the same as that
given in §29.
With the point V as centre, and with radii equal to r and r/n
eben^ ' J*BuscH: Die Lehre von dcr Brechung und Farbenzeretreuung des Lichts an
Fbechen und in Prismen in mehr synthetischer Form dargestellt: Fogg. Ann.,
c^. (X862). 241-262.
„j^^j^ Radau: Bcmerkungenueber Prismen: Pogg. Ann., cxviii. (1863). 452-456. The
^™>oa wstt obtained independently by Radau and it is often called by his name; but he
in Caals Rep.,iv, (1868). p. 184, acknowledges Reusch's priority in the matter.
76
Optics, Chapter IV.
I§(i6.
(where r may have any v-alue), describe the arcs of two concentric
circles. Through V draw a straight line parallel to the given ind*
dent ray Li-Bj meeting the circumference of circle tin in a point G,
and through G draw a straight line perpendicular at £ to thcf first
face of the prism (produced, if necessary), and let H designate the
Fio. 33.
Construction op Path op Rat Rsfractbd thkouoh a Prism in a Prdtcxpai.
Caae when ns' "^ n\ and n i' < n\,
position of the point where this straight line meets the circumf(
of the circle r. Then the straight line B^B^ drawn parallel t<
straight line VH will show the path of the ray within the prism.
if «! = Z N^BiLi and a| = /.OB^B^ denote the angles of ind
and refraction at the first face of the prism, then, by the law
fraction:
ni'sin «! = n(sin a|.
According to the construction, we have:
sin /.EGV _ VH
sin /.EHV " VG
and since, by construction, ZEG V= ai, it follows that Z EHV
and hence the path of the ray within the prism must be parallel t:<
Again, from the point H let fall a perpendicular on the seco:
78
Geometrical Optics, Chapter IV.
[§68.
ray Lj^i is varied, the vertex H of this angle will move along the cir-
cumference of the circle of radius r, the sides of the angle having the
fixed directions of* the normals to the prism-faces. The two extreme
positions of this point H which are reached when one or other of the
sides of the Z G HJ is tangent to the circle of radius rjn (which can occur
only when n is greater than unity, because then only will the point E
lie outside the circle of radius tin) are shown in Fig. 34. The two inci-
dent rays which correspond to these two extreme positions of the point
H are the ray ^i-Bj, which, entering the first face of the prism at
''grazing" incidence {a^ = 90®) at the point S|, and traversing the
prism as shown in the figure, emerges as the ray y^, and the ray z^^
which, entering the prism at the point -Bj, and arriving at the second
face at the critical angle of incidence (§ 27), emerges only by "grazing''
this face. In order that a ray incident at the point B^ may not b^
totally reflected at the second face of the prism, it must lie within th^
68. When the point H (Fig. 32) lying on the circumference of ther
circle of radius r has such a position that the sides of the Z GHJ inter*
cepted between the two
concentric circles are equal,
that is, HG^ HJ, the diag-
onal VH of the quadrilat-
eral VGHJ is normal to
the bisector of the refract-
ing angle of the prism. The
special positions of the
points G, H and / in this
case may be designated by
Go, Ho» and Jg (Fig. 35).
The ray which traverses
the prism parallel to the
straight line VH^ is sym-
metrically situated with re-
spect to the two faces of
the prism, so that the tri-
angle VBiB2,o is isosceles, and the angles of incidence at the first face
and emergence at the second face are equal.
The angle / VG, denoted by e, between the directions of the inci-
dent and emergent rays is called the angle of deviation, and it may be
shown that when the ray traverses the prism symmetrically, this
angle has its least value. Let H (Fig. 35) designate the position of a
Pxo. 35.
Path op the Rat op Minxicuic Dbviatioh.
80 Geometrical Optics, Chapter IV. I § 70.
say, the angle at £,, is a right angle, there will be no deviation at
emergence, but at the other incidence-point Bi the ray will be bent
away from the prism-edge. And, finally, if one of the angles of the
triangle VB^B^ at B^ or B^ is obtuse, for example, the angle at B^
the deviation at emergence will, it is true, be towards the prism-edge,
but this will not be so great as the previous deviation at S| which was
away from the edge ; as will be easily seen by examining a diagram for
this case. So that in every case, provided n > i, the total deviation
will be away from the prism-edge.
If w < I, all these effects are reversed.
ART. 22. ANALYTICAL IHVESTIGATIOH OF THE PATH OF A RAT RBFRACTBD
THROUGH A PRISM IIT A PRINCIPAL SECTION.
70. The angles of incidence and refraction at the first and second
faces of the prism, denoted by a^ a[ and a,, a^, respectively, are, by
definition (§ 14), the acute angles through which the normal to the
refracting surface at the incidence-point has to be turned in order to
bring it into coincidence with the incident and refracted rays at each
face of the prism. The angle of deviation or the total deviation of a
ray refracted through a prism, denoted by the symbol «, is the angle
between the directions of the emergent and incident rays, or the angle
through which the emergent ray must be turned around the point D
(Fig. 32) in order that it may coincide with the incident ray in both
position and direction. Thus, e = /LJVG,
Assuming that the prism is surrounded by the same medium on
both sides, we have obviously the following system of equations:
sin a^ = n-sin aj, nsin a^ = sin a,;
a| — a, = /3;
€ = «! — aj + «£ ~ «2 =* «i — "a — P^
i
Combining these formulae, we obtain:
(*s)
sin a^ = sin a^cos /3 — sin j8- l/n* — sin^ai; (26)
whence, knowing the relative index of refraction (n = w|/«i = »i/«j)
and the refracting angle (j8) of the prism, we can complete the angle
of emergence (aj) corresponding to any given value of the angle of
incidence (aj) at the first face of the prism.
The total deviation (e) of a ray refracted through a I>^m Ae^^^
( 71.] Refraction Through a Prism or Prism-System. 81
only upon the values of the magnitudes a^, fi and n: for given values
of these magnitudes, the angle c will be uniquely determined by for-
mulae (25). So long as n is different from unity and fi is different from
zero, the value of e cannot be zero. On the other hand, to each value
of c there corresponds always two values of the angle cxi ; for a second
ray incident on the first face of the prism at an angle equal to the
ai^e of emergence of the first ray will evidently emerge at the second
face at an angle equal to the angle of incidence of the first ray at the
first face; and hence it is obvious that these two rays will undergo
equal deviations in traversing the prism. For example, suppose that
the values of the angles of incidence and emergence of the first ray
were ai = ^, ai = ^: a second ray incident on the first face of the
prism at the angle aj = — d' will emerge at the second face at an angle
04 = — ^, and both of these rays will have the same deviation, viz.,
t = e - ^ - /3.
71. Analytical Investigation of the Case of Minimum Deviation.
We have just seen that there is always a pair of rays for which the
deviation (je) has a given value. One pair of rays for which the
deviation is the same are the two identical rays determined by the
idation:
In this case the course of the ray through the prism is symmetrical
^th respect to the two faces of the prism ; that is, the ray crosses
*t right angles the plane which bisects the dihedral angle P between
^ two faces of the prism.
Inasmuch as the deviation (c) is a symmetrical function of a^ and
"■ Oj, it must be either a maximum or a minimum when the ray within
^ prism is equally inclined to both faces of the prism {a^ = — aj).
We shall show that as a matter of fact the deviation in this case is a
muumum.
For a critical value of the angle €, we must have de/dui = o. Differ-
entiating the prism-formulae above, we obtain:
, dai dam . , da^
cosai 8 n-cosa, -,— , n-cos aj-r— = cos a--^ — 1
dui cUxi * dai
da\ doj d€ da^
dai dai" ' da^ da^
r-f- 1 fbese latter give;
d€ cos a. • cos a.
= 1 f / »
dai cos cXi ' cos a,
82 Geometrical Optics, Chapter IV. [ S 71*
and, hence, putting dt/da^ = o, we have:
cos cxj cos a,
J ss •
COS cX| COS a^
Now the first side of this equation is a function of a| and n, whereas
the second side of the equation is the same function of a, and n; and
therefore we must have:
«! = ± a,.
The upper sign is inadmissible here, as the value ai = + a^ would
make the refracting angle of the prism equal to zero, which cannot be
in the case of a prism. Hence, the critical value of the angle c occurs
when we have:
«i = - «i;
and, consequently, also:
«! = — a,.
As we saw above, this was the value of a^ when the ray crossed the
prism symmetrically.
In order to determine whether this critical value of 6 is a maximum
or minimum, we shall have to investigate the sign of the second deriva-
tive of €. Differentiating the formula above for the first derivativCr
we obtain:
cos'
ai • cos ttj -T~% == cos «! • cos a, I cos aj; sm a^ -; f- sm a^- cos a^j
cos a| • sm a, -^ h cos a^ • sm a^ -^ — I-
Now when dtjda^ = o, we have:
da^ dai coscx, da2 , . ,
-J— = 3 — = » -3— = If «!=—«• and a. =8 — OtJ
dai da ncosa, da^ * * 1 ^»
moreover,
«i = - «i = /5/2;
hence, substituting these values in the above, we obtain finally, when
n-cos aj-cos a^ -j-^ = (n — i)-sm /3.
Since all the values of the angles a^ and a[ are comprised between
+ tt/z and — 7r/2, and since the angle P is positive, the sign of if c/daj
Refraction Through a Prism or Prism-System. 83
nds on the value of n. If n > i, d^^/dal will be positive, and
e for €Xi ^=^ — €X2 the deviation c has a minimum value. But if
\,d^€/dot\ 'will be negative, so that we obtain the rather unexpected
It that, under these circumstances, the deviation has a maximum
tte. The explanation is apparent; for if we recall that the angle €
negative -when n < i, as will be seen by an inspection of Fig. 33,
is evident that a maximum value of c in this case corresponds to a
imimuni absolute value.
It the critical value of the angle € is denoted by the symbol €(,, we
lave, therefore, for the Position of Minimum Deviation the following
Bet of equations:
«! = — ttj, «! = — ttj,
. P + eo
sin
n =
sin -
2
(27)
The last of the above formulae is the basis of the pRAUNHOFER-method
of determining the refractive index n, the angles P and €q being capable
of easy measurement.
72. Other Special Cases. If the emergent ray is normal to the
second face of the prism, we must put a, = a, = o; and, thus, for the
<^3se of perpendicular emergence at the second face, we have a[ = /3,
*i = l3 + e, so that we obtain:
sin 03 + 6)
sin /S ^ ^
^ also is a convenient formula for the experimental determination
of the value of the refractive index n. The procedure is described in
^tises on physics.
Case of a Thin Prism {Prism with very small Refracting Angle).
** the refracting angle of the prism is so small that we may put
^i8 = /3, cos /3 = I, the deviation € will also be a small angle of the
s^roe order of m2^;nitude. In this case, therefore, since
«2 = «i — 03 + €),
we have:
sin ttj = sin aj — (j8 + c) cos a,.
84 Geometrical Optics, Chapter IV. [ § 72.
Moreover, since
sin a'j = n- sin cr,, cr, = aj — /9,
we obtain
sina, = n (sincxj — /3cosa|) = sincxi — n-fi-cosai.
Therefore, equating these two values of sin a,, we obtain in the case
of finite value of the incidence-angle a^:
(cos a! \
n ^-i ).
COSCXi /
In case the angle of incidence ai is also a very small angle, we obtai
the following approximate formula for the deviation:
6 = (n - i)i8. (2:
In these formulae the angles are all measured in radians. Accordir
to (28), for small values of both fi and a^, the deviation € is propo
tional to the refracting angle fi, and is independent of the inddeno
angle a^.
The Case of Total Reflexion at the Second Face of the Prism.
the angle of emergence at the second face of the prism is a right angl<
that is, if ai = — 90**, the emergent ray B^Qf will proceed along ((
"graze") the second face /x,. When this occurs, we have «, = —-/
where the symbol A, defined by the formula:
smi4 = -7 = ", (n > i),
denotes the magnitude of the so-called "critical angle" (§ 27) for tl
two media whose relative index of refraction = n. If the absolut
value of the angle a, is greater than this critical angle A, the ray wi
be totally reflected at the second face of the prism, and there will t
no emergent ray. We proceed to discuss this case in some detai
For a prism of given refracting angle (/3), there is a certain limilit
value t of the angle of incidence aj at the first face of the prism f(
which we shall have a^— — A and a^ = "" 9^** »' so that a ray whii
is incident on the first face of the prism at an angle less than this limitit
angle i will not pass through the prism, but will be totally reflected at tl
second face. Putting ai = i, a, = i4 in formulae (25), we obtain tl
formula:
sin I = n- sin 03 — i4) ; (2<
whereby the limiting angle of incidence (i) for a given prism can 1
572.1
Refraction Through a Prism or Prism-System.
85
computed. Examining this formula, we derive the following con-
dudons:
(i) If /3 > 2il, then (since sin A =i/«) sin i is greater than
umty; which means that for such a prism there is no limiting
angle i. Accordingly, if the refracting angle of the prism is more than
toice as great as the '^critical angle' ^ {A), it will be impossible for any
ray whateoer to be transmitted through the prism. For instance, for a
crown-glass prism in air, the angle A = 40® 50', and hence a prism of
this material with a refracting angle greater than 81® 40' will not per-
mit any ray to emerge at its second face.
(2) If j5 = 2i4, we obtain, by formula (29), i = 90®. In this case
the limiting ray I^B^ (Fig. 36) will "graze" the first face of the prism.
Pio. 36.
'I'Ucniio Arolb of Prism equal to
^f*KBTHB CRmcAi« Anolb. Thc only ray
^ Ctt PUB throiiflrh the prism is IiBiBn^t.
Fio. 37.
Refraction of a Ray through a Prisbc.
I^imitiniT Ray in the case when 2A>fi> A
TUsisthe only ray that can pass through the prism, and it will emerge
*t B, and proceed along the second face of the prism in the direction
^1^. Since here we have aj = i = 90** = — aj, evidently this is also
^ray of minimum deviation (§71) for this prism (cq = j8).
(3) If the refracting angle j8 is greater than A, but less than 2 A
(that is, 2i4 > j8 > -4), the value of t, as determined by (29), will be
^^prised between 90® and o**. Hence, for a prism with a refracting
^gle such as this, the limiting ray I^B^ will have a direction between
the directions Mi^i and N^B^\ that is, the Z.VB^Iy (Fig. 37) will be
^ obtuse angle.
(4) If fi ^ A, we find 1 = 0; so that for such a prism the limiting
iDddent ray 7iJBi will be in the direction of the normal at B^ (Fig. 38).
86
Geometrical Optics, Chapter IV.
[§71
In this case, therefore, Z VBJi = 90''. The ray which "grazes" the
first face of this prism will meet the second face normally.
(S) Finally, Up < A, formula (29) gives in this case a negative value
of the angle i; and, hence, for a prism with such a refracting angle the
-^/
^^
PlO. 38.
Rbfraction op a Ray through a
Prism. Case when fi= A (it > 1). I«imit-
ing incident ray meets first face of prism
normally.
Pio. 39.
Path op LnorxNO Rat for Prisx
OF &BFRACTZNO ANOUB fi<A (» > l).
limiting incident ray will lie on the same side of the normal at Bi
the vertex V of the prism, so that now Z VBJ^ is an acute
(Fig- 39)-
In all these cases incident rays which meet the first face VpLi
the prism at the point JBi, and which are comprised within the ang^ ^
IiBifjLi will be transmitted through the prism; whereas all rays incS^
dent at B^ and lying within the angle VBJi will be totally reflected
at the second face of the prism.
If j8 = o**, the prism is a Slab with Parallel Faces, and then we hav^
t = — 90® and Z VBJi = o®. All incident rays will be transmitted^
through such a slab.
In Kohlrausch's method of measuring the relative index of re-^
fraction (n), the prism is adjusted so that the incident ray "grazes*^
the first face of the prism ; in which case the value of n will be giveiL
by the formula:
, /"^ cos 3 — sin a- / «x
sin/3 ' ^ 1 ^ '
The Total-Reflexion Principle is made use of also in the so-called
Total Refractometers of Abbe and Pulfrich for the determination of
the refractive index.
IS.]
Refraction Through a Prism or Prism-System.
87
73. For convenience of reference the following collection of formulae
ir calculating the path of a ray refracted through a prism in a prin-
ipal section are placed here.
Prism FoRMULiG.
sin «! = cos /S- sin aj + sin /S l^n' — sin' ai ;
sin a, = cos /3- sin Oi — sin /S Vn* — sin' a^ ;
cr, = a,' — /S; 6 = a| — a^ — /S.
Minimfim Deviation:
at ^ ^ a
It
«! = -«, = -;
P
an tti = n- sin -; ^ = 2ai — /J.
Grosing Incidence:
«i = 90^» sin ai = cos /S — sin /S l/n' — i ;
a| = i4, cr, = i4— /S, 6 = go*' — ai — /S.
Gromg Emergence:
«i = — 90^» sJ^ «i = sin /S l/n* — i — cos /3;
a[ = /S — i4, a, = — i4, € = 90® + aj — /S.
iVormoJ Incidence:
«! = o, sin a, = — n*sin/3;
a( = o, a, = — /S, € = /3 — a,.
iVormo/ Emergence:
aj = o, sin a^ = n* sin /9;
a, = o, a| = /S, 6 = «! — /3.
The subjoined table gives the results of the calculations of the
^ues of these angles for a prism of flint glass, designated in the glass
catalogue of Schott u. Gen., Jena, as O.103, the refractive index of
^ich for rays of light corresponding to the Fraunhofer P-line is
* = 1.620 2. The refracting angle of the prism is taken as 30®.
Tfce value of the critical angle A (= sin"^ i/n) is 38** 6' 45'.
FlntFaoe.
Second Face.
Angle of
laddfiMT.
Angle of
Ref taction.
Angle of
Incidence.
Angle of
Refraction.
Deviation.
c
90« o' 0^
54 6 20
«4 47 34
000
-'3 13 I
38<» e^ 45^^
30 0 0
15 0 0
000
-8 6 45
8<» 6^ 45^
000
—15 0 0
—30 0 0
-38 6 45
,30,3, ,//
000
—24 47 34
— 54 6 20
—90 0 0
460 46' 59^'
24 6 20
19 35 8
24 6 20
46 46 59
88
Geometrical Optics, Chapter IV.
[§74.
ART. 23. PATH OF A RAT REFRACTED ACROSS A SLAB WITH PARALLEL FACES.
74. If the two plane refracting surfaces of the prism are parallel
(j3 = o), we no longer call it a prism, but a slab or plate with paraM
faces. The path of a ray refracted across such a slab may evidently
be constructed as follows:
Around the incidence-point B[ (Fig. 40), where the incident ray
meets the first surface of the slab, describe in the plane of incidence
CONSTRUCTXOir OP PATH OF RAT RBFKACTBD ACROSS A SLAB WITH PULNB PARALZJO. PA(
three concentric circles of radii r, nir/n[, f^'%fln[y where the radius r
has any arbitrary length, and where n^, n[ and n, denote the absolute
indices of refraction of the first, second and third medium, respectively.
Let G designate the point where the incident ray L^B^ meets the drde
of radius nir/n[, and draw GE perpendicular to the first face of the
slab at the point £, and let this perpendicular, produced if necessary^
meet the circle of radius r in the point H; then the straight line EBj
will evidently give the direction of the ray after refraction at the firet
face. Moreover, if / designates the point where the straight line G£
meets the circle of radius n2r/n[, and if B2 designates the inddenoex
point of the ray at the second face of the slab, the straight line B^^
drawn parallel to JB^ will give the direction and path of the ^™^rto^5^\
ray. ^^^
In the special case when the last medium is identical with t!k\j^
90
Geometrical Optics, Chapter IV.
[576.
medium on both sides. For example, if we interpose in front of the
object-glass of a telescope pointed towards a star a plate of glass with
plane parallel sides, the image of the star will not be deviated thereby.
This fact is employed in a simple method of testing with a high d^;iee
of precision whether or not two faces of a plate of glass are accurately
parallel.
ART. 24. REFRACTION, THROUGH A PRISM, OF AN INFINITKLT NARROW
HOMOCENTRIC BUNDLE OF INCIDENT RATS, WHOSE CHIEF RAT
LIES IN A PRINCIPAL SECTION OF THE PRISM.
76. If an infinitely narrow homocentric bundle of incident rays
is refracted through a prism, and if the chief ray lies in the plane of
a principal section of the prism, the meridian sections of the incident
and refracted bundles of rays will lie in the principal section which
contains the chief rays of the bundles, and which is the plane of inci-
dence of the chief incident ray «, ; whereas the planes of the sagittal
sections of the bundles of incident and refracted rays will intersect io
straight lines parallel to the edge of the prism.
We give, first, the construction of the I. and n. Image-Points coir^
'• • « "•
Fig. 41.
Refraction, through a Prism, op an Infinitely Narrow Bundle of Rats whose CHnr
RAY lies in a Principal Section op the Prism. Construction of the I. and II. Image-Pointt
S»', St' on the chief emersrent ray mi' correapondini? to Object-Point Si on chief incident ray m*
Spending to a homocentric object-point. Let 5| (Fig. 41) be the radiant
point or homocentric object-point of the bundle of incident rays, the
§ 76.] Refraction Through a Prism or Prism-System. 91
chief ray of which (1*1) is incident on the first face of the prism at the
point Bi. The path of this ray within the prism and after emergence
from it is constructed by Reusch's Construction (§ 66). We shall
assume that the medium of the emergent rays is identical with that of
the incident rays, so that n^n[ jn^ = n[ jn^. In the figure VG = VZ[ /n.
The straight line VG is drawn through V parallel to the chief inci-
dent ray u^; the straight line GZ[ is drawn through G normal to the
first face /*! of the prism. The path within the prism of the chief
ray u[ of the astigmatic bundle of rays refracted at the first face will
be along the straight line B^ JS, drawn from B^ parallel to the straight
line VZ[. And if Z[J is normal to the second face of the prism, the
ddef ray u^ of the astigmatic bundle of emergent rays will be along
the straight line B^u^ parallel to VJ.
On the ray VG which is incident on the first face of the prism at
the point V where the principal section intersects the edge of the
prism and which, by construction, is parallel to the incident ray 1*1
there is a point Z^ to which the point Z[ on the ray VZ[ resulting
from the refraction of the ray VG at the first face of the prism, cor-
responds as I. Image-Point. This point Z^ may be constructed accord-
ing to the second of the two constructions given in § 64, as follows:
Through Z[ draw a straight line perpendicular to VZ[, and let C/j, C/,
designate the positions of the two points where this straight line meets
|ke prism-faces /ij, /i,, respectively. Let X designate the point of
intersection of the normal at V to the first face of the prism with the
straight line UiG. The point Zj is the foot of the perpendicular let
fall from X on the incident ray VG. In the same way the I. Image-
'^oint Zj corresponding to the point Z[ on the ray VZ[ incident on
"le second face of the prism at V will be found to lie on the emergent
'^y yj at a point which is determined by drawing UJ to meet at
^ ^e normal at V to the second face of the prism, and dropping
irom Fa perpendicular on VJ, the foot of which will be the required
point 2;.
Hence, according to the relations of affinity which were shown in
8 ^3 to exist between the object-points and the image-points in the
^ of the refraction of parallel rays at a plane surface, the I. Image-
"omt 5,' corresponding to the homocentric object-point Si on the chief
"^y ^'i of the bundle of incident rays will be the point of intersection
<^ the straight line drawn through S^ parallel to Z^Zl, with the chief
^y «i of the astigmatic bundle of rays refracted at the first face of
^e prism. This point S[ is the vertex of the pencil of meridian rays
^ter refraction at the first face of the prism. Considered with re-
92 Geometrical Optics, Chapter IV. [ S 77.
spect to the refraction at the second face of the prism, this point is
the vertex of the pencil of meridian rays which are incident on this
face; so that it might also be designated as the point 52- From S[
draw a straight line parallel to Z[ Z\ meeting the emergent chief ray
u\ in the point S\^ which is accordingly the vertex of the pencil of emer-
gent meridian rays of the bundle, and which is therefore the I. Image-
Point on the emergent chief ray u'^, corresponding to the object-point
5i on the chief incident ray u^.
Again, the normal to the first face of the prism drawn through the
object-point 5i on the chief incident ray 1*1 will meet the chief ray
u\ of the astigmatic bundle of rays refracted at this face in the II.
Image-Point S| ; which is the vertex of the pencil of sagittal rays after
refraction at the first face of the prism. This point may also be desig*
nated as the point 5, by regarding it as the vertex of the pencil ot
sagittal rays which are incident on the second face of the prism. Andt
finally, if through S\ we draw a normal to the second face of the prisnr»»
this normal will meet the chief emergent ray u\ in the point 5i whicl*
is the II. Image-Point on the chief emergent ray u\ corresponding t^^
the object-point 5, on the chief incident ray u^.
Applying here the results of § 63, we can say:
Corresponding to a range of homocentric object-points P^ Qi^ l?i, • • ^
on an incident chief ray w,, which is refracted through a prism in a prin^^
cipal section, we have a similar range of I. Image-Points Pj* Q't» ^t ' " "
and a similar range of II. Image-Points P,, ^2, 71^, • • • both lying an tk^
emergent chief ray u\,
77. Formulae for Calculation of the Positions on the Chief Emer-^
gent Ray of the I. and n. Image-Points. Let a„ a| and a,, a, denote
the angles of incidence and refraction of the chief ray of the bundle
at the first and second faces of the prism, respectively; so that if «,,
n\ and n, denote the absolute indices of refraction of the three media^
traversed by the ray in succession, we shall have:
njSin a( = n^sin aj, Wj-sin a, = »i-sin a,.
Referring to the figure (Fig. 41), let us employ the following symbols* "=
x>iOi = 5i, -^i^i = 5p ^\^\ ~ ^i» I^%^% ~ ^2» -^i^t ^^ ^a»
then, by formulae (19) and (21), we have:
For the Meridian Rays:
n[ ' cos' a[ ^ _ ^2 • cos* aj - ^,
^ njcos ai *' n^cos «i
94 Geometrical Optics, Chapter IV. [ S Sa
are called the Convergence-Ratios of the Meridian and Sagittal Rays,
respectively. Applying formulae (20) and (22) to the refractions at
the two faces of the prism, we obtain immediately:
Meridian Rays:
da^ nj cosovcosa,.
Z^ = ,— = — r 7 ?f (32;
* tfa, n, cos «! • cos a^ ^
SagitkU Rays: _
79. If the prism is surrounded by the same medium on both sides,
so that n, = n^ the formulae above (30), (31), (32) and (33) may be
simplified by putting n = n[/ni = n[/n2. In this case the convergence-
ratio of the sagittal rays will be equal to unity for all directions oi
the chief incident ray. Moreover, if when n, = nj the chief incident
ray has the direction of the ray of minimum deviation, so that (§ 71)
cos «! • cos ttj = cos a[ • cos ai, the convergence-ratio of the meridiai^
rays will likewise be equal to unity; that is, 2^,o ~ ^•
In general, therefore, the image of a luminous point as seen througl^
a prism, viewed either by the naked eye or through a telescope, will no^
be a point. Depending on how the eye or telescope is focussed, the int^
age of a point-source of light will appear through the prism as a small
straight line parallel to the prism-edge (I. Image-Line), or a disc of
light, or, finally, a small straight line lying in the plane of the prin-^
cipal section of the prism (II. Image-Line). See description of L-
Burmester's Experiment, § 85. In a prism-spectroscope the source of
light is usually a narrow illuminated slit with its length parallel to
the prism-edge. If, as is usually done in this case, we focus the tele-
scope on the II. Image-Line, the slit-image, except near its ends, will
be clear and distinct, so that here we encounter practically no serious
disadvantage on account of astigmatism (§ 86).
80. The Astigmatic Difference. If the bundle of incident rays is
itself astigmatic, instead of a homocentric object-point, we shall have
a I. Object-Point S^ and a II. Object-Point "S^; and the astigmatic
difference (see § 61) of the bundle of incident rays will be:
where s^ = B^S^.s^ = B^Si; and the astigmatic difference of the cor-
responding bundle of emergent rays will be :
•^•)»»^9 ^^ ^2 ■" S^m
'2*^2
96 Geometrical Optics^ Chapter IV. [ § 8L
JB2 and the 11. Image-Point JB,, respectively, corresponding to the
homocentric object-point JBi on u^.
(2) If the object-point is a point Zj lying on an incident chief ray
2i which meets the refracting edge of the prism, so that the ray goes
through the point V in the principal section of the prism, in this
limiting case the ray-length within the prism is vanishingly small, and,
hence, putting 5^ = o in formula (34), we find:
"^^ n^Vcos a^-cos cr, / *
(3) The condition that the astigmatic difference shall be indepen-
dent of the distance of the homocentric object-point 5, from the point
Bi where the chief incident ray meets the first face of the prism, that
is, the condition that the astigmatic difference shall be independent
of the magnitude 5i, is evidently:
cos CX| * cos a, = cos a| * cos oe,;
which, in the general case, leads to an equation of the eighth d^;ree for
calculating the value of the angle of incidence d| in order to ascertaofl
what must be the direction of the chief incident ray. In the sp&dsX
case, however, when the prism is surrounded on both sides by tb€
same medium (nj = nj), the equation above will be recognized 3*
the condition that the ray shall traverse the prism with minimum
deviation (§71). Accordingly, in case the chief ray 11^ j of the homO^
centric bundle of incident rays has the direction of the ray of mitti^
mum deviation, we have here the following special formulae:
Minimum Deviation (» = »i/»i == wj/n,):
In this special case, since la^ -f- /S = o, the formula for the magnitui
of the astigmatic difference in the case of minimum deviation may
written also in the following form:
5:5; = ^ (n" - I) tan* ^ ;
which shows clearly that the magnitude of the astigmatic ^ffere::;::::^
which in ever>* case depends on the length of the ray-path witlu:
prism, is in the special case of minimum deNaation oi ^^ ch\<
directly proportional to this magnitude 6^. The neax^t to tKj
98 Geometrical Optics, Chapter IV. [ § &3.
Burmester/ this can occur; although until the publication of Bui-
mester's investigations on this subject, the laws of the homocentric
refraction of light through a prism appear not to have been clearly
formulated except for certain special cases.'
In the following discussion we shall show how the main results
obtained by Burmester by purely geometrical methods may be de-
duced from the general formula (34) for the astigmatic difference, as
is done by Loewe;^ and we shall give also an outline of the el^;ant
geometrical method used by Burmester himself.
83. Analytical Method. In the first place we may remark that
when the incident rays are an infinitely narrow bundle of parallel rays
(^1 = ^)i the ratio (5, — Ji)/^i will be vanishingly small; that is,
the astigmatic difference will be practically equal to zero in com-
parison with the distance from the prism of the point-source of light.
This is the essential advantage of using parallel incident rays in work-
ing with a prism-spectroscope.
The condition that the astigmatic difference 3^5^ of the bundle of
emergent rays shall vanish, that is, that the I. and II. Image-Pdnts
on the emergent chief ray Uj shall coincide in a single point Zf* ^
found immediately by putting ^i — 5i = SjSj = o in formula (34), and
is as follows:
n^S^ cos' ai (cos' o^ - cos' a.) .
* 1*1 cos tti-cos a, — cos ttj-cos a,
where Si designates the homocentric Object-Point on the incident chief
ray u^ to which corresponds the homocentric Image-Point D, on the
emergent chief ray Wj. This distance JBjSi is determined by this equa-
tion as a unique function of a^ and 5^; the two magnitudes whidit
for a given prism, define completely the incident chief ray «|. Henoet
equation (35) shows that:
On every incident chief ray u^, refrcLcted through a prism in a prindpA
section, there is in general one, and only one, Object- Point 2)i to wUiA
on the emergent chief ray u^ there corresponds a homocentric Imagj^
Point Sj.
Moreover, for a given value of the angle of incidence a„ the lei^tfa
' L. Burmester: Homocentrische Brechung des Lichtes durch das Prisma: ZfUf*
Math. u. Phys., xl. (1895), 65-90.
' See H. Helmholtz: WissenschafUiche Abhandlungen, Bd. II (Leipzig, 1883), S. 167.
A. Gleichen: Ueber die Brechung des Lichtes durch Prismen: Zft. f. Math. u,Pkys„ zxziT.
(1889), 161-176. J. Wilsing: Zur homocentrische Brechung des Lichtes durch diS
Prisma: ZfL f. Math, u, Phys., xl. (1895), 353-361.
'F. Loewe: Die Prismen und die Prismensystcme: Chapter VIII of DU TheorUim
optischen InstrumenU, Bd. I (Berlin, 1904), herausgegcben vonM. von Rohr. See p. 433*
100 Geometrical Optics, Chapter IV. I § 8i
of I. Image-Points
Pl.ii Ol.i» -'^.it • *:
and a similar range of II. Image-Points
PL, If ^ If Ru, u • • :
both lying on the chief ray u[ of the bundle of rays refracted at the
first face of the prism. And so corresponding to a range of Object-
Points on each incident chief ray of the system of parallel rays we shall
have two similar ranges lying on the corresponding chief ray of the
bundle of rays refracted at the first face of the prism. This system
of Object-Points lying on parallel incident chief rays, such as m,, Vp etc-»
may be referred to as a whole as the system iji ; and to this system of
Object- Points i/j there corresponds, as explained in §63, a system o^t
I. Image-Points ri[ and a system of II. Image- Points ri[ lying on th^
parallel chief rays of the bundles of rays which are refracted at th^2
first face of the prism. Each of these systems ii[ and ri[ is in affinU^i^
with the system of Object-Points ri^.
Again, corresponding to the system i7|,wehave a system of I. Imag^—
Points rj2 which lie on the rays of the pencil of parallel emergent chi^^
rays, which is likewise in affinity with the system lyj. And, similari^"*
corresponding to the system ?|, we have a system of II. Image-Pointss
7J2 which lie on the rays of the pencil of parallel emergent chief ray^«
which is likewise in affinity with the system ri[.
Since the system of Object-Points rji is in affinity with the systefl*-^
ri[ and ^[, and since rj[ is in affinity with rj^, and ri[ is in affinity with 5f^ •
it follows that the system rj^ is in affinity with both ly, and rj'^; ao^*
hence, also the systems rj^^ rj^ ^^ in affinity with each other.
The three systems ryp rj[ and ^J, which are in affinity each with t^^
other, have a common affinity-axis, viz., the straight line VB^ in whiC^
the plane of the principal section meets the first face of the prism.
The straight line VB2 in which the plane of the principal secti^^
meets the second face of the prism is the affinity -axis of the two s/^
tems rj[ and rj^; and this straight line is also the affinity-axis of tt^^
two systems ri[ and ^j- The point V in which the plane of the pri<^
cipal section meets the refracting edge of the prism is on both of the^
affinity-axes; so that for each pair of the five systems i/p lyj, ^J, i|J, ^i
which are in affinity with each other the point 7 is a self-correspondii^^
point.
The three points 5,, S[ and 5^ on the corresponding chief rays ^^
u[ and tt, (Fig. 41), or the three points Z^ Z| and Zi on the correspond*
.02 Geometrical Optics, Chapter IV. I § 84.
^2 which are in affinity each with the other; for the corresponding
points of 77p ri[ lie on the normals to the first face of the prism, and the
corresponding points of ri[, ^j lie on the normals to the second face of
the prism.
Since the corresponding points of the two systems i;i, 5, lie on the
rays of the pencil of parallel emergent chief rays, the affinity-axis of
this pair of systems must go through the double-point V; and along
this straight line, which we shall denote by a,, must lie the self-cor-
responding points of ri2 and ^2» or the Homoceniric Image-PoirUs of
the pencil of parallel emergent chief rays.
This affinity-axis may be constructed by determining the point of
intersection of two corresponding straight lines of the systems lyj and
rj'^. In the figure (Fig. 42) the points Zj, Z^ lying on the incident
and emergent chief rays Zj, Zj* respectively, to which on the chief ray
z[ of the bundle of rays refracted at the first face of the prism corre-
sponds the I. Image-Point Z[, are constructed exactly as was described
in § 76 (see Fig. 41); as are also the two pairs of I. and II. Image-*
Points S[t 5[ and 5^, Sj corresponding to the object-point Si on the?
chief incident ray u^ drawn parallel to the chief incident ray S|, whicbi
latter ray meets the refracting edge of the prism.
To the point Bi where the chief incident ray Ui meets the first face
of the prism, regarded as an object-point on this ray, there correspond
on the emergent chief ray Ut (as was also explained in § 81) the I. anc
II. Image-Points -Bj, JJj, which are constructed by drawing from B
two straight lines, one parallel to the straight line ZjZj and meetifl?
ttj in B'2, and the other perpendicular to the second face of the prisr
and meeting u^ in E^.
The II. Image-Point Zj corresponding to the Object-Point Zi '
the chief incident ray z^ may be found by drawing from Zj a strain
line perpendicular to the first face of the prism meeting the ra'
in the point ZJ, and from 2[ a straight line perpendicular to the sec
face of the prism meeting the corresponding chief emergent rs
in Zj. The points 5,, Sj and Z^, Zj, are two pairs of correspor
points of the systems ijj, ^2 J ^md hence J32'^2> ^2^2 ^tre a pair (^
responding straight lines of these systems; which must therefore
sect in a point fi^ lying on the affinity-axis a^ ot the two systems
Accordingly, the affinity-axis aj is the strai^t line VH^. Ins'
using here the pair of corresponding points B^ B'„ we may
any other pair as ^j, S2 on w,, in conjuncti^V^ti^tVvtVve p^ i
Zj, which will determine some other poin^t-^^'^oivtVie afflvtut:
The point Sj where the emergent ray "^--^^^s^ ^^ amu'
104 Geometrical Optics, Chapter IV. [ § 85.
prism, to which on the rays of the pencil of parallel emergent rays
correspond Homocentric Image-Points, are ranged along a straight line
Fifli which goes through the refracting edge of the prism; and the
corresponding Homocentric Image-Points are ranged likewise along
a straight line Fa, which may be regarded as the emergent ray cor-
responding to the incident ray Va^. All these results are in agree-
ment with those found in § 83. These laws were distinctly formulated
first by BuRMESTER, and for a further account of his investigations
the reader is referred to his original paper on this subject.
85. In order to verify his results, Burmester employed a glass
prism of refracting angle 60**, for which the value of n for the Fraun-
HOFER D'line was n =1.7. The prism is shown in section in the
diagram (Fig. 43) which gives the disposition of the apparatus. On
o
^ £
Atic^0*<*/»m
0/ass C»S^
Fig. 43.
SHOWIIfO THB Plan op BURMBSTBR*8 BXPSRXirBNT.
a certain incident ray Uy^B^ the Object-Point S^ to which corresponds
a homocentric Image-Point Sj on the emergent chief ray u^ was con-
structed; and, moreover, the I. and II. Image-Points S\ and 5^ cor-
responding to an arbitrary Object-Point 5, on Wj were also constructed
by the methods given above. The prism was supported on a blockt
which was movable in parallel guides in a direction parallel to the
straight line ttj. On this same block was placed a glass cube with
two of its parallel faces perpendicular to the incident chief ray fi|,
the face nearest to the prism being at the distance B^x from it. This
face was covered with lamp-black except at the point Sx where a small
opening was made with a needle. A sodium-flame F was placed on
the block with the prism and the glass cube. Finally, an Abbe'»
Focometer A, for which the distance from the objectw^ 0 of a dis-
tinctly visible object is equal to iio mm., was placed, vcv.^^^ poa-
tion with its axis coinciding with the emergent eVv\^i -^'^^^^^^^^ ^€
Geometrical Optics, Chapter IV.
ent from that of the slit; for, since
cos a, ■ cos a
(«; = «i).
it appears that the value of Z. will depend on the angle of inci
a,. If, for example, either a', or a'^ = 90°, the value of Z, w
infinite, and hence db' = 00. On the other hand, if one of the t
in the numerator, for example, a, = 90° (case of so-called "gi
incidence"), we have Z. = o and, therefore, also db' = o. Thu
image of the slit may appear infinitely broad or infinitely narron
may have any apparent breadth between these two extremes de
ing on the value of the angle of incidence a,.
When the rays proceed through the prism with minimum devi
(«i = «i. cos a, cos a, = cos a|-cos a^), we have Z, = i; so that
both the apparent height and breadth of the slit-image are eqi
the apparent height and breadth of the slit itself.
AST. 27. ASTIOHATIC KBFRACTIOIT OF ntFINITBLT NASSOW, H01l<
TBIC BDITDLB OF IRCIDERT RATS ACKOSS A SLAB WITH
PLANE PARALLEL FACES.
87. As we have seen, a Slab or Plate, with parallel plane refra
faces, may be treated as a prism whose refracting angle is eqi)
zero; so that the methods and formulae of the preceding article
be adapted to this problem by treating the slab as a special ci
the prism. For the sake of generality, let us assume that the nie<
the incident and emergent rays are different; and let us denot
absolute indices of refraction of the three media, in the order in i
they arc traversed by the rays of light, by n,, n[ and %. We
give, first, Ihe construction of the I. and II. Image-Points corres\
ing to an Objeci-Painl on a given incident chief ray «,.
In the figure (Fig. 44) the plane of the paper represents the
of incidence of the incident chief ray m, which meets the first fa
the slab at the point 5,. With 5, as centre and with radii eqi
T, njjn't, and n'^rjn^ (where r denotes any arbitrary length) de«
the arcs of three concentric circles; and through the point G i
the circle of radius nj/n[ meets the incident chief ray «,, di
straight line normal to the first face of the slab, and let this stn
line meet the circles of radii r and nlr/n', in the points 5[ and J, rei
tively. Then, exactly as in S 74, the straight line 5jS, will deter
the path of the ray S,5j or u[ after refraction at the first fa
the slab, and the path of the emerRent ray «i is determined by
ing through B, a straight line parallel to JB^.
108
Geometrical Optics, Chapter IV.
[§».
Meridian Rays:
$2 = B^2 ~ ~
Sagittal Rays:
fin —
n\' sin' g| f $1
»; |ni- cos*al""n;*-
?- = BJSm = Wj ( 7 )•
#»;«,
S • 2
nj" sin «!
► •
(36)
(37)
In these formulae, ^, = 5i5„ 5x = Sj-B,.
Similarly, by specializing formulae (32) and (33), we obtain for the
Convergence-Ratios of the meridian and sagittal rays in the case of a
slab with plane parallel faces:
Meridian Rays:
Axi n, cos ttx . , g^
Z„ =
Sagittal Rays:
dai n, cos a^
z =
n
(fX| ft
1
7»
(39>
In the special case when we have the same medium on both sides Off
the slabf the formulae above may be simplified by putting n = nj/i
= nj/nj, in which case, in addition to the condition a{ = a,, weha^
also a, =
^2 = "r
Thus, we obtain:
= ^i-
^f
52 = 5i -
cos* «! 5i
2 ^ ""'
COS «! n
— f
n
Z„ = Z„ = I.
In case the slab is at the same time very thin, so that B^B^ is pr
cally negligible, we have approximately 5j = ^2 — ^r
89. Astigmatic Difference in Case of a Slab. The formula £
astigmatic difference of the bundle of emergent rays correspond
an infinitely narrow homocentric bundle of incident rays refc
across a slab with plane parallel faces may be obtained by co
formulae (36) and (37), or, perhaps more simply still, by in
in formula (34) the condition a[ = a2; thus, we obtain:
-,, , / -. n^fcos^a^ \ n^fcos^a^ V
O2O2 = ^2 "" ^2 ~ I 2 I Hi > I 2 ^ rv
In general, therefore, the astigmatic difference for a given
dent ray will depend on the position on this ray of the rad\a.tvi
no Geometrical Optics, Chapter IV. [ §90l
which the Object-Point and its Homocentric Image-Point coincide with
each other. If the slab is surrounded by the same medium on both
sides, this point 0 lies at infinity.
When a luminous point Jlf, is viewed normally through a trans-
parent slab with parallel plane faces, the displacement in the line of
vision of the Homocentric Image-Point M^ with respect to the Object-
Point Ml is:
and if the slab is surrounded by the same medium on both sides
(n = n[/ni = njn^), we obtain:
a formula which, according to a method suggested by Due DE Chaui^*
NEs (1767), is employed for the determination of the relative index of
refraction (n), the lengths M1M2 and d^ being both capable of easy
measurement.
90. Exactly as in the case of refraction through a prism (§ 84), w6?
can construct on every incident chief ray 11, of a narrow bundle of
incident rays refracted across a slab with parallel plane faces the
Object-Point 2i to which on the chief ray u'^ of the bundle of emergent
rays there corresponds the Homocentric Image-Point S^. Thus, draw-
ing through 5| (Fig. 44) a straight line perpendicular to the first face
of the slab and meeting the straight line S[S2 in the point W^ we find
the Homocentric Image-Point S^ at the point of intersection of BiW
with the emergent chief ray ttj- A straight line drawn through Zt
perpendicular to the first face of the slab will determine by its inter-
section with the incident chief ray ttj the Object-Point Sj which cor-
responds to the Homocentric Image-Point 2^.
The formula for the determination of the position on a given inci-
dent chief ray u^ of the Object-Point Si which has a Homooentric
Image-Point can be obtained from formula (41) by equating to zero
the right-hand side of this equation. Thus, writing here BjSi in place
of 5p and also employing the relations:
fti • sin ay = n[' sin a|, n[ • sin a, = fjj * sin a,,
we obtain*
nv _^i ^2 ""**» cos ofi
•^i*-i — ,s ,1" ~, i~", "i»
Wj fK — ni cos «!
112
Geometrical Optics, Chapter IV.
m
case that every other medium of the series is air, and almost invariabi;
the first and last media are air, so that Wj = n^ = n^ = • • • =if^; bu
.a
Fio. 45.
Path op a Ray in a common principal section of ▲ ststek of
refractino edges all parallel.
PUSlia WITH TKI
for the sake of generality we shall not assume here (except in sped
cases) that any two of the series of media are the same.
The points in the diagram where the refracting edges of the prisn
meet the common principal section are designated as V„ V,, etc
thus, the point where the refracting edge of the ifeth prism (that i
the straight line in which the two refracting planes n^ and /i*+i inte
sect) meets the principal section will be designated as the point V^.
92. Construction of the Path of the Ray. The path of a ra
through a system of prisms can be constructed geometrically by r
peated applications of the construction of the path of a ray throug
a single prism (§ 66). Thus, with centre at the point V^ and wit
radii fj, n{rjn[ and n^r^Jny^ (where rj denotes any arbitrary lengtl
describe the arcs of three concentric circles; and through F, draw
straight line parallel to the given incident ray L^By meeting the circ
of radius n^rjn[ in the point G,. Through Gj draw a straight lii
perpendicular to the first refracting plane Mi and meeting the circ
of radius Tj in the point Hi\ and from H^ draw a straight line perpe
dicular to the second refracting plane /i2 ^^^ meeting the circle
radius n^filn^ in the point J^\ and draw the straight lines V^Hy^ ai
V^J^. Similarly, with V^ as centre and with radii r^, n\r^\n^ ai
"4^2/^3 describe the arcs of three concentric circles, ^nd through
114 Geometrical Optics, Chapter IV. [ §!
mon to all the prisms, we have the following system of equation
I II III
n[sina| = ni-sina|, / ^ €,=ai— a(, 0*i)
Wj-sinag^^i'sina,, / €2 = a2~«2t (M2)
a3 = «2~P2»
«* = «*-! -ft
n^sina^ = n;k_,sina;k * *"* ^*"" €4=a;k-«». (mJ
Total Deviation = c = ^ €4=ai— a^— ^ /J^.
•• li
0
Here the term 2*21"* ft = angle between the first and last (or mi
refracting planes; and if we denote this angle by Q, we can write:
€ = «i - «m - 0-
94. Condition that the Total Deviation shall be a Minimum. 1
total deviation € of a ray refracted through a given system of pris;
will be a minimum when the ray is incident on the first refracti
plane at an angle a^ determined by the condition d^jda^ = 0; it bei
assumed that the conditions d^e/da] > o and € > o are also fulfills
According to the equation above, the condition d^jda^ = o is equii
lent to:
and in order to express da^ as a function of dap we employ the eqi
tions in columns I and II of the system of equations (43). Th
differentiating each of these equations, we obtain:
, nj cos «!
* Wj cos aj *
, nl cos a, - .
da^ = ~ > »«•, aa2 ~ ^»
* Wj cos a2
, , nj.^1 cos aj^ . - , /
da. = - V V daj,, daj, = daj^x,
fif^ cos ajfc
> See A. Glkichen: Ueber die Brechungdes Lichtes durchPrismen: Z/r. /. Mai
Phys., xxxiv. (1889). 161-176. Also. S. Czapski: Theorie der opitschen It^^um^
Abbe (Breslau. 1893). S. 137. H. Kayser: Handbuch der Spearoscopf^, Bd,l Q^
1900). S. 272. F. Loewe: Die Prismen und die Prismena^^slcme: ^^^^J;[*"
Theorie der optischen Instrumente. Bd. I. herausgegeben vonU. vos KOHR ^iseru^
S. 421.
118
Geometrical Optics, Chapter IV.
We shall assume that the system of prisms is formed by t
refracting surfaces and that the edges of the (m — i) prisms
parallel.
According to formulae (19) and (20) of § 59, we have for
The Meridian Rays After Refraction at the kth Plane:
§
Su =
' 2 ^A» \* — 1» 2f
• *»).
da^ n^^i cos a^
Uu cos a.
(A = 2, 3, •••,#»),
(* = I, 2, •••,#»).
In the first and third of these formulae we must give h in sue
all integral values from ifc = i to i = m; and in the second all i
'A » '^v
r^^^::::
FlO. 47.
SHOWIIfO THE PATH OF THE CHIEF RAY THROUGH THE *TH PRISM OF A STSTEK C
values from jfe = 2 to * = m. It may be observed also that
Eliminating Sj, from the first two of equations (45), and abt^
by writing:
n; cos' «;
nfc_iCOS af,
122 Geometrical Optics, Chapter IV. (J 99.
ogous to those which were obtained in the case of a single prism (§§8o
and foil.).
99. Homocentric Refraction through a System of Prisms. If the
astigmatic difference of the bundle of emergent rays is equal to zero,
the I. and II. Image-Points 5^ and iS^ will coincide in a single point
S'„ on the chief emergent ray u^ ; and in this case the image of a point-
source 2i on the chief incident ray ttj will be a point ^'^. Thus, ex-
actly as in Art. 25, where the special case of homocentric refraction
through a single prism was investigated, the condition that the astig-
matic difference shall vanish is found by putting 5^ — 5jj^ = o in for-
mula (51); whereby we obtain:
,, ".TCiii.(^)-l] ,„
M \cos* a J
where Sj designates the Homocentric Object-Point on the chief inci-
dent ray Wj to which corresponds the Homocentric Image-Point "im
on the chief emergent ray u^. This formula gives the distance 5,Si
as a unique function of the angle of incidence a^ of the chief inddeO*
ray Wj and the ray-lengths hj^ from one surface to the next; so tha-*
precisely as in the case of a single prism (§ 83), we have also here tb^
following statement:
On every incident chief ray «,, refracted in a principal section through
a system of prisms with their edges all parallel^ there is, in general, one^
and only one, Object- Point Sj to which on the chief emergent ray «1 ther^
corresponds a Homocentric Image- Point Z'^.
It is easy to show likewise by the same methods as were used in
the case of a single prism (§§83, foil.) thsit Object- Points, lying on
parallel incident chief rays, refracted, in a principal section, through a
system of prisms with their edges all parallel, to which on the paralld
emergent chief rays correspond Homocentric Image- Points, are ranged
along a certain straight line a^; and the Homocentric Image- Points are
rajiged also along a straight line a^, which may be regarded as the emer-
gent ray corresponding to the incident ray a^.
The construction of the Homocentric Image-Point X'^ on the chief
emergent ray u^ and of the corresponding Homocentric Object-Point
2i on the chief incident ray «, is performed by a method entirely
analogous in every detail to the method given for the case of a single
prism (§ 84). The system of Object-Points P«,i, Pr.i, etc., lying on
parallel incident chief rays «„ v^, etc., may be denoted as the system
126
Geometrical Optics, Chapter IV.
[§102.
surrounded by the same medium on both sides, then
whence it follows that the arcs z^A and Bz^ are equal, and the right-
angled spherical triangles ACz^, BCZf
are congruent; and therefore arc AC
= arc CB, arc z^C = arc Cz'^. From the
right-angled triangle ACz^ we have:
cos (arc «iC) ■» cos (arc A C) • cos (arcilsj,
or
D
E
cos — = COS- -cos fit*
2 2 "
(57)
FiO. 49.
Deviation (Z?) of a b.ay ».»•
fr acted obliquely through a
PRISM. ^ Eia the projection oi ^ D
on the plane of a principal section
This equation, together with (56), enables
US to compute the total deviation P of a
ray obliquely refracted through a prism-
It will be seen that the angle P is always
greater than the angle £.
The angle £ will have its minimuiA
value £0 at the same time that the angte
D has its minimum value P^; and
condition that E shall be a minimum
given by the equations:
7i = - 72 =
of ^^e p^'rll^'"' "^^ "'"'''°' "^ which are derived exactly in the
way as the analogous equations (27)
the case of an actual ray traversing the prism symmetrically ixx
plane of a principal section were obtained; only, we must obi
that here for the so-called "projected ray", instead of n, we haV'
''artificial" relative index of refraction (see § 33):
n, = «
so that
cos IJi
cos IJi
(n = »i/ni = ni/»i);
103.] Refraction Through a Prism or Prism-System. 127
rhich b analogous to the formula :
sm
n =
sm-
2
vhich is the third of formulae (27). Here e^ denotes the minimum de-
viation of a ray traversing a principal section of the prism. Hence, we
obtain:
sm— ^^
2 ^n,
sm
2
In case we have n > i, then 1;'^ > ri^ and, therefore, n, > n; hence,
. Eq+P . Cp + jS
sm > sm
2 2
Therefore, the angle £0 (which is the projection on the plane of a
prindpal section of the angle D^of the minimum deviation of a ray ob-
Kqudy refracted through the prism) is always greater than the angle of
nunimum deviation 6,, of a ray incident on the prism at the same angle aj
but lying in a principal section; and, hence, Dq itself is greater than Cp.
Of all the rays which go through a prism, that one which, lying in a prin-
^pol section, traverses the prism symmetrically will be the least deviated.
The case when n < i may be discusssed exactly in the same way as
^^done when the ray was in the principal section (see § 71).
IM. The formulae given in this article for the path of a ray ob-
liquely refracted through a prism may properly be attributed to
o^AVAis,* although the same results, in a more general form, were
^terwards derived by geometric methods by Reusch* and Cornu*
^di analytically, by Stokes* and Hoorweg.*
'a. Bkavais: Notice sur les parhelies qui sont situ^s k la mtoie hauteur que le soleil:
^^'ie TU. polyi^ xviii., cah. 30 (1845), 79. M6moire sur les halos, etc.: Joum. de
^^•^Polyi., xviii.. cah. 31 (1847). 27.
^' Rkusch: Die Lehre von der Brechung und Farbenzerstreuung des Lichts an ebenen
'**'*en und in Prismen in mehr synthetischer Form dargestellt: Pogg. Ann., cxvii.
f»W2). 241-284.
A-CoRNu: De la refraction k travers un prisme suivant une loiquelconque: Ann. ic.
■""■•• (2) I. (1872), 255-257.
*G. G. Stokes: In a " Note " on a paper by Th. Grubb: Proc. Roy. Soc, xxii. (1874),
*J- L. Hoorwbg: Ueber den Gang der Lichtstrahlen durch ein Spectroscop: Fogg.
it«».. div. (1875). 433-430.
^ also. A. Anderson: On the maximum deviation of a ray of light by a prism:
Camh. Ptoc., ix. (i896-*8). I95-I97.
128 Geometrical Optics, Chapter IV. [ { li
The explanation of the curvature of the lines of the spectrum,
observed through a prism-spectroscope, which appears to have be
remarked for the first time in Gehler's Physikalisches Woerterbu
is to be found in the fact that the function denoted above by «, <
pends on the inclination (r?,) of the incident ray to the principal si
tion of the prism. Bravais^ derived a formula for the radius of ci
vature at the vertex of the image-line which is given in Kayse;
Handbuch der Spectroscopies Bd. I (Leipzig, 1900), Art. 321; also,
F. Loewe's treatise Die Prismen und die Prismensysteme.^
ART. 31. HOMOCEirTRIC REFRACTIOir THRGUOH A PRISM OF AN DTI
NITELT NARROW, HOMOCENTRIC BUNDLE OF OBUQUELT
INCIDENT RATS.
104, We propose now to investigate the conditions that must
satisfied in order that to a narrow, homocentric bundle of obliqw
incident rays refracted through a prism there shall correspond a hoii
centric bundle of emergent rays. The solution of this problem n
given first by Burmester,* whose geometrical method is the one giv
here. An analytical deduction of the same results, based on Hel
HOLTz's formulae for the passage of light through a prism as given
his Handbuch der physiologischen Optik, has been given by WiLSiN'
When a ray of light is refracted through a prism, the plane of in
' A. Bravais: Mfynoiie siir les halos, etc.: Journ, de Vic. pdlyt,, zviii., cah. 31 (18^
z-380.
' See Die Theorie der optischen Instrumente: Herausgegeben von M. von Romu B(
(Berlin, 1904), p. 429.
In the same connection, see also the following:
L. Ditscheinbr: Ueber die Kruemmung von Spectrallinien : Wien, Ber., H,
(1865), 368-383. Notiz zur Theorie der Spectralapparate: PoGG. Ann,, czziz. (i8(
336-340.
J. L. Hoorweg: as cited above.
H. V. Jettmar: Zur Strahlenbrechung im Prisma; Strahlengang und Bild von lew
enden zur Prismenkante parallelen Geraden: 35. Jahresb. ueber das k. k, Staaisiymn.
Bez. Wiens, 1885.
J. V. Hepperger: Ueber Kniemmungsvermoegen und Dispersion von Prismen: W
Ber,, xcii., II. (1885), 261-300.
A. Crova: Etude des aberrations des prismes et deleur influence aur les observati
spectroscopiques: Ann. chim. et phys., 5, xxii. (1881), 513-520.
W. H. M. Christie: Note on the curvature of lines in the dispersion spectntm, c
Monthly Notices of the Roy. Astr. Soc, xxxiv. (1874), 263-5.
W. Simms: Note on a paper by Mr. Christie: MoniMy Noi., xxxiv. (1874), 363-
See also Kayser's Handbuch der Spectroscopies Bd. I (Leipzig, 1900), Arts. 260. 2
332 and 323.
'L. Burmester: Homocentrische Brechung des Lichtes durch das Prisma: i^
Math. u. Phys., xl. (1895), 65-90.
^ J. Wilsing: Zur homocentrischen Brechung des Lichtes durch das Prisma: i^
Math, u, Phys., xl. (1895), 353-36i.
{ 104.] Refraction Through a Prism or Prism-System. 129
dence at the first face and the plane of emergence at the second face
will, in general, not be coincident; in fact, this will be the case only
when the incident ray lies in the plane of a principal section of the
prism, as we have seen. To a homocentric bundle of incident rays
emanating from an Object-Point 2^ on the chief incident ray Ui there
corresponds within the prism an astigmatic bundle of refracted rays
whose chief ray is designated by the symbol «!, and which, therefore,
we may speak of as the "bundle tt'i". If the incidence-point of the
chief ray at the first prism-face is designated by 5|, the II. Image-Plane
of the astigmatic bundle u[ will be the plane of incidence UiByu\^
and the I. Image-Plane will be the plane which contains u\ and which
is perpendicular to the plane UiByu\.
On the other hand, let us consider an astigmatic bundle of rays
within the prism whose chief ray may be designated as the ray rj, and
which, therefore, we shall call the "bundle v\'\ Let 5, designate the
incidence-point of this chief ray v\ at the second prism-face. More-
over, let us assume that the bundle of emergent rays corresponding to
^ bundle v'l is a homocentric bundle of rays with its vertex at a point
ittignated by S^. The II. Image-Plane of the astigmatic bundle v\
coinddes with the plane of incidence v[ B^'2 of the ray v\ at the second
Jace of the prism, and the I. Image-Plane of this bundle is the plane
^idi contains the ray v\ and which is perpendicular to the plane
Now, if these two astigmatic bundles u\ and v\ within the prism are
*^ntical, then the point Sj is the homocentric Image-Point on the
^f emergent ray u\ which corresponds to the homocentric Object-
Point Sj on the chief incident ray u^. Now in order that these two
^gmatic bundles of rays shall be identical, it is necessary, in the first
Pl^, that the I. and II. Image-Planes of the two bundles shall be co-
I ^''adent; which may happen in either of two ways: (i) The I. Image-
"anes of the two astigmatic bundles of rays may be identical, and
^ the II. Image-Planes; in which case the chief rays will lie in the
plane of a principal section of the prism; which was the case investi-
gated in Art. 25; or (2) The I. Image-Plane of one bundle of rays may
OJindde with the II. Image-Plane of the other bundle, and this is the
<^ that interests us at present. In this latter case, if also the I.
Image-Point of one bundle of rays coincides with the II. Image-Point
of the other bundle, and vice versa, the two astigmatic bundles of rays
«i and v\ will be identical (provided we neglect infinitesimals of the
*cond order, as is here assumed). Therefore, in order that, corre-
sponding to an Object-Point lying on a chief incident ray which is ob-
10
w
130 Geometrical Optics, Chapter IV. [ § 105.
Hquely refracted through the prism, we shall have on the chief emex
gent ray a homocentric Image-Point, it is necessary, first of all, tha
the planes of incidence and emergence sJmll be at right angles; that Li
if ttj, U2 designate the chief incident ray and the corresponding chi€
emergent ray, respectively, and if the straight line B^B^ represents th
path of the chief ray from the first face of the prism to the secon
face, the two planes u^B^B^ and B^B^u'^ must be perpendicular.
105. In the accompanying diagram (Fig. 50) the refracting edge c
the prism is represented by the vertical straight line Vy lying in th
plane of the paper, which, as in the similar diagram (Fig. 48), is su|
posed to be the plane of the second face of the prism. From the poir
By in the first face of the prism draw the straight line B^M normal fl
this face and meeting the second face in the point designated by 2
and the straight line B^ N normal to the second face at the point A^
ignated by N\ so that B^MN will be the plane of the principal se«
tion of the prism which is passed through the point B^. On th
straight line MN as diameter, describe in the plane of the paper
circle, only half of which is shown in the figure; and in the circunc
ference of this circle take any point -Bj, and draw the straight line
MB^, NB^, ByBz- If the straight line B1B2 represents the path withS
the prism of the chief ray of a bundle of rays, to which corresponds tla
chief incident ray UyBy and the chief emergent ray J?2^2» then ByBfli
will be the plane of incidence at the first face, and B^B^ will be thi
plane of emergence at the second face; and these two planes will b
at right angles to each other, according to the essential conditio
which was found in § 104 above.
In order to construct the chief incident ray tt| and the chief emer
gent ray u^ corresponding to a ray ByB^ (or u\) within the prism, yt
proceed almost exactly as in § 100. First, we revolve the triangle
BiBJSit around the straight line MB^ as axis until it comes into th<
plane of the paper; so that the point B^ falls at a point C in the straight
line NB2 whose real distance from M will depend on the scale of th€
oblique parallel projection. In the figure as here drawn, the real
length of B^N is twice its length as actually shown. With the point
C as centre, and with radius equal to n^- CBjn\, describe in the plane
of the paper the arc of a circle meeting the straight line drawn from B|
parallel to the straight line MC in a point designated by £, and let fl
designate the point of intersection of the straight lines CE and JlfBi;
then the straight line BJ) will give the direction of the chief inddent
ray Wi to which within the prism corresponds the ray ByB^.
Again, revolve the triangle B^NB^ around the straight line NB^ti
J
Refraction Through a Prism or Prism-System.
131
until it comes into the plane of the paper, and let the point des-
ted by 0 be the impression in this plane of the point B^, so that
is the real length of the straight line B^ N, and B^O = B^C; and
a
y
\ \
\
\
V^
1 *
.l\\
1
Vi\ ^•^
\*^,
\>>
e
V
3
8
3
o
3
<
S
o
n
H
the point O as centre and with radius equal to nj-OJJj/ni, describe
e plane of the paper the arc of a circle meeting the straight line
Refraction Through a Prism or Prism-System. 133
be straight line drawn through X[ parallel to the straight line
ill determine by its intersection with the chief incident ray Wj
jject-Point S^ to which corresponds the Homocentric Image-
results of this investigation may be summarized as follows:
Toery incident chief ray that meets the first jace of the prism at a
5| and that is refracted through the prism along a generating line
conical surfcu^e B^c (where c designates the circle described on
raight line MN as diameter — see Fig. 50), there is one single
Point to which on the emergent chief ray there corresponds a
"xntric Image- Point.
. The analytical expression for the position of this unique
t-Point 2)| on such an incident chief ray Ui may be easily obtained
lows:
x)rding to formulae (19) and (21), we have:
* * «i cos «! * * * * ni * '
J put
btain:
7»| cos CKj ni
BJr^ = —7 — 2 ( — • -Si-Si ~ Sj I ;
* ' n, cos ^2X^1 /
„p._nicos'a; n^Si
Wi cos ttj * * n,
ice we find :
ni\cos a, cos tti/ ' * Wj \cos ^2 /
ag IJ^ri = o, we obtain finally:
n^ cos' g, (cos' a^ - cos^ 0^;)
X/i-6*i — / 2 ~* 2 2 2 ' ' »'1» V50/
«! cos Ofj-COS Oj — cos tti'COS ttj
CHAPTER V.
»>•
^^
Pxo. 51.
Rat Incident on a Sphbbtcat. Su&facb.
REFLEXION AND REFRACTION OF PARAXIAL RAYS AT A SPHBRI
SURFACE.
ART. 32. INTRODUCTION. DEFINITIONS, NOTATIONS. STC.
108. In nearly all forms of optical apparatus the reflecting
refracting surfaces are spherical; for a plane may also be regBi
as a spherical surface of infinite radius. In our diagrams the ce
of the reflecting or refracting sphere will be designated by the U
C (Fig. si). The straight line determined by this point C and ano
point M is called the ax\
^^. ' ^ the spherical surface with
spect to the point M,
the point A where the stra
line MC meets the refrac
M (or reflecting) surface is ca
the vertex of the surface ^
respect to the point Jf. ]
dently, aspherical surface
be symmetrical with res
to such an axis, and the p
of the diagram which contains the axis is a meridian section <A
spherical surface.
Consider now an incident ray lying in this plane, and crossiiig
axis, either really or virtually (see § lo), at the point M. If the p
M is situated in front of the vertex A (that is, to the left of A]
in the flgure, the intersection of an incident ray with the axis at
point M will be a "real" intersection; whereas if the point M
beyond A (in the sense in which the incident light is propagated, nH
in our diagrams is represented always as being from left to right),
intersection of an incident ray with the axis at the point M wil
a "virtual" intersection. If B designates the position of the p
where the ray meets the spherical surface, and if on the straight
CB we take a point N in the medium of the incident ray, the angi
incidence, deflned as in § 14, will be Z.NBM = a. In the figure
plane of the paper represents the plane of incidence, and after lefles
or refraction at the point B, the path of the ray will still lie in
plane.
134
} 108.] Reflexion and Refraction of Paraxial Rays. 135
It will be convenient to take the vertex A of the spherical surface
as the origin of a system of plane rectangular co-ordinates; the axis
of the spherical surface, defined as the straight line AC, being taken
as the X-axis, and the tangent to the surface at its vertex A , in the
incidence-plane, being taken as the y-axis. The positive direction oj
ih^ X-axis is the same as the direction which light would pursue if this
line were the path of an incident ray (see § 26). The positive direction
of the y-axis is found by rotating the positive half of the x-axis about
the point A through an angle of 90® in a sense opposite to that of the
motion of the hands of a clock; so that in our diagrams where the
^*axis is represented as a horizontal line with its positive direction
from left to right, the positive direction of the y-axis will be vertically
upwards.
The abscissa of the centre C, which we shall call the radius of the
spherical surface, will be denoted by the symbol r; thus, AC '^ r.
The radius r is positive or negative according as the centre C lies
beyond or in front of the vertex A ; and according as the sign of r is
positive or negative, the spherical surface is said to be "convex" or
concave •
From the incidence-point B draw BD perpendicular to the jc-axis
at the point D; the ordinate h = DB is called the incidence-height
of the ray which meets the spherical surface at the point B, It will
be poative or negative according as the incidence-point B is above
or below the x-axis.
The slope of the ray is the acute angle through which the x-axis
has to be turned about the point M in order that it may coincide in
position (but not necessarily in direction) with the rectilinear path
of the ray. This angle will be denoted by the symbol d; thus, in the
figure LAMB = B. Here, as always in the case of angular magni-
tudes, counter-clockwise rotation is to be reckoned as positive. The
»gn of the angle B may always be determined from the following
• relation:
The acute angle at the centre C of the spherical surface subtended
by the arc il 5 will be denoted by the symbol <p. This angle is defined
as the angle through which the radius drawn to the incidence-point B
nust be turned in order that the straight line B C may coincide with
AC\ thus, tp s= ZBCA. According to this definition, we shall have
always sin ^ » A/r.
136 Geometrical Optics, Chapter V. [ § 109.
From the diagram, we derive at once the following important re-
lation:
a ^ e + (p. (60)
109. From the diagram, also, we obtain easily the following re-
lations:
DM DC+CA + AM ricosip -i) + AM
BM
cos B cos B cos B
In the special case when the point 0} incidence B is very near to the vtrt^^
A of the spherical surface, the angle of incidence a will be corresponds
ingly small, as will be also the angles denoted by B and ^. Now ^
these angles a, 6 and (p are all so small that we may neglect the secor*-^
and higher powers thereof, and write therefore in place of the sin^
of these angles the angles themselves and also put
cos a = cosB = cos ^ = i,
obviously, we obtain in this case BM =^ AM. Under these drcuiB^
stances, the ray MB is called a Paraxial Ray.
A Paraxial Ray is a ray which proceeds very near to the aocis of
Spherical Surface, which, therefore, meets this surface at a point
close to the vertex and at nearly normal incidence; the angles a, B and 4
being all so small that we can neglect the second powers of these angled*
The ray which proceeds along the x-axis is called the axial ra^'
In this chapter, as well as in several chapters following, we shall
be concerned with the special case of paraxial rays only; that is, w^
shall consider only such rays as proceed within a very narrow cylin-
drical region immediately surrounding the axis of the spherical sur-
face which is also the axis of the cylinder. The only part of the spher-
ical surface that will be utilized for reflexion or refraction will be the
small zone which has the vertex A for its summit. We may imagine,
therefore, that, physically speaking, the rest of the spherical surface .
is abolished entirely, or that it is rendered opaque and non-reflecting
by being painted over with lamp-black; or we may suppose that a
screen with a small circular opening is placed at right angles to the
axis with the centre of the opening on the axis just in front of the ver-
tex A of the spherical surface, so that only such rays as proceed throu^
this opening and are incident on the spherical surface at points very
close to the point A will undergo reflexion or refraction at the spheri-
cal surface.
J
S 110.]
Reflexion and Refraction of Paraxial Rays.
137
I. Reflexion of Paraxial Rays at a Spherical Mirror.
A&T. 33. CONJUGATE AXIAL POINTS IN THE CASE OF RELEXION OF PAR-
AXIAL RATS AT A SPHERICAL MIRROR.
110. In the accompan3ang diagrams (Figs. 52 and 53) the axis of
the spherical mirror is shown by the straight line MC, The straight
line MB represents an incident paraxial ray meeting the spherical re-
flecting surface at the point B. In Fig. 52 the spherical surface is
convex, and in Fig. 53 it is concave. At the point B the ray is reflected
-ff/^/V
tk»-
Fio. 52.
Rbfuezion of Paraxial Rays at a Sphbrxcal Mirror. Convex Mirror.
AAf^'u, Ahr='u\ AC^r, ^F=W2=— /. FM'^x, FM' = x',
IAMB'*: ZAJtrB'^r, IBCA^^. I NBM-^ IWBN=^,
» a direction BW, such that Z NBM = Z WB N, where BN is the
'^^^'nial to the surface at the point B drawn in the medium of the inci-
^t and reflected rays. Designating by JIf ' the point where the re-
"^cted ray crosses the axis, either really (as in Fig. 53) or virtually
(as in Fig. 52), let us denote by the symbols u and u' the abscissae,
*ith respect to the vertex A as origin, of the two points M and M'
I 'iere the ray crosses the axis before and after reflexion, respectively;
thus AM = u, AM' = u\ Also, as in § 108, AC = r.
Since the normal B N bisects the (interior or exterior) angle at B
of the triangle MBM'^ we have:
CM M'C.
BM" BM''
and since the point B is very close to A^ this proportion may be
written:
CM^ M'C
AM" AM''
irhere, as we saw above (§ 109), magnitudes of the second order of
138
Geometrical Optics, Chapter V.
[§1
smallness are neglected/ Now
CM=^ CA + AM --u-r, M'C = M'A + AC^r-u';
and, therefore,
w — r r — w
u
W
or
u u' r
c«
Thus, knowing the mirror as to both size and form (which means tl
we know both the magnitude and sign of r), and being given the pc
tion of the point M where the ray crosses the axis before reflexioii
the mirror, we can determine the abscissa of the point Af' where 1
ray crosses the axis after reflexion.
According to formula (6i), any paraxial ray which crosses the a
before reflexion at the point M, will cross the axis after reflexioa
«^
X
Pxo. 53.
Rbflbxion of Paraziax. Rats at a Sphsrxcai. Mduior. Concave Mirror.
AM^u, AM' = u\ AC=r, AF'^fl2='—A FM=^x, FM'^x',
IAMB=-^, lAArB^9\ l.BCA"^, L NBM '•^ I WEN ^^ ^
the point M', Thus, a homocentric bundle of paraocial rays incii*
on a spherical mirror remains homocentric after reflexion at the mirr
According as the sign of the abscissa u' is positive or negative, t
point M' will lie to the right or left of the vertex A. In the foni
case the image at M' is a virtual image (Fig. 52), whereas in the latt
case we have a real image at M' (Fig. 53); see § 44.
Moreover, since the formula is symmetrical with respect to tt audi
so that the equation remains unaltered when we interchange the letti
u and u'f it follows that if Jkf ' is the image of Af , then M will also
* In writing this proportion, we must be careful that the two memberB of It ifaall h
like signs. Thus, in the diagrams, as here drawn, CM and AM have the same dbectk
80 that for these diagrams the ratio CM/ AM is positive. Hence, if the t^aoItClA
is equal to CM /AM, it must be positive also; that is. M'C and AM' must Ukeviie b
the same directions.
111.] Reflexion and Refraction of Paraxial Rays. 139
he image of M'l which is merely, of course, an illustration of the
reneral law of Optics known as the Principle of the Reversibility of the
;^ight-Path (§ 1 8). But the symmetry of the equation implies more
than is involved in this principle. It indicates also that, in the case
of Reflexion, Object-Space and Image-Space coincide completely : the
paths of the incident and reflected rays both lying in front of the
mirror; so that an Object- Ray and an Image- Ray are always so re-
lated that when either is regarded as the Object-Ray, the other will
be the corresponding Image-Ray.
The magnitudes denoted by u and u' are the radii of the incident
and reflected wave-fronts at the moment when the disturbance arrives
at the vertex A of the mirror; and hence the relation given by formula
(6i) may also be expressed as follows: The algebraic sum of the curva-
iwes of the incident and reflected waves at the instant when the disturb-
anu arrioes at the vertex of the spherical mirror is equal to twice the
cvmiure of the mirror.
The convergence of paraxial rays after reflexion (or refraction) at
^spherical surface is said to be a "convergence of the second order";
whidi means that the second and higher powers of the incidence-angle
« are neglected. When we neglect magnitudes of this order, the spheri-
cal surface will coincide with every surface of revolution which has
Sesame curvature at the vertex; so that the formula (6i) applies to
Ae reflexion of paraxial rays at a surface of revolution of any form,
vhere r denotes the radius of curvature at the vertex of a meridian
action of the surface.
111. Since CM.AM^ M'C:AM\ it follows that
CM^ AM- _
CM'' AM'" ^'
"*^t is, the anharmonic or double ratio of the four axial points C, A,M
andJIf'is
{CAMM') = - i; (62)
f consequently, the points C, A, M, M' are a harmonic range of points,
^ Object-Point M and its Image-Point M' being harmonically sepa-
rated by the centre C and the vertex A of the spherical mirror. Ac-
^^^ngly, we have the following simple construction of the Image-
Faint M' due to the reflexion at a spherical mirror of paraxial rays
emanating originally from an axial Object-Point M:
On any straight line, supposed to represent the axis of the spherical
mirror, take three points ^4, C, Af (Fig. 54), ranged along the line in
140 Geometrical Optics, Chapter V. [ § \\'
any order whatever; the letters A and C designating the positioi
of the vertex and centre, respectively, of the spherical mirror, and tl
letter M designating the position of the given axial Object-Point. C
any other straight line drawn through the point M take two points
and J; and draw CH and A J intersecting in a point P and AH 3:
CJ intersecting in a point Q. The straight line connecting the poit
P and Q will meet the axis in the Image-Point M' conjugate to t
Object-Point JIf. In making this construction a straight-edge is t
only drawing-instrume
that will be needed. T
proof of the constnicti
is obtained at once frc
the complete quadrani
A CHJ. If the points
and C in the diagram 2
interchanged, the poii
M and Af ' will evideni
Reflexion of Paraxial Rats at a Spbbbjcal be a pair of COnjUgS
MiRXOR. Construction of Conjugate Axial Points M, NT, r>^\r\\ts. olcsn Tini-ti rocrKvH-
Centre of Mirror at C vertex at A. pomxs aiSO WlUl rCSpCCl
this new spherical surfa<
thus, if the points if , M' are a pair of axial conjugate points wi
respect to a spherical mirror with its centre at C and its vertex
Ay these same points will be conjugate to each other with respect
a spherical mirror of the opposite kind with its centre at A and :
vertex at C
112. Focal Point and Focal Length of Spherical Mirror. In t
special case when the axial Object-Point M coincides with the infinite
distant Object-Point £ of the x-axis, the conjugate point M! will
this case be situated at a point £^ such that
AE
{CAEE')^-^,^^xx
that is,
AE = E'C,
Hence, a cylindrical bundle of incident paraxial rays parallel to t
axis will be transformed by reflexion at a spherical mirror into a hoffl
centric bundle of rays with its vertex at a point E lying midw
between the vertex and centre of the mirror.
On the other hand, if the Image-Point M' coincides with the :
finitely distant Image-Point F' of the x-axis, the corresponding C
{ 112.] Reflexion and Refraction of Paraxial Rays. 141
]ect-Point M will be situated on the axis at a point F, such that
{CAFF) =-;^= - i;
or
AF = FC\
that is, this point F, which is the vertex of a bundle of incident par-
axial rays to which corresponds a cylindrical bundle of reflected rays
ail parallel to the axis, is also situated midway between the vertex
and the centre of the spherical mirror. Thus, in the case of a spheri-
cal mirror the two points F and E' are coincident.
The points designated by F and E' are called the Focal Points of
the optical system.
The Focal Length of a Spherical Mirror may be defined as the
absdssa of the vertex A with respect to the Focal Point F\ thus, if
the Focal Length is denoted by the symbol /, we have:
FA^f^^-r/z.
If the abscissae, with respect to the Focal Point F, of the conjugate
axial points Af, M' are denoted by x, x\ respectively; that is, if we put
FM^x, FM'-=x\
^ we have at once:
u = x-f, u' = x'-f,
^ substituting these values in formula (6i), we obtain:
xx'=/; (63)
a naost convenient and simple form of the abscissa-relation of conjugate
^ points, which contains the whole theory of the reflexion of par-
*^ rays at a spherical mirror.
According as the Focal Length / is positive or negative, the mirror
*s convex or concave. Thus, in a concave mirror the Focal Point F
"^ in front of the mirror, so that incident paraxial rays parallel to the
^ will be converged by reflexion at a concave mirror to a real focus
^^ f\ whereas in a convex mirror the Focal Point F lies beyond the
'^or (to the right of the vertex ^4), so that a bundle of incident par-
^ rays which are parallel to the axis will be transformed by reflexion
at a convex mirror into a bundle of rays diverging as if they had come
irom a virtual focus at the point F.
Whether the mirror is convex or concave, and whether the bundle
^ incident rajrs is convergent or divergent, the conjugate cLxial points Jlf,
i
142 Geometrical Optics, Chapter V. [ § H
M' lie always on the same side of the Focal Point of the Spherical Mirrc^.
as is readily seen from formula (63).
ART. 54. EXTRA-AXIAL CONJUGATE POINTS AND THE LATERAL MAGET
FICATION IN THE CASE OF THE REFLEXION OF PARAXIAL
RATS AT A SPHERICAL MIRROR.
113. Graphical Method of Showing the Imagery by Paraxial Rajn
Let Jlf, Jkf ' designate the positions on the axis of a spherical mirror of
pair of conjugate points, constructed according to the method given 1:
§111; and connect both of these points by straight lines with a poin
V on the surface of the reflecting sphere. In the plane of these line
draw Ay tangent to the sphere at its vertex A, and let B and G desig
nate the points where the straight lines MV, M'V meet the straigh
line Ay. Also, join the point B with the point M' by a straight line
If the point V were very close to the vertex i4, then the straight line
MV would be the path of an incident paraxial ray proceeding from
My and the path of the corresponding reflected ray would be VM', In
this case, however, the points designated here by the letters F, B and
G would all be so near together that, even when we cannot regard V
as coincident with A, we can regard F, B and G as coincident with
one another; and therefore we might take the straight line BM' as
representing the path of the reflected ray.
In the construction of diagrams exhibiting the procedure of paraxial
rays a practical difficulty is encountered due to the fact that, whereas
in reality such rays are comprised within the very narrow cylindrical
region immediately surrounding the axis of the spherical surface, it is
obviously impossible to show them in this way in a figure, because we
should have to take the dimensions of the flgure at right angles to the
axis so small that magnitudes of the second order of smallness in such
directions would no longer be perceptible. On the other hand, if we
were to represent these magnitudes as larger than they actually atei
the relations which we have found above would no longer be true ia
the case of such lines; thus, for example, the rays in the drawing wouM
not intersect in the places demanded by the formulae.
In order to overcome this difficulty, Reusch suggesi|:ed a method of
drawing these diagrams which has been very generally adopted, ana
which in large measure is entirely satisfactory. Without altering d*
dimensions parallel to the axis, the dimensions at right angles to th^
axis are all magnified in the same proportion. Thus, for exampte
if the ordinate h = DB (Fig. 51) is a magnitude of the order i/i, it*
shown in the figure magnified k times; whereas an ordinate of magfli'
Reflexion and Refraction of Paraxial Rays. 143
order i /ife*, that is, of the second order as compared with A,
^ magnified diagram would be shown as a magnitude of the
so that if k is infinite, such ordinates as A, which are of the
of smallness, will be shown in the figure by lines of finite
sreas magnitudes of the second order of smallness will dis-
ipletely in the magnified diagram.
2, one effect of this lateral enlargement will be to misrepre-
le extent the relations of the lines and angles in the figure,
sample, the circle in which the spherical surface is cut by the
leridian section will thereby be transformed into an infinitely
ellipse with its major axis perpendicular to the axis of the
irface, that is, into a straight line Ay tangent to the circle
X A. The minor axis of this ellipse remains unchanged and
le diameter of the circle, and, moreover, the centre of the
ains at the centre C of the circle. The most apparent
I be in the angular magnitudes which will be completely
Tius, for example, every straight line drawn through the
iaily meets the circle normally, but in the distorted figure
II be the only one of such lines which meets normally the
le which takes the place of the circle. Angles which in
equal will appear unequal, and vice versa. However — and
11 is the really essential matter — the relative dimensions of
;es and the absolute dimensions of the abscissae will not be
all; and, therefore, lines which are really straight lines will
traight lines in the figure, and straight lines which are really
[1 be shown in the figure as parallel straight lines. The
the point of intersection of a pair of straight lines as it
the figure will be the real abscissa of this point.
a figure, therefore, any ray, no matter what slope it may
low far it may be from the axis, is to be considered as a
y. The meridian section of the spherical surface will be
I in the figure by the straight line Ay (the y-axis), and the
the centre C with respect to the vertex A will show whether
is convex or concave.
we suppose that the axis of the spherical mirror is rotated
:entre C through a very small angle MCQ so that the axial
oves along the infinitely small arc of a circle to a point Q^
ite axial point Af' will likewise describe an infinitely small
icentric circle, and will determine a point Q' on the straight
I Q with C, such that if U designates the point where the
e QC meets the spherical surface, the points Q, Q' will be
144
Geometrical Optics, Chapter V.
[§115.
harmonically separated (§ iii) by the points C, U; that is, (CUQQ^
= — I. Thus, the point Q' is evidently the image-point conjugate
to the Extra-Axial Object-Point Q. If the Object-Points lie on the
element of a spherical surface which is concentric with the reflecting
sphere, the corresponding Image-Points will likewise be found on an
element of another concentric spherical surface, and any straight line
going through the centre C will determine by its intersections with
this pair of concentric surfaces, of radii CM and CM\ a pair of con-
jugate points such as Q, Q\ If, as we assume here, the angle MCQ
is infinitely small, the arcs MQ^ M'Q' may be regarded as very short
straight lines perpendicular to the axis at JIf, JIf', respectively. Ac-
cordingly, on the supposition that the only rays concerned in the pro-
duction of the image are such rays as meet the reflecting surface at
very nearly normal incidence, the following conclusions may be drawn:
(i) The image t in a spherical mirror , of a plane object perpendicukf
to the axis is likewise a plane perpendicular to the axis; (2) A slrai^
line parsing through the centre of the spherical mirror intersects a pair^
such conjugate planes in a pair of conjugate points; and (3) To a homO'
centric bundle of incident paraxial rays proceeding from a point Qi»^
plane perpendicular to the axis
of the spherical mirror Hun
corresponds a homocentric iutt
die of reflected rays with itt
vertex Q lying in the conjufi^
Image- Plane.
115. In order to constmd
the Image- Point Q^ of t^
Extra-Axial Object-Point 0»
we have merely to find the
point of intersection after t^
flexion at the spherical mifl*
of any two rays emanatiflf
originally from the point G-
The two diagrams (Figs. 55
and 56), which are drawn ac-
cording to the method dc
scribed above (§ 113), exhibit this construction in the case of both*
concave and a convex mirror. Of the incident rays proceeding fro*
Q it is convenient to select the following pair for this construction'
the incident ray QC which proceeding towards the centre of tl*
mirror C meets the spherical surface normally at the point Uf wbenc^
C^fr90g Mirror
Fio. 55 and Fio. 56.
Reflexion of Paraxial Rays at a Spherical
Mirror. Construction of point (/ conjusate to
extra-axial Object-Point Q. In the diasTams the
meridian section of the mirror is represented by a
straight line Ay perpendicular to the axis of the
mirror at the vertex A. The straight line Af(/ per-
pendicular to the axis is the imasre of the straight
line AfQ also perpendicular to the axis.
i6.] Reflexion and Refraction of Paraxial Rays. 145
is reflected back along the same path, and the incident ray QV
lich proceeding parallel to the axis and meeting the mirror in the
int designated by V is reflected at V along the straight line joining
with the Focal Point F, The Image-Point Q' will be the point of
tersection of this pair of reflected rays. Moreover, having located
le position of Q', we can draw QM and Q'M' perpendicular to the
ds at Jf and M\ respectively; then M'Q' will be the image of the
laight line MQ perpendicular to the axis at Af. In Fig. 55 the case
shown where the image M'Q' is real and inverted; whereas in Fig.
6 the image M'Q' is virtual and erecL Whether the image is real or
irtual and erect or inverted will depend on the position of the object
ith respect to the mirror as well as on whether the mirror is convex
r concave.
116. The Lateral Magnification. If the ordinates of the pair of
to-axial conjugate points Q, Q' are denoted by y, y', respectively,
hat is, if MQ = y, M'Q' = y', the ratio y'/y is called the Lateral
^(ipiification at the axial point M. This ratio will be denoted by Y;
hus, F = y/y. The sign of this function Y indicates whether the
inageis erect or inverted; if Y is positive, as in Fig. 56, the image
rill be erect; whereas if Y is negative, as in Fig. 55, the image will
« inverted. The absolute value of Y depends on the relative heights
f the object and its image; it will be greater than, equal to, or less
^, unity, according as the height of the image is greater than, equal
^ or less than, that of the object. A very simple investigation shows
ow F is a function of the abscissa of the axial point M. Since the
"angles MCQ, M'CQ' (Figs. 55 and 56) are similar,
M'Q':MQ-=M'C:MC;
ikI since
M'C=-r-u', MC^r-u,
^by formula (61), we have:
r ^ u' u'
r — u u
^ derive the following formula for the Lateral Magnification in the
^ of a Spherical Mirror :
y' v!
F = 5L=«!L. (64)
y u ^
% in case we wish to obtain F as a function of the abscissa x
n
146
Geometrical Optics, Chapter V.
1 1 117.
(* = FM), we obtain from the diagrams directly:
M'Q' Wq FM'
MQ
AV
FA
and putting FM' = *', FA =/, and using also formula (63), we haw:
(65)
which of course is likewise easily deducible from (64).
Either of the two pair of formulae (61) and (64) or (63) and (65)
determine completely the Imagery in the case of the Reflexion of
Paraxial Rays at a Spherical Mirror.
117. If the axial Object-Point M is supposed to travel along tbe
axis of the spherical mirror, and if at the same time the point Q is
supposed to travel with an equal velocity along a line parallel to tbe
axis, the corresponding manoeuvres of the image M'Q' will be easily
perceived by an inspection of the diagram (Fig. 57), which shows tbe
* *
Fio. 57.
Reflexion of Paraxial Rays at a Spherical Mirror. The numerals 1. 2. 3. ctc^
from left to ri^ht alonff a straight line parallel to the axis of the mirror indicRte the
positions of an object-point, and the numerals l'. 2^, 3'. etc.. show the cmiCBDonding poiltlaMi' "
the imaffe-point ranged along the straight line VF. The case shown in the fisure is lor a OmOM j*
Mirror. The straight lines 11'. 22f, 33'. etc.. all intersect at the centre Cof the mlrrar. If At
object-point is virtual (as at 7 or 8). the image in a concave mirror will be real.
case of a concave mirror with its Focal Point F in front of the nurror.
Let us suppose that the Object moves from left to right starting froa ^
an infinite distance in front of the mirror. The numerals I, a, 3, etc^
are used to designate a number of successive positions of the Object- .
Point Q, whereas the same numerals with primes show the correspond*
L8.] ^ Reflexion and Refraction of Paraxial Rays. 147
; positions of the Image-Point Q'. Evidently, all the straight lines
\ 22', 33', etc., will pass through the centre C of the mirror. So
\% as the Object MQ lies in front of the Focal Point F the image
''C in the concave mirror is real and inverted. As MQ advances
)wards the centre C, the Image M'Q' proceeds between F and C also
wards the centre C, and Object and Image arrive together in the
lane perpendicular to the axis at C, the Image being then of the same
ize as the Object, but inverted. As the Object proceeds past C
awards F, the real and inverted Image proceeds in the opposite direc-
ion towards infinity; so that when the point Q arrives at the point
marked 4 in the Focal Plane, the point Q! is the infinitely distant point
)f the straight line VF. As the Object continues its journey from the
Focal Point F towards the vertex A of the mirror, the Image, which is
WW virtual and erect, travels from infinity towards the vertex i4,
ind Object and Image arrive together at the vertex and coincide with
5adi other there. If the Object proceeds beyond the vertex, we shall
bve then the case of a virtual Object, to which therie corresponds a real
srect Image lying between the vertex A and the Focal Point F. The
'roage, it will be observed, travels always in a direction opposite to
that taken by the Object; which is a characteristic property of re-
'^on. Moreover, it will be noted that Object and Image lie always
Dn the same side of the Focal Plane.
n. Refraction of Paraxial Rays at a Spherical Surface.
^•15. CONJUGATE AXIAL POINTS IN THE CASE OF THE REFRACTION
OF PARAXIAL RATS AT A SPHERICAL SURFACE.
U8. In the diagrams (Figs. 58 and 59) the plane of the paper repre-
*nts the meridian section of a spherical refracting surface separating
i^o isotropic optical media of absolute refractive indices n and n'.
'iRg. 58 the centre C lies in the second medium (n'), so that the
pherical surface is convex; whereas in Fig. 59 the centre C lies in
l^e first medium (n), and the spherical surface is concave. The axis
f the refracting sphere is the straight line xx which joins the centre C
ith the vertex A. The letters in these figures have the same mean-
gs as in the corresponding diagrams for the reflexion of paraxial
ys at a spherical mirror.
An incident ray meeting the spherical surface will be refracted in
iirection such that, if a and a' denote the angles of incidence and
Faction, then, by the Law of Refraction :
n-sin a = n'-sin a'.
148
Geometrical Optics, Chapter V.
[§118.
If Mj M' designate the points where the ray crosses the axis before and
after refraction at the spherical surface, and if BN is the normal to
Pio. 58 and Pio. 59.
RSFRACTION OF PARAXIAL RAYS AT A SPHERICAL SXTRPACB 8BPA&ATXNO TWO MB0I* ^
IMDXCBS n. n\
AM-^u, AAr=-y^, AC=-r, FA^'A JE'/f=/, FM^x, ^Af-*'. DB^k,
LNBM^^, LCBAf='^\ JLAArB = 9, JLAAtB='9^, LBCA^^,
the spherical refracting surface at the incidence-point B drawn frtw**
B into the first medium, then in the figures:
LNBM^a, LCBM' -^a'\
also if tp denotes the angle subtended at the centre C by the arc 13%
then, according to the definition of tp given in § io8, ABC A = ^.
In the triangles MBC and M'BC we have:
CMiBM = sina:sin^, CM'iBM' «= sina':8in^,
and, hence, dividing one of these equations by the other, we obtain:
CM BM ^n'
CUf'BM'" n'
Since the incidence-point B is supposed to be so near A that we cu
neglect magnitudes of the second order of smallness, we may write A
in this equation in place of B; and thus we obtain for the refiractioft
of a paraxial ray at a spherical surface:
Cilf AM ^n'
CM'' AM'" n'
Reflexion and Refraction of Paraxial Rays. 149
{CAMM') = -; (66)
w
s, The Double {or Anharmonic) Ratio of the four aocial points C, i4,
V is constant, and equal to the relative index of refraction of the two
r
r*
US, for a given spherical refracting surface, the axial point M'
spending to a given axial point Jlf is a perfectly definite point,
.ccordingly we derive the following result:
o homocentric bundle of incident paraxial rays with its vertex lying
e axis of the spherical refracting surface there corresponds also a
centric bundle of refracted rays with its vertex lying on the axis.
MS, if M designates the position of an axial Object-Point, its
e produced by the refraction of paraxial rays at a spherical sur-
will be at a point M' on the axis. In Fig. 58 we have at M' a real
je of the Object-Point M; whereas in Fig. 59 the image is virtual,
four points Jlf , M', A and C may be ranged along the axis in any
a* whatever, depending on the form of the spherical refracting
ace and on whether n is greater or less than n\ If the incident
J converge towards a point M lying on the axis beyond (or to the
tof) the vertex A, the point M will be a virtual Object-Point;
in this case, as in all cases, the corresponding Image-Point M' can
oimd by formula (66).
[oreover, if {CAMM') =^ n'/n, then also (CAM'M) = n/n\
s, if a ray proceeding from an axial point M in the first medium
ses the axis after refraction at the spherical surface at the point
in the second medium (see § 10), then also a ray proceeding from
point M' in the second medium will be refracted at the spherical
ice so as to cross the axis at the point M, This is in accordance
the general Principle of the Reversibility of the Light- Path (§18).
' b the image of Af , M will be likewise the image of M\
D. Construction of the Image-Point M' conjugate to the Axial
Ct-Point M. The following is a simple method of constructing
mage-Point M' corresponding to an Object- Point M lying on the
)f a spherical refracting surface. Through the centre C draw any
;fat line, and take on it two points so situated that their distances
C are in the ratio »' : ». Instead of drawing any straight line
gh C, it will be convenient to take this line perpendicular to the
it C, as is done in Fig. 60. Let G, G' designate the positions on
ine of two points whose distances from C are such that we have:
150
Geometrical Optics, Chapter V.
[§120.
CG : CG' = n' :n. Through the vertex A of the spherical refracting
surface draw the straight line Ay parallel to the straight line dravrn
through C; if this latter is perpendicular to the axis, the straight line
Fio. 60. .
Refraction of Paraxial Rays at a Spherical Sxjrfacb. Coiistmction of axial V^^f_^
conjugate to axial Object-point AT. Cia centre. A is vertex and xx axia of Spherical RefndiC^^
Surface. CC^'C^ris perpendicular to axia; CG:CG'='n*:n, ria infinitely diatant point of /«d^
Ay will be tangent to the spherical surface at the point A. Jointlie
axial Object-Point M with the point G by a straight line, and let Y.
designate the point where this line meets Ay; then the straight line
joining the points Y^ and G' will meet the axis in the required point if.
For, evidently, the point Y^ is the centre of perspective of the two
projective point-ranges C, -4, Af, ilf' and C, T, G, G', where T desig-
nates the infinitely distant point of the straight line which intersectB
the axis at C; and since, by construction,
_^CG^ TG _CG n^
it follows that we must have:
{CAMM') = — »
in accordance with formula (66).
120. The Focal Points F and E' of a Spherical Refracting Stufice.
Evidently, the vertex A and the centre C of thfe spherical refracting
surface are two self-corresponding points of the two projective rangv
of Object-Points and Image-Points lying along the axis. Let us dis-
tinguish these two ranges of corresponding points by the letters x and
x', and let E and F' designate the infinitely distant points of x and Jt',
respectively. Thus E is the infinitely distant axial Object-Point and
F' is the infinitely distant axial Image-Point. In order to find the
Image-Point £' conjugate to the infinitely distant Object-Point B^
J
§120.] Reflexion and Refraction of Paraxial Rays. 151
we must draw through the point G (Fig. 60) a straight line parallel to
the axis meeting the straight line Ay in the point designated by F,;
and then the straight line Yfi' will determine by its intersection with
the axis the required point E\ Similarly, in order to find the position
oa the axis of the Object-Point F corresponding to the infinitely dis-
tant Image-Point F', we must draw through the point G' a straight
line parallel to the axis and meeting the straight line Ay in a point Y/,
and then the straight line Gly. will determine by its intersection with
the axis the required point F.
Thus, a paraxial ray which before refraction is parallel to the axis
of the spherical surface will, after refraction, cross the axis (really or
virtually) at the point designated by £'; and, also, a paraxial ray
which before refraction crosses the axis (really or virtually) at the
point designated by F will, after refraction, be parallel to the axis.
These points F and E' are called the Focal Paints; the point F is
called the Focal-Point of the Object-Space or the Primary Focal Point,
and the point £' is called the Focal Point of the Image-Space or the
I Secondary Focal Point. The two Focal Points of an optical system
I are always of the highest importance.
j A mere inspection of the diagram (Fig. 60) shows that the Focal
Points F and E' of a spherical refracting surface are situated so that
FA = CE\ E'A = CF\ (67)
and, hence, we have the following rule:
The Focal Points of a Spherical R^racting Surface are so situated
an the axis that the step from one of them to the vertex A is identical
wiih the step from the centre C to the other one.
This result may also be stated in a different way; for, since
FA = CE' = Ci4 -h AE\
we have also the following relation :
AF+AE'-^ AC; (68)
that is, The algebraic sum of the distances of the Focal Points from the
vertex of the spherical refracting surface is always equal to the distance of
ike centre from the vertex.
Another useful relation, obtained from the two similar triangles
AYf Fand AY^ E' is the proportion:
FA CG'
AE' " CG '
152 Geometrical Optics, Chapter V. [ { 12
or
which may be put in words as follows: The two Focal Points Fand E^
of a spherical refracting surface lie on opposite sides of the vertex^ and (^t
distances from it which are in the ratio n : »'.
The answer to the question, Which of the two Focal Points lies i«
the first medium, and which in the second medium? will depend on
each of two things, viz.: (i) Whether the spherical surface is convex
or concave, and (2) Whether »' is greater or less than n. Thus, lor
example, if the rays are refracted from air to glass {n'/n = 3/2), we
find from formulae (68) and (69) AF = iCA, AE' = ^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
^^ "^"^
ZQT
Pzo. 61.
Refraction op Paraxiai. Rats at a Spsb&icai. Surfacb. Constmcdoii of ttie VooA. Fol>>*
FsaoABf, In I and U the rays are refracted from air to glass. In III and I V the rays are icCractBi
from fflass to air. In I and ni spherical refracting surface is convex. In II and IV sphcffcil
refracting surface is concave. In I and IV incident rays parallel to the asda aze ouuteifBd to'
real focus at £f ; whereas in II and III ^ is a virtual focus.
the step A C three times, we can locate the Secondary Focal Point &•
The two diagrams I and II (Fig. 61) show the positions of the Focal
Points in the case when the rays are refracted from air to glass at *
convex and at a concave spherical surface. It will be seen that for
this case the Primary Focal Point of the concave surface lies in tie
second medium (virtual focus), whereas the Primary Focal Point of
the convex surface lies in the first medium (real focus). On the othtf
hand, in case the rays are refracted from glass to air {n'jn =■ 2/3)1
we have AF = ^^AC, AE' = iCA, and now the Primary Focal Pij^*
of a convex spherical refracting surface will lie in the second mediu*
and the Primary Focal Point of a concave surface will lie in the fir*
medium, as is shown in the diagrams III and IV (Fig. 61).
22.] Reflesuon and Refraction of Paraxial Rays. 153
r. 36. REFRACTION OP PARAXIAL RATS AT A SPHERICAL SURFACE.
EXTRA-AXIAL CONJUGATE POINTS. CONJUGATE PLANES. THE
FOCAL PLANES AND THE FOCAL LENGTHS.
121. To an Object-Point Q lying not on the axis, but very near to
evidently there will correspond an Image-Point Q' lying on the
aight line joining Q with the centre C of the spherical refracting
rface, the position of which is determined by the equation
(CUQQ') = »7n,
here U designates the point where the self-corresponding ray QQ'
aeets the spherical surface. Employing here exactly the same reason-
ng as was used in § 1 14 in the similar case of Reflexion at a Spherical
Mirror, we may copy verbatim the results which were obtained there,
merely changing the words ''mirror", "reflexion", etc., to adapt the
statements to the case of refraction at a spherical surface. Thus:
(i) The image of a plane object perpendicular to the axis of a spherical
Tefracling surface is likewise a plane perpendicular to the aocis; (2) A
^aight line drawn through the centre of the spherical refracting surface
^intersect a pair of such conjugate planes in a pair of conjugate points;
^ (3) To a homocentric bundle of incident paraonal rays proceeding
frm a point Q in a plane perpendicular to the oms of the spherical
^fffdcting surface there corresponds a homocentric bundle of refracted rays
^ Us vertex Q' lying in the conjugate Image- Plane.
122. The Construction of the Image-Point Q' Corresponding to the
Kxtia-Axial Object-Point Q may be performed also by a process pre-
Qsdy similar to that used in § 115. Thus, in the diagrams (Figs. 62
^63), which are drawn according to the plan explained in § 113, the
points designated by the letters A and C represent the vertex and
^tre, respectively, of the spherical refracting surface. In Fig. 62
"* surface is convex, and in Fig. 63 it is concave. If the positions
^ the Focal Points F and E' are not assigned, they can be determined
^^^y by the relations given in formulae (68) and (69). Both of the
^"^rams show the case when n' is greater than ».
Tke incident ray proceeding from the point Q towards the centre
5^1 meet the spherical surface normally and will continue its route
^to the second medium without change of direction. Thus, as was
"^^ also above, the corresponding point Q' must lie on the straight
^joining Q with C. To the incident ray QV proceeding from the
Object-Point Q parallel to the axis and meeting the straight line Ay
^ the point V there corresponds a refracted ray which passes (really
* virtually) through the secondary Focal Point £'. Thus, the Image-
154
Geometrical Optics, Chapter V.
[§12.
Point Q' will be at the point of intersection of the straight lines Q
and VE.
The intersection of any pair of refracted rays emanating originall
from the Object-Point Q will determine the position of the Imag»
Point Q'. Thus, for example, instead of one of those used above, kp
Cotfcavt Sytfmf
Fio. 62 and Fxo. 63.
Rbfraction op Paraxial Rats at a Sphbricai. Surpacb. Conatmctioii of Image-Foin^ ^
correspondinff to extra-axial Object-Point Q, The points A and C designate the positioai d ^
vertex and centre of the spherical refracting surface, and FwaA Ef designate the positioHflf ^
Focal Points. In Pig. 62 the surface is convex, in Pig. 63 it is concave ; for both diagxami ■'^ '
In Pig. 62 NtQf is a real, inverted image of MQ\ whereas in Pig. 63 the image is virtual and eicc
might have employed the ray which proceeding from the Object-Potf*
Q towards the primary Focal Point F and meeting the straight line Al
in the point designated in the diagrams by W is refracted paralld t<
the axis of the spherical surface.
If My M' designate the feet of the perpendiculars let fall from ft ^
respectively, on the axis, then M'Q' will be the image, by paraxi^
rays, of the infinitely small straight line MQ, In Fig. 62 this imap
is real and inverted, whereas in Fig. 63 it is virtual and erect.
123. The Focal Planes of a Spherical Refracting Suxface. If tb^
Object-Point Q is the infinitely distant point of the straight line Q^
(Fig. 64), it will be a point of the infinitely distant plane of the Object-
Space to which is conjugate a plane perpendicular to the axis at4<
Focal Point £' of the Image-Space. This plane is called the Focal
Plane of the Image-Space or the Secondary Focal Plane. Its trace io
the plane of the paper (which shows a meridian section of the sphcricil
surface) is the straight line e' which we may call the secondary Focal
Line. Thus, we can say:
To a bundle of parallel incident paraxial rays there corresponds^
§124.]
Reflexion and Refraction of Paraxial Rays.
155
Jurntocentric bundle of refracted rays with its vertex lying in the secondary
foc^ plane of the spherical refracting surface.
Similarly, the plane perpendicular to the axis at the primary Focal
Point F is called the Focal Plane of the Object-Space or the Primary
^ r 4
Pxo. 64 (a).
iLKFmAcnoN OP Paraxial Rays at a Spherical Surface. Incident Parallel Rays intersect
•iter refrKtion in a point of the focal plane of the Image-Space, the trace of which in the plane of
tbe paper is the focal line /.
Focal Plane, and its trace in the plane of the paper (Fig. 64) is the
straight line/, which we may call the Primary Focal Line in the plane
of this meridian section. The Image- Plane conjugate to the Primary
Focal Plane is the infinitely distant plane of the Image-Space; and,
hence, if the Object-Point Q lies in the Primary Focal Plane, the cor-
Fio. 64 {b),
^"'Uctioh of Paraxial Rays at a Spherical Surface. Incident Rays emanatinsr from a
•^itof the Focal Plane of the Object-Space (the trace of which in the plane of the paper is the
'^ Uae /) are made parallel by refraction.
'Ending Image-Point (^ will be the infinitely distant point of the
«raight line Q C. Thus:
To a homocentric bundle of incident paraxial rays, with its vertex
¥ni in the Primary Focal Plane of the spherical refracting surface,
^e corresponds a bundle of parallel refracted rays,
124. The Focal Lengths / and e' of a Spherical Refracting Surface.
In Fig. 65 the points designated by Af, M' are the points where a
paraxial ray crosses the axis, before and after refraction, respectively.
156
Geometrical Optics, Chapter V.
[§12
at a spherical surface, and the point B is the incidence-point of thS
ray. The vertex of the spherical surface is at the point marked i"^,
and the Focal Points are at F and E', Let
ZAMB^e, ZAM'B^e',
where 6, 6' denote the slope-angles (§ io8) of the ray before and aft^
refraction, respectively. Through the Primary Focal Point F dr^w
FK' parallel to the incident ray MB and meeting the straight li«c
Ay in the point designated by X', and through the Secondary Focral
Point £' draw GE' parallel to the refracted ray BM' and meeting ^J
Pio. 65.
Refraction of Paraxiai. Rays at a Sphbrxcai. Sxtrfacb. The focal Irngthi off the q»heri^
refracting surface are :
tantf tan^
where FR — g, EfS' * H denote the intercepts on focal planes of incident ray MB and cotif^H**^
infir refracted ray BM* , and Z AMB - •. Z AHitB « ^ are the slopes of incident and i ef ractei n^^
in the point designated by G. Through the points G and K' drai/'
straight lines parallel to the axis of the spherical refracting surface;
the former meeting the incident ray MB in the point designated by
i?, which is the Object-Point corresponding to the infinitely distant
Image-Point B! of the refracted ray BM'\ and the latter meeting the
refracted ray BM' in the point 5', which is the Image-Point cone-
sponding to the infinitely distant Object-Point 5 of the incident ray
MB. The point R will He in the Primary Focal Plane, and the pcnnt
S' will lie in the Secondary Focal Plane. Let us put
FR = AG = g, E'S' = AK'^k\
Evidently, we have then the following relations:
g
tan e
-,^E'A,
V
tan e
FA\
so that whatever be the slopes of the incident and refracted rays, the
Reflexion and Refraction of Paraxial Rays. 157
ts g and k' will always be such that the above ratios have
t values. If we denote these constant values by / and e', that
put
[nations can be written:
k' g
istants denoted here by the symbols / and e' are called the
^ and Secondary Focal Lengths^ respectively, of the spherical
ig surface. The proper definitions of the Focal Lengths (see
re given by formulae (70); thus:
Primary Focal Length (/) is equal to the quotient of the distance
optical axis of the point where a refracted ray crosses the Secondary
^lane by the tangent of the slope-single of the corresponding inci-
V and similarly:
Secondary Focal Length (e') is equal to the quotient of the distance
optical axis of the point where an incident ray crosses the Primary
^lane by the tangent of the slope-angle of the corresponding re-
'ay,
e special case where the optical system consists of a single
J refracting surface, the Focal Lengths may also be defined as
Focal Lengths of a Spherical Refracting Surface are equal to the
I of the vertex A with respect to each of the Focal Points; that is,
, ^ = E'A, as above stated.
Focal Lengths / and e' of a spherical refracting surface may
e expressed in terms of the radius r = AC. Thus, in Fig. 60,
le two pairs of similar triangles AFY^ G'Yfi and E'AY,^
we obtain the following proportions:
FAiYjG' ^AY^iG'G, E'A :G%-- AY.iY^Y,;
ce
= i4C = r, AY=G'G^CG-CG\ CG.CG' ^n'ln,
ve immediately the following formulae for the magnitudes of
al Lengths in terms of the radius r :
/=r7-^— r, «'=-3~:r; (71)
n — n n — n
158 Geometrical Optics, Chapter V. ( § 125.
whence also we find :
»'/ + ne' = o, (7a)
which is equivalent to formula (69) ; and also:
f+e' + r^o; (73)
which is equivalent to formula (68).
ART. 37. THE IMAGE-EQUATIONS IN THE CASS OP THE REFRACTIOH OT
PARAXIAL RATS AT A SPHERICAL SURFACE.
125. The Abscissa-Equation in Terms of the Constants n, n' and r-
If the vertex A of the spherical refracting surface is taken as origin
of a system of rectangular axes whose jc-axis is the optical axis deter-
mined by the centre C and the vertex A, the co-ordinates of an Object-
Point Q may be denoted by «, y and of the corresponding Image-Poiat
Q' by u\ y'\ thus:
AM^u, AM'^u\ MQ^y, M V = /•
The problem is to determine «', y in terms of «, y.
Since
CAf = Ci4 + ilAf = tt - r, CM' = CA + AM' -^u' -f,
equation (66) may be written in the following form:
u -
- r
•
•
— r
u
u''
•
' n '
n'
«'
n
u
n'
— n
r
or, finally:
^/ ^ ^t ^
(74)
To every value of u comprised between w = — 00 and « = + *f
we obtain by this equation a corresponding value of the abscissa nfi
thus, to every axial Object- Point M there corresponds one, and onlyof^
cucial Image- Point M'. This linear equation connecting the abscissae of
conjugate axial points in the case of the refraction of paraxial rays at
a spherical surface is one of the most important formulae of Geometri-
cal Optics. It is entirely independent of the special law of refraction
known as Snell's Law; for if the angles of incidence and refraction
a, a' are connected by any equation of the form /(a, a') = o, whercitt
it is assumed that the angles denoted by a, a' are small, it is easy to
show that the limiting value of the ratio aja! will be a constant whidi
may be denoted by n'/w; in which case we shall derive formula (74)
Reflexion and Refraction of Paraxial Rays. 159
lost general expression of the relation between conjugate points
laraxial ray which passes through the centre C of the spherical
In a supplement to this chapter it will be shown that this
Q is the analytical expression of Central Collineation in a Plane.
The so-called Zero-Invariant. If according to the convenient
of notation, introduced by Abbe, we denote the difference of
Lies of an expression before and after refraction at a spherical
by the symbol A written before the expression, formula (74)
written also in the following abbreviated form:
n I , ^
A - = - A». (7S)
u r ^
Diagnitude
las the same value before and after refraction at the spherical
, is called the ^^Zero-Invariant'' or the invariant in the case of
action of paraxial rays at a spherical surface. This magnitude
1 here by / plays an important part in the Theory of Spherical
dons, and the following formulae, all easily derived from (76),
found useful in the investigations of that theory. For example,
lin:
^-= -•^^-' (77)
»:
vire find:
u n
A— = -A--/A— 2- (78)
nu r n n '
A -2 = 7* A —2 A - . (79)
u n r n
ling formulae (77) and (79), we obtain:
A-2 = 7*A-2+-A-; (80)
u n r u
abining formulae (78) and (79) :
/A— = -A--A-V (81)
nu r n u
over, if B, 6' denote the slopes of a ray before and after re-
at a spherical surface, and if a, a! denote the angles of incidence
160 Geometrical Optics, Chapter V. ( § 127.
and refraction, and, finally, if ^ denotes the central angle {tp = IBCA,
Fig- S0» then, as in formula (60):
a- e = a! - e' = ip.
In the case of Paraxial Rays where these angles are all so small that
we may neglect powers above the first, we have (see § 108):
^ = - - , ^' = - -, ^ = - , (82
u u r
where h = DB (Figs. 58 and 59) denotes the incidence-height of the
ray. From these relations we obtain easily:
« = — . «'=-T. (83)
n n
127. The Lateral Magnification. The ratio Y » y'ly is called the
Lateral Magnification of the spherical refracting surface with respect
to the axial Object-Point Af. Referring to Figs. 62 and 63, veaee
that we have the proportion:
M'Q' : MQ = CM' : CM,
and, consequently:
y' v! — r
y u — r
This equation, together with formula (74), enables us to write the
transformation-formulae between Object-Space and Image-Space a»
follows:
n'ru , nry ,^ 1
whereby, being given the co-ordinates u, y of the Object-Point 0» *•
can find the co-ordinates u\ y' of the corresponding Image-Point C-
The formula for the Lateral Magnification Y may also be written
as follows:
y = ?^ = ^; (85)
y nu
whence we see that the Lateral Magnification F is a function of the
abscissa u, and that it is independent of the absolute magnitude of
the ordinate y. For a given pair of conjugate planes at right anj^
to the axis of a spherical refracting surface, the ratio denoted by ^
is constant, but it is different for different pairs of conjugate planeii
29.] Reflexion and Refraction of Paraxial Rays. 161
128. The Image-Equations in Terms of the Focal Lengths /, e'.
, by means of formulae (72) and (73), we eliminate », n' and r from
le formulae (84), the Image-Equations for the Refraction of Paraxial
ays at a Spherical Surface may be obtained also in the following
>rms:
- + -=-!, - = 7^' (86)
^herein the constants which determine the spherical refracting surface
re the two focal lengths/ and e'.
If, instead of taking the vertex A as the origin of abscissae, both in
he Object-Space and in the Image-Space, we take the two Focal
'oints F and E' as origins for the Object-Space and Image-Space, re-
pectively, we may put:
FM = X, E'W = x'\
5 that the co-ordinates of the conjugate points Q, Q' referred to axes
ith origins at F, £' will be :c, y and x\ y\ respectively. Evidently,
* ilAf = i4F+ Filf = :r -/, tt' = AM' = AE + E'M' ^ x' - e'\
^ substituting these values in place of u and u' in equations (86),
'^obtain the Image-Exiuations in their simplest forms, as follows:
v' f x'
xx'^fe\ F=^ = ^ = ^. (87)
^ » y X e' ^ '^
IW. The case of the Reflexion of Paraxial Rays at a Spherical
MuTor, which was treated at length in Arts. 33 and 34, may be re-
B^rded as a special case of the Refraction of Paraxial Rays at a Spheri-
^ Surface. Thus, according to the general principle explained in
■26, we have merely to put »' = — n in the formulae of Arts. 35-37
^"1 order to derive at once the corresponding formulae of Reflexion.
*1qs, for example, if we put »' = — n in formulae (71), we obtain / = e'
* '/i; which shows that the two Focal Points F, £' coincide in the
^of a spherical mirror (§112).
^ Another interesting special case that may be remarked here also
* obtained by putting r = 00 ; in which case we shall obtain the
■^ulae for the Refraction of Paraxial Rays at a Plane Surface; thus
^find:
n y
*ittch will be recognized as the same as the results obtained in § 53.
12
162 Geometrical Optics, Chapter V. [ § li
The last of these equations is of special interest, for it shows that tl
Focal Points F and £' of a refracting plane are themselves the ia
nitely distant points of the two ranges of conj ugate axial points. Henc
to a bundle of parallel incident rays refracted at a plane there corr
sponds a bundle of parallel refracted rays. Any optical system whi<
treats parallel incident rays in this way is called a Telescopic Syste
— a, name which is derived from the fact that the Focal Points of
telescope are both at infinity.
III. Supplement: Containing Certain Simple Applications c
THE Methods of Projective Geometry.
ART. 38. central COLLINBATION OP TWO PLAKE-PIELDS.
130. In the investigation of the refraction (or reflexion) of paraxi
rays at a spherical surface, we have seen that the imagery is idea
so long at least as the rays of light are supposed to be monochromati
so that the refractive indices », n' have fixed values. Thus, to a honn
centric bundle of Object-Rays there corresponds always a homocentr
bundle of Image-Rays, and to each point of the Object-Sf)ace, withi
the region of the paraxial rays, there corresponds one, and only on
point of the Image-Space. This unique point-to-point correspoi
dence by means of rectilinear rays between the Object-Sf)ace and tl
Image-Space, which is the fundamental and essential requiremei
of Optical Imagery, is called in the modem geometry " CoUineation'
Thus,
Two spdce-sysiems S and S' are said to he **collinear** with each othe
if to every point P of X there corresponds one, and only one, point P i
S', and to every straight line of S which goes through P there correspom
one straight line of 2' which goes through P'.
In the theory of optics these two spaces 2) and S' are designati
as the Object-Space and the Image-Space. They are not to be thoug
of as separate and distinct parts of space; they penetrate and indu
one another, so that a point or a ray may be regarded as belonging
either of the two space-systems.
Inasmuch as the problem of the refraction of paraxial rays affoi
a simple and at the same time a very useful application of the degs
methods of the modern- geometry, it is proposed to give here a spe<
investigation of it from this point of view; especially, too, beca
this study will prove a good introduction to the general theory
Optical Imagery which is treated at length in Chapter VII.
Since the optical axis of the spherical surface is an axis of symme
\ 131.] Reflexion and Refraction of Paraxial Rays. 163
for both the Object-Space and the Image-Space, it will suffice, as we
have seen, to investigate the imagery in any meridian plane of the
spherical surface; that is, in any plane containing the optical axis.
In this plane in space we have two collinear plane-fields^ one belonging
to the Object-Space and one belonging to the Image-Space, which
correspond with each other point by point and ray by ray. The total-
ity of all the points and straight lines situated in an infinitely extended
plane is what is here meant by the term * 'plane-field".
The distinguishing characteristics of the kind of ColHneation which
we have in the case of the Refraction of Paraxial Rays at a Spherical
Surface may be said to be two in number, although, indeed, one is a
consequence of the other. These characteristics are contained in the
following statements:
(i) If Q, (^ are a pair of corresponding, or conjugate, points the
straight line QQ^ passes through the centre Cof the spherical refracting
surface; or, the straight lines joining pairs of conjugate points all inter-
M in one point {C).
(2) Since an incident ray and its corresponding refracted ray meet
^ the spherical refracting surface, and, moreover, since we are con-
^^cnied here only with paraxial rays, which, therefore, meet the spheri-
cal surface at points infinitely near to its vertex i4, so that the straight
Iwe (y) in the meridian plane which is tangent to the spherical surface
*t A may be regarded as the section of the surface made by this plane
WI113); It follows, therefore, that any pair of corresponding rays
^ Oie two collinear plane-fields will meet in this straight line (y).
When two collinear plane-fields are so situated relative to each other
"^t they have in common a self-corresponding range of points, we have
^ special case of the " Central ColHneation'' of two plane-fields.
*w straight line (y) which corresponds with itself point by point is
<^ the '*Axis of ColHneation' \ The point C through which every
•Wght line joining a pair of corresponding points passes is called the
"Centre of CoUineation". This point C is a ^'double point'* or self-
cwresponding point of the two collinear plane-fields. Hence, every
straight line drawn through C contains two double points, viz., the Centre
of 0)Ilineation itself and the point where the straight line intersects
the Axis of CoUineation.
131. Projective Relation of Two Collinear Plane-Fields. If P, Q,
JZ, 5 (Fig. 66) are a range of four points lying on a straight line 5 of
one of the plane-fields, the points P\ Q\ R', 5' conjugate to P, Q, R, 5,
respectively, will be ranged along the corresponding straight line s*
•A the collinear plane-field, and it is easy to show that we have the
164
Geometrical Optics, Chapter V.
[{
following relation :
{PQRS) = {P'Q'R'S') ;
FlO. 66.
Central Collineation op Two Plane-
Fields. The centre of collineation (C) and the
axis of collineation {y) are the centre and axis of
perspective; so that if «. / are a pair of corre-
sponding rays,
{PQRS) - iP'C/if'S').
that is, two collinear plane-fields are in ** projective'' relation to each oi
The proof of this is especially simple when we have Central Collij
tion of the two plane-fields;
then every straight line joii
a pair of corresponding po
passes through the centre
collineation C, and hence
two plane-fields in this <
will be in perspective; whc
it follows that any two cc
sponding straight lines s, s*
intersect a pencil of rays \
its vertex at C in two projec
ranges of points.
In case, however, the ray s itself passes through the centre C
that s, s' are, therefore, a pair of self-corresponding rays (Fig. 67),
above proof of the projective relation of 5, s' will not be applica
In such a case we may proceed as follows:
Through the point C draw any other straight line, and take 0
a point 0. Connect 0 by straight lines with the Object-Points P
R, S ranged along the straight line s. The straight lines joining
corresponding Image-Points P', Q', R\ S' ranged along the strai
line s' with the points where the straight lines PO, QO, RO, SO, re^
tively, intersect the axis of collineation (y) will all pass through
point 0' conjugate to 0, which is a point of the straight line jai
0 and C Since, therefore, the Object-Ray OC and the correspoiM
Image-Ray 0' C coincide in the straight line joining O and CX, the )
cils of rays 0, 0' are in perspective with each other; so that /or
jugate points P, Q, R, S and P', Q\ R', 5' of a central or sdf-corresp
ing ray s (or s') we have also the projective relation, characterized b^
equation :
(PQRS) = (P'Q'R'S').
The self-corresponding ray at right angles to the axis of collinei
(y) coincides with the optical axis of the system. This ray wi
designated as the ray x of the Object-Space and the ray x* o:
Image-Space. And the point A where it crosses the axis of coil
tion will be selected, in the special case of Central Collineation, a
i 132.1
Reflexion and Refraction of Paraxial Rays.
most convenient point for the origin of a system of rectangular co-ordi-
nates, the ajtes whereof are the optical axis and the axis of collineation.
■.Anon or Two Puutb-Fields. CoDitructlon of CoiijUB>(e PolnU of a (df-
ilnl iBvi (orj'}. The«nCreof collineatianUatC: the uiaof collineation.
I* Uie itraitrbt line dniEnaleil byj/. O. (T'areB fair of conjuiBtepointiOD
IlinepaaiinBthrotish C. The points /■.(?. ^,.Sor the rantre of Obiect-PoiDtai are iyt>-
vencil of rayi fiom O. and the coniuafate points P'. (f . If. S' of the racfre oT Imaffe-
• [nojected br a pencil of raya from <f which is ia penpeclive with the pencil O. The
ue points of i'. i conjugate to the InSnitely distant points /, J' of i. Z, reapectivelr,
X (orj/) which meets the axis of collineation at^ at risht anelea is
Ills: and the itislcht lines parallel to theazisof collineation which (redrawn IhrouEh
id which meet the optical axis al light anglea at ^ond f. respectively , are the two focal
132. Geometrical Constructions.
(i) (inwn tiie axis of collineation (y) and the centre of coUineation ( C),
•(^ with the positions of two conjugate points P, P': it is required to
*»wfr«( the Image-Point Q' of a given Object-Point Q.
Through the two given Object-Points P, Q draw the straight line s
•weting the axis of collineation in the double point B. Suppose (i)
tint the straight line s joining P, Q does not pass through the centre C,
M in Fig. 66. This is the genera) case. The straight line s' corre-
■pondii^ to s will connect B with the given point P', and this straight
line must also pass through the point Q' conjugate to Q. But Q"
most also lie on the self-corresponding ray which goes through Q and
the centre C; and hence the Image-Point Q' will be uniquely deter-
nuned by the intersection of the straight lines BP' and QC. Again
nippose (ii) that the straight line s joining P, Q passes through the
teittre C, as in Fig. 67; so that s (or s') is a self-corresponding ray.
166 Geometrical Optics, Chapter V. [ § 132.
This is a special case of great importance. In this case the above con
struction fails, and we may proceed, therefore, as follows: From J
and C draw two straight lines intersecting in a point 0. Join by *
straight line the point where PO meets the axis of collineation with th
given point P' conjugate to P, and let 0' designate the point where thi
straight line meets the straight line CO. Join QO by a straight iio
and from the point where QO meets the axis of collineation dra^i
through 0' a straight line, which will meet s' in the Image-Point Q
conjugate to the given Object-Point Q.
(2) Construction of the so-called **Flucht'' Points J and I' of am
central, or self-corresponding^ ray s (or 5') ; being given, as before, th
axis of collineation (y), the centre of collineation (C) and the pairc
conjugate points P, P'.
The Image-Point /' conjugate to the infinitely distant Object-Poin
/ of the pencil of parallel object-rays of which the self-correspondin
ray is the central ray s (Fig. 67) will be a point on s' which may b
constructed exactly as was explained above. For example, knowin
the positions of P, P', we can locate the positions of a pair of conjugaC
points 0, 0\ as was done above. A straight line drawn through ^
parallel to 5 will go through the infinitely distant point / of s. Tta
straight line joining the point where 01 meets the axis of collineati<^
with the point 0' will intersect s' in the Image-Point /' conjugate t:
the infinitely distant Object-Point /. German writers on geometr"
call this point /' the **Flucht'' Point of the ray s'.
Similariy, the 'Tlucht" Point / of the Object-Ray s is that poia
of this range which corresponds with the infinitely distant point f c
the Image-Ray s\ It may be constructed in a way precisely analogoi'
to the construction of /' above, in the manner indicated in the diagiaiKi
The *Tlucht'' Points / and /' are, in general, actual, or finite, poin*:
of the projective ranges of points 5 and s\ respectively. In particular'
the *Tlucht" Points, designated by Pand £', of the self-correspondiiii
ray x, x\ which coincides with the optical axis, are the points caBe^
the Focal Points of the optical system (§ 120).
(3) Given the axis of collineation (y), together with the positions afit^
*' Flucht'' Points, J and P, of any central ray s, s\ to construct the Imag^
Point P' corresponding to a given Object- Point P of s.
Take any point 0 (Fig. 67), and through O draw the straight Kfl^
01 parallel to s; and draw the straight line joining with I' the poW
where 01 meets the axis of collineation. Draw the straight lin» Jft
PO, and from the point where JO meets the axis of collineation di«*
straight line parallel to s' meeting in 0' the straight line drawn throoil^-
§ 132.1
Reflexion and Refraction of Paraxial Rays.
167
J'. The straight line which joins with 0' the point where PO meets
the axis of collineation will meet s' in the Image-Point P' conjugate
to the Object-Point P.
In particular, knowing the positions of the two Focal Points F
and £' on the optical axis, and knowing also the position of the axis
ot collineation, we may, as above, construct any pair of conjugate axial
points M , M\
(4) Given the axis of collineation {y) and the centre of collineation (C)>
tofett«r with the positions of two conjugate points P, P' it is required to
construct the image-ray v' corresponding to a given object-ray v.
Let E (Fig. 68) designate the double point where the given object-
ray meets the axis of collineation. Through the given Object- Point
PlO. 68.
^^''TSAL COLLXNXATXOif OP Two Planb-Pxblds. Construction of Imaze-Ray f/ conjuffate to
•**™ ObjecMlay v ; also, construction of the " Flucht " I«ines or Focal I^incs/, /,
'draw any ray s meeting the given ray » in a point Q and the axis of
^^eation in a point B. The point of intersection of ^C and BP'
^determine the Image-Point Q' conjugate to the Object- Point Q\
^ hence the straight line HQ' will be the image-ray v' conjugate to
^ given object-ray v.
(s) // the given object-ray in {4) is the infinitely distant straight line e
^tke Object- Plane^ we can construct the conjugate straight line e' of the
^^It-Plane^ as follows:
The point of intersection of the infinitely distant straight line e of
^ Object-Plane with the axis of collineation (y) is the infinitely dis-
^t point T (Fig. 68) of y; and hence e' will be parallel to y. Any
'^y 5 drawn through the given Object-Point P will meet the infinitely
&tant straight line e of the Object-Plane in the infinitely distant
point S of 5. If the object-ray s meets y in 5, the corresponding
168 Geometrical Optics, Chapter V. [ § li
image-ray s' will be the straight line BP\ and a straight line drau
through the centre C parallel to 5 will determine by its intersecti<
with 5' the Image-Point 5' conjugate to the infinitely distant Objec
Point 5. The straight line drawn through 5' parallel to y will, ther
fore, be the image-ray e' conjugate to the infinitely distant object-ray
This straight line e' which is conjugate to the infinitely dista:
straight line e of the Object- Plane is called in Optics the Focal Line
the Plane-Field of the Image-Space (see § 1 23) . Since e' passes throuj
the point 5', which is the 'Tlucht'* Point of any ray of the plane-fie
of the Image-Space, it follows that the Focal Line e' is the locus of i
'*Flucht'' Points of all the image-rays in this plane-field.
In a precisely similar way, we can construct also the straight line
in the plane-field of the Object-Space which is conjugate to the infinite
distant straight line f of the plane-field of the Image-Space^ and whi<
may, likewise, be defined as the locus of the ''Flucht'' Points of all l
rays in the plane-field of the Object-Space.
The focal lines /, e' are, in general, actual, or finite, straight line
They are both parallel to the axis of collineation, and perpendicula
therefore, to the optical axis.
133. The Invariant in the Case of Central Collineation. Sim
all the rays of the pencil C are self-corresponding, each of these ra)
is the base of two projective ranges of points, a range of Object-Point
and a range of corresponding Image-Points. Moreover, to each i
these self-corresponding rays belongs a pair of double, or self-correspondin^
points (§ 130) ; one of these double points being the centre of collines
tion itself and the other the point where the ray crosses the axis of co
lineation.
Similarly, each point on the axis of collineation is the commo
vertex of two projective pencils of rays, viz., a pencil of object-ra)
and a pencil of corresponding image-rays; and eocA pair of such pend
of corresponding rays contains two self-corresponding rays, of which tl
axis of collineation itself is one, and the ray joining the common ve
tex of the two pencils with the centre of collineation is the othc
Let P, P' (Fig. 69) and Q, Q' be two pairs of conjugate points of i
self-corresponding ray 5, s\ and let U designate the double point whc
this ray crosses the axis of collineation (y). Since the ray s, s' is t
common base of two projective ranges of points, the double ratio
the four Object-Points C, C/, P, Q on s is equal to the double ra
of the four corresponding Image-Points C, C/, P', Q' on s'; that
(CC7P<2) = (CC7P'<20;
(113.1
Reflexion and Refraction of Paraxial Rays.
169
whence it follows immediately that we have also :
{CUPP') = {CUQQ') ;
and, consequently, the double ratio of any pair of conjugate points P, P'
with the two self-corresponding points C, U of the two projective ranges
of points which have the common base PP' has a constant value y which
is independent of the positions of the conjugate points P, P\
Let Jf, Jf' be any other pair of conjugate points not situated on
the straight line PP'; for example, it will be perfectly general if we
Fio. 69.
Cbntral Collin BATioif op Two Plan b-Piblds.
{CUPP') - {CUQ(/) - {CAMM*) - {CALL') - {CAEE') - {CAFF*).
^ the points Jf , Jf ' on the optical axis x, x' which crosses the axis
^collineation (y) at the point A. Let the two corresponding rays
*^» M'P' intersect in a point H on the axis of collineation. It is
^ous immediately that the two ranges of points C, C/, P, P' and
C A, Jf, M' are in perspective, since they are both sections of the
pencil of rays which has its vertex at H. Hence, the double ratios
a/^each of these ranges of four points are equal; and if we denote the
value of this double ratio by the symbol c, we have the following re-
BiBrkable relations:
c = (CUPP') = {CUQQ') = etc..
iCAMM')
(CAFF')
(CAEE')
(CALL') = etc.,
CF : AF,
AE' : CE'-,
(88)
170 Geometrical Optics, Chapter V.
where, as heretofore, F and £' designate the positions of the two
Points, and F' and E designate the infinitely distant points of
X, respectively.
The most striking characteristic of the Central CoUineation <
plane-fields consists, therefore, in the fact which we have hei
covered, that it has an invariant:
The Double Ratio of any pair of conjugate points of a self-corresp
ray and the two double points of the ray has the same value for a
rays.
The value of this invariant, as above stated, is:
CF AE\
^^ AF' CE''
accordingly,
CA + AF AC+CE' _ AE' _
AF " CE' "CA + AE''^'
which gives:
FA = CE\ E'A ^CF, JJ = - c.
These relations are likewise characteristic of Central CoUim
The first two of formulae (89) — ^which may be derived also d
from the equation ( CFA E) = ( CF'A £')— are identical with fo
(67) which were obtained for the special case of the Refract
Paraxial Rays at a Spherical Surface; whereas the third equatic
responds with the relation given in formula (69).
134. The Characteristic Equation of Central CoUineation. 1
ticular, if M, M' designate the positions of any two conjugate
of the optical axis, the relation
{CAMM') = c
may be written in the following form :
c \ c — I
w' w ~ r *
where the symbols «, u' and r denote the abscissae, with respect
point A as origin, of the points M, M' and C, respectively;
u^AM, u'^AM', r^AC.
This equation, which expresses for the case of any Central CoUio
the relation between the abscissae of conjugate axial points, is
§ 134.] Reflexion and Refraction of Paraxial Rays 171
fectly general expression of the one-to-one correspondence of two pro-
jective ranges of points lying upon the same straight line. The cases
which occur in Optics are comparatively restricted; we shall proceed
to examine them.
If the sign of the invariant (c) is positive, the conjugate points Jlf , Af'
are not "separated", in the geometrical sense, by the axis of col-
lineation (y) and the centre of colHneation (C). That is, for c > o,
the points M and M' are either both situated between C and A , or
neither of them lies between C and A. In other words, the points
C, A, M, M' are what is called a "hyperbolic throw", (CAMM') > o.
This case occurs always when the rays are refracted from one medium
to another; so that in Optics a positive value of c indicates Refraction;
whereas, on the contrary, whenever the light-rays are reflected at a mirror^
ike imagery is of a kind that corresponds to a negative value of c (c < o);
in which case one of the points M or M' will lie between C and A , but
not the other point. In this latter case the points C, ^4, Af, M' are an
"elliptical throw", (CAMM') < o.
Case I. Refraction of Paraocial Rays; c > o.
A. Suppose, first, that r = AC is not equal to zero; that is, that
the centre of colHneation (C) does not lie on the axis of colHne-
ation (y).
This is the case of the Refraction of Paraxial Rays at a Spherical
Surface^ which has been specially treated in this chapter. The in-
vanant c in formula (90) is identical in value with the relative index
I of refraction (n'/n) from the first medium to the second medium,
t while the other constant r denotes here the radius of the spherical sur-
^^^» as wiU be seen by comparing formula (90) with formula (74).
^"^ points A and Care, therefore, identical with the vertex and centre,
'^P^tively, of the spherical refracting surface.
^veral special cases included under this head may be jbriefly
noticed:
^') If c = + I (the value of r, as above specified, being different
Tottx zero), the relative index of refraction is equal to unity (n' = n).
^^hiscase equation (90) gives u' = u; and, hence, Object-Space and
°J^e-Space coincide point by point; in fact, the two spaces are identi-
• When n' = n, there is no optical difference between the first
"^^ium and the second medium.
v^) The case when r = 00. An infinite value of r in this case means
°^ly that the centre C is at an infinite distance away in the direction
^^ a line at right angles to the axis of colHneation (y) ; so that now
172
Geometrical Optics, Chapter V.
[§1
the refracting surface is a plane surface. Formula (90) becomes nc
tt = — w,
n
which is the abscissa-relation for the case of the Refraction of Parax
Rays at a Plane (§53 and § 129). Since the centre of coUineation (
is at an infinite distance in a direction perpendicular to the refracti
plane, the trace of which in the plane of the diagram (Fig. 70) is t
Fio. 70.
CBimiAL COLLINEATXON OP TWO Pi:.ANE-PlBLDS FOR THB CASB WHBN C>0AIfO r««.
diairnun shows the case when m « c > 1. This case (c > 0, r « •) U the case of the Refractki
Paraxial Rays at a Plane Surface. The double point C is the infinitely distant point of the opi
axis xx', and the two Focal Points F, Ef both coincide with C.
axis of collineation {y), all straight lines joining pairs of conjug:
points are parallel to the abscissa-axis. In this case the infinit
distant straight lines of the two collinear plane-fields must pass throi
the infinitely distant double point C; and, therefore, the two infinit
distant straight lines must be a pair of self-corresponding rays, au
accordingly, the five points designated by C, £, £', F, F* must
be coincident. In the modern geometry two collinear plane-fidds «
said to be in affinity with each other when their infinitely distJ
straight lines are conjugate to each other. Hence in this case the t
focal lines / and e' are coincident with the infinitely distant straij
lines e and /', respectively. In Optics this type of imagery is cal
telescopic (§ 129).
B. A case of Central Collineation which is of much importance
Optics is the case when the invariant c = + i o.nd the other consk
r = o. If r = o, the centre of collineation (C) is situated on the axis
collineation (y), so that the two double points A and C of the opti
axis coincide. In this case we find :
FA = AE\ or f+e'^o;
where / = FA, e' = E*A denote the focal lengths of the optical s
tem. This type of imagery is characterized, therefore, by the f
{ 134.] Reflexion and Refraction of Paraxial Rays. 173
that the two focal points are equidistant from the axis of collineation,
and on opposite sides thereof. In the following chapter it will be seen
that this is the imagery obtained by the refraction of paraxial rays
through an Infinitely Thin Lens^ or through any number of such lenses
in successive contact with each other.
In this special case, the expression on the right-hand side of formula
(90) becomes illusory. This leads us to remark here that we can ob-
tain the abscissa-relation of Central Collineation in another form,
which is characteristic not only of Central Collineation, but of the col-
Unear relation in general. Thus, since
(MAFE) = {M'AF'E'),
we derive the equation :
xx' =fe\
where x = FM, x! = EM' denote the abscissae of Jlf , Jlf ' referred to
the Focal Points F, £', respectively, as origins. In the special case
here under consideration for which we have «' ~ ~/» this formula
takes the form :
XX' ^ -/.
Case II. Reflexion of Paraxial Rays; c <o.
The only negative value of c that has any practical significance in
Optics is the value c = — i . For this value of c we have :
{CAMM') = -*i;
80 that each pair of conjugate points is harmonically separated by the
^^ (C) and the axis of collineation (y). This formula will be recog-
'"^ immediately as the formula for the Reflexion of Paraodal Rays
^ ^Spherical Mirror {% III).
Since {CAMM') = - i = {CAM'M), the two ranges of points
'y'^upon any central ray are in "involutory position"; so that, if,
w example, a point M' of one range x' is conjugate to a point M of
™c conlocal range x, the same point M regarded now as a point of
*^^11 be conjugate to M' regarded as a point of x (see §110). We
find here:
FA = E'A, or f ^ e' = o;
^ ftat, as was pointed out in § 1 1 2, the Focal Points F, E' of a spherical
°""t)r are coincident.
Finally, if r = 00, we have, for c = — i, «' = — w, which (see § 50)
^ *e formula for Reflexion at a Plane Mirror.^
hi connection with Art. 38, see J. P. C. Southall: The geometrical theory of optical
^**fy: Astropkys. Joum,, xxiv. (1906), 156-184.
CHAPTER VI.
REFRACTION OF PARAXIAL RAYS THROUGH A THIN LENS. OR THR0UC3
A SYSTEM OF THIN LENSES.
ART. 39. REFRACTION OF PARAXIAL RATS THROUGH A CENTERED STSTS:
OF SPHERICAL SURFACES.
135. Centered System of Spherical Surfaces. Nearly all optic::
instruments consist of a combination of transparent isotropic medii
each separated from the next by a spherical (or plane) surface; th
centres of these surfaces lying all on one and the same straight lin-'
called the * 'optical axis" of the centered system of spherical surface
which is an axis of symmetry. The spherical surface which the raj
encounter first is called the first surface of the system ; in our diagranL
in which the light is represented as being propagated from left ■
right, the first surface will be the one farthest to the left. The t^
media separated by this surface will be called the first and secois
media, respectively, in the sense in which the light travels. If tt"
number of spherical surfaces is m, the number of media will be m +
the (w + i)th medium being the last medium into which the ra3
emerge after refraction (or reflexion) at the mth surface. The aM
solute index of refraction of the first medium will be denoted by ^
(= no); and, generally, the index of refraction of the kth media J
(where k denotes any positive integer from o to m) will be denoted b
nj^_i. Thus, the index of refraction of the last medium will be d^
noted by n^. The centre of the *th spherical surface will be deij
nated by Cf^, and the point where the optical axis meets this surfac*
called the vertex of the surface, will be designated by il^. The cex^
tered system of spherical surfaces is completely determined provide^
we know the index of refraction of each of the successive media an<
the positions on the optical axis of the centres and vertices of tb<
spherical surfaces.
136. To a homocentric bundle of incident paraxial rays there oat'
responds a homocentric bundle of rays refracted at the first surface-
The image-point or vertex of this bundle of refracted rays may be
real or virtual; but in either case it is to be regarded as lying in the
second medium, even though the actual position of this point inspatf
may lie in a region which is occupied by the material of some one oi
the other media (see § lo). This bundle of rays refracted at the firtt
174
136.1
Refraction of Paraxial Rays Through a Thin Lens.
175
surface will be a homocentric bundle of paraxial rays incident on the
second surface, to which, therefore, there corresponds a homocentric
bundle of rays refracted at this latter surface, with its image-point
lying in the third medium. Proceeding thus from surface to surface,
remaining always a bundle of homocentric rays, and producing a point-
image in each successive medium of the series, the rays emerge finally
into the last medium and form there a point-image, which, with re-
spect to the entire centered system of spherical surfaces, is the point
conjugate to the Object-Point in the first medium from which the
rays originally came. Thus, precisely as in the case of the refraction
of paraxial rays at a single spherical surface, we have also for the re-
fraction of such rays through a centered system of spherical surfaces
strict collinear correspondence between Object-Space and Image-Space.
The accompanying figure (Fig. 71) represents a centered system of
three spherical refracting surfaces; the sections of the surfaces made
•.
La
A
(
V
V
/\^'
^
^
SJir.
A. ,-'
A
r,
A.
^!^
..S<n:
k^
'*.
n.'
^
^'^
Fig. 71.
IXAOSkY BT RBPRACTXOlf OP PARAXIAL RAYS THROUGH A CENTBRBD SYSTEM OF SPHBRICAL
*^^CTQiQ Surfaces. In the diaffram. the spherical surfaces are represented by the straight
™^''^*>i,etc.; in the fi8:ure aU the surfaces are represented as convex, with no two centres Ci,
7'^.ia the same medinm. Moreover, each imaire is represented as a real imaire formed between
^tie of oQe surface and the vertex of the next followinjr. The diajp-am is thus drawn merely
**"«irite of simplicity
AiU\ - Ki, A\Mii - «i'. A\C\ - n. A\A\ ■» rfi, ^iM' — «f, A^il - «/,
AiC% - n. AtA% " <ft. AkMk'-\ - Uk, AkMil - «*', AkCk " Mk.
°ya plane containing the optical axis (the plane of the diagram) being
**own by the tangent-lines 3^1, y^, etc., in accordance with the graphi-
^ method explained in § 1 1 3 . Consider a ray Af , B^ lying in the plane
^ the diagram which crosses the optical axis at the point designated
^^x and meets the first spherical surface at the incidence-point B^.
^ter refraction at this surface this ray crosses the axis in the second
J'Wiuni at the point designated by Af^, which is, therefore, the axial
^'^^point in this medium conjugate to the Object-Point Af,. Inci-
^tat jB, on the second surface, the ray is again refracted, and again
^^'^osses the axis at a point Af^ which is the image-point in the third
176 Geometrical Optics, Chapter VI. [ J 1
medium conjugate to the axial Object-Point Afj in the first mediu
Any one of these image-points may be real or virtual, depending
circumstances. If the number of spherical surfaces is m, the point J
where the ray crosses the axis after refraction at the last surface v
be the image-point which, with respect to the entire system of si
faces, is conjugate to the axial Object-Point M^,
The diagram shows also the path of a ray which, emanating from
Object-Point Qi near the optical axis, but not on it, traverses t
centered system of spherical surfaces. The actual ray whose path
drawn is the ray which in the first medium is directed from Q^ towai
the centre C^ of the first surface, and which, meeting this surfs
normally, proceeds into the second medium without change of din
tion; so that the point Q\ in the second medium which is conjugs
to ^1 must lie, therefore, on the straight line connecting Q^ and i
If the extra-axial Object-Point Qi is a point on the straight line p
pendicular to the optical axis at Af^, the point Q\ will lie on the straig
line perpendicular to the optical axis at JIfj, and the straight line Ifj
will be the image in the second medium of the short Object-Line J/ j
in the first medium. The image of Q\ produced by the second refra
tion will be at a point Q2 in the third medium, which is the point
intersection of Q\ Cj with the perpendicular to the optical axis at A
thus, M'iQ't will be the image in the third medium of the Object-Li
M\Qv The point Q'^ in the last medium will be the Image-Poii
with respect to the entire system, of the extra-axial Object-Point (
and -MlQl will be the image, produced by the refraction of paraxi
rays through a centered system of m spherical refracting surfaces,
a small Object-Line MiQi in the first medium perpendicular to t
optical axis.
Thus, exactly as in the case of a single spherical refracting surfa
any plane of the Object-Space perpendictdar to the optical axis ofacentei
system of spherical refracting surfaces will be imaged by paraxial ft
by a plane of the Image-Space also perpendicular to the optical axis.
137. The abscissae, with respect to the vertex A^ of the Ath si
face, of the points Jlfl_i, Jlf^ where a paraxial ray crosses the optii
axis before and after refraction at this surface will be denoted
u^f u[, respectively; thus,
^k*
where k denotes any integer from i to m. For ife = i, we lu
i4|Afi = Wj, since we write here M, instead of Af^. The radius of 1
Jfeth surface is denoted by r^, and is defined as the abscissa, with resp
137.] Refraction of Paraxial Rays Through a Thin Lens. 177
► the vertex A^, of the centre C^. Moreover, the abscissa of the
»rtex Ai^i of the {k + i)th surface with respect to the vertex Aj^
: the Ath surface, called the thickness of the {k + i)th medium, is
»noted by d^. Thus,
rhus, for the ifeth Spherical Refracting Surface, we have, according to
lormula (76) :
•^'""*(^-i) = "-'(^.-4)'
(91)
wherein, since
Aj^iM^i + Af^^iA^ = A^^iAj^t
the value of Uj^ is determined by :
tt* = ttl-i- d*-i- (92)
In these formulse (91) and (92) we must give k in succession all integral
values from A = i to Jfe = m, where m is the total number of spherical
^aces (Note that ^q = o). Thus, provided we know the magnitudes
^oted here by n, r and d, that is, provided we are given the centered
•y^tem of spherical surfaces, we can, by means of these recurrent
lonnulse, eliminate in order the magnitudes denoted by m, and thus
^in the final value u^ corresponding to a given value of u^ ; that is,
T^ine the position of the Image-Point Jli^ corresponding to a
9^ position of the axial Object-Point Afj.
The Focal Point E' of the Image-Space of a centered system of
fi*^cal refracting surfaces is the point where a paraxial ray, which
''^ the first medium is parallel to the optical axis, crosses this axis after
'^'Hction at the last, or mth, surface. If in the above equations we
Ptttu, s 00^ then u'^ == A^E' will be the abscissa of the point £' with
^^^9^ to the vertex -4^ of the last spherical surface. We shall have
1 '^*-"i)equationswith (2m— i) unknown magnitudes, viz., Wg* «3» * • • ^m
^ % «2f • • • u'm' Accordingly, by successive substitutions we can
^ u^. Similarly, the Focal Point F of the Object-Space is the point
^^iere a ray crosses the optical axis in the first medium which emerges
^ the last medium parallel to the optical axis. In order to locate
to point F, we must put «! = 00 and find the value of the abscissa
«i « i4,F of the Focal Point F with respect to the vertex i4, of the
Srst spherical surface.
u
178 Geometrical Optics, Chapter VI. [ § 1» •
138. The Lateral Magnification F. Putting JlflQl = yu, m^
making use of formula (85), we obtain the following equations:
Multiplying these equations together, we obtain:
yi "" ftk Ui'U2'"Uk
The ratio
is called the Lateral Magnification of the centered system of spheriw
surfaces with respect to the axial Object-Point Mi. Thus, accordiTS
to the formula above, we have :
where the symbol 11 means the continued product of the terms (^
tained by giving k in succession all integral values from ife = i to ft = -3i
Y is evidently a function of u^.
139. The Principal Points of a Centered System of Spheric^
Surfaces. The pair of conjugate planes perpendicular to the optics
axis for which the Lateral Magnification has the special value F =* +
so that for this pair of planes object and image are equal both as ^
magnitude and sign, were called by Gauss* the Principal Planes ^
the optical system; and the two conjugate axial points, designate
here by the letters -4,-4', where the Principal Planes were cut fc
the optical axis, were called similarly the Principal Points. PuttiE3
F = + I in formula (93), we have:
which, together with the equations (91) and (92), gives us 2m equatioi>
with 2m unknown quantities, whereby we can determine the absdss^
«! = -4i-4 and u^ = -4^-4', and thus ascertain the positions of tb*
Principal Points -4,-4'.
The earlier writers on Geometrical Optics proceeded by computiol
the values of w, w' from surface to surface. MoEBius and, espedaOfi
Gauss strove to derive general formulae for finding the podticm d
the image-point conjugate to a given object-point, without invdvifl|
' C. F. Gauss: Dioptrische Uniersuchungen (Goettingen, 1841). p. 13.
I.] Refraction of Paraxial Rays Through a Thin Lens. 179
tedious process of tracing the path of the ray from surface to
ace. It was Gauss who introduced the notion of the so-called
•
dinal Paints of the optical system. These are certain distinguished
s of conjugate axial points, the most important of which are the
idpal Points A, A', which are briefly referred to here. We may
ark that Gauss's method marked a great advance in the science
ieometrical Optics, and led to very simple and elegant formulae,
e recently, Abbe (as we shall see in the following chapter), with-
employing the Cardinal Points at all, has obtained the so-called
ige-equations" by a still simpler method depending only on the
acteristics of the Focal Points of the optical system. Abbe's
ry of optical imagery will be explained at length in the following
)ter; where will be found also a more extended reference to the
linal Points of the system (§ i8o).
he formulae which have been obtained will be applied in this chap-
to the problem of the refraction of paraxial rays through an infi-
ly thin lens.
ART. 40. TTPSS OP LENSES; OPTICAL CENTRE OP LENS.
W. A centered system of two spherical refracting surfaces con-
utes what is known in Optics as a Lens. In practice the Lens is
ally surrounded by the same medium on both sides, and we shall
iMne in this chapter that such is the case. We may denote the
iices of refraction of the two media by the symbols n and n'; thus,
/ / /
lice m = 2, we obtain from equations (91) and (92) the following
nnulae for the refraction of paraxial rays through a Lens surrounded
'the same medium on both sides:
n n
n
— n
U, Ui
M, = m'i - d,
n n
n
— n
1 "^
=
*
M, «,
u J
(94)
ere here we use d instead of d, to denote the thickness -4,^42 of the
IS. Thus, if we know the radii r,, r^ of the two surfaces of the Lens
the index of refraction of the Lens-medium relative to the sur-
iding medium (n'/«)» together with the thickness d of the Lens,
an find the position of the Image-Point M\ conjugate to the axial
180
Geometrical Optics, Chapter VI.
[514L
Object-Point M^. The positions of the Focal Points F and E' may
be determined by putting, first, 1*2 = °^ and solving for u^ = A^?^
and, second, w, = 00 and solving for Mj = A^\
The Lateral Magnification with respect to the Object-Point M^ is
obtained at once by putting m = 2 in formula (93); thus, we have:
yi ti['U^
(95)
141. Lenses may be conveniently divided into two main classes,
as follows:
(i) Lenses which are thickest along the optical axis. In this group
are included, therefore, such forms of lenses as are shown in the figure
(Fig. 72), viz., the Bi-convex Lens, the Plano-convex Lens and the
Convexo-concave (or Concavo-convex) Lens with a shallow concavity
(the so-called "Positive Meniscus").
(2) Lenses which are thinnest along the optical axis. To this group
belong the Lenses shown in Fig. 73, viz.: the Bi-concave Lens, the
Stt>^ttr4»
PiaM»'Con^*t
Mtrthctf
M0MI4CV*
TL c
Pic. 72 and Pio. 73.
Types op I«bnses. In Fiff. 72 the lenses are " convergent ** or positive : in Piff. 73 tbe
" divergent *' or negative : assuming that the lenses are glass lenses surroimded by air
thick.
A\A%^d, yfiCi — ri. AtCt^rt,
Plano-concave Lens and the Concavo-convex (or Convexo-concave)
Lens with a deep concavity (the so-called "Negative Meniscus")-
A bundle of incident parallel paraxial rays falling on a Lens of the
first group — supposed to be a moderately thin glass lens surrounded
by air — will be converged to a real focus on the far side of the Lens;
whereas, under the same circumstances, a beam of parallel rays wiD
be made divergent by passing through a Lens of the second groupi
On account of this characteristic treatment of incident parallel rayii
the Lenses of the first group are sometimes called "Convergent"
^ ^ \ 142.1 Refraction of Paraxial Rays Through a Thin Lens. 181
Unses, and those of the second group are called "Divergent" Lenses.
But this property depends essentially on the thickness of the Lens
and on the relative index of refraction.
142. Optical Centre of Lens. Any ray, whether paraxial or not,
irhich leaves the Lens (supposed to be surrounded by the same medium
on both sides) in a direction parallel to that of the corresponding inci-
dent ray, will have passed, within the Lens, (either really or virtually)
through, a remarkable point on the optical axis called the Optical
Cenire of the Lens. In order to prove this statement, and at the
Fxo. 74.
OmcAX. Cbxttrb of I,bn8 at thb point marked O* Any ray passinsr throturh O emersres from
tiie lens in ^ direction parallel to the direction of the incident ray : the lens beinff surrounded on
bodi skies t»y tl>^ same medium.
A\C\ - n. A\C\ - n, A\A% - rf, I CiBiO - aj' - Z CtBtO = a,.
same time to determine the position of this point, let us draw through
the centres C|, C, of the two Lens-surfaces any two parallel radii
C B f CJS^ (Fig- 74) • then the point 0 where the straight line ByB^
crosses the optical axis is a fixed point. For in the similar triangles
OCiBx and OCfit we have:
Cfi CiBi
Cfi " C^,'
Cfi - Cji4,'
182 Geometrical Optics, Chapter VI. [ § 1.
and, hence:
And, since
Afi = A^Ay^ + Afi = Afi - A^A^ = i4|0 — <i,
where i ^ A^A^ denotes the thickness of the Lens, we obtain final
Afi^—'—i. fe
fi - r.
Thus, for a Lens of given form and thickness, this equation enables
to determine the abscissa, with respect to the vertex Ay^ of the fij
surface of the Lens, of the point 0, which is a fixed point on the optic
axis, since its position is independent of the inclination of the pair
parallel radii C^B^ and C,5j. If, therefore, B^B^ represents the pa
of a ray within the Lens going through this point 0, the directions
the corresponding incident and emergent rays must be parallel, sin
the angle of refraction d^ at the first surface is equal to the angle
incidence a, at the second surface. The optical centre 0 will be reoc
nized as the internal centre of similitude (or perspective) of the tt
circles in the plane of the diagram which have C^, C, as centres aJ
fp r, as radii, respectively.
In the figure, as drawn here, the incident ray QB^ crosses the a3
virtually at the point designated by iV, and the emergent ray B^
which is parallel to QB^ crosses the axis virtually at the point desi
nated by JV'. If the ray is a paraxial ray, the points JV, N' will be
pair of axial conjugate points — the so-called "Nodal Points" of t
Lens.
In case one of the surfaces of the Lens is plane, the optical cent
0 will coincide with the vertex of the curved surface, as is evide
from formula (96). When the curvatures of the two surfaces of tJ
Lens have the same sign, as is the case with either the positive
negative meniscus, the optical centre does not lie within the Lens at a
ART. 41. FORMULA FOR THE REFRACTION OF PARAXIAL RATS THROOG
AN INFINITELY THIN LENS.
143. When the Lens is so thin that we may neglect its thickne
(d) in comparison with the other linear magnitudes which are measun
along the optical axis, we have the case of an Infinitely Thin Leu
In comparison with the other dimensions the thickness of the Lens
often quite small, but an Infinitely Thin Lens is, of course, unreaU
able, so that such a Lens is sometimes called an "ideal Lens*'. If 1
144.]
Refraction of Paraxial Rays Through a Thin Lens.
183
>ut AiA^^ d = o, this is equivalent to regarding the vertices -41,-4,
IS coincident, and the Lens-surfaces as, therefore, in contact with each
3ther. The approximate formulae that are obtained under these cir-
cumstances are often of very great utility, especially in the preliminary
de^gn of an optical instrument; and in many cases such formulae are
quite sufficient to enable us to form a proper idea of the behaviour
and general characteristics of a real Lens of not too great thickness.
144. Conjugate Axial Points in the case of the Refraction of Paraxial
^ys through an Infinitely Thin Lens. In accordance with the graphi-
cal mode of representation explained in § 113, an infinitely thin lens
may be represented in a diagram by a straight line perpendicular to
the optical axis. The point A (Fig. 75) where this straight line crosses
the a3ds is not only the common vertex of the two spherical surfaces,
but it is also the position of the optical centre of the Lens; for, ac-
cording to formula (96), when d = o, the optical centre coincides with
the common vertex of
the Lens-surfaces. The ^
form of the Lens is
shown in the figure by
the positions of the cen- a
tres Ci, C, of the two
spherical surfaces. If
^ the second of formulae
(94) we put d = o, we
have «2 s ttj. Impos-
^ this condition, and
^ding the two other
^uations, and at the
**nie time writing here
« and u' in place of u^
Pio. 75.
Refraction op Paraxiax^ Rats through iNFiinTBLY
TBnr I«BNS. M, Af are a pair of conju^rate axial points.
The points designated in the diagram by M, M\ C\, Ct and
A may be ransred alonar the optical axis in any order what-
ever, dei)endinsr on the form and optical properties of the
lens and on the direction of the incident ray AfB. The lens
represented in the diagram is a Biconvex I^ens. the lens-
medium beinir more hij^hly refracting than the surtoundinir
medium.
ACi='ri, ACt^tt. AAf=u, AAf^'i/.
ttj, respectively, we obtain the useful abscissa-relation for the
'fraction of paraxial rays through an infinitely thin Lens in the ioU
'o^'form:
(97)
^expression on the right-hand side of this equation, involving only
™e Lens-constants, fj, fj and n'/«» has for a given Lens a perfectly
^*^te value. If we denote this constant by i//, so that
I _ n' — n fi ^ i\
7" n Kti" rJ
(98)
I
184 Geometrical Optics, Chapter VI. [ § M
the formula above may be written as follows:
III
Thus, having determined by means of formula (98) the value of tl
magnitude denoted by /, or else being given its value directly, we os.
ascertain the position of the Image-Point M' corresponding to a giv<
axial Object-Point Af ; that is, knowing u, we can find u\ and w
versa.
It may be remarked that equation (99) is symmetrical with respe^
to u and — u'\ that is, if — w be written in place of u! and — «'
place of w, the equation will not be altered. Hence, if the Object-Pol 1
M is situated on the axis at the point {u, o) and the Image-Point 21
at the point {u\ o), and if the Object-Point is then supposed to 1
transferred to a new position (— u\ o), the new Image- Point ^^
have the position (— w, o). Or, if we adjust the Lens so as to prodx^
at a given point on the axis the image of a fixed Object-Point, we c^
find two positions of the Lens which will accomplish the purpose, vi^
a position for which the Object-Point has the abscissa u and the ImagS
Point the abscissa w' and a second position for which the Object
Point has the abscissa — u' and the Image-Point has the abscissa —
145. The Focal Points of an Infinitely Thin Lens. Putting u-^
in formula (99), we obtain:
where E' designates the position on the optical axis of the Secondaifl
Focal Point of the Infinitely Thin Lens. Similarly, putting n' =■ o""
we find :
where F designates the position of the Primary Focal Point of the Len*
Thus, the two Focal Points F and E' of an Infinitely Thin Lens a-
equidistant from the Lens^ and on opposite sides of it.
The imagery of an Infinitely Thin Lens is completely detennins
so soon as we know the positions of the three points A, F and S
and, since the point A lies midway between the Focal Points Fand S
Lenses may also be divided into two classes, as follows:
(i) Lenses in which the points F, A, E' are ranged along the optica
axis in the order named in the sense in which the light is props^te^
(therefore, in our diagrams from left to right) ; so that for Lenses CJ
this type incident rays which proceed parallel to the axis will be com
verged to a real focus at the point E' beyond the Lens, as shown i^
§ 145.1
Refraction of Paraxial Rays Through a Thin Lens.
185
the first diagram of Fig. 76: and, hence, such Lenses are called Con-
vergent Lenses. They are also called Positive Lenses, because FA = /
is positive, if we take the direction along which the light is propagated
as the positive direction of the ray. Assuming that n' > » (as, for
y
^^V^
/'^-••^
A
^ir\
E'
T::^^
Fio. 76.
^^^^Ctaoi on I^ft represents a Convergent I^ns (/> 0) ; diagram on Riffht represents a Divergent
^*»»«(/<0).
/"FA, ^^EfA, /-— ^.
^'^mple, in the case of a glass lens in air), the sign of/, according to
'^rmula (98), is the same as the sign of (i/fi — i/^J* ^^ the Bicon-
^^ Lens (fi > o, r^ < o), the Plano-convex Lens (r^ = oo, r^< o,
^ ''i > o, fa = 00) and the Positive Meniscus (r, > fi > o) — that is,
'^^ all Lenses which are thicker in the middle than towards the edges— r
^^ sign of (i/fi — i/fj) is positive, and, therefore,/ > o; and, hence,
^ already stated (§ 141), such lenses (provided n' > n) are convergent.
(2) Lenses in which the order of the above-named points is £', A, F.
'^r lenses of this class incident rays which proceed parallel to the
^^^ are made divergent by passing through the Lens, and emerge
^ if they had come from a virtual focus at the Secondary Focal Point
•E', lying in front of the Lens, as shown in the second diagram of
*^g. 76. Accordingly, such Lenses are called Divergent or Negative
J^^nses, since here FA = / is negative. In case »' > n, the sign of/, as
^bove stated, agrees with the sign of (i /fi — i /fj). In the Biconcave
l-ens {r^ < o, r, > o), the Plano-concave Lens (fi = 00, r^ > o or fj < o,
^ = 00) and the Negative Meniscus (r^ < r^ < o) — that is, for all
lenses which are thinner in the middle than they are at the edges —
^ sign of (i/fi — 1/^2) is negative; and, hence (provided n' > n),
«ttch Lenses are divergent.
Several special forms of Lenses may be mentioned here, viz. :
The Equiconvex and the Equiconcave Lens, for which rj = — ^1,
which we have, therefore, / = nr,/2(»' — n). In the case of the
^^nvex Lens fj > o, and, therefore (assuming n' > n), we have
'^0; whereas for the Equiconcave Lens r, < o, and, therefore,/ < o.
^e Plano-convex and the Plano-concave Lens : Assuming that the
^ surface of the Lens is the plane surface, we have here rj = 00 ;
§ 148.] Refraction of Paraxial Rays Through a Thin Lens. 187
Since the Focal Lengths of an Infinitely Thin Lens are equal to
the distances of the Lens from the two Focal Points, the theory of
the refraction of partial rays through such a Lens is very similar to
that of the refraction of paraxial rays at a spherical surface; only, in
the case of the Lens the theory is simpler, because the Focal Points
are equidistant from the Lens.
147. Putting tt, = u\ in formula (95), and writing «, u' in place
of ii|, «2» lespectively, we obtain for the Lateral Magnification of an
Infinitely Thin Lens :
F = ^- = ~; (102)
that is, the ratio of the linear dimensions of the Object and Image is equal
^ ^he ratio of the distances of the Object and Image from the Lens.
If X, x' denote the abscissae, with respect to the Focal Points F, £',
of the conjugate axial points Af, Jlf ', respectively, that is, if
FM = X, E'M' = x\
then
u = AM = AF + FM ^ X -/,
u' = AM' = AE' + E'M' ^ x' " e'\
^^d substituting these values in formulae (99) and (102), we obtain
the so-called "Image-Elquations" of an Infinitely Thin Lens in the
following simple and convenient forms:
y x
"^c absdssa-equation is the same as the characteristic equation of
y^e Central CoUineation of two plane-fields for the case when the
^Variant c = + i (see § 134). It may be derived at once from the
P^jective relation :
(MAFE) = (M'AF'E'),
y^nere E and F' are the infinitely distant points of the two correspond-
>^ ranges of Object-Points and Image-Points, respectively, lying upon
™c optical axis of the Lens.
148. Construction of the Image Formed by the Refraction of
^^^noial Rays through an Infinitely Thin Lens. In the diagrams
(Rgs. 77 and 78) MQ represents a very short Object-Line perpen-
dicular to the optical axis at the axial Object-Point M. The Infi-
ffitely Thin Lens is itself represented by the straight line y perpendicu-
br to the optical axis at the point designated by A . Fig. 77 shows the
188
Geometrical Optics, Chapter VI.
[§
case of a Convergent Lens, and Fig. 78 shows the case of a Divergent
Lens. Since the point A where the optical axis meets the Infinitc^li^
Thin Lens is also the optical centre of the Lens ( §144), any ray
Cp/y^«/ye/f/ leu*
■* T^V
Pio. 77 and Pio. 78.
Refraction op Paraxial Rays through an Inpinitblt Thin I«BNa. Constmction of Ii
towards A will emerge from the Lens without change of directio
and, hence, the straight line joining any pair of conjugate points ^
Q' will go through this point A. Thus, we see that the Object-Spa^
and Image-Space of an Infinitely Thin Lens are in perspective relati
to each other with respect to the point A as centre of perspectiv—""^"^
This is obvious also from formula (102). As was remarked abo
(§ 146), the imagery in the case of an Infinitely Thin Lens is qui
similar to that of a single spherical refracting surface, where the cen
of the surface is the centre of perspective of the Object-Space
Image-Space.
Knowing the positions of the axial points A^ F and E' of an In
nitely Thin Lens, we may easily construct the Image M'Q' conjugau-
to MQ. All that we have to do is to locate the position of the poi
Q\ and then draw M^Q' perpendicular to the optical axis at M\
point of intersection of any pair of emergent rays emanating original^ J^
from the Object-Point Q will suffice to determine the correspondii^iBr
Image-Point Q\ In the diagrams (which need no farther explan^'^
tion) three such rays are shown, any two of which are sufficient.
The imagery in the case of the Refraction of Paraxial Rays througf*
an Infinitely Thin Lens is exhibited in the two diagrams, Figs. 79 and
80, the first of which shows the case of a Convergent Lens and the
i
Refraction of Paraxial Rays Through a Thin Lens.
189
the case of a Divergent Lens. The numerals 1,2,3, ^tc, desig-
irious successive positions of an Object- Point, which, starting
ifinite distance in front of the Lens, is supposed to travel towards
ns along a straight line parallel to the optical axis. The cor-
ding positions of the Image-Point on the straight line connect-
e point V with the Secondary Focal Point £' are designated
liagram by the same numerals with primes. Thus the straight
i', 22', etc., connecting each pair of conjugate points, will, if
re drawn, all pass through the perspective-centre A. In both
)f Lens the Object-Point and Image-Point coincide with each
It the point V on the Lens itself, and hence the two Principal
(§ 139) of an Infinitely Thin Lens coincide with each other at
int A. If the Object-Point lies beyond the Lens (that is, to
;ht of the Lens in the diagrams), it is a virtual Object- Point.
ong as the Object is in front of the Primary Focal Plane of a
:rgent Lens (Fig. 79), we have a real, inverted Image lying on
0!¥€ry*nt Lmn*
Fio. 79 and Fio. 80.
*cnoif OP Paraxial Kays through Infinitely Thin I<en9. Imagery of Ideal I^ens.
^'^fnis 1. 2, 3, etc.. show a number of selected positions of an object-point supposed to move
t to rigfht alonff a straight line parallel to the optical axis. The numerals with primes show
i^xmdinff positions of the imaire-point on the straight line Ef V.
' Side of the Secondary Focal Plane; and when the Object is in
imary Focal Plane, the Image is at infinity. If the Object lies
n the Primary Focal Plane of a Convergent Lens and the Lens
J 151 J
Refraction of Paraxial Rays Through a Thin Lens.
193
passings
section .
ame nr.
through all three of the lenses. In the diagram the inter-
Lt L] is represented as virtual, and that at L, is real. In the
y L[ and L^ designate the positions of the points where the
1; -&; ^
Pio. 81.
fMTJiXVSMD DC DBDUdNO COTB8*9 THEOREM. AlPi, Atl\, A%Pu represent three infinitely thin
^e^g%. lA tlae diagrmm these lenses are all represented as concave or divergent lenses. MQ is
«tiied.-tiM Perpendicolar to optical axis of ssrstem of lenses. QFiPUhU is the outermost ray pro-
mrtfaftwo^ the endrpolnt Q of the object and traversins all the lenses. The eye is supposed to
Uvksecdod thcaxlAmt L^. KU is the " appaient distance ** of the object from the eye.
Tay crosses the axis after passing through the iirst and second lenses,
respectively: both of these intersections, as shown in the diagram, are
^'^rtual. From the points Q, P^ and Pj draw straight lines parallel
to the optical axis and produce them until they meet the straight line
^^snnined by the emergent ray PsLi, and from each of these points
^ intersection let fall perpendiculars on the axis at the points desig-
nated by K, J and H, respectively.
Tke "apparent distance" of A^P^^y regarded as an object viewed
^^^^^^ the Lens A^ is HL,; and from the similar right triangles
« the figure we obtain the following proportions:
tlL^ A2P2 A2L2,
^ consequently :
A^L^ AjfPi
A,
7»
^'' - ^^' (■ + 7^)
Since L,, L, are a pair of conjugate axial points with respect to the
Lens Ap we have, according to formula (99) :
1 II.
A^2 A^^ /j
/j denotes the Focal Length of the Lens A^. Introducing this
14
194 Geometrical Optics, Chapter VI. [ § 15L
value of i/ilgLj in the above, we obtain:
HL', = 4,l; -
/,
In the same way, the "apparent distance" of AiP^ regarded as an
Object viewed through the Lenses A^ and A^ is JL^. Here we have
the proportions:
or
JK - «i;G + ^)-
In the same way, also, we have here:
I i_ I _ I A^
A^,''Aj:,'f,''HV,A^
and hence:
,_ , AjAfAtL', AiAjAj/^ , AiAtA,AiAtI^
JL,-A,L,- ^^ - ^^ + yj^
Again, the "apparent distance" of the Object-Line MQ viewed
through the three Lenses A^, Aj and A^ is KL'^; and, as before:
JL^ A iPi A |X»i
or
I i_ I _ HL'i / 1 ?^ _ i
I (A,L', HL'^\ 11 ( Aj:, HL'\ i
Also,
\
4
I
i
i
so that
and, finally we obtain the following formula for the "apparent <&• i
§ 152.] Refraction of Paraxial Rays Through a Thin Lens. 195
tance" of the Object viewed through the Lenses A^, A^ and A^\
MA^'A^A^'A^V^ MA^'A^AyA^L'^ MA^A^A^-A^L'^
/i/i /i/s /2/s
MAy^' A^Az' ^2-^3* A^L'ii
/1/2/3
(107)
This is CoTEs's Formula for the case when the system is composed
of three lenses A^, A^, A^\ but the law of the formation of the terms
is apparent, and the formula can be immediately written for a system
of any number of Lenses. Thus, if we observe that the piece of the
optical axis included between the Object at M and the eye at L,
may be considered as divided by the Lenses at Ay, A^^ A^ into two,
three and four segments in the following ways:
ML'^ = MA, + ^,L; = MA^ + ^^; = MA^ + AJL'^
= MA, + A,A^ + 4^; = MA, + A,A^ + AJL'^
= MA^ + ^^a + AJL'^ = MA, + A,A^ + A^A^ + AJL^
it will be seen that the members of each of these groups when multi-
.plied together form the products which are the numerators of the
fiacrtions on the right-hand side of equation (107), while the denomi-
nators are the products of the Focal Lengths of the Lenses which
in the numerators; the signs of the fractions being positive or
;ve according as the number of factors in the denominator is
even or odd. The "apparent distance" is equal to the real distance
added to the algebraic sum of the set of fractions whose numerators
^Tu\ denominators are formed according to the rule just explained.
A general proof of Cotes's Theorem was given by Lagrange,^
who "was evidently acquainted with Smith's work on Optics, as he
fefers to it in his paper.
152. Lord Rayleigh in the article above-mentioned (§ 150) quotes
at length several of the corollaries which Smith derives from Cotes's
Theorem, the first of which is as follows:
''WTiile the glasses are fixt, if the eye and object be supposed to
change places, the apparent distance, magnitude and situation of the
object will be the same as before. For the interval ML[ being the
> J. L. DK Lagsangb: Sur la thforie des lunettes: M (moires de VAcad. de Berlin (1780).
196
Geometrical Optics, Chapter VI.
I § 152.
same, and being divided by the same glasses into the same parts,
will give the same theorem for the apparent distance as before."
Thus, in Fig. 8i, if we suppose that the axial point of the Object
is at L3 and that the centre of the pupil of the eye is on the axis at
the point designated by ilf, then A^P^ will be proportional to the
breadth at the object-glass -4., of the bundle of incident rays from the
axial Object-Point L3, and MQ will be proportional to the breadth of
the corresponding bundle of emergent rays where they enter the eye
at M, and from the figure we have evidently:
MQ ^ xl;
whence is derived Smith's Second Corollary, which he states as fol-
lows:
"When an object MQ is seen through any number of glasses, the
breadth of the principal pencil where it falls on the eye at i,, is to
its breadth at the object-glass A^, as the apparent distance of the
object, to its real distance from the object-glass; and consequently
in Telescopes, as the true magnitude of the object, to the apparent"
This very striking result can be put in a different form. Thus»
from the figure, we obtain:
MQ^^MQ^ A^ A^^ML, AJ^ AJ^
A^i AiFi A2P2 A^^ AiLi AJ^i A^^
and therefore:
KL'ji ^ AiL'^'A^L^'A^Ll
MLi AiLi ' A^L'i • A^L'2
The expression on the right-hand side of this equation, according to
formula (105), is the value of the Lateral Magnification y^lyi at the
conjugate axial points L^, L^, so that we have:
KL',
Moreover,
yi
yx
ML,
XL; tan Z^tLiPt
MLi " tan /lA^L'^P^
and hence we derive the formula:
tan^i
tan^l
^'^tan^; = y,-tan^i;
§ 152.] Refraction of Paraxial Rays Through a Thin Lens. 197
or, if the system consists of m Lenses:
yl,' tain ei-^y^' tan e^. (io8)
This formula, which was given by Lagrange* more than fifty years
sifter the publication of Smith's Optics, is a particular case of the gen-
eral formula usually known in Optics as the Helmholtz Equation
(see § 194).
> J. L. DK Lagrangb: Stir une loi g^n^rale d*optique: Mhnoires de VAcad. de Berlin,
CHAPTER VII.
THE GEOMETRICAL THEORY OF OPTICAL IMAGERY.
I. INTRODUCTION.
ART. 43. ABBB*S THEORY OF OPTICAL UCAOSRY.
153. The function of an optical instrument is to produce an image
of an external object. Elach point of the object is the base (or vertei)
of a bundle of rays, of which, in general, only a part is utilized in the
formation of the image. These object-rays which are affected by tto
instrument are called the ''incident" rays. Within the apparatus these
rays undergo a series of refractions (or reflexions) at the plane or curved
boundary-surfaces of suitably disposed optical media; and, thusmofr
fied, they **emerge" into the last medium and form there a more of
less perfect image of the object, which may be "real" or "virtual i
etc. ; the nature of the image in the several respects of position, dimeft'
sions, orientation, etc., depending primarily on the peculiarity 9sA
design of the instrument itself. Proceeding from any point P (rf the
Object, a bundle of incident rays ''enters" the opticaJ instrumenti
and emerging therefrom, a portion of these rays at least, if not all »
them, will intersect ("really" or "virtually") in the corresponding!^ \
"conjugate", point P' of the Image. In the case of an ideal, orpo-
metrically perfect, image, all of the emergent rays correspondiflg ^
the rays of the bundle of incident rays P will intersect in the Imag^
Point P'; so that a homocentric bundle of object-rays will be (astb^
German writers say) "imaged" {ubgebildet) by a homocentric buttfl^
of image-rays.
154. Until comparatively recent times the method of investigate
of the relations between image and object in Optics was to advance
by a process of mathematical induction from simple special cases tO
more complex general cases of homocentric imagery. This mcdiod
was used with conspicuous success by Roger Cotes (§150), W
Plumian Professor of Astronomy in Cambridge University, ^Ao*
brilliant and original contributions to optical science were cut short
by his untimely death (1716) at the age of thirty-four years. The
same method was employed also by C. F. Gauss in his famous Diop'
trische U titer suchungen (Goettingen, 1841), who developed compktd]
the theory of the refraction of paraxial rays through a centered «>•
198
The Geometrical Theory of Optical Imagery. 199
f infinitely thin lenses. By substituting in place of the original
such as the radii, refractive indices, etc., certain constants of a
more general kind. Gauss obtained remarkably simple formulae,
marked a great advance in optical theory and added a new
ragement to such investigations. But even Gauss, with his
)rdinary insight and rare gift of analysis, seems not to have dis-
i that the general laws of optical imagery are independent of all
I assumptions as to the particular mode of producing the image.
EBius,* indeed, came nearer to the real and essential idea of
tical image when he pointed out that the unique connection be-
Object-Point and Image-Point in the case of the refraction of
ial rays at a spherical surface is equivalent to the expression
relation of CoUinear Correspondence between Object-Space and
i-Space; and that if this is true in the case of a single spherical
ting surface, it must be true also for the relation between object
aage in the refraction of paraxial rays through a centered system
erical refracting surfaces; and, hence, finally, that all the formulae
ig the relation between object and image in such a case as this
ledudble from the thgory of Collinear_Correspondence. This
itment was quickly seized by other investigators (asFTLiPPiCH,*
CK* and H. Hankel*) who, following the lead of Moebius, and,
im, employing the methods of projective geometry, extended
lea of optical imagery to less simple cases. Thus, for example,
*PICH* showed that there is also collinear correspondence of ob-
ad image in the case of infinitely narrow bundles of rays incident
pherical refracting surface at finite slopes. Yet neither Moebius
If nor any of his followers in this mode of treating the matter
ble to discard entirely the idea that some kind of Dioptric action
»ential for the production of an optical image. At least not one
m stated distinctly that a purely geometrical assumption was all
?as necessary, viz., that an optical image is produced by rays.
.. A remarkable paper **0n the General Laws of Optical Instru-
" was contributed in 1858 by James Clerk Maxwell to The
F. Moebius: Entwickelung der Lehre von dioptrischen Bildern mit Huelfe der
itioiw-Vcrwandschaft: Leipziger Berichte, vii. (1855), 8-32.
Lippich: Fundamentalpunkte eines Systemes centrirter brechender Kugelflaechen:
ungen dts naturwissenschaftlichen Vereines fUr Steiermark, ii. (1871), 429-459.
Bcck: Die Fundamentaleigenschaften der Linsensysteme in geometrischer Dar-
: Zfl. /. Math. u. Phys., xviii. (1873), 588-600.
Hankkl: Die EUmente der projekiivischen Geomeirie in synthetiscker Behandlung
. 1875).
Lippich: Ueber Brechung und Reflexion unendlich duenner Strahlensysteme an
tecben: Wiener Denksckr., xxxviii. (1878), 163-192.
200 Geometrical Optics, Chapter VII. [ § 155.
Quarterly Journal of Pure and Applied Mathematics^ ii., 233-246; it
is reprinted in the collection of Maxwell's Scientific Papers, vol. i.,
271-285. In an introduction to this article, Maxwell describes the
undertaking as follows:
"The investigations which I now offer are intended to show how
simple and how general the theory of optical instruments may be ren-
dered, by considering the optical effects of the entire instrument, with-
out examining the mechanism by which these effects are obtained. I
have thus established a theory of 'perfect instruments', geometrically
complete in itself, although I have also shown that no instrument
depending on refraction and reflexion (except the plane mirror) can
be optically perfect."
A "perfect instrument" is one which is free from "certain defects
incident to optical instruments"; thus, according to Maxwell, "a
perfect instrument must fulfil three conditions:
"I. Every ray of the pencil, proceeding from a single point of the
object, must, after passing through the instrument, converge to, or
diverge from, a single point of the image. The corresponding defect
when the emergent rays have not a common focus, has been appro-
priately called (by Dr. Whewell) Astigmatism.
**II. If the object is a plane surface, perpendicular to the axis of
the instrument, the image of any point of it must lie in a plane perpen-
dicular to the axis. When the points of the image lie in a curved
surface, it is said to have the defect of curvature.
*'III. The image of an object on this plane must be similar to the
object, whether its linear dimensions be altered or not; when the
image is not similar to the object, it is said to be distorted.**
Assuming that the image is free from these three defects, and &
therefore a "perfect image". Maxwell derives formulae for the rela-
tive positions and magnitudes of the object and image which are pr^
cisely equivalent to the formulae obtained by Gauss ; but the difference
consists in the fact that, whereas Gauss's investigations are based
on certain physical assumptions not only in regard to the Law o»
Refraction of light-rays, but also as to a centered system of spherical
surfaces and paraxial rays, the modus operandi is left out of conader-
ation entirely by Maxwell, who shows that an optical image, how-
ever it may be produced, provided it is free from the geometrical
* 'defects" above enumerated, must have certain perfectly definite
geometrical relations with the object. This very important idea seems
to have been clearly perceived and distinctly stated by Maxweix
first of all.
The Geometrical Theory of Optical Imagery. 201
. The most notable contribution in recent years to the litera-
)f Geometrical Optics is Dr. S. Czapski's Theorie der optischen
menu nach Abbe, the first edition of which was published in
lu in 1893. In this brilliant work, recognized immediately as
och-making book, was set forth for the first time a complete and
riy exposition of the remarkable theories of Professor Abbe, of
JE, without a knowledge of the investigations of Moebius and
VELL, discerned even more clearly than they that the physical
y or mechanism which was employed in the actual formation
optical image was in no wise involved in the geometrical theory
ical imagery; so that without any special assumptions whatever
the construction or constitution of the optical apparatus, and
rithout reference to the physical laws of reflexion and refraction,
iuced in the simplest and most direct way all the laws concerning
lative positions, dimensions, etc. of the object and image.
IS, the fundamental and essential characteristic of optical im-
is a point-to-point correspondence, by means of rectilinear rays,
en object and image; and from this one assumption — at once
lost natural and the most obvious — Abbe, in his celebrated
rsity lectures, used to deduce the general laws of optical images.
t advantage of this is that in investigating an actual image pro-
l by an optical instrument it will be possible to separate what
) laws of this image depends on the general fundamental laws of
J imagery and what is due to the particular mode of producing
Tiage. Moreover, although to-day a certain optical instrument
be a mechanical impossibility, it is possible to say whether such
tem is theoretically practicable; so that the geometrical theory
oint the way of future inventions.
the modem geometry this unique point-to-point correspondence
ans of rectilinear rays between image and object is called ** Col-
ion" — a term introduced by Moebius in his great work entitled
irycenlrische Calcul (Leipzig, 1827).
E THEORY OF COLLINEATION. WITH SPECIAL REFERENCE TO ITS
APPLICATIONS TO GEOMETRICAL OPTICS.
ART. 44. TWO COLLINEAR PLANE-FIELDS.
. Definitions. In this treatment we shall employ the beauti-
l appropriate methods of projective geometry. As some readers
lot be entirely familiar with the terms here employed, a brief
uction may be required.
202 Geometrical Optics, Chapter VII. [ § 158.
The totality of points and straight lines which are contained in a
plane is called a '' plane-field* \ and the plane is then said to be "the
''base "of this system of points and lines. All the points lying on a
straight line of the field, taken together, form a **range of poitarls'\
the straight line itself being called the '*base" of the point-range- A
straight line considered as a whole (that is, without reference to the
points which lie on it) is called a *'ray*\ All the straight lines of a
plane-field which go through one point form a ^*pencU of rays*\ aJid
the common point of intersection of these rays may be regarded as
the "base" of the pencil.
Two plane-fields w and t' are said to be collinear, if to every poiTti P
of T there corresponds one point P' of tt', and to every straight liTie P
of v which goes through P there corresponds a straight line p' of t' %s^h^^
goes throng P\
The totality of rays which go through a single point 0 in sl>^^
is called a ^'bundle of rays'\ so that a bundle of rays consists c^f ^
infinite number of pencils of rays. We speak also of a **buw^^ ^
planes*\ meaning thereby the totality of planes which pass thrc:^^^
one point 0. In either case the point 0 which is common to all ^^
elements of the bundle is the "base" of the bundle. A **she€^ ^
planes'' is the term applied to the totality of planes which all hav^ *^™J
common line of intersection: thus, in a bundle of planes are comp^^*^^
an infinite number of sheaves of planes. The common line of i^*-^^'
section is the "base" of the sheaf of planes.
A plane-field t and a bundle of rays 0' are said to be coUinear ^t^
each other if to every point P of t there corresponds a ray p' of ff^ ^^
to every straight line I of t that goes through P there corresponds a pt^^
\' of the bundle 0' that contains the straight line p'.
And, again:
Two bundles of rays 0 and 0' are said to be collinear with each af^^'
if to each ray p of 0 there corresponds a ray p' of 0\ and to each pJ^^
\of 0 that contains p there corresponds a plane X' of 0' that contains P '
158. Projectiye Relation of Two Collinear Plane-Fields.
Two collinear plane-fields t and t' are also called **projective*\ i^
cause to each harmonic range of four points of t there corresponii (^
harmonic range of four points of v\
Thus, if P, Q, Ry S (Fig. 82) are a harmonic range of four pointe
of the plane-field tt, and if P', Q', R\ S' are the four correspon^ng
points of the collinear plane-field tt', in the first place, since the pointi
P, Q, R, S all lie upon a straight line 5, the points P', ^, J?', y must
all likewise lie upon a straight line s' which is conjugate to 5. Let
§ 159.] The Geometrical Theor>' of Optical Imagery. 203
A £ CD he any quadrangle of the plane-field v, such that the two
opposite sides A B and CD intersect in the point P, and the other two
opposite sides AD and BC intersect in the point Q, while the two
diagonals BD and AC go through the points R and 5, respectively.
To this quadrangle of t there will correspond a certain quadrangle
A'B'C'D' of x', such that the two opposite sides A'B' and CD' inter-
sect in the point P', the other two opposite sides A'D' and B' C
^^'
Fio. 82
PROjBcnvB Relation op Two Collinbar Plane-Fields.
ntersect in the point Q\ and the two diagonals B'D' and A'C go
^ough the points R' and 5', respectively. Accordingly, the points
•P', Q, R\ 5' are also a harmonic range of points; and this is the
condition that the two plane-fields tt and t' shall be projective.
By a similar method we can show also that two collinear bundles
of rays or a bundle of rays and a plane-field in collinear relation are
projective to each other.
159. The so-called "Flucht" Points of Conjugate Rays. Let 5
^Jwi s! denote two conjugate rays of the collinear plane-fields t and
"^^ Since, as has just been shown, the point-ranges s, s' are projective,
H follows that the Double Ratio {PQRS) of any four points P, Q, R, S
^ s is equal to the Double Ratio (P'Q'R'S') of the four corresponding
prints P\ Q', P', 5' of s\ That is,
PR PS _P^ P^
QR' QS" Q'R' • O'y*
If we suppose that P, Q and R are three fixed points of s and that
SissL variable point, the Double Ratio (PQRS) will vary in value
as the point 5 moves along s; and if the point S moves away to an
infinite distance until it coincides with the infinitely distant point*
'Ev-ery actual straight line contains one (and only one) infinitely distant (or ideal)
point, and all rays having in common the same infinitely distant point are parallel.
204 Geometrical Optics, Chapter VI I. [ § 1^
/ of 5, we shall have :
PR
(PQRI) = (P'Q'RT) = ^ .
where /' designates the point on s' which corresponds to the infinit^^^'
distant point / of s. Since P, Q and R are three actual, or finit
points of s, no pair of which are supposed to be coincident, the vali
of the ratio PR : QR is finite ; and hence the point /' conjugate to tm^^ '^
ideal point I of s is a determinate and^ in general^ an acttml^ or fini^^^^ **'
point of s\
Similarly, if /' designates the infinitely distant point of s\ we sha
have:
iPQRJ) = (P'Q'R'J') = ^;
SO that the point J which corresponds to the infinitely distant, or ic
point J' of s' iSy likewise, a determinate and, in general, an actual^ c^ o,
finite, point of s.
In general, therefore, the points J and /', corresponding to tl—^Be
infinitely distant points /' and / of s' and s, respectively, are actus il,
or finite, points having perfectly determinate positions on s and
respectively. In the German treatises the points J and /' are call*
the *'Fltu:ht'* Points of the two projective point-ranges 5 and s'.
It will be remarked that we are careful to say that the so-call^sd
**Flucht" Points are "in general" actual, or finite, points; for inc>^«
special case, viz., when
PR P' R'
(PQRJ) = (P'Q'RT) = ^ = ^'
the "Flucht" Point r will coincide with the infinitely distant poi***
/' of s\ and the 'Tlucht*' Point J will, likewise, coincide with tb^
infinitely distant point I of s; and in this particular ccLse the infinUdy
distant points I and J' of the projective point-ranges s and s' vrill also ^
a pair of conjugate points,
160. The so-called 'Tlucht*' Lines (or Focal lines) of Conjiig^
Planes. In the plane-field ir consider now a quadrangle A BCD (fig-
83) such that the two pairs of opposite sides form two pairs of paraM
straight lines. The two parallel sides A B and CD intersect in die
infinitely distant point P, and, similarly, the other two parallel sidtt
AD and BC intersect in the infinitely distant point Q\ so that if H
and 5 designate the infinitely distant points of the two dis^^onals BD
and A C, respectively, the four points P, Q, R, S are a harmonic
I
1
The Geometrical Theory of Optical Imagery.
205
X of points of the infinitely distant, or ideal, straight line i of the
: T.
the collinear plane-field t' the ray A'B' conjugate to ^45 will
irough the point P' conjugate to the infinitely distant point P
e ray AB; so that the point P' is therefore the *Tlucht" Point
Fio. 83.
' Flucht** I«inb i' OP THB Planb-Fibld v' corrbspondino to the Infxxitbly Distaitt
BT I«IKB OP THB COIXIlf BAR PLANB-FiELD V.
5 ray A'B\ Obviously, the point P' is also the '*Flucht" Point
e ray CD' conjugate to CD, Precisely in the same way, the
(/, conjugate to the infinitely distant point Q of the parallel
AD and BC, is the common "Flucht" Point of each of the rays
and 5'C' conjugate to the rays AD and 5C, respectively. Let
id 5' designate the positions of the **Flucht" Points of the rays
(conjugate to BD) and A'C (conjugate to AC), respectively,
^ce (§ 158) the points P\ Q\ R\ S' area harmonic range of points,
all lie on a certain definite straight line i' of the plane-field t'\
tiw straight line i\ which is conjugate to the infinitely distant straight
I of the plane-fi^ld t, is the locus of the **Fluchi'* Points of all the
of the plane-field t' collinear with t.
milarly, there is a certain straight linej of the plane-field t, conjugate
^ infinitely distant straight line j' of the collinear plane-field t\
^ is the locus of the ** Flucht*' Points of all the rays of v.
erman writers call these two straight lines j and i' the *' Flucht''
Ul the infinitely distant points of a plane are assumed to lie in an infinitely distant,
ll. straight line. The ideal line of a plane must be a straight line, because every
I straight line of the plane meets it in only one point — the infinitely distant point
tline; whereas a curved line may have in common with a straight line more than one
Just aa a pencil of parallel rays determines one infinitely distant point common
the rays, so a sheaf of parallel planes determines one infinitely distant straight line
on to all the planes.
206 Geometrical Optics, Chapter VII. [§161
Lines (or ''Gegenaxen'') of the two projective plane-fields. We shall
designate them hereafter, from the stand-point of Optics, as the Focd
Lines of the two conjugate planes t and t\
If two plane-fields are collinear, then, in general (that is, except in
one particular case considered in § i6i below), to the infinitely distant^
or idealy straight line of one field there corresponds an actual, or finite,
straight line of the other field, the so-called ''Focal Line'' of thatfidd.
To a pencil of parallel rays in one plane-field there corresponds
therefore a pencil of rays in the other field which all intersect in a
point situated on the Focal Line of that field; or, as we might say,
the Focal Line of one plane-field is the locus of the bases of pencils of
rays which are conjugate to pencils of parallel rays of the coUvmv
plane-field.
161. AflSnity of Two Plane-Fields. The exceptional case men-
tioned above cannot be passed over without some explanation. The
points P\ Q\ R y 5' of t', corresponding to the infinitely distant
points P, Q, Ry S oi V are, in general (as was stated), actual, or finite,
points, and determine, therefore, an actual, or finite, straight line ♦';
except in the one particular case when the quadrangle A'B'CJy^^
well as the quadrangle A BCD, has each pair of its opposite sides
parallel. In this special case the points P', Q', R', S' will be ranged
along the infinitely distant straight line j' of the plane-field x', aO"
the Focal Line V will therefore coincide with the infinitely distant
straight line f.
This special case, in which the two Focal Lines j and V are also <fe
infinitely distant straight lines i and f of the collinear plane-fields t
and tt', respectively, is the so-called case of ** Affinity'' of ihe two pfa**"
fields.
This extremely important special case will be met with again.
Here we merely call attention to it.
ART. 45. TWO COLUNEAR SPACB-STSTEMS.
162. Two Space-Systems 2 and 2' are said to be collinear with oaA
other if to every point P of ^ there corresponds one {and only one) povA
P' of S', and to every straight line p of 2, which goes through P, ft^
corresponds one straight line p' of S' which goes through P\
It is not necessary to think of 2) and S' as two separate and distinct
regions of space; they are to be regarded rather as completely iIlte^
penetrating one another, so that any point, ray or plane in space may
be considered, according to the point of view, one time as belonging
to the system S and another time as belonging to the system 2'. to
L] The Geometrical Theory of Optical Imagery. 207
when we say that two space-systems are collinear with each
r, we mean that the whole of space is in collinear relation with
f; or in the language of the modem geometry, the whole of space
le common **base" of the two space-systems 2, S'.
i the geometrical theory of Optics the two space-systems 2, 2'
distinguished as the Object-Space and the Image-Space^ respect-
r. and the points, rays and planes of space, according as they are
ided as belonging to the one or the other of these two space-
ems, are called Object-Points, Object-Rays and Object-Planes or
ge-Points, Image-Rays and Image-Planes. Since the relation be-
m the Object-Space and the Image-Space is perfectly reciprocal,
€ is no essential difference between them ; whence is deduced at
5 the theorem known as the Principle of the Reversibility of the
^Paih (§ i8).
direct consequence of the unique point-to-point and ray-to-ray
^pondence between Object-Space and Image-Space is plane-to-
»e correspondence; so that to every plane v of the Object-Space
^corresponds a definite plane v' of the Image-Space, and vice versa,
IS, using the language of the modern geometry, we may say:
'» two Collinear Space-Systems to every plane-field there corresponds
^near plane-field; to every bundle of rays or planes, a collinear
ik of rays or planes; and to every point-range, a projective point-
M. Two Space-Systems 2 and 2' may be placed in collinear cor-
X)ndence with each other by taking any two bundles of rays A
B of 2 and associating them with any two bundles of rays A'
B' of 2' in such fashion that the rays AB, A'B' common to the
pairs of bundles are corresponding rays, and the sheaf of planes
' of 2 corresponds with the sheaf of planes A'B' of 2'. For if
correspondence is established, and if P designates a point of 2,
pair of rays AP, BP determine a certain plane 17 of the sheaf of
les i4 5 to which corresponds in 2' a plane 17' of the sheaf of planes
^and corresponding to the rays AP, 5P of 2, which intersect
', there will be two rays A'P\ B'P' of 2', which determine by
r intersection the point P' of 2' corresponding to any point P
. Moreover, corresponding to any ray 5 of 2 projected from A
B by the planes As and Bs, respectively, there will be a ray 5'
'which is determined by the intersection of the two planes A's'
BW corresponding to the planes ^4^ and Bs, respectively. And,
ly, if X denotes any plane-field of 2 whereby the two bundles of
A and B are in perspective with each other, with the ray AB
208 Geometrical Optics, Chapter VII. [ f 164.
common to the two bundles, the two corresponding bundles of rays,
A' and B\ being also in perspective relation with each other, with the
ray A'B' in common, will, accordingly, determine a plane-field t' of
2' collinear with the plane-field v of 2. Therefore, the two Space-
Systems 2 and 2' are placed in complete collinear correspondence.
From this we derive immediately the following rule:
// we take any five points of one Space-System, no four of which Ue
in one plane, and associate them as corresponding with Jive such points
of the other SpcLce-System, the two Space-Systems will be completely col'
linear to each other.
Thus, suppose we take five points A, B, C, D, E of 2, no four
of which lie in one plane, and associate them with five such points
A\ B\ C\ D\ E' of 2', then the two bundles of rays AB, AC, Ad,
AE and BA, BC, BD, BE of 2 correspond to the two bundles of
raysil'jB', A'C, A'D', A'E' and B'A', B'C, B'D', B'E', respectivdy,
of 2', and to the sheaf of planes ABC, ABD, ABEol^ corresponds
the sheaf of planes A'B' C, A'B'D', A' B'E' of 2'; and we see, accord-
ingly, that the rule given above is equivalent to the method which
we gave first.
Since each point of a space-pentagon may have a 3 -fold infinitude of
positions, it is obvious that two Space-Systems may have a is-fold in-
finitude of collineations.
164. The so-called ''Flucht*' Planes, or Focal Planes, of Two Col-
linear Space-Systems. In two collinear Space-Systems 2 and t kt
T and v' designate two corresponding plane-fields, wherein KlM^
and K'L'M' N' are two corresponding quadrangles. The quadrangt
XLJIfiV determines a harmonic range of four points P, Q, R, 5 which
all lie on a straight line s of the plane-field t; and, similariyi the
quadrangle K'L'M'N' determines also a harmonic range of fo^^
points P', Q', R', S', which are conjugate to P, Q, R, S, respectivdyi
and which all lie on a straight line s' of t' which is conjugate to *•
From a point i4 of 2, lying outside the plane-field x, this field is prt>"
jected by a bundle of rays or planes, and from the correspoodhig ,
point i4' of 2' the plane-field v' will be projected by a bundle of lajf*
or planes which is projective with the bundle A ; so that, for exampfc
the four rays AP, AQ, AR, AS and the four corresponding rays A' Ft
A'Q', A'R', A'S' form two harmonic pencils of rays. The compfctt
quadrangles KLMN and K'L'M' N' are projected from A ami At
respectively, by two complete four-edges.
Now suppose that the two pairs of opposite sides KL, If iV and
LM, NK ot the quadrangle KLMN are two pairs of paralld straight
iS.J The Geometrical Theory of Optical Imagery. 209
s, so that the four points P, C R, S are a harmonic range of
nts all lying on the infinitely distant straight line i of the plane-
d T. In the collinear plane-field t' the two pairs of opposite sides
the quadrangle K'VM' N' will, in general, not be pairs of parallel
aight lines, so that the harmonic range of points P', Q\ R\ 5' con-
nate to the infinitely distant points P, Q, R, S will, in general,
termine a finite straight line i\ the so-called "Flucht" Line, or
cal Line (§ i6o) of the plane-field t' corresponding to the infinitely
tant straight line i of the plane-field t.
To the plane At parallel to the plane-field tt corresponds the plane
i' of 2', which, in general, will not be parallel to the plane-field t'.
now the point A is itself an infinitely distant point of the Space-
stem 2, the corresponding point A' will be the common "Flucht"
int of all the rays of tlie bundle conjugate to the bundle of parallel
fsotZ whose direction is determined by the infinitely distant point
; and, in general. A' will be a determinate and actual, or finite,
int of the Space-System 2'. In this case the plane At will be the
initely distant plane* e of 2, and the corresponding plane A'i' is
eso<alled ''Fluchr Plane t' of 2'. It contains the 'Tlucht" Lines
all the planes and the "Flucht" Points of all the rays of 2'.
Similarly, there is a certain plane ^ of 2, conjugate to the infinitely
stant plane ip' of 2', in which are contained the "Flucht" Lines of
I the planes and the "Flucht" Points of all the rays of 2.
These two planes ^ and c' are the so-called *Tlucht" Planes of the
iro Space-Systems 2 and 2', respectively. In the geometrical theory
' optical imagery they play a very important part, and are called the
^Planes of the Object-Space and Image-Space. Hence:
^/a« have two collinear Space-Systems 2 and 2', which, in the language
(^^ometrical Optics, we shall call the ''Object-Space'' and the ''Image-
P^*\ respectively, then (except in the so-called case of Telescopic
'^sgery, referred to below) to the infinitely distant (or ideal) plane of
^ system there will correspond a finite {or actual) plane, the so-called
f^ucht" Plane or Focal Plane, of the other system.
Thus, to a bundle of parallel rays in one space there will correspond
bundle of rays in the other space which all intersect in a point of
c Focal Plane of that space.
165. AflSnity-Relation between Object-Space and Image-Space.
the exceptional case when the quadrangle K'VM'N', as well as
The infinitely distant points and lines of space are assumed to lie in an infinitely
lot or ideal surface, which, since it is intersected by every actual straight line in only
point and by every actual plane in a straight line, must be a plane surface — the in.
iy distant plane of space.
15
210 Geometrical Optics, Chapter VII. [ § 1
the quadrangle KLM N, has each pair of its opposite sides parall
so that the points P', Q\ R\ S\ corresponding to the infinitely diste
points P, Qf Rf S of the plane-field tt, are themselves also infinib
distant points lying on the infinitely distant straight Hnej' of the plai
field tt', the plane A'f conjugate to the plane At is parallel to the jJa
tt'. And if also the two corresponding points A, A' are infinitely d
tant points of the Space-Systems 2, S', respectively, then the two Fo
Planes ip and t' are also the infinitely distant planes € and iff of the Objt
Space and the Image-Spa^e^ respectively. This case, which actua
occurs in certain optical systems, is called in geometry the case
''Affinity of the two Space-Systems. In Optics it is the imports
case known as ''Telescopic Imagery''.
ART. 46. GEOMETRICAL CHARACTERISTICS OF OB JBCT-SPACB AHD
IMAGE-SPACE.
166. Conjugate Planes. The two Focal Planes tp and t' of 1
Object-Space and Image-Space, respectively, not only from the opti<
but from the geometrical stand-point as well, are the most distinguisli
planes of the Space-Systems to which they belong. The exceptioi
case of Telescopic Imagery, alluded to in § 165, in which the Fa
Planes tp and t' are themselves the infinitely distant planes of t
Space-Systems 2 and 2', respectively, will be treated specially a
in detail in a separate division of this chapter. Therefore entin
excluding this case for the present, and assuming that (he Focal Plan
tp and €' are finite, or actual, planes, we proceed to enumerate the m(
striking general characteristics of the collinear correspondence of t'
Space-Systems 2 and 2'.
1. In general, to a sheaf of parallel planes of one of the two Sp^
Systems there will correspond a sheaf of non-parallel planes of the 06
Spcu:e'System.
The axis of the sheaf of parallel planes is an infinitely distant strai(
line of the Space-System 2 to which this sheaf is supposed to bdo
and the axis of the conjugate sheaf of planes will be, therefore
straight line lying in the Focal Plane of the other Space-System
Generally speaking, this straight line will be a finite, or actual, Un
2^ and such a line can only be the base of a sheaf of non-par
planes.
However, there is one very important exception to the above st
ment, viz. :
2. The two sheaves of parallel planes to which the Focal Planes i
selves belong are conjugate sheaves.
1161.1 The Geometrical Theory of Optical Imagery 211
The infinitely distant straight lines of the Focal Planes ^ and t'
are the bases or axes of these two sheaves of parallel planes, and since
these infinitely distant straight lines are the lines of intersection of
the Focal Planes with the infinitely distant planes of their respective
Space-Systems, they are a pair of infinitely distant conjugate straight
lines, in fact (the case of Telescopic Imagery being excluded) the only
sudi pair of conjugate lines. Hence, the two sheaves of parallel
planes which have these two infinitely distant conjugate straight lines
as axes are conjugate sheaves of planes.
Tm conjugate planes a and a' which art parallel to the Focal Planes
9 and t\ respectively y are in (he relation of ^'Affinity to each other,
because their infinitely distant straight lines are a pair of conjugate
straight lines. Since, therefore, to each infinitely distant point of
one such plane there corresponds also an infinitely distant point of
the plane in "affinity" with it, it follows that:
Parallel straight lines of the plane a correspond to parallel straight
^ of ike plane a'; so that a parallelogram in the Object-Plane <r
^ be "imaged" by a parallelogram in the Image-Plane <r\
Moreover:
Any range of points r of the Object-Space, which is parallel to the
focal Plane ip, will be **iniaged^' in a ''projectively similar''^ range of
t^ntsr' of the Image-Space, which is likewise parallel to the Focal Plane t'.
167. The Focal Points and the Principal Axes of the Object-Space
ttd the Image-Space.
/i'To a bundle of parallel rays in the Object-Space will correspond,
* l^neral, a bundle of non-parallel rays in the Iniage-Spcu:e, the vertex
^vhick lies in the Focal Plane of that space; and vice versa.
The particular point of the Focal Plane which will be the vertex
w the bundle of non-parallel rays will depend on the direction of the
bnndle of parallel rays. If, for example, the bundle of parallel rays
^ ooe space meets the Focal Plane of that space at right angles, the
^^^^ of the corresponding bundle of rays in the other space will
TOnnine a certain definite point in the Focal Plane of that space, viz.,
^so-called Focal Point of that space. The Focal Point of the Object-
^Poce, designated by F, is the vertex of the bundle of object-rays to
wiifch corresponds a bundle of parallel image-rays which cross the
' The peculiarity of " projectively similar " ranges of points is that the lengths of cor-
fOpoDding segments of them are in a constant ratio to each other. Thus, for example.
Iff, /are two projective ranges of points whose infinitely distant points W, W' correspond
to CBcfa other, and if A, A'; B, B'; C, C are any three pairs of conjugate finite points of
f./, then, once (ABCW) — {A'B'CW), we have immediately :
AC:BC = A'C': B'C\ or A'C . AC= B'C : BC,
212 Geometrical Optics, Chapter VII. [ § 168.
Focal Plane of the Image-Space at right angles; and, similarly, the
Focal Point of the Image-Space, designated by £', is the vertex of the
bundle of image-rays to which corresponds a bundle of parallel object-
rays which cross the Focal Plane of the Object-Space at right angles.
The two straight lines drawn through the Focal Points F and B*
perpendicular to the Focal Planes tp and e' are called the PrincipBl
Axes of the Object-Space and the Image-Space, respectively. This
pair of straight lines will be designated as the axes of x and x'. Since
the ray x' passes through the Focal Point £' (which corresponds to
the infinitely distant point E of x) and also through the infinitdy
distant point F' (which corresponds to the Focal Point F likefFise
situated on x), it follows that the Principal Axes x and x' are a pai^
of conjugate straight lines and, in fact, this is the only pair of conjufflti
rays which are at right angles to the Focal Planes.
168. Axes of Co-ordinates. An immediate consequence of the faC
that X and x' are a pair of conjugate rays is the following:
To the sheaf of planes in the Object-Space which has for its axis th
X-axis corresponds the sheaf of planes in the Image-Space which hasft^
its axis the x'-axis.
Of these two projective sheaves of so-called "Meridian Planes'
there is, according to an elementary law of projective geometry, oi*
pair of Meridian Object-Planes at right angles to each other to whid
corresponds a pair of Meridian Image-Planes which are also at v^'
angles to each other. In each space this particular pair of Mendiac
Planes at right angles to each other, together with a third plane per-
pendicular to the Principal Axis, and, therefore, perpendicular to earfj
of the two Meridian Planes, will determine by their intersections «
set of three mutually perpendicular straight lines. Hereafter, whcii
we come to derive the Image-Equations, we shall find it convenieirt
to select these two sets of straight lines as the axes of two systems €^
rectangular co-ordinates, one in the Object-Space and the otho* il*
the Image-Space. One of these straight lines is, of course, the Priif
cipal Axis x or x' of the Space-System. But, whereas jc, x^ will always
be a pair of conjugate straight lines, the other two pairs of strad^
lines, designated after the manner of Analytic Geometry, as the J-
axis and z-axis in the Object-Space and the y'-axis and x'-axis in flu
Image-Space, may, or may not, be pairs of conjugate straight linei
This will depend on whether the >f2-plane and the 3rV-plane are \
pair of conjugate planes.*
* Strict consistency in the matter of notation, which is eminently desirable,
in Geometrical Optics, cannot, however, alu-ays be observed without sacrificiiig
[69.] The Geometrical Theory of Optical Imagery. 213
ART. 47. METRIC RELATIONS.
169. Relation between Conjugate Abscissas. Let 5, 5' be a pair
)f conjugate rays of the two collinear Space-Systems S and S', and
et / and /' designate the points where these rays cross the Focal
Planes <p and c', respectively. Moreover, let / and J' designate the
infinitely distant points of s and s' conjugate to /' and 7, respectively.
Finally, if P, P' and Q, Q' are two other pairs of conjugate points
of 5 and 5^ we shall have:
{PQJI) = {P'Q'rV),
or
PJPI_P21 Pill
whidi, ance / and J' are the ideal points of 5 and s\ reduces to the
foUowing:
PJ QT
QJ " PT '
TTiis equation may be written :
JPVP' = JQ'VQ' = a constant.
Stated in words this Characteristic Metric Relation of Optical Imagery
oay be expressed as follows:
Tkt product of the '^abscisses" of two conjugate points, P and P\
^respect to the so-called ''Fluchf' Points, J and /', of two conjugate
^ytsand s' which go through P and P\ respectively, is constant.
In this statement the term "abscissa" is employed (for lack of a
wtter word) to describe the position of a point on a ray with respect
to the "Flucht" Point of the ray as origin. Thus, for example, the
Asdasa" of the point P of the ray 5 is JP, which means the segment
^^ the ray included between J and P, and reckoned from J to P, that
^ redconed always in the sense indicated by the order in which the
feters are written. (See Appendix, Art. 4.)
The product of the "abscissae" of pairs of conjugate points of any
«ie pair of conjugate rays 5, s' is constant, but the magnitude of
of fraiter importance. Thus, according to the system of notation employed in this chapter
BBd very generally throughout this book, the designation " y's'-plane " would naturally
■pisr a plane in the Image-Space conjugate to the yz-plane in the Object-Space. But
vni when these two co-ordinate planes are not conjugate, we shall continue to designate
be plane in the Image-Space as the y's'-plane rather than complicate and, perhaps, con-
wt things by introducing a pair of entirely new letters. As a matter of fact, except in
it important case when these planes ys, y'z' are the two Focal Planes ^, c', they are gen-
■II7 a pair of conjugate planes.
170.1
The Geometrical Theory of Optical Imagery.
215
ordinate axes y and %\ and, similarly, there will be determined in the
Image-Space in the same way a pair of perpendicular straight lines
of f/ conjugate to those of <r which are parallel to the co-ordinate
axes i and z' of the Image-Space. If, therefore, y, z and y\ z'
denote the co-ordinates of a pair of conjugate points of cr and <r', so
that y, i and 3, «' in this sense are used to denote the lengths of cor-
responding segments of point-ranges of <r, c' which are parallel to the
axes y, / and the axes z, «', respectively, the Magnification-Ratios
for rays of <r and a' parallel to these axes will be y' jy and ?! jz.
These ratios y^ ly and z' jz are called the Lateral Magnifications for
the pair of conjugate planes a and a\
For a given Object-Plane a parallel to the Focal Plane ^, the two
Lateral Magnifications have perfectly definite values. Thus, for ex-
ample, the value of y/y for the plane <r may be denoted by the symbol
F.sothat
y
whidi states that the value of Y is independent of the actual magni-
tudes of y and y\
As origins of the two systems of rectangular co-ordinates of the
Object-Space Z and the Image-Space S' let us select the two Focal
Pio. 84.
CoLLDTKATiON OF TWO 8pacb-St8TBScs X. V : flhowiuff how the I<ateral Maffnification of Conju-
fite Planes panllel to the Focal Planes depends on the distances from the Focal Planes. The ranges
>. /, as drawn in this fifforc, are " oppositely projective " — a case that does not actually occur in
optied imagery ; but that fact is immaterial so far as the question under consideration is concerned.
Points F and £' (Fig. 84), respectively. The Principal Axes will be
the axes of x and x\ Let it be observed that the Focal Planes which
are the planes yz and yV are not conjugate planes, as the notation
would imply.
216 Geometrical Optics, Chapter VII. [ § 1
Take any point P of the Object-Space, whose co-ordinates ^
X = FMj z = MN, y = iVP, and through P draw the ray p parallel
the ^-axis. Let P' designate the point in the Image-Space conjugs
to P, and let x' = E'M\ z' = M' N\ y' = N'P' denote the coK)n
nates of P'. Corresponding to the object-ray p going through t
Object-Point P, we shall have an image-ray p' which connects t
Image-Point P' .with the Focal Point £'. The pair of conjugs
planes perpendicular to the Principal Axes x, x' at the points Af, j
will be designated by <r, cr', and the value of the Lateral Magnificat!
for this pair of planes and for rays which are parallel to y and / ^
be denoted by F; so that Y = y' jy.
If the Object-Point P is supposed to move along a straight li
parallel to the Focal Line y, it is obvious that the Image-Point
must also traverse a straight line parallel to the Focal Line y*
such fashion that the Lateral Magnification y' jy = N'P' jNP =
shall be constant.
Again, if the Object-Point P is supposed to move along the ra>
which is parallel to the jc-axis, the Image-Point P' will travel alo
the conjugate ray p' which connects P' with the Focal Point E\
that as the ordinate y = NP remains constant as to both magnitu
and sign, its image y' = N'P' assumes all values from — oo to + '
Thus, it appears that the Lateral Magnification Y has different wrft
for each pair of conjugate planes cr, cr' which are parallel to the Fa
Planes ^, e'. That is, the Lateral Magnification Y is a function
the abscissa x.
It is obvious that the same thing is true also in regard to the Lata
Magnification z'/z in the direction perpendicular to the Focal line
171. The Image-Equations. We proceed, therefore, to asoerta
in what way the Lateral Magnification Y depends on the abscissa
We shall continue to employ the same symbols as in § 170, and sb
use the same diagram (Fig. 84). In addition to the pair of conjuga
planes <r, a' parallel to the Focal Planes ^, e' and containing the cc
jugate points P (jc, y, 2), P'(^', y', 2')» respectively, consider al
another pair of such planes (t,, a[ perpendicular to the Principal A)
X, x' at the points M^, ATj, respectively. And let F, Fj denote t
values of the Lateral Magnification for these two pairs of conjugJ
planes <r, a' and (r,, <r[, respectively. Let the object-ray p para
to the jc-axis cross the plane <r, at the point Q whose co-ordinates
FMi = ^1, Af, iVi = 2, = 2, NiQ = yi =^ y, Similariy, in the Ixna
Space let the ray p' conjugate to the object-ray p meet the plane 9
the point C whose co-ordinates are E'Afj =^ x[, M*iN[ = z[,N'i(/ ^
The Geometrical Theory of Optical Imagery. 217
e point R of the Focal Plane tp is the "Flucht" Point of the
ray p, then, by the abscissa-relation of § 169, we have:
RP'E'P' = RQE'Q',
ice RP = FM = X, RQ = FM^= jCj, and since from the figure
ealso:
— *»
E'P' N'P' y
f write the relation above as follows:
I'^Yi'yi = ^ry» and Y = y^/y\ accordingly, we obtain finally:
Y X,,
Yr'x'
F-x = a constant = b (say).
«re see that the Lateral Magnification Y is inversely proportional
bscissa x.
precisely the same process we should find that the Lateral
ication z'/z is also inversely proportional to the abscissa x,
rdingly, we are able now to express the co-ordinates jc', y\ z'
point P' of the Image-Space in terms of the co-ordinates jc, y, z
corresponding point P of the Object-Space. Thus, taking the
^oints F and E' as the origins of the two systems of rectangular
nates, and therefore using equation (109) together with the
which we have just obtained, we can write the Image- Equations
•ws:
^ = z. y="» 2'=", (no)
XXX
hich we infer that the most general case of optical imagery, as
by these equations, involves at least three constants a, b and c.^
SKI, in his celebrated book, derives the Image-Equations entirely by the methods
ic Geometry. Taking as the basis of his mathematical investigation the plane-
iXMTespondence which is characteristic of the collinear relation of the Space-
218 Geometrical Optics, Chapter VII. [ § 172.
III. COLLINEAR OPTICAL SYSTEMS.
ART. 48. CHARACTERISTICS OF OPTICAL IMAGERY.
172. Signs of the Image-Constants a, b and c. Up to this point
we have developed the theory of Optical Imagery from the stand-
point of pure geometry, and on this account, while keeping steadily
in view the application to the theory of optical instruments, we have
purposely avoided introducing in this general treatment any of the
physical properties of optical rays whereby the problem would become
Systems S and S' and denoting the co-ordinates of any point P of 2, with respect to an
arbitrary system of rectangular axes in S, by x, y, s. and the co-ordinates of the conjugate
point P', also with respect to an arbitrary system of rectangular axes in 2', by x', y\ «'.
CzAPSKi shows that the following equations, involving 15 independent constants (cf. end
of i 163). are the anal)rtical expression of coUinear correspondence between £ and 2':
, __a\X -t- biy 4- CjZ + rfi
._ a«g 4- ft^y -f CfS -f <<t
OiX -f ft^y 4- Ci,z 4- d^'
From this system of equations we may obtain, in general, also a second system which
may be written as follows:
_ «|g^ 4- Ot/ 4- «3g^ 4- «♦
* "" \x' 4- ^w 4- V 4- ^;
_ py 4- ^0/4- P^ 4- P,
' " <Ji*^ 4- ^t/ 4- V 4- <54 *
In each of these two sets of equations it will be remarked that the right-hand members
are fractions with linear numerators and denominators, and that the denominators of the
fractions are identical for all three equations in each group. It is obvious that
«4« -f ft^y 4- c«» 4- d^=o,
c^ij/ 4- <^o^ 4- V -f '^i = o
are the equations of the " Flucht " Planes or Focal Planes ^, e' of the two Space-Ssrstems
2, 2', respectiN-ely.
Having thus obtained the equations above. Czapski proceeds to show how by a suitable
choice of axes of co-ordinates the equations may be reduced finally to the simpler forms
given in equations (no), where, instead of as many as 15 independent constants in the
case of arbitrar>' s\'stem$ of co-ordinates, the number of independent constants is only 3.
See CzAPSKi: Tkeorie der optiscken Instrumente nach Abbe (Breslau. 1894), pages 27-33.
See also E. Wandersleb: Die geometrische Theorie der optischen AbbQdung nach
E. Abbe: Chapter III of Die TkeorU der optischen InstrumenU, Bd. I (Berlin, 1904)
edited by M. \x>n Rohr.
Also: JamesP. C. Solthall: The Geometrical Theory of Optical Imagery: Astropkfs.
Jomrn., xxiv. U9o6^. 156-1S4.
§ 172.]
The Geometrical Theory of Optical Imagery.
219
more or less specialized. But having obtained the Image-Equations
(§ 171), we shall find it convenient now to call attention to the mani-
fest singularity which distinguishes optical rays from the rays of
ordinary geometry. Along each optical ray there is one direction,
viz., the direction which the light follows, which is the obvious, or
natural, direction of the ray. First of all, therefore, we may take
advantage of this property by agreeing to define the positive direction
of an optical ray as that direction along the ray which the light takes.
In the case, therefore, of two conjugate ranges of points s, s\ there
are two possibilities. Thus, for example, if P^ Q, R,- - - is a series of
points of 5 which are traversed by the light in the order named, the
series of conjugate points P', Q\ R\- - - lying on s' will be traversed
either in the same or in the reverse order. In the former case (when
the direction of the ray s' is therefore the same as the direction P'Q'),
we shall call 5 and s' a pair of directly projective ranges of points (Fig.
85) ; and in the latter case (when the direction of the ray s' is opposite
to that of P'Q'), we shall call s and 5' a pair of oppositely projective
ranges of points (Fig. 86). Obviously, in optical imagery we can have
only directly projective ranges of points, and, consequently, so far as
our purposes are concerned, we may leave out of account altogether
oppositely projective ranges.
If, therefore P, P' are a pair of conjugate points of the directly
projective ranges of points 5, s' (Fig. 85), and if /, I and /', /' desig-
FiO. 85.
DniBCTLT PROJBCnVB RANOBS OF POINTS ; SUCH AS WE HAVE ALWAYS IN OPTICAL IMAGERY.
O, (/ are a pair of conjugate points, from which the point-ranges P.Q,R,"- and P', (/ ,Rf ,-" lyins:
on the atrai^rht lines *, /, respectively, are projected. The points P,Q,R,'- and P' ,(/,Bf ,-•• are
traTCTsed by the liirht in the order in which the points are named. / and /' are the " Flucht"
Points and / and f the infinitely distant points of s. /. respectively. (However, the rays in the
diagram which are desisnated as j and ^ do not here correspond to the " Flucht" I«ines of the
Plane-Fields « and v'.)
nate the "Flucht" Points and the Infinitely Distant Points of s and
/, respectively, then, as the point P is supposed to travel along s
from / to / in the direction of the ray 5, P' will travel along s' from
220
Geometrical Optics, Chapter VII.
[ § 172.
*>^
J'
The ransre P* , (/. I^, "' of points lyinsT on / is
oppositely projective with the rang-e P, Q, R, •••
lyinff on s (see left-hand side of Piff. 85). This case
cannot occur in optical imagery.
J' to /' in the positive direction of the ray 5'; so that, supposing, for
example, as is represented in the figure, that the point P lies on the
negative side of the *Tlucht"
Point J {JP < o), the point
P' will lie on the positive
side of the "Flucht" Point /'
(/'P' > o), and vice versa.
Hence, in the case of two di-
rectly, or, as we might say,
''optically", projective point-
ranges, the "abscissae" (§ 169)
JPj V P' always have opposite
signs. Accordingly, recalling
the Abscissa-Relation derived in § 169, we may say:
In the case of Optical Imagery, the product JP - I'P' = a constant
is always negative.
The value of this constant for the two projective point-ranges lying
along the Principal Axes jc, jc' was denoted by a; hence, provided the
positive directions of the axes of x, jc' are defined as the directions which
light pursues along these rays, the Image- Constant a is negative in ail
cases of optical imagery; that is,
a < o.
In the case, however, of a ray which is parallel to the Focal Plane,
the positive direction of the ray, as defined above, is indeterminate,
for the light may be supposed to traverse such a ray equally well in
either of the two opposite directions of the straight line to which the
ray belongs.
With regard, therefore, to the two systems of rectangular co-ordi-
nates of the Object-Space and Image-Space (§ 168), the positive direc-
tions of the Principal Axes jc, jc' have been clearly defined ; but nothing
whatever has been done towards choosing the positive directions of
the secondary axes y, z in the Object-Space and y\ 2' in the Image-
Space. So far as our previous investigation goes, the positive direc-
tion of each one of these axes is entirely arbitrary; and, accordingly,
the signs of the two constants b and c which enter into the Image-
Equations (no) may be positive or negative and like or unlike, de-
pending only on the choice of the positive directions of the axes of
y, z and of y, 2'.
It makes no difference which directions we choose as the positive
directions of the axes of y, 2 in the Object-Space; but, having chosen
§ 174.] The Geometrical Theory of Optical Imagery. 221
these, let us contrive so that the positive directions of the axes of
y\ 2! in the Image-Space shall be thereby determined. Accordingly,
we have merely to make, for example, the following agreement:
The positive directions of the axes of y and y' are to he chosen relative
to each other in such manner that the constant b shall be a positive number.
And in the same way, the positive directions of the axes of z and z'
are to be chosen with respect to each other so that the constant c shall be
a positive number.
Thus,
6 > o, c > o.
Hence, assuming the positive directions of the secondary axes to be
determined according to these considerations, it follows that the Lateral
Magnifications y'/y and z'/z always have the same sign, viz., the sign
of the abscissa x.
The signs of the three constants a, b and c which enter into the Image-
Equations are dependent, therefore, only on the choice of the positive
directions of the axes of co-ordinates. If these directions are defined
as above, then the signs of these constants are as follows:
a < o, ft > o, c > o.
173. So long as we do not assume any definite position-relation
between the Object-Space and the Image-Space, we shall define the
positive directions of the axes of co-ordinates in this way, so that a
is negative and ft and c are positive. For the entirely general case
this is the best choice to make. But when the optical system consists
of a centered system of spherical refracting surfaces, as is usually
the practical case, the corresponding axes of the two systems of co-ordinates
are parallel, and then it will generally be more convenient to define the
positive directions in such a way that the positive directions of cor-
responding, or parallel, axes will be the same. If this method is used,
the signs of the Image-Constants may be different, in some cases, from
the signs which they have above. It is important to bear this in
mind, as the student may be puzzled when he finds that the signs of
the Image-Constants are sometimes different from the signs as given
above; merely because the positive directions of the axes of co-ordi-
nates have been determined by different considerations (see § 176).
174. Symmetry around the Principal Axes. In the most general
case of optical imagery, defined by equations (no), which involve
at least as many as three constants a, b and c, the imagery is not
symmetrical with respect to the Principal Axes of the Object-Space
222 Geometrical Optics, Chapter VII. [ § 174-
and Image-Space; that is, in general, the two Magnification-Ratios
y jy and z' jz have different values corresponding to the same value
of X, However, in most actual optical systems, in fact almost with-
out exception, the Principal Axes are axes of symmetry; and, since
we are concerned primarily with the applications of these laws to the
theory of optical instruments, it will be assumed hereafter that this
is the case. Thus, we shall put
c = 6;
in which case the Image-Equations become:
XXX
80 that the character of the imagery will be defined now by the two
constants a and 6. The Principal Axes being axes of symmetry,
every pair of Meridian Planes of the Object-Space which are at right
angles to each other has a corresponding pair of Meridian Planes of
the Image-Space also at right angles to each other; so that the choice
of the axes of y and z is now indeterminate.
Moreover, in the case of symmetry with respect to the Principal
Axes of X, x\ when we have c = 6, the collinear plane-fields <r, a'
parallel to the Focal Planes ^, c', respectively, are not only in affinity
with each other (§ i66), but they are also similar; so that \i A, B^ C
are three points of a not in the same straight line, and A'^ B\ C
the three corresponding points of <r', then
A'W _ S'r _ A'C\
AB " BC " AC'
and, consequently, corresponding angles of two similar plane-fields are
ec|uul.
The Himplest, and at the same time the most periect, tdnd of optical
image would be one which was geometrically exactly similar to the
()l)jcct; 8() that it would always be possible to conceive the object
and image oriented with respect to one another in such fashion that
all corresponding lines were parallel. This case of complete geomet-
rical similarity between an object and its optical image may not, in
general, he realized l)y any optical apparatus, although it is possible
in special cases. But if the object is a plane figure lying in a plane
parallel to the Focal Plane of the Object-Space, the image will be a
completely similar figure lying in a plane parallel to the Focal Plane
of the Image-Space — assuming that we have collinear correspondence
§ 175.] The Geometrical Theory of Optical Imagery. 223
between Object-Space and Image-Space, and that the Principal Axes
are axes of symmetry.
175. The DijSerent Types of Optical Imagery.^ In coUinear
bundles of rays there are always two corresponding rectangular three-
edges. Thus, at any point 0 of the Object-Space, let OA, OB, OC
be three rays mutually at right angles to each other, to which there
correspond three rays 0'A\ 0'B\ Cf C meeting at the conjugate
point Cy of the Image-Space, and also mutually at right angles to
each other. Let us suppose that the three edges OA, OB, OC oi the
octant O'A B C form a canonical or right-screw system of axes (so that,
if a right-screw was turned in the direction from OB to OC, the point
of the screw would advance along OA). Two cases may occur, as
follows: (i) The system 0'A\ 0'B\ O* C may also be a right-screw
system; or (2) The system 0'A\ 0'B\ O'C may be a left-screw
(or acanonicat) system of axes.
In the first case, we can see how it might be possible, by placing
the points 0 and 0' together, to fit one of the octants into the other
in such fashion that the directions of the three pairs of corresponding
edges of the two conjugate octants OA, 0'A'\ OB, 0'B'\ OC, 0' C
agree with one another; so that except for the fact that the pairs of
corresponding points A, A'', B, B'-, C, C will not, in general, be
superposed on each other, we should have ^'^ congruence'' of the two
rectangular comers 0-A B C and 0^-A'B' C.
In the latter case, when one system is canonical and the other
acanonical, no such "congruence" would be possible. Thus, for
example, if in this case we place the points 0 and O' in coincidence
with each other, and if we orient the two octants relative to each other
so that the directions of two pairs of conjugate edges, say, OB, O'b'
and OC, Cf C are the same, the directions of the third pair of edges
OA, O'A' will be exactly opposite to each other. Instead, therefore,
of a so-called "congruence" of the two conjugate octants O-ABC
and C/'A'B' C, such as was possible in the first case, we shall have
here a certain ^'symmetry" of the two octants; although here again
we are employing a term in a sense somewhat different from the pre-
cise meaning attached to it in geometry. Strictly speaking, both
"congruence" and "symmetry" involve the idea of the equality of
corresponding line-segments, which is by no means necessarily implied
in the employment of these terms in the present connection.
' The following discussion is based on the admirable treatment of this matter in E.
Wandbrslbb's article on " Die geometrische Theorie der optischen Abbildung nach E.
Abbb", which is Chapter III of Die Theorie der optischen Instrumente, Bd. I (Berlin*
1904), edited by Dr. M. von Rohr. See pages 92. foil.
224 Geometrical Optics, Chapter VI I. I § 175.
By virtue of the Principle of Continuity, it is obvious that if we
have ''congruence*', or ''symmetry", between one pair of conjugate
octants, we shall have "congruence", or "symmetry", between all
pairs of conjugate octants. It is true that possibly at the Focal
Planes (which are sometimes called the "Discontinuity Planes") of
the two Space-Systems, the imagery might change from "congruence"
to "symmetry", or vice versa. That this change does not occur in
crossing from one side of the Focal Plane to the other, we shall now
proceed to show.
Let us assume that the two pairs of conjugate points 0, 0' and
A, A' are situated on the Principal Axes x, x'; so that 05, 0'5' and
0 C, 0' C are parallel to the co-ordinate axes y, y' and «, «', respectively.
Moreover, let us assume also that the points A, B, C are all infinitely
near to 0; and, consequently, the points A\ B\ C will also be
infinitely near to C. Hence, if
X = FO, x' = E'O'
denote the abscissse of the points 0, 0' with respect to the Focal Points
F, £', respectively, as origins, we shall have:
OA = dx, OB = dy, OC = dz\
O'A' = dx\ O'B' = dy', O'C = &'.
And, finally, let us suppose that the directions 0^4, 05, OC agree
with the positive directions of the axes of jc, y, «, respectively (no
matter how these directions may have been defined), so that the magni-
tudes denoted here by dx, dy, dz are all positive. If we write the
Image-Equations (no) in the differential form, as follows:
d^c' = — -2 dx, dy' = dy, dz' = - dz,
X XX
we see that, as the point 0 is supposed to cross the Focal Plane at
F, whereas the abscissa x changes its sign, the sign of dx' remains the
same; but the signs of dy' and dz' both change with change of the
sign of jc. Consequently, the octant O'-A'B' C remains of the same
type, so that if it was "congruent" (or "symmetric") with the octant
O'A B C when the point 0 was on one side of the Focal Plane, it will
remain "congruent" (or "symmetric") with it when the point 0 is
taken on the other side of the Focal Plane.
Accordingly, from this purely geometrical standpoint, and entirely
without reference to the actual signs of the Image-Constants a, 6, c,
§ 175.]
The Geometrical Theory of Optical Imagery.
225
it appears that there are these two essentially different types of optical
imagery, which may be conveniently distinguished as follows :
1. Right-Screw Imagery — the case when the two conjugate octants
are "congruent"; in this case the image of a right-screw will be a
right-screw, but, in general, distorted; and
2. Left-Screw Imagery — the case when the two conjugate octants
are "symmetric"; in this case the image of a right-screw will be a
left-screw, although, in general, distorted.
These two types of imagery may be exhibited by diagrams (Figs.
87, 88 and 89) as follows:
In the Object-Space parallel to the x-axis draw two pairs of rays,
viz., two rays 6, b (Fig. 87) in the x>r-plane at equal distances from,
► ^
Fio. 87.
Ttpxs of OpncAi, IXAORRT : OmRCT-SPACB. This fiflTure shows a series of equidistant axial
Object-Points (h, 0%, etc. and two series of equidistant Object-Points ^i. B%, etc. and C\, Ct, etc..
lyins in the planes xy and xx, respectively, on the straight lines b and c parallel to the Principal
Axis ix) of the Object-Space.
and on opposite sides of, the x-axis; and, similarly, two rays c, c
drawn in the same way in the jcz-plane. In the Image-Space (Figs.
88 and 89), corresponding to the two pairs of object-rays ft, ft and c, c,
we have two pairs of image-rays 6', 6' and c', c' all passing through the
Focal Point E', The pair of rays ft', 6' will lie in the jc'y-plane,
and the pair of rays c', c' will lie in the jc'«'-plane, and the jc'-axis
will bisect the angles at E' between each of the pairs of rays ft', ft'
and c', c'. In Fig. 87 Oj, Ojt ^^c. represent a series of equidistant
Object-Points ranged along the jc-axis. Through each one of these
points draw a pair of lines parallel to the axes of y and z, and con-
sider the segments of these lines comprised between 6, 6 and c, c.
The image of one of these rectangular crosses made by such a pair of
line-segments will be a rectangular cross with its arms parallel to the
16
226
Geometrical Optics, Chapter VII.
[ § 175,
axes of y' and 2'; the end-points of these arms lying in the pairs of
rays h\ V and c\ c\ as shown in Figs. 88 and 89. The points 0\, 0\,
etc., corresponding to the axial Object-Points Oj, 0„ etc. (Fig. 87),
PlO. 88.
Typbs of Optical ImAob&t : Ixaob-Spacb. This figure is to be taken In connection with Fiff.
87. It shows the case of Riffht-Screw Imagery.
will be ranged along the jc'-axis, and will lie nearer to the Focal Point
£' in the same proportion as the object-points O^, 0,, etc. are farther
from the Focal Point F, and vict versa.
In Figs. 87 and 88 the imagery is right-screw imagery; whereas in
Figs. 87 and 89 the imagery is left-screw imagery. The directions of
the line-segments are shown by the arrow-heads. In these diagrams
•-5
FlO. 89.
Types of Optical Imaobkt : Iicao»»Spacb. This fiffnie is to be taken in connectloa with. Ffff.
87. It shows the case of I^ef t-Screw Imagery.
the positive directions of the axes are chosen so that the signs of the
Image-Constants (§ 172) are given by the following relations:
a < o, 6 > o, c > o;
and, consequently, corresponding to positive values of the co-ordinates
X, y, 2, we shall have x' negative and y' and z' positive; whereas if
y and z are both positive, but x negative, x' will be positive, and y
and z' both negative.
§ 176.] The Geometrical Theory of Optical Imagery. 227
One of the most obvious and characteristic features of optical
imagery is the symmetry of the imagery with respect to the two Focal
Planes. Each of the two space-regions is divided by its Focal Plane
into two equal halves, and to each half of the Object-Space corre-
sponds one of the two halves of the Image-Space.
176. In order not to affect the generality of our results, up to
this point we have purposely nowhere assumed any definite position-
rdaiion between the Object-Space and the Image-Space. As a matter
of fact, however, practically all optical instruments consist of a
centered system of spherical refracting {or reflecting) surfaces, so that
the system is perfectly symmetrical with respect to the optical axis
(§ I3S)» or straight line along which lie the centres of the spherical
surfaces. In such a system the Principal Axes x, x' of the Object-
Space and the Image-Space are both coincident with the optical axis.
A ray lying in a Meridian Plane of the Object-Space must in its transit
through the system continue to lie always in this same plane in space,
so that a Meridian Object- Plane and its conjugate Image-Plane are
the same plane in space. Thus, for example, the two Meridian Planes
of the system of co-ordinates of the Object-Space, viz., the planes xy
and xz, are coincident with the planes x'y' and x'z\ respectively, of
the Image-Space. Hence, the axes of y and z in the Object-Space are
parallel to the aoces of y' and z\ respectively, in the Image-Space. This
being the case, it is usually found convenient to select the positive direc-
tions of the axes of x\ y' , z' so that these directions shall be the same as
the positive directions of the axes x, y, 2, respectively. Thus, while we
shall always select the positive direction of the x-axis as the direction
taken by the incident light along that line (§172), the positive direc-
tion of the jc'-axis may, or may not, be the direction pursued by the
light along it. And, therefore, the constant a may in a case of this
kind be either positive or negative, depending on which direction of
the aZ-axis is the positive direction (see § 173).
In an optical system composed of a centered system of spherical
surfaces, it is important to emphasize the fact that the positive direc-
tion along the optical axis is always the direction of the incident light;
so that, for example, if one of the spherical surfaces is a reflecting
surface whereby the original direction of the light along the optical
axis is reversed, notwithstanding, we must continue to reckon as
positive that direction which was originally the positive direction;
and all axial line-segments, irrespective of any subsequent change of
the direction of the light, are to be reckoned as positive or negative
according as they have the same direction as, or the opposite direction
to, the incident axial ray (see §§26 and 108).
228 Geometrical Optics, Chapter VII. [ § 176.
So also in regard to the other Image-Constant 6 = c: since here
we do not, as in the general case, choose the positive directions of
the secondary axes of y' and z' so that 6 = c shall be positive, the
Image-Constant 6 = c niay, therefore, be positive or negative. In
brief, in these special circumstances, the two systems of axes are
chosen with respect to each other so that a mere displacement along
the optical axis of the origin of co-ordinates of the Image-Space is
all that is needed in order to bring the axes of co-ordinates of the
Image-Space into coincidence with the axes of co-ordinates of the
Object-Space.
If we write again the Image-Ekiuations in their differential forms,
viz.:
dx' = \dx, dy = ~dy, dz' = -dz^
XXX
and assume always that dx, dy, dz are positive, we may consider the
following cases:
I. As to the sign of the constant a:
(i)Ifa<o, then whatever may be the sign of the abscissa x, the
sign of dx' must be positive. The signs of dy' and dz' are always
either both positive or both negative, depending on the sign of x.
Consequently, the two conjugate octants which have dx, dy, dz and
dx', dy' , dz' as corresponding edges are "congruent", and, hence, when
a < o, we have Right-Screw Imagery (§175).
(2) When a > o, the sign of dx' must be negative for both positive
and negative values of x; whereas, as before, the signs of dy and dz'
are either both positive or both negative, depending on the sign of x;
so that the two conjugate octants which have dx, dy, dz and da/, dy',
dz' as corresponding edges are "symmetric" (§ 175). Hence, when
a > o, we have Left-Screw Imagery.
II. As to the sign of the constant b = c:
(i) When 6 > o, the signs of dy' and dz' are the same as that of x.
Accordingly, for positive values of x, we have erect images, and for
negative values of x, we have inverted images. An optical S3rstem of
this kind is called a convergent system,
(2) When 6 < o, the signs of dy' and dz' are opposite to that of x;
so that the f)ositive half of the Object-Space is portrayed by inverted
images, whereas the other half (the negative half) is portrayed by
erect images. This case is, accordingly, precisely opposite to the one
above, and a system of this kind is called divergent.
These results may be summarized as follows:
§ 177.1
The Geometrical Theory of Optical Imagery.
229
A centered system of spherical refracting {or reflecting) surfaces is
convergent or divergent according as the Image- Constant b > or < o;
and the Imagery is Right-Screw or Left-Screw Imagery according as the
other Image- Constant a < or > o.
ART. 49. THB FOCAL LBNGTHS, MAGNIFICATION-RATIOS, CARDINAL
POINTS, ETC.
177. Analytical Investigation of the Relation between a Pair of
Conjugate Rays. Let the Focal Point F (Fig. 90) be the origin of
'^w:^
Fio. 90.
Relation of Oejbct-Rat and Conjuoatb Imaob-Ray. The figure shows only the object-ray ;
a ftiinilar diaffram. with letters suitably changed, may be imaarined for the ima£:e-ray. /X? (or s)
represents an object-ray which crosses the Pocal Plane yz at the point desiirnated by R. FR — g,
this distance beinff reckoned positive or nesrative accordinsr as ^ is above or below the :r^-plane.
The anirlc 9 may have any value between v/2 and — v/2 : the sisrn of this angle beinar always the
same as that of the anffle <fr = Z FLU» where PUis the projection of PQ on the 4r^-plane.
the system of rectangular co-ordinates of the Object-Space, the Prin-
cipal Axis of the Object-Space being the x-axis, and the Focal Plane
ip being the yz-plane. Similarly, in the Image-Space (not represented
in the figure) the Focal Point E' is the origin of a system of rectangular
axes, the Principal Axis of the Image-Space being the x'-axis, and the
Focal Plane t' being the y's'-plane. Consider a pair of conjugate
rays, an Object-Ray (5), which crosses the co-ordinate-planes xy, xz,
yz at the points designated in the figure by the letters P, Q, R, re-
spectively, and the corresponding Image- Ray (5'), which crosses the
co-ordinate-planes x'y\ xV, y'z' of the Image-Space at the points
N\ (y, S', respectively. Draw R U perpendicular to the y-axis and
S'V perpendicular to the y'-axis; then the straight lines PU and
N'V meeting the jc-axis in the point L and the x'-axis in the point
230 Geometrical Optics, Chapter VII. [ § 177.
M\ will be the projections on the jcy-plane and the ^V-plane of the
conjugate rays 5, s\ respectively.
The co-ordinates of the points R and S' where the object-ray (5)
and the image-ray (5') cross the Focal Planes ip and c', respectively,
will be:
(o, FU, UR) and (o, E'V, V'S') ;
and, hence, if /, m, n and /', m', n' are the direction-cosines of the
straight lines 5, s', respectively, the Cartesian equations of these
straight lines will be:
X
1
y
-FU
tn
z
-UR
n
y
-E'V
tn'
=
«' - V'S'
n'
respectively.
Assuming that the imagery is symmetrical with respect to the
Principal Axes x, x\ so that 6 = c (§ 174), we can express the relations
between the co-ordinates ^, y, z of an Object-Point and the co-ordinates
x\ y\ z' of the conjugate Image-Point by means of the Image-Equa-
tions (ill); in consequence whereof the second pair of the above
equations may be written as follows:
E'V am^ _V^ x^Vl
^" b ^'^b y ' *- b *■*■*/'•
Comparing this pair of equations with the first pair above, we obtain
immediately the following relations for the co-ordinates of the two
points R and 5' in the Focal Planes yz and yV, respectively:
o tn d fi
£'r = fty, rs'^bj.
Let us denote the focal distances of the points i?, S' where 5, s' cross
the Focal Planes ip, t' by g, i', respectively; that is, FR^g^ ES^V\
and, moreover, let ^, B' denote the angles of inclination to the axes
ar, x' of the conjugate rays 5, 5', respectively. Squaring and adding
the two equations in the top line, and doing the same for the two
§ 177.] The Geometrical Theory of Optical Imagery. 231
equations in the lower line, and introducing the symbols which we
have just defined, at the same time remarking that we have also:
? + m' + «' = /'' + m'' + «'' = I
and
/ = cos e, I' = cos e\
we obtain inunediately the following results:
Thus, we find:
g* = ptanV, ife'* = 6*tan*tf.
g =»=*:- -tan ^, ife==fc6tantf.
In order to avoid ambiguity of signs in this pair of equations, it
is necessary to define more precisely the linear magnitudes g and k'
and the angular magnitudes B and B\
1. As to the signs of the linear magnitudes g and k'l The focal
distances g and V are to be reckoned positive or negative according
as their projections FU and E'V on the y-axis and y'-axis, respec-
tively, are positive or negative. Thus, according as the point U lies
on the positive or negative half of the y-axis, the sign of g will be plus
or minus; and, according as the point V lies on the positive or
negative half of the y-axis, the sign of V will be plus or minus.
2. As to the angular magnitudes B and B'\ If through the point
P where the object-ray meets the xy-plane a straight line is drawn
parallel to the x-axis, in the same direction as the positive direction
of the X-axis, the angle B is the acute angle through which this straight
line has to be turned about P in order to bring it into coincidence
with the straight line PQ. The s'gn of this angle may be positive or
negative, its value being comprised between B = t/2 and ^ = — t/2.
The sign of the angle B can always be ascertained by the following
rule: If ^ = /.FLP denotes the acute angle through which the x-axis
must be revolved about the point L in order to make it coincide in
position with the projection PL of the object-ray PQ on the xj^-plane,
and if the sign of the angle ^ is determined by the relation
FU
tan^=-^,
let us agree that (he signs of the angles here denoted by B and yp shall
always be (he same. Thus, for example, in the figure, as it is drawn,
232 Geometrical Optics, Chapter VII. [ § 178.
both F U and FL are positive, since their directions are the same as
the positive directions of the axes of y and x, respectively; and hence
the angle B in the figure is negative.
The angle B' in the Image-Space is defined in an entirely similar
way.
If the pair of conjugate rays lie in a pair of conjugate Meridian
Planes, we shall find, on investigation, that it will not be necessary
to extract a square-root, as it was in the general case above, and that,
with the above definitions of the magnitudes denoted by g, k\ B^ ^,
the positive sign in the two formulae is alone admissible. Thus, the
ambiguity disappears, and we must write :
g = T-tan^', jfe' = 6tan^. (112)
From these formulae we derive the following:
To object-rays, whose inclinations (B) to the Principal Axis (x) are
all equal, correspond image-rays which cross the Focal Plane («') of the
Image-Space at equal distances (k') from the Focal Point E' ; and,
similarly, to object-rays, which cross the Focal Plane (^) of the Object-
Space at equal distances (g) from the Focal Point F, correspond image-
rays whose inclinations (B') to the Principal Axis (x') are all equal.
We had already perceived (§ 167) that to a bundle of parallel rays
of one Space-System, say, S, there corresponds a bundle of non-
parallel rays of the other Space-System, the vertex of which lies in
the Focal Plane of S'. We see now that this fact is merely a particu-
lar case of a more general law of optical imagery, as given in the above
statement. The absolute value of the focal distance of the point R
or S', where the object-ray or image-ray crosses the Focal Plane ip
or c', depends only on the magnitude of the inclination ^ or ^ of the
conjugate ray to the x'- or x-axis, respectively.
178. The Focal Lengths / and e'. Equations (112) obtained in
the last section, which may be written:
g a k'
= 0,
tan d' 6 ' tan ^
afford us a new way of defining the Image-Constants a and 6. Thus,
the constant b may be defined as the ratio of the Focal distance k'
of the point where an image-ray crosses the Focal Plane of the Image-
Space to the tangent of the angle of inclination B of the corresponding
object-ray to the Principal Axis x of the Object-Space; and, similarly,
§ 178.] The Geometrical Theory of Optical Imagery. 233
the magnitude ajb may be defined as the ratio of the Focal distance
g of the point where an object-ray crosses the Focal Plane of the
Object-Space to the tangent of the angle of inclination B' of the cor-
responding image-ray to the Principal Axis x' of the Image-Space.
From the equations above, as well as from the I mage- Equations them-
selves (§ 174), it is apparent that the dimensions of the Image-Constants
a and b are different; thus, whereas b denotes a length, a denotes an
area. For this and other reasons it is convenient to introduce at this
point a new pair of symbols / and e' instead of a and 6, and to write :
/=6, «'=j. (113)
Thus, the Ims^e-Constants denoted by / and e' will be defined by
the following formulae:
k' g
The constants / and e' are called the Focal Lengths of the optical
system. According to Abbe, the proper definitions of these character-
istic constants of the optical system are given only by formulae (114).
So soon as the magnitudes denoted by / and e' are ascertained, the
optical system may be regarded as completely determined.
The definition of the Focal Lengths of a system of lenses, as given
by Gauss/ is essentially the same as Abbe*s definition by means of
the above equations; thus:
The Focal Length of the Object-Space (denoted here by 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
corresponding infinitely distant object; and
The Focal Length of the Image-Space (denoted by e') is equal to the
ratio of the linear magnitude of an object lying in the Focal Plane of
the Object-Space to the apparent magnitude of its infinitely distant image.
Introducing the Focal Lengths / and e\ we may now write the
Image-Equations (iii) as follows:
V z' f x'
xx'^fe\ ^ = - = ^ = -. (us)
'* y z X e ^ J/
Provided we adhere to the choice of the positive directions of the
axes of co-ordinates which was made in § 172 (where we had a < o,
' See S. CzAPSKi: Theorie der optischen Instrumente nach Abbe (Breslau, 1893), p. 40.
234 Geometrical Optics, Chapter VII. [ § 179.
b > 6), we shall have always:
/ > o, e' <o.
179. The Magnification-Ratios and their Relations to one another.
1. The Lateral Magnification F. This, as has been already defined
(§ 170), is the ratio of conjugate line-segments lying in planes at right
angles to the Principal Axes. Thus,
y f x'
F = ^ = ^ = ^; (116)
y X e'
whence we see that the Lateral Magnification Y may have any value
from — 00 to + 00, depending on the value of the abscissa x.
2. The Axial or Depth-Magnifircation X, By differentiating the
abscissa-equation
xx' = fe\
we obtain for the ratio of infinitely small conjugate line-segments dx,
dx' of the Principal Axes:
dx' fe^ x'*
This ratio, denoted by X, is called the Axial or Depth-Magnifica-
tion. It is inversely proportional to the square of the abscissa x.
If we choose the positive directions of the axes of co-ordinates so that
/ > o, e' < o (see § 178), then X will be necessarily positive, and may
have any value comprised between o and -f- 00.
Comparing formulae (116) and (117), we obtain the following rela-
tion between the Axial Magnification {X) and the Lateral Magnifi-
cation (7):
y2=~y; (118)
and, hence, we can say: At each point the Axial Magnification is pro-
portional to the square of the Lateral Magnification.
3. The Angular Magnification Z. Let Af, M' (Fig. 91) designate
the positions of two axial conjugate points, whose abscissae with respect
to the Focal Points F, E' are denoted by x, x', respectively; so that
FM = X, EM' = x\
Let the straight line MR represent an object-ray crossing the Focal
Plane of the Object-Space at the point jR and making with the Prin-
cipal Axis X of the Object-Space an angle xMR = B. Let 5' designate
§ 179.1
The Geometrical Theory of Optical Imagery.
235
the point where the conjugate ray S'M' crosses the Focal Plane of
the Image-Space, and let ^' = Z E'M'S' denote the inclination of
this image-ray to the Principal Axis x' of the Image-Space. Putting
FR = g, E'S' = *',
we have, in accordance with our agreement in § 177 concerning the
signs of the angles By $':
tand=--, tand'= ;.
X X
The Focd Lengths/ and e', by definition, are given by the formulae:
/=
*'
tan^'
e' =
g
tan e' '
And, hence, if Z denotes the ratio of the tangents of the angles of
inclination to the Principal Axes of a pair of conjugate rays in any two
conjugate Meridian Planes, we have:
Z =
tan^
X
/ »
(119)
whence it will be seen that Z is independent of the values of 0, B'
themselves; so that for a given value of x, the ratio denoted by Z
Fig. 91.
ANGULAR Magnification.
tBai9^-~FRIFM, tani^^ ='-Ef^lEfM', 2r= tan •'/tan •.
2r denotes the angular magnification for the conjugate axial points 3f, M*.
has a constant value. Thus, for all rays which pass through the axial
point ilf , the ratio tan 0' : tan 0 is constant.
This ratio denoted by Z is called the Angular Magnification, or
the " Convergence-Ratio", and is an important magnitude in the theory
of optical instruments.
236 Geometrical Optics, Chapter VII. [ § 180.
Comparing the values of the Magnification-Ratios X, Y and Z,
as given by formulae (ii6), (117) and (119), we have the following
relations between them :
180. The Cardinal Points of an Optical System. As we see from
formulae (116) and (119), the Magnification-Ratios Fand Z may have
any values comprised between — 00 and + 00, depending on the
value of the abscissa xi whereas the Depth-Magnification X, as is
shown by formulae (117), may have any value between o and + 00;
since we assume in this discussion that the positive directions of the
axes of X, x' are so chosen that the Focal Lengths /and e' have always
opposite signs. Each of these ratios is a function of the abscissa jc,
so that by assigning any particular value to one of these ratios, we
shall thereby determine at least one pair of conjugate axial points.
Those pairs of conjugate axial points for which one or other of the
magnitudes denoted by X, F, Z has the absolute value unity are all
of more or less interest, and certain of them are especially distin-
guished in the theory of optics. They may be enumerated in the
following order:
I. The two pairs of conjugate axial points for which the Depth-
Magnification X has the value -}- i ; for, since X is a function of x^
we shall obtain always for a given value of X two equal and opposite
values of the abscissa x. Thus, putting X = -f i in formulae (117),
we find:
x= =fc V-ie\ x' = =Fl/-/e';
so that there are two pairs of conjugate points on the Principal Axes
of the optical system for which an infinitely small displacement dx
of the object-point will correspond to an equal displacement dx^ of
the image-point. Moreover, it will be remarked that the Focal Points
F and E' are midway between the two axial object-points and the two
axial image-points, respectively. However, these two pairs of axial
conjugate points are of slight importance, and need not detain us any
longer, except merely to add that the Lateral and Angular Magnifi-
cations at these points are equal. Thus, we have:
Y^Z^ ^ Vf/e'.
2. The most important and the most celebrated of all these pairs
§180.]
The Geometrical Theory of Optical Imagery.
237
of conjugate axial points is the pair named by Gauss * the Principal
Points (see § 139) of the optical system, which in our diagrams will
be designated by the letters A and A' (Fig. 92). The Principal
Pio. 92.
Cardihai. Ponrrs op Optical Ststbm. Focal Points F, £f; Prlndpal Points A, A'; Nodal
Points AT, ^.
FA=-N*Ef^f\ EfA'^NF^/i NA^N'A': AV^A'V', I ANV=-9^ I A'N* V* =-V .
Points are defined by the value F = + i. Putting Y = y' \y = + i
in the equations (116), we obtain for the abscissae of the Principal
Points:
x^ FA ^f, x' = E'A' = e'; (121)
and, hence, The Focal Lengths /, e' of an optical system may also be
defined as the abscissa^ with respect to the Focal Points F, E\ of the
Principal Points A, A', respectively.
If the positive directions of the Principal Axes are determined by
the directions pursued by the light in traversing these lines, then, as
has been repeatedly stated, / will be positive and e' negative (see
§ 178); hence, the Primary Principal Point A will lie always on the
positive half of the x-axis, and the Secondary Principal Point A' will
lie on the negative half of the x'-axis.
The pair of conjugate planes at right angles to the Principal Axes
at the Principal Points A, A' were likewise named by Gauss the
Principal Planes of the system. These planes are characterized by
the fact that to any point V in the Principal Plane of the Object-
Space there corresponds a point V in the Principal Plane of the Image-
Space, such that AV = A'V; so that an object lying in the Primary
Principal Plane will be portrayed by an image lying in the Second-
ary Principal Plane, which is equal to the object in every particular.
We may remark also that at the Principal Points x = f, x' = e'
we have also:
^ Z f
'C. F. Gauss: Dioplrische UnUrsuchungen (Goettingen, 1 841), J 7.
238 Geometrical Optics, Chapter VII. [ § 180.
The points A, A' are sometimes called also the Positive Principal
Points in order to distinguish them from another pair of axial conju-
gate points called by Toepler ^ the Negative Principal Points, which
are defined by the value F = — i. These points are, however, of
no particular importance.
3. The conjugate axial points N,N\ for which the Angular Magni-
fication has the value Z = tand' : tan ^= +if were named by Listing*
the Nodal Paints of the system. These points, which are next in
importance to the Principal Points, are characterized by the following
property:
To an object-ray crossing the x-axis at the Primary Nodal Point N
at an inclination B there corresponds an image-ray crossing the x'-axis
at the Secondary Nodal Point N' at an inclination 0* = 6,
Putting Z = -h I in formulae (119), we find for the abscissae, with
respect to the Focal Points F, £', of the Nodal Points iV, N':
jc = FiV = - e', x' = E'N' = -/;
or (Fig. 92) :
FA = N'E' =/; E'A' = JVF = e'. (122)
Moreover, since
AN = AF+ FiV = - (/ + O, A'N' = A'E\+ E'N' = - (/+ O,
we have:
AN ^ A'N'. (123)
Hence, the two Nodal Points are equidistant from the Principal Paints;
and, since the abscissae of JV, N', with respect to ^4, -4', respectively,
have the same sign, the Nodal Points lie always either both to the
right or both to the left of the corresponding Principal Points. And
if i4 iV = o, then A' N' — o also.
For Z = -|- I, we have:
e
The planes perpendicular to the Principal Axes at the points JV, N^
are called the Nodal Planes of the system. Toepler likewise dis-
tinguished a pair of Negative Nodal Points defined by Z = — i.
These distinguished pairs of conjugate axial points are called the
^ A. Toepler: Bemerkungen ueber die Anzahl der Fundamentalpuncte dnes bdiebigen
Systems von centrirten brechenden Kugelflaechen: POGG. Ann., cxlii. (1871), a3a-a5i.
2 J. B. Listing: Beitrag zur phjrsiologischen Optik: GoeUinger Studien, 1845. See
also article by Listing on the Dioptrics of the Eye. published in R. Wagnsr's Handwoer'
ierbuch d. Physiologic (Braunschweig, 1853), Bd. iv., p. 45 1«
§ 181.] The Geometrical Theory of Optical Imagery. 239
Cardinal Points of the optical system; and some writers include also
under this designation the Focal Points F, E\ Knowing the positions
of one pair of the Cardinal Points, and knowing also the Focal Lengths
of the optical system, we can determine completely the character of
the imagery.^
181. The Image-Equations referred to a Pair of Conjugate Axial
Points. It will be convenient sometimes, and always in the case of
Telescopic Imagery (Art. 50), to select as origins of the two systems
of co-ordinates some other pair of axial points besides the Focal Points
which have been used hitherto for this purpose. Thus, suppose we
take two conjugate axial points 0, 0' as origins, and let the co-ordi-
nates of the conjugate points Q, Q' with respect to 0, 0' be denoted
as follows:
OM = f, MQ^ y, O'M' = {', M'Q' = / ;
where Af , M' are the feet of the perpendiculars let fall from Q^ Q' on
the axes of ^, x\ respectively. Moreover, let
FM = X, EM' = x\ FO = Xo, £'0' = x[ ;
so that
XX ^^ XijXq ~^ JC •
Now
« = ^0 + f , X =Xo + ^,
and, therefore:
(Xo + O(xo + 0 =x^o'f
which may be written:
f + ^ + i=o; (124)
which is the required relation between the abscissae { and f. The
constants Xq, Xq which occur in this formula are the distances of the
origins O, (y from the Focal Points F, £', respectively.
^ Id connection with this subject, the following writers may be consulted (in addition
to those aheady named):
C. G. Nbumann: Die Haupl- und Brenn-Punkie eines Linsen-Sy stems. Elementare
Dawstdtung der durch Gauss begruendelen Theorie (Leipzig. 1866).
J. A. Grunert: Ueber merkwuerdige Puncte der Spiegel- und Linsen-Systeme: Grun,
Arch. f. Math. Phys., xlvii. (1867). 84-105.
F. Lippich: Fundamentalpunkte eines Systemes centrirter brechender KugeWiaechen:
MiU. des naiurw. Ver,f. Steiermark, ii. (1871). 429^459-
L. Matthibssbn : Grundriss der Dioptrik geschichUter Linsensysteme. Mathematische
EinUUung in die Dioptrik des menschlichen Auges (Leipzig, 1877). Also. Ueber eine
Methode zur Berechnung der sechs Cardinalpuncte eines centrierten Systems sphaerischer
Lmaen: Zft. f. Math. u. Phys., xxiU. (1878). 187-191. Also. Bestimmung der Cardinal-
imncte eines dioptrisch-katoptrischen Sjrstems centrirter sphaerischer Flaechen, mittels
Kettenbruchdetenninanten dargestellt: Zft.f. Math. u. Phys., xxxii. (1887). 170-175-
The above is only a partial list of the writers on this subject.
240
Geometrical Optics, Chapter VII.
[ § 181.
For the Lateral Magnification at Jlf, M' we obtain:
F = ^ =
/ *; + f '
*«+{
(125)
The Angular Magnification, in terms of the abscissae f , {', is given
as follows:
tan d' x^ + i f
Z =
tan^
/ , . / .
(126)
If Fp denotes the Lateral Magnification at the points O, O', then
Y^ «. fjxQ = XqIc- Hence, if we choose, we may eliminate the con-
stants jCo, Xq in the above equations by putting:
thus,
/Fo e'Y, + r
Z- -
e'Y,
_ _ / i' _L
c'F. + r r ••
(127)
In [>articular, if the origins of the two systems of co-ordinates are
the pair of conjugate axial points A, A' called the Principal Points
(S 180), and if for this special case we denote the abscissae of the points
Q, Q' by u, u', so that
AM = «, A'M' = «',
then, writing «, «' in place of $, {', respectively, in formulae (127), and
putting Fy = I, we obtain the Image- Equations referred to the Prin-
cipal Points as origins, as follows:
- + -> + i =0,
u u
y ^ _J_ ^ c^+m; ^
y f+u e'
tan^ f+u _
tantf e' ~
F=-'=
fu^
e'u'
f
u
t' + v! u
r
(128)
§ 182.]
The Geometrical Theory of Optical Imagery.
241
182. Geometrical Constructions of Conjugate Points of an Optical
System.
I. Construction of Conjugate Axial Points, The equation
- + -, + 1 =o
u u
suggests a simple method of construction of the pair of conjugate axial
points M, M\ provided we know the positions and directions of the
Principal Axes x, x\ the positions of the two Principal Points A^ A\
and the magnitudes of the Focal Lengths /, t' of the Optical System.
For since/ = FA^ e' = E'A\ the equation above may be written:
AF . A'K'
u
+
w
= i;
and, hence, if we suppose the two Principal Points A, A' (Fig. 93) are
placed in coincidence with each other so that the positive directions
of the Principal Axes x^
x' make with each other
at A (or A') any angle
xAx' different from zero,
and if through the Focal
Points F and F! we draw
straight lines parallel to
x' and X, respectively, in-
tersecting each other at a
point O, then any straight
line drawn through O will
make on the axes x^
xf intercepts AM ^ u^
A'M' « «', respectively,
which will satisfy the
above equation. In fact the point 0 is the centre of perspective of
the two point-ranges x^ x\
2. Construction of Conjugate Points Q, Q' not on the Principal
Axes. Suppose that the optical system is given by assigning the
positions and directions of the Principal Axes x, x' (Fig. 94), the posi-
tions of the two Focal Points F, £' and the Focal Lengths /, e\ The
Principal Points A, A' may be located at once, since FA =/, E'A' = e'
(§180); and the planes through these points perpendicular to the
Principal Axes are the Principal Planes. The point Q' is the vertex
17
Fig. 93.
COirSTRUCTION OF CONJUGATE AZIAL POINTS Af, At
(or Lt Lf) OF AS Optical Systbm.
FA=-f', EfA''^^', AM=^u\ A*M' = h\
242
Geometrical Optics, Chapter VII.
[ § 182.
of the bundle of image-rays corresponding to the bundle of object-
rays whose vertex is the Object-Point Q. If, therefore, we can deter-
mine two of the rays of the bundle of image-rays, they will suffice to
determine by their point of intersection the Image-Point ^. Thus,
for example, to the object-ray Q V which is parallel to the x-axis
there corresponds an image-ray which goes through the Focal Point
£'; and moreover, this image-ray will cross the Principal Plane of
the Image-Space at a point V conjugate to the point V where the
Fio.94.
Construction op Conjuoatb Points Q^ (/ of an Optical Ststbic. ^and Ef are the Fbcal
Points : A and A' are the Principal Points ; and ^and N' are the Nodal Points.
FA^N'Ef^f\ E'A»=-NF='^; AN'^A'N'x
/ ANU^' LA*N*(f\ AV^'A'V', AU^A*lf\ AlV^A'W.
f;
corresponding object-ray crosses the Principal Plane of the Object-
Space, determined, according to the property of the Principal Planes
(§ i8o), by the fact that AV ^ A' V\ Again, the object-ray QW,
which goes through the Focal Point F, and which meets the Principal
Plane of the Object-Space at the point W, must correspond to an
image-ray, which, crossing the Principal Plane of the Image-Space at
a point W such that AW = A'W\ proceeds parallel to the Principal
Axis x'; and the intersection of this ray with the other image-ray
V'E' will determine the Image-Point Q' conjugate to the Object-
Point Q.
If we know the positions of the two Focal Points F, E\ and if we
know also the Focal Lengths /, «', we may locate the Nodal Points
iV, iV' (§ 1 80). To an object-ray QN meeting the Principal Plane
of the Object-Space at a point U there corresponds an image-ray,
which, crossing the Principal Plane of the Image-Space at a point IT
such that A U = A' U\ has the same inclination to the x'-axis as the
object-ray QN has to the x-axis; that is, ZANU^ LA'N'V*.
Thus, the point Q' may be determined as the point of intersection of
any pair of the three image-rays V'Q\ U'Q' and W'Q\
§ 183J The Geometrical Theory of Optical Imagery. 243
ART. 50. TELESCOPIC IMAGERY.
183. The Image-Equations in the Case of Telescopic Imagery.
In the special and singular case of Telescopic Imagery (§ 165), the
Image-Equations (no) referred to the Focal Points F, £' are not
applicable, because in this case the Focal Planes <p and c' are no longer
actual, or finite, planes, but they are the infinitely distant planes c
and tp' of the Object-Space S and the Image-Space 2', respectively.
The infinitely distant planes are not only the Focal Planes, but they
are also a pair of conjugate planes; so that, if we were consistent in
our notation, and if we designated the Focal Plane of the Object-
Space by ipy in the case of Telescopic Imagery we should designate
the Focal Plane of the Image-Space by tp'. In the language of geom-
etry the two Space-Systems S, S' are said to be in ''affinity" with
each other. Each pair of conjugate plane-fields ir, t' of S, 2' are
also in affinity with each other, because to every infinitely distant
straight line of 2 there corresponds an infinitely distant straight line
of 2'. Thus, also, each pair of conjugate point-ranges of 2 and 2'
are "projectively similar", so that corresponding segments of them
are in a constant ratio to each other (§ 166). Hence, the image in
2' of a parallelogram of 2 will likewise be a parallelogram; and so,
also, the image of a parallelepiped will be a parallelepiped. To a
bundle of parallel object-rays will correspond a bundle of parallel
image-rays.
Since corresponding point-ranges are "projectively similar", we
can say:
In the case of Telescopic Imagery, the Magnification-Ratio has the
same value for all parallel rays.
This fundamental characteristic of Telescopic Imagery will enable
us to deduce the Image-Equations immediately. Thus, selecting as
origins of the two systems of co-ordinates any pair of conjugate points
O, Cy, let us take as axes of «, y, z any three straight lines meeting in
O, and let the three straight lines conjugate to x, y, z which meet in
(y be selected as axes of x', y', z\ If the magnification-ratios for the
three bundles of parallel object-rays to which the axes of x, y, z belong
are denoted by the constants p, q, r, we may write the Image- Equations
for ike case of Telescopic Imagery, as follows:
x' = px, y' = qy, z' = rz, (129)
If we select a set of rectangular axes in the Object-Space, the axes of
jc', y, 1! in the Image-Space will in general be oblique. But in the
i44 Geometrical Optics, Chapter VII. [ § 184.
two t>f\>jei^tive bundles of rays 0, (7^ there is always one set of mutually
V)triV*eiKiicuIar ra> s oi O xq which corresponds also a system of three
tiiuiuall> ji^rpendicuiar ruys o£ (X; so that if we choose these two
ivitiKuiar >*ics^ oi currtjspunding raj's as axes of co-ordinates of the
i\%o >i>acxf<>>5>C!eins» che equatkxis abo\'e will be the general Image-
K^uaiK>us> ^elem^i tu recmngnbr axes, for the case of Telescopic
U v%tU l>^ ^^tMrkl^i that ia the general case of Telescopic Imagery
.tK liiki^iie^H^^iMkiioiis tix\x)tve at least three independent constants.
V >iHi;jii^atlfef^«K>? l>etwifi«i Telescopic Imagery and Optical Imagery
u ^^tK«cu :^ :v b« iKKiiid ta the fact that in the former there are no
^>t4K^^HU Vm«^ ?o thdt k is merely a matter of preference which of
uV xvt^^>4 s»*sx%iuuc!ie$ k selected as the axis of x.
Hc%«v^. ti ,^*^ i;s^ (KOMrtkally nearly always the case) the Imagery
jfL. ^%M»M%*f»Ha» >«»di n^sfKCt to one pair of the conjugate axes of the
i^K> >>^«^M^ >M nectayn^utar co-ordinates, it is usual to select these
A*. ',Jk .wvrt^ v4 Jt and x\ In this case putting r = g, we may write
.iK liUiii^s^^KN|Mautfiiis^ as (otlows:
*' - ^» y = 2y» «' = fi2. (130)
liMk QtemcMiBtks of Telescopic Imagery. In the case of Tele*
HrV^HC liua^;^r>\ both the Lateral Magnification Y and the Depth-
Mi^U4tK«itkMi A* aiv constant. Thus:
v' ^ ^ dx' x' ^ ., ,
y-' -«-n. X = -7-=- =p = Xo, (131)
V
dx
\n icxarU to the Focal Lengths / and «', defined as in § 178, it is
v<>viou5^ th%u w^ have here:
/ = «' = 00.
Kui ^> tociuula Ui*) we have X/F* = — e'/f; hence, here:
(132)
V vv'ivUui^K , ^v"^ im^v say that the characteristic of Telescopic Imagery
N ;m., v^/K'4^v tk^ FoiijU Lengths f and t' art infinite, the ratio of the
Vs » -'%e^**l.}< is\ initio*
11 «v\iusa4|^ iHin liuitv ratio as one of the image-constants and the
U . s^.^: V«y;uaK\^livm V " (J " Fq as the other constant, we may
w.^^s -^ tiuv^^ Kv^v^iK^^* for the case of Telescopic Im2^;ery as
§ 185.1 The Geometrical Theory of Optical Imagery. 245
follows:
The Angular Magnification Z is given by the formula:
tan_^_ F
tan^ "A"
Hence, for the case of Telescopic Imagery:
Thus, the Angular Magnification in Telescopic Imagery is constant
also. It may be remarked that the Angular Magnification is an
especially important magnitude in this kind of imagery; for when we
are considering the infinitely distant image of an infinitely distant
object, the Angular Magnification is the only kind of magnification
that conveys any meaning. If in the Image-Equations we introduce
Z = q/p = Zq and the ratio e'/f = — p/q^ as the two image-con-
stants, these equations may also be expressed as follows:
x^ e' Zy y' e' Z'
2, .. ./ ^ • (i3S)
0
ART. 51. COMBINATION OP TWO OPTICAL SYSTEMS.
185. The Problem in General. A series of Optical Systems may
be so arranged one after the other that the Image-Space of one system
is at the same time the Object-Space of the following system, and so on.
The resultant effect of all these successive imageries will be an imagery
which may be regarded as due to a single optical system which by
itself would produce the same effect. An optical instrument, whose
function is to produce an image of an external object, is, in fact, nearly
always a compound system or combination of simpler systems. Pro-
vided we know the Focal Lengths and the positions of the Focal
Points and of the Principal Axes of each of the component systems,
it is always possible to ascertain the Focal Lengths and the positions
of the Focal Points of the compound system. In case the system is
composed of spherical refracting (or reflecting) surfaces with their
centres ranged along a straight line, the Principal Axes of each of
the elements of the system will coincide with the ''optical axis*' (§ 135).
This is usually the case in an actual optical instrument, and the
246
Geometrical Optics, Chapter VII.
[§185.
problem is greatly simplified by this condition. However, following
CzAPSKi,^ and supposing at first that we have only two component
systems, we shall consider here a rather more general case than the
one above-mentioned. Thus, the only restriction which we shall
make is the following:
The Principal Axis {x[) of the Image-SpCLce of System (/) shcUl be also
the Principal Axis (x^ of the Object-Space of System (11).
(Since this condition is usually satisfied in the case of imagery by
means of narrow bundles of rays inclined to the Principal Axes at
finite angles, the results which we shall obtain here will be directly
applicable also to this case, as we shall have occasion of seeing in
a subsequent chapter. See § 248.)
Let Fi, E[ and Fj* ^'2 (Fig. 9S) designate the positions of the Focal
Points of the systems (I) and (II), respectively; and let /j, e[ and /j, e'^
Fio. 95.
Combination of Two Opticai. Ststbms. Fi and Ei' and J^, Ei mark the poddtiofis of the Focal
Points, and A\, A\* and A%, Ail the positions of the Principal Points of ssrstems (I) and (n). re-
spectively ; and F, E^ mark the positions of the Focal Points of the compound ssrstem (I + n).
/i — F\A\, e\' = Ei'Ai', and/fl =- F%A%, et' = Et'Ail denote the Focal I^en^ths of systems (I) and (n).
respectively; whereas/. / denote the Focal ]>nffths of the compound system. The "interval**
between the two systems (I) and (II) is E\'Ft — A.
denote the Focal Lengths of the two systems. Since we have assumed
that the Principal Axis (:Ci) of the Image-Space of (I) is coincident
with the Principal Axis {x^ of the Object-Space of (II), the Focal
Planes at E\ and F^ will be parallel. The relative position of the
two component systems may be assigned by giving the distance of
the point F^ from the point Ej, that is, the abscissa of F, with respect
to E\. This magnitude, usually denoted by writers on Optics by the
symbol A, that is,
a=£;f„
and reckoned positive or negative according as the direction from JBj
^ S. CzAPSKi: Theorie der optischen InstrumenU nach Abbe (Breslau. 1893), pagn a6»
foil.
§ 185.] The Geometrical Theory of Optical Imagery. 247
to F, is the same as or opposite to that along which the light travels,
is called the interval between the two systems.
The Focal Points of the compound system will be denoted by the
letters F and £' (without any subscripts) ; and, .similarly, the Focal
Lengths of the compound system will be denoted by the symbols /
and e\
The problem is, therefore, with the data above-mentioned defining
the component systems, to determine the Focal Points and the Focal
Lengths of the compound system.
In the first place, it is obvious that the Focal Planes at the points
F and Fi are parallel, as is also the case with the Focal Planes at the
points E' and E'^. For in the system (I), to the sheaf of planes which
are parallel to the Focal Plane at Fy there corresponds a sheaf of
planes which are parallel to the FocaLPlane at E[ (§ i66), and which
are therefore, by hypothesis, parallel likewise to the Focal Plane at
F,; and to this sheaf of planes in the Object-Space of system (II)
there corresponds a sheaf of planes which are parallel to the Focal
Plane at E,* Hence, to planes which are parallel to the Focal Plane
at Fi there correspond planes which are parallel to the Focal Plane
at -^. But we have seen that in an optical system, in general, the
only two sheaves of parallel planes which are conjugate sheaves of
planes are the sheaves to which the. Focal Planes belong. Hence,
the Focal Planes at F and £' are parallel to the Focal Planes at Fj
and £i, respectively.
Moreover, the Principal Axis (x^) of the Object-Space of system (I)
is also the Principal Axis (x) of the Object-Space of the compound
system; and, similarly, the Principal Axis {X2) of the Image-Space of
system (II) is also the Principal Axis (:*:') of the Image-Space of the
compoimd system. For since Xi and x' are obviously a pair of conju-
gate rays with respect to the compound system, and since the Principal
Axes of an optical system have been defined (§ 167) as that pair of
conjugate rays which meet at right angles the Focal Planes of the sys-
tem, it follows that Xi and X2 must coincide with x and x', respectively.
We proceed, in the next place, to ascertain the positions of the Focal
Paints F, E' of the compound system. Consider, for example, an object-
ray proceeding in the direction QVi parallel to the Principal Axis
(x) of the Object-Space. Emerging from the first system, this ray
will go through the Focal Point E[ of the Image-Space of this system,
and, traversing the second system, will finally cross the Principal Axis
(xO of the Image-Space of the compound system at the point E';
so that, with respect to system (II), the points E[ and E' are a pair of
[§185.
:•: irstof formulae (ii5):
. :his equation enables us
.. the Image-Space of the
^ :>.rough the Focal Point F
. -. . •:. This ray, after travers-
..•■■■, to the Principal Axis (.r')
-, ::i. And, therefore, it must
■: :he Object-Space of system
• .>: l>o a pair of conjugate axial
\,vordingly, in the same way
_ • . .
.. :V:::t F of the Object-Space of the
.. v:h respect to the position of the
- i":* of the Focal Points F, E' of the
;• determine the Focal Lengths /, e'.
•\xm1 Lengths as given by formula?
tan^' •
:^s£ which we have already employed.
^•^xN .v*rallel to the Principal Axis (.v) of
.xx V. system, crosses the Focal Plane sp
..^ .;!:or traversing the entire system,
^ "*• '^,r;.ul Axis (.v') of the Image-Space at
.,^»,* ^ = /.A'.EiV., so that c' = g/tam e\
- .. V Am!^ C^'i) ^^^ ^he Image-Space of sys-
.5^ .'c\ :k> that
. . :,m IA\E[V\,
§ 185. J The Geometrical Theory of Optical Imagery. 249
And since E\ and £' are a pair of conjugate axial points with respect
to system (II), the ratio tan B' : tan Z.A\E\Vx is the value of the
Angular Magnification (Zj) of system (II) for this pair of conjugate
points. Applying, therefore, formula (119), we obtain:
tan 6' F^[ A
tan ZiljEjF'i e, e^'
Thus, the formula e' = g/tan 0' becomes:
^ " A '
whereby the magnitude of the Focal Length e' of the Image-Space of
the compound system is determined in terms of the known magnitudes
ej, €2 and A.
Similarly, if AA^FX^ = ^ is the inclination to the jc-axis of the
object-ray QX^, which goes through the Focal Point F ol the Object-
Space, and which, after traversing the entire system, emerges in the
direction -X'20' parallel to the :c'-axis, and if k' denotes the height at
which the emergent ray crosses the Focal Plane c' of the Image-Space,
then / = k'/tSLTi 0. This ray crossed the Principal Axis {x^ of the
Object-Space of system (II) at the point F^, so that
jfe' = /j • tan ZA2F2X2*
The points F and Fj ^ire a pair of conjugate axial points with respect
to system (I), and the value of the Angular Magnification (Zj) of this
system for this pair of conjugate points is given by the ratio
tan Z ^42^2-^2 • tan 0.
Thus, by formula (119), we obtain:
tan ZA2F2X2 fi^ - _-^
tan^ " £'i/^s"" A'
Accordingly, we find:
•^" A •
whereby the magnitude of the Focal Length / of the Object-Space of
the compound system can be determined in terms of the known
constants /„ /j and A.
The formulae for the Focal Lengths / and e' may be obtained also
by considering the ray, which, proceeding from the Object-Point Q,
250
Geometrical Optics, Chapter VII.
[§185.
goes through the Focal Point F^ of the Object-Space of system (I),
and which, therefore, emerges from system (II) so as to cross the jc'-axis
at the Focal Point £' of the Image-Space of system (II). Thus, from
Fig- 9S» we have:
^' tan ZA,F,W, ' ^^ " tan Z A'^E'^W^ '
Now A'lW'i = AzWzf and therefore:
tan Zi4;£;ty; _ /,
tan ZA^F^W^ "" e
f »
But since Fi and £2 ^^e conjugate axial points with respect to the
compound system, we have by formula (119):
Hence:
tan Zi4;£;Ty;_ J_
tan ZA^F^W," FfE^
FF,
fi
PPx
e:e\
J »
and, therefore, as before
/ /
The formulae derived above may be collected as follows:
Positions of the Focal Points F, £';
Magnitudes of the Focal Lengths /, e':
f —
A •
« =
(136)
The influence of the interval A between the two systems, which
forms the denominator of the right-hand member of each of these for-
mulae, is at once apparent. Two given systems (I) and (II) may be
combined in an infinite number of ways by merely altering the inter-
val A either as to its magnitude or as to its sign or as to both. So
long as this magnitude A is different from zero, and none of the Focal
Points of the component systems are situated at infinity, we shall have
a compound system with finite Focal Lengths/, e\
§186.]
The Geometrical Theory of Optical Imagery.
2J1
186. Special Cases of the Combination of Two Optical Systems.
We may consider several special cases of the combination of two optical
systems as follows:
I. The Case when the *' Interval'' is zero (A = o), the Focal Lengths
/i, e[ andfi, e^ of the component systems (I) and (II) being all finite.
In this case the compound system will be telescopic, since according
to equations (136), we have here / = e' = 00, whereas the ratio
fje' = — /i/a/^i^a ^s finite by hypothesis (see § 184).
In a telescopic system the three magnification-ratios Xy Y and Z,
as we saw in § 184, are all constant; let us denote their values here
by the special symbols X^, Y^ and Zq, respectively. So soon as we
know the values of any two of these magnitudes, the system will be
completely determined. In fact, since we know already the value of
the finite ratio of the Focal Lengths of the Telescopic System, it will
be sufhcient if we know also only one of the magnitudes denoted by
•^o» Yi
0) ^0*
The diagram (Fig. 96) represents the case of the combination of
two non-telescopic systems into a telescopic system, which is the case
*.«
Pio. 96.
TSLBSCOPICSTSTBlff RBSULTING FROM THE COMBINATION OF TwO NON-TSLBSCOPIC SYSTEMS
PLACED TOOBTHBR SO THAT A « 0.
now under consideration. The letters in this figure have the same
meanings as they have in the preceding figure, so that they do not
need to be explained again.
The LatercU Magnification is evidently :
V — ^ — —1—? — V
If we wish to obtain the value of the magnitude Fq in terms of the
Focal Lengths of the component systems, we have from the definitions
of the Focal Lengths (§ 178) :
*' " tan ^A[E[V, ' •'» tan Z A^F.Vt '
252 Geometrical Optics, Chapter VII. [ § 186.
and since A =
o,
so
that the points E\ and F^ are coincident, it follows
that
^2^% fi
A,V,-e['
and, hence:
Similarly, the Angular Magnification Z = Zq, in terms of the Focal
Lengths of the component systems, may be obtained as follows:
Consider the axial points F^ and Eg* which, with respect to the com-
pound system, are a pair of conjugate points; evidently, we have:
tanz^Xir;
^^ tanZ^jFiTT, '
and since
AW A'W*
tanz^;£;Tr; = ^^4!^% t^n Z.A,F,W,^^^^, a^w^^a\w,.
we find:
z = z = ^*
And, finally, for the Axial Magnificaiionj X = dx'/dx = x'/x = X^,
we have, since, by the last of formulae (120), X = Y/Z,
2. Combination of Telescopic System (I) with Non-Telescopic^ or
Finite f System (11).
Let the Telescopic System (I) be given by the values of the constant
axial and lateral magnification-ratios X^, Y^ respectively, and by the
positions of the conjugate axial points JIfi, Jlfj (Fig. 97). Here, as
in the preceding case, we assume that the Principal Axis (x[) of the
Image-Space of system (I) and the Principal Axis (xj) of the Object-
Space of system (II) are coincident. If Fj designates the position of
the Focal Point of the Object-Space of system (II), the relative
positions of the two systems may be assigned by the value of the
abscissa of the point Fj with respect to the point M[. Let us denote
this abscissa by the symbol a, so that At^F^ = a.
A ray proceeding parallel to the Principal Axis (jc,) of the Object-
Space of system (I) will also be parallel to the Principal Axis {x[)
of the Image-Space of this system, and, emerging finally from system
§186.1
The Geometrical Theory of Optical Imagery.
253
(II), will cross the Principal Axis {x') of the Image-Space of the
compound system at the Focal Point £i of the Image-Space of system
(II), which is likewise also the Focal Point £' of the Image-Space of
the compound system. And since the position of £2 ^s given, we know,
therefore, the position of E\
The position of the other Focal Point F of the compound system
will be determined if we ascertain its position with respect to the
--^-•'■7^
Fio. 97.
COMSZZCATION OF ▲ TBLSaCOPZC WITH A NON-TBLBSCOPIC SYSTBM UTTO A NON-TBLBSCOPIC
Compound System.
given axial Object-Point Mi, An image-ray which emerges from the
compound system in a direction parallel to the x'-axls corresponds
to an object-ray which crosses the :x:-axis at the required point F, and
which must also have passed through the Focal Point F, of the Object-
Space of system (II). Hence, the points F and Fj must be a pair of
conjugate axial points with respect to the telescopic system (I). And
dnce, by the second of equations (131),
z.=
we obtain immediately:
M,F " M,F'
M,F^Y'*
whereby the position of the Focal Point F is ascertained.
It only remains therefore to determine the magnitudes of the Focal
Lengths/ and e'. From the figure we obtain:
A 2X2 =/2-tan ZA2F2X2,
/ =
-4 2-^2
/jtan /.A2F2X2 - „
^ =/2'A;
tan Z M^FQ tan Z M^FQ
where Z, denotes the constant value of the Angular Magnification of
the telescopic system (I). Since Z, = YjX^, we have, therefore:
/ =
4Jj
254 Geometrical Optics, Chapter VII. [ § 186.
where / is determined in terms of the known constants /j, Xi and F^.
Again,
\fn ^^^ - ^2^2 ggtan Zi4^2^a
tan Zi4^2^'
and, hence:
whereby the Focal Length e' of the Image-Space of the compound
system is determined in terms of the given constants e^ and F^.
Since the magnitudes denoted by Xj, Fj, /^ and €2 are all finite,
it is evident from the formulae here obtained that the combination
of a telescopic system with a non-telescopic system is a non-telescopic
system. If the system (II) were the telescopic system, the procedure
would be entirely similar to that given above.
3. Both Systems Telescopic.
Let us suppose that the two component telescopic systems are given
by the values of their constant AxiaJ and Lateral Magnification-Ratios
Fio. 96.
Combination op Two Tblbscopic Systems into a Tblbscopic System.
Xi, Fp and Xj, Fg; and that the relative position of the two systems
IS given by the positions of a pair of conjugate points L, L[ (Fig. 98)
of system (I) and the positions of a pair of conjugate axial points
Afj, M' of system (II). Let us write a = LjJIfj.
To begin with, it is obvious that the compound system is also tele-
scopic. Thus, the Lateral Magnification of the compound system is:
L'Q' L'Q' L\Q\
^ LQ^I^.Tq^^'-^'^^ constant;
and the Axial Magnification of the compound system is:
^ L'AT L'Af L\M, ^ ^
^ = LM = L[M,'Tm = ^''^^ = * constant.
Here the letters M and L' designate the positions of the points, which,
with respect to the compound system, are conjugate to the given
§ 187.] The Geometrical Theory of Optical Imagery. 255
points M' and L, respectively. The positions of the points M and
L' may be determined as follows:
Since L\M^ : LM = Xj, and L\M = a, we find:
a
whereby the position of the point M is determined relative to that
of the given point L. Again, since M* ll\M^L\ = Xj, we haver
whereby the point V may be located with respect to the given point M'.
ART. 52. GENERAL PORMULiB FOR THE DETERMINATION OP THE FOCAL
POINTS AND FOCAL LENGTHS OF A COMPOUND OPTICAL SYSTEM.
187. We shall suppose that the compound system consists of m
component systems, and we shall assume that the Principal Axis (:ci)
of the Image-Space of the kth system is likewise the Principal Axis (:c^+i)
of the Object-Space of the (k + i)th system. In this statement the
symbol k denotes an integer, which is supposed to have in succession
every value from Jfc=itoJfc = w— i.
In the diagram (Fig. 99), Fj^ and £l designate the Focal Points of
the jfcth system, and Af^ and Ak designate the Principal Points of this
Xk %r-^ ^^ -^ **•'
FlO. 99.
DBTBXlfflNATION OP POCAL PonTTS (F, £*) AND FOCAL LBNOTHS (/. ^) OF A COMPOUND
Optxcai. Systbm. The diagram shows the Principal Axes {xk, Jct'). the Principal Points (At, Ak')
and the Focal Points {Fk, Eu') of the ^th member of the system, and the path of a ray traversing
this component.
FkAk-^fk, Ek'Ak'^ek', ^*'/5*+i = A».
FkM»''i=-xk, Eh'Mk'^Xk', AkBk = hk = Ak*Bk' = hk\
I Ak'-iAfk'^\Bk''i =» •»'-!, ^ Ak'Afk'Bk' = •»'.
system. To an axial Object-Point M^ lying on the Principal Axis
(jCi) of the Object-Space of the first system there corresponds an axial
Image-Point Af^ lying on the Principal Axis (:c^) of the Image-Space
of the last, or mth, system. A ray proceeding originally from the
point Jlf, will cross the Principal Axis {xj) of the Object-Space of the
ith system at the point Jlf*_i and the Principal Axis {x'k) of the
250 Geometrkai Optics, Chapter VII. [ § 187.
lni<i^e-v>pcict» of this system at the point Jlf*. Moreover, let Bj, and
Bk Jesig^iiace the points where this ray crosses the Principal Planes of
ihv ^th j>ysteiu» and let us write
aiiU, also;
FMk-x = ^*» KM'u = Xj,.
rhv slop^ of the ray at JfLi is:
Z. A'k^M'k^iBk-i = B'k-\ = LAkM'k^iBk.
The Focal Lengths of the *th system will be denoted by /*, «i; thus,
iUiU, finally » the interval between the ith and the (fe + i)th systems
will be denoted by
E\ ideally » w^ have the following two systems of equations :
**3f; - fifil, (* = I, 2, • • • , m), (137)
viud
^-x^i + A», (*= I, 2, •••,m- i). (138)
I . t>i^A^riiHiialHm 0/ the Positions of the Focal Points F, E' of the
bivMU th^ two systems of equations (137) and (138), we obtain by
p^vH,v«4^ v>l[ ^Micveaaive elimination :
^ _ fi^i M__ _ ^.^
w»|^ X f — */ — etc.
\\^\\ it ihv i.>biect'Point M^ coincides with the Focal Point F of the
vK^vvV Sh^vv v4 the compound system, the Image-Point At^ will be
ilw vuluuivly dUtant point of the Principal Axis (x') of the Image-
"^s^^vv v4 Vhv w*upound system; and in this case:
X, = FiF, xl = 00.
Uvi^\\UwvU|i vKc«*t values, we obtain the abscissa of the Focal Point
^ x^ vUv' vN^H^'^^Hi 9>*8tem in the form of a continued fraction as
S 187.]
The Geometrical Theory of Optical Imagery.
257
F,F =
/i«i
A.+
SA
(139)
A,+ ":
A,-,+
On the other hand, if we suppose that the Image-Point 1/t^ coincides
with the Focal Point E' of the Image-Space of the compound system,
the Object-Point Jlf ^ will be the infinitely distant point of the Principal
Axis (:c) of the Object-Space of the compound system; so that in
this case we shall have:
Xx =00, x« = 1S!JE1\
and, by a process precisely analogous to the above, we obtain:
K^--
A.-I +
fm-l^m-rl
(140)
A^2+-:
A2 +
whereby the abscissa of the Focal Point £' of the compound system
with respect to the Focal Point E'^ of the wth system is expressed
also in the form of a continued fraction. _
The continued fractions which form the right-hand sides of equations
(139) and (140) may be expressed in the form of determinants. Thus,
writing
FiF = | and K^^§.
we have for the numerators A^ A' and the denominators 5, B' the
following determinant-arrays:
A =
/l
0
0
0
0 • • •
0
0
0
0
0
0 • • •
0
0
0
/.
A,
0 • • •
0
0
0
0
/,
A,
- «4
0
0
18
258
Geometrical Optics, Chapter VII
A'
B
B' =
/.
o
o
o
I
o
o
o
o
I
o
o
o
o
o
o
o
o
/^_a A^_, -
o
o
o
o
o
A|
o
o
o
o
A, -Uj
o
o
o
o
^ -«i
o
fm-l
O
O O
O O
K-.2 -<
o
o
o
'w-S
[ S 187.
o
o
o
/a
o
o
o
o
Jmr-\
O
O
o
o
o
o
o
o
o
o
o
A.-1
o
o
o
o
A mere inspection of the last two arrays will show that the denomi-
nators B and B' are equal.
2. Determination of the Focal Lengths f^ e' of the Compound System.
In order to determine the Focal Length e' of the Image-Space of
the Compound System, let us consider, an object-ray parallel to the
Principal Axis (x) of the Object-Space, to which corresponds, therefore,
an image-ray which goes through the Focal Point E' of the Image-
Space. Accordingly, for this ray:
By the definition of the Focal Length e', we have :
S 187.]
The Geometrical Theory of Optical Imagery.
259
where C = -^ K^K-
This equation may be written:
K
tan(?: '
e' =
hy tan^j tan ^2 ^^^^^'m-\
tan^'i tan^i tan^i tan^i^, '
and since e^ = hj ta,n d[, and since, also, by formula (119):
tangjfc
tan ^i_i
we have:
«'=(-!)
,-1 g| • g» • • • <«
(141)
Putting Xi = 00, we obtain from the two systems of equations (137)
and (138):
X,- X, •
.(-.)-i,(A.+«)(
'^-^f
= (— I)**" Ri' R2' ' ' Rnt^V
where i?^ is used to denote the ifeth term of this product of continued
fractions. Writing
R -^»
we may express each of these continued fractions as the quotient of
two determinants as follows:
p.
A,
A
o
o
0
0
0
Vi
-e'k-i
0
0
/^.
V,
— 4-f
0
0
/^,
A*-s
-ei-s
• • •
• « •
o
o
/2
o
o
o
o —
62
.•i^-.:r VU.
[§187
— e.
^ 2 *<
[nants shows that we have
. - r- = (- 1)— p._,.
•
f^
(142)
...^ 'X' Fvval Length / of the Object-Space
...a*u\;*' in image-ray parallel to the Prin-
,:<s.^i.or: to which corresponds an object-
:xi *\xm1 Point F of the Object-Space.
- *' v .r« = 00.
^ cv^ - ^^ilth /, we have :
* tan e[ '
"Ss^ ^^;uarion may be written :
.-vr /, tan 0,n-i Am
,4.tf. tan ^|„_2 tandln^i*
•
,-«x»iV& ,uo\ we have:
Mn ^'k_
/*
v' '
5 187.1
The Geometrical Theory of Optical Imagery.
261
we obtain here:
/=(-!)
»— 1 fl'Ji
(H3)
Putting xi[^ = 00, we derive from the two systems of equations (137)
and (138):
A,+ r
where JR* is used to denote the ifeth term of this product of continued
fractions. Writing
we may express each of these continued fractions as the quotient of
two determinants as follows:
Ai — 4+1 o
/*+i A*+i — ek+2
p; =
o
o
o o
o o
/k+a ^k+i " ^*+s O
o /k+s Afc+s — ejk+4
o
o
o
o
o
o
o
o
0; =
o
A*+i
fk+i
o
o
I
A*+2
o
o
A*+s
o
o
— e
*+4
o
o
o
o
/m-l
O
"" ^m-1
A.-1
o
C-1 A^-i
Here also it is evident that:
accordingly,
p; = ei-, ;
262 Geometrical Optics, Chapter VII. [ § 187.
since we must put Q^_i = i. Thus, we find:
/=(-i)>-'^'-^'^;-^-'. (144)
Comparing the two determinant-arrays denoted by P^ and P*,
we see that P^_i and P[ are equal; if, therefore, we write:
formulae (142) and (144) may be written:
Consequently, also:
7/ = (- 1) ;^~7 7"- Ci4^)
If, therefore, we know the determination-constants and the rela-
tive positions of the members of the compound system, the formulae
which we have here obtained will enable us to determine the Focal
Points and the Focal Lengths of the compound system/
* In regard to the literature dealing with the subject of Art. 51. the following is a partial
lilt of the writers:
With reference to the matters treated in {{ 185 and 186, consult: S. Czapski: Tkeone
dtr optischen Instrumente nach Abbe (Breslau, 1893), pages 46-51. Also, vrith reference
to { 187. see Czapski, pages 51-53- See also E. Wandbrsleb: Die geometriscfae Theorie
der optischen Abbildung nach E. Abbe, Chapter III of Vol. I of Die Theorie der opHscken
JnsirumeHte (Berlin. 1904). edited by M. von Rohr; pages 112-121. Also. Dr. J. Clas-
sen's Maihematische Optik (Leipzig, 1901), Art. 46.
With special reference to S 187:
A. F. MoEBius: Beitraege zu der Lehre von der Kettenbruechen, nebst einem Anhang
dioptrischen Inhalts: Crellbs Joum., vi. (1830), 215-243.
F. W. Bessel: Ueber die Grundformeln der Dioptrik: Astr. Nach., xviii. (1841),
No. 415. pages 97-108.
F. Lippich: Fundamentalpuncte eines Systemes centrirter brechender Kugelflaechen:
MiU. naturw. Ver. Sleiermark, ii. (1871). 429-459.
S. Guenther: DarsUllung der NaeherungswerU von Kettenbruechen in independenter
Form : Habll.-Schr. Erlangen. 1873.
O. Roethig: Die Probleme der Brechung und Reflexion (Leipzig. 1876).
F. Monover: Th4orie g^n^rale des systtoes dioptriques central Paris. Soc. Pkys.
Stances, 1883. 148-174.
L. Matthiessen: Allgemeine Formeln zur Bestimmung der Cardinalpunkte eines
brechenden Systems centrierter sphaerischer Flaechen; mittels Kettenbruchdetermin-
anten dargeatellt: Zft. f. Math. u. Phys,, xxix. (1884). 343-350.
CHAPTER VIII.
IDEAL IMAGERY BY PARAXIAL RAYS. LENSES AND LENS-SYSTEMS.
ART. 53. UfTRODUCTIOir.
188. The geometrical theory of optical imagery, as developed in
the preceding chapter, is entirely independent of the physical laws of
Optics. The fundamental and single assumption on which the theory
rests is that of point-to-point correspondence, by means of rectilinear
rays, between Object-Space and Image-Space. With regard to the
angular apertures of the bundles of rays employed in the production
of the image, as also with regard to the dimensions of the object to be
portrayed, absolutely no conditions were imposed. In that chapter
we were not at all concerned with the mechanism whereby an image
may be realized; we merely assumed that such imagery was possible
arid investigated the laws thereof. Whatever practical difficulties may
lie in the way of realizing the geometrical condition of coUinear cor-
respondence, we have not yet encountered them, as we shall have to
do hereafter.
The investigation of the Refraction of Paraxial Rays of monochro-
matic light through a centered system of spherical refracting surfaces
had prepared the way for the geometrical theory of optical imagery;
for in this special, and, to be sure, more or less impractical, case we
saw that there was strict coUinear correspondence between Object-
Space and Image-Space. Hence, here at any rate, the formulae of the
preceding chapter are immediately applicable. The theory of the re-
fraction of paraxial rays through a centered system of lenses was first
fully worked out by Gauss^; and, hence, the imagery which we have
under these circumstances is frequently called *'GAUSsian Imagery".
The determination-data of a centered system of spherical surfaces
are usually the refractive indices of the successive isotropic media,
the radii of the spherical surfaces, and the distances between the con-
secutive vertices. If we introduce these constants into the general
formulae of the preceding chapter, we shall obtain not only all the re-
sults which for the case of Paraxial Rays we have previously obtained
by independent methods, but also a number of new and useful formulae,
'C. F. Gauss: Diopirische Uniersuchungen (Goettingen, 1841). See also paper by
F. W. Bessbl, entitled " Ueber die Gnindformeln der Dioptrik " {Astr, Nach., xviii., 1841,
No. 415* poses 97-108).
263
«- •
VIII. [ § 190,
Leogths and the Magnification-
■z ' ^^ . - - 'nr*^ Sptteriad Surface, of radius r, separating
-.^r. v« »xiu:s^ t- t • we found (§ 124) that the focal
C3LC > ^<;ti :> 3e uilowing formulae:
n
' "^' 7^^ -Z-^' (147)
- .i m --n e n
.-^nift^cf s- -ic the point designated by A , and if
LOA i-^^Mitts jnr designated by F, E\ then/ = FA^
Ma«sH««Ettu> 5iSo), the Principal Points coincide
. 8* ^crvesL ^. The Nodal Points, evidentiy, coin-
r :i9!i4|pamce the positions of a pair of conjugate
6 csiAw<«^=i& 5arace, and if, according to our previous
.*^ *«t "tic J.}£ « ii, AM' = u\ we obtain at once
4
*
+ 1 =
0,
•^
-
~e'u-
nu'
~ n'u'
^»ir
u
^
4
"* SJtfl
i#
"«'•
(148)
V •
^ » Kft^ %;rtMto will be recognized as identical with
!«*. -'<NaN!^ ^tfswns or a cbhtersd system of spherical
: x.-- %^ ^^*««cd bow to determine the positions of the
•*. w *«*; 5^ 'i'^' A CMitered system of spherical refracting
>wv ssri^ c'^ '^'i'* sj-stem of notation as was employed
^ .'WN> ^«$ v>f equations of the following types:
X * ■
* NX -•N-sA cN^ * i* succession all integral values from * = i to
vs.^ M^s^"^ tliAt < " O- The diagram (Fig. 100) shows the
^ V V A 5A>" l^<^««<* *e *th and the (fe + i)th surfaces of
NV ^^ *
5190.1
Ideal Ims^ery by Paraxial Rays.
265
the centered system of spherical surfaces; and we have:
Also, in accordance with our previous notation, let us write:
Here A^ and C^ designate the vertex and centre of the ifeth spherical
surface, and Ml designates the
point where the paraxial ray
crosses the axis after refraction
at the ifcth surface.
In order to determine the
Focai Length t' of the Image-
Space of the system of m spher-
ical surfaces, in terms of the
magnitudes denoted by r, n and
d, let us consider a ray which in
the Object-Space is parallel to
the optical axis (Wi = oo), and
which, therefore, in the Image-Space crosses the optical axis at the
Focal Point E\vL == A^^E'). By the definition of the Focal Length
e' given in § 178, we have:
K
ni
^*t
X*
^^^
Ck
-mT^
^^^
^k*t
•4-,
n^h
^^
^k>t
Pio. 100.
Path op a Paraxial Ray bbtwbbn kth and
(^ + l)th SURFACES OF A CSNTSRBD SYSTBM OF
Spherical Surfaces.
AkMk' = Uk\ Ak+iMk' = ««*fi, AkAk¥\ = dk,
AkCk = rk, AhBk = hk, Ak^iBM.'^ hini.
«' =
*l
tan e'n, '
where B'^ = AAmE'Em. Thus, we may write:
h\ h%
h% hs
km-i hm
*m
tan d.
By the diagram we have obviously:
hk Uk
hk
hk+i ttfc+i
tan dk = r
u'k'
and, accordingly, we derive the following formula:
e = EfA^ •
(ISO)
where the magnitudes Uk, «* can be determined in terms of the known
constants by means of the (itn — i) equations (149).
Opcks. Chapter VIII. [ § 193.
-i'»uiu: :x>t: ixmi ifCf may be written in the following
.^^ -c*! • !c Jicmthmic computation of the path of a paraxial
Mi^^i . TMiw^^ -x;?aan of spherical surfaces:
I I I
i .'Ai I - djuj) • • • (i - d^^i/u'in^i)
(ISO
% ■ \««Mk«J^ ^
^t»^ !.<.» \ i^i'v ^j- ^vett a numerical illustration of the use of
X ;i v.ucuiauii^ the value of u^ = AlnE\ When the focal
i« o^k^vtttu i;^ itvm. the last of these formulae gives a very
, .ka> » oitfckxii^ the logarithmic computation.
Length / of the Object-Space may be determined in
>> vvu^kkring a second ray for which Ui = A^F,
e >adfcl obtain a similar formula for /, as follows:
• » FAy ■- ? (152)
•v'i. v^kcwvi. *K* :iid^mtudes denoted by Wk, wi will, of course,
u. «x***iic ^*mU<s jtf^ tbey have in formula (150), because here
^ « ^ ^4;.ivu(wv A> a Jidferwt ray.
s. - N^«.»** * >^)a*.HrKdl surfaces may be regarded as a compound
.4<w > tK^ cvuibioation of tn spherical surfaces; thus, if
K 'CVHi Lengths of the ftth spherical surface, we must
«,a^ ,v iK :hirJ of formulae (147):
s *»v •'
A ^*-i
tfi. n
k
^ ,> ,^^ OkOi^u;^ .\4f3) At the end of Chapter VII, we obtain
•t *l ^m-\
/ .
»l Hi «m
V ...XV. -J s '<><^ :ittAi?fical surfaces are refracting surfaces,
•^
(iS3)
S 194.[ Ideal Imj^ery by Paraxial Rays. 267
but if the system consists of both refracting and reflecting surfaces,
and if the number of the reflecting surfaces is add, the formula will be:
4=+^. (iS3a)
e n^ V jj /
In connection with these formulae, it is well to remind the reader
again, as was stated in § 176, that in the case of a centered system of
spherical surfaces, the positive direction along the optical axis is deter-
mined by the direction of the incident axial ray; and no matter if the
direction of this ray should be reversed by one or more reflexions, the
positive direction of the optical axis remains unchanged. Thus, the
Focal Lengths, Radii, etc., are to be reckoned positive or negative,
according as they are measured in the same direction as, or in the
opposite direction to, the direction of the incident axial ray (see § 26).
The useful result, which we have just found, may be stated as follows:
In any centered system of spherical surfaces, the absolute value of the
ratio of the two focal lengths is equal to the ratio of the indices of refraction
of the first and last media. This ratio is negative, except in the case
when an odd number of the m spherical surfaces are reflecting. In
this exceptional case the ratio //e' is positive.
In particular, if the media of the Object-Space and Image-Space
are identical in substance, that is, if n^ ^ n^, the absolute values
of the focal lengths are equal. In this case, which is so often realized
in optical instruments, if we suppose that all the surfaces are refract-
ing, or that an even number of them are reflecting, we shall have
/ = — e', and, therefore, the nodal points N, N' will coincide with
the principal points A, A', respectively; for, according to § 180, we
have:
FN^A'E'-^-'e'^f^FA, and EN' ^ AF -^ - f ^ e' ^ E'A' .
ART. 55. SEVERAL IMPORTANT PORMULJB FOR THE CASE OP THE RE-
FRACTION OF PARAXIAL RATS THROUGH A CENTERED SYSTEM
OF SPHERICAL SURFACES.
194. Robert Smith's Law.
According to the second of formulae (120), we found that in an optical
system the product of the Lateral Magnification Y = y'/y and the
Angular Magnification Z = tan d'/t3.n B is constant; that is,
or
y ' tan e' _ __ /
y-tdLTiB e"
r ::i. [§194.
"*• tjj jnd Image-Space are
..J :z} and (153a) may be
^ .^a::on. we obtain one of the
... ;:e:os. as follows:
•^n :^, (154)
^ ■- ^- (1540)
•..t< '.^nly in case we have an odd
^ v!s formula is called the *'La-
ltv interesting account of the
>* :< given in a note at the end of
-;v.Tuag der optischen Abbildung",
^ ..!!:o of Die Theorie der optischen
\ A Berlin, 1904). Culmann con-
t .iltxi the "Helmholtz Equation",
••: .'."jation in the form in which it
>. \.. i: might be called the "Smitii-
. v;: himself attributed the law to
^^:. .1 sjxvial case of the law." But
. .■'lac RoHiCRT Smith,* with whose
' .u\]uainted. had enunciated the
-.; '. *i infinitely thin lenses as early as
•^o rc:n i,see Art. 42, esp)ecially § 152).
I .1 very masterful way, and recog-
♦v Anisequences of the law, although
•i :ho most general case. The for-
-x,r.:v. of spherical surfaces, was given
vs AHH^r **Zur Dioptrik," etc., pub-
i Isironomische Nachrichten (pages
< lis jxiper.
. .», ^..Licii Optik (1867), p. 50.
,..'.;. I V d'optique : MSmoires de I'Acadimie de
\ o">. vhii-riy Historical, on some Fundamental
NX <Sv^\ pp. 466-476.
• H .«.< vCani bridge. 1738); Book II, Chap. V.
§ 195.] Ideal Imagery by Paraxial Rays. 269
195. Formulae of L.Seidel. The following formulae, due to L.Seidel,*
will be frequently employed in the Theory of Spherical Aberration.
Let Ml, Ml designate the positions of two points on the optical
axis of a centered system of spherical refracting surfaces, and let
AiMi = «!, A^Mi = Ui, where A^ designates the vertex of the first
surface. Consider two Paraxial Rays, which, before refraction at the
first surface, cross the optical axis at M^, Afj, and which, before re-
fraction at the jfeth surface, cross the optical axis at the points desig-
nated by M'k^u A'iLi, and which are incident on the kth surface
at points designated by B^, B^^ respectively. In agreement with our
previous notation, we shall write:
^*^*-i = «*. AJtf, = ul A^^ = A^, Z At.Atjfi^ = el,
Since, by formulae (149)1
%-i - % = «*-i - «* = ^k-V
and combining the two equations thus obtained, we derive the first
of Seidel's formulae, as follows:
' L. Sbidel : Zur Dioptrik : Astr, Nachr., xxxvii. (1853). No. 871. pages 105-120.
we obtain:
Also, since
and
we find :
270 Geometrical Optics, Chapter VIII. [ § 195,
If we introduce here the so-called * 'zero-invariant" (see §126), and
write
«(i/r - i/«) = «'(i/r - i/u') = J,
n(i/r - i/u) = n'(i/r - i/u') = 7,
the formulae above may be written as follows:
= kMA-Ji)- dss)
Moreover, as can be seen easily from Fig. lOO, we have:
^*~i "" K _ ^*=.i ^*-i "" ^k _ ^ti} .
**-i %-i* '•a-i ***-!*
and, hence, combining these equations, and using formula (155), we
obtain :
/»jk_i njk_, njfc_i ^»-r ^*-r '»»-i
or
diving k in succession the values 2, 3, • • • i, and adding all the
c(|uations thus obtained, we derive the second of Seidel's Formulae,
as follows:
?-'-^- *,*.(/. -y.)!:---^^^
Moreover, since
(156)
and
we can write:
§ 196.] Ideal Imagery by Paraxial Rays. 271
or
and, hence, by (156), we derive also the following:
= h'AU, - J,)
which is the third of Seidel's Formulae.
The expression
-A-^Z--%V1; (157)
which occurs frequently, especially in Seidel's optical formulae, may
be transformed as follows:
rf*-i
^ i_ /f^ _ «i\ , ^ f!^«!lf "^ J +i- /"-^ _ ^V-^ ^
Aj \»i «i/ hi \n[ n^ ) h\ \n\^^ nj njt\ n^h\
u'k _ «i -lV ^ f ^* _^\\
and, finally:
2^ -^^ — P — r = "^ — ^ "" 2^ "ia^ ~ ^ ( ~ ) J (158)
where Ajs the symbol of operation introduced in § 126; so that here
A(i/n);k= iK- iK-i-
196. In order to illustrate some of the uses of these formulae, we
shall apply them, as Seidel does, to determine the positions of the
Focal Points F, E' and of the Principal Points A, A' and the magni-
tudes of the Focal Lengths/ = FA, e' = E'A' of a centered system of
m spherical refracting surfaces.
We shall suppose that we know completely the path of a certain
paraxial ray through the system, for example, the ray which in the
object-space crosses the optical axis at the point designated by M„
and which meets the first spherical surface at the point designated by
270
GeonietriiMJ > ?
_j, Liiapier
VIII.
[ § 196.
If we introduce here the
write
nil I y -
the formula; above in;-
KKUh - J,) - •
Moreover, as can !
and, hence, combirv
obtain:
K_ ^k ^
or
Giving k m m;
equations thii- ■
as follows:
• rnii'wtion-constants of the optical
^. rnickne^ses !(/}, we shall suppose
, 1 this rav, so that the system
I •^fie ray on the optical axis
.. ;5 ;jertectly possible, since the
.:: ierms of the latter, as follows:
<« i ■
Moreover, ^il
and
we can wrii
= Ui. — u
1+1
_ ve ?iiall. chiefly for the sake of
,,^L .»! the optical system, denoted
c expression:
(IS9)
.-*^-
(159a)
.^jcu gi^'^n in formula (158).
^.;;* 5. £' can be found by means of
^ i'Maid we put, ist, u^ = A^F, ul = oc,
%t jbtain
(160)
^ ^>4c:0QS of the Principal Points A, A'
j^ixui surfaces, we recall (§139) that this
»t^-jcwrized bv the condition:
^ which, since
:;*«*
1^«-
hi u
u,
35>
1 197.[ Lenses and Lens-Systems. 273
may be escpressed by the following relation:
In formula (iS7)i therefore, we must, ist, substitute njijnjai for
h^ju^, solve for Uj, and put u^^A^A; and, 2nd, substitute nji^ln^u^
for hjuit eliminate (/j — Ji) by (iSS)» solve for u^, and, finally, put
ul^=^ A^A\ Performing both of these op< rations, and making use of
formula (159) each time, we obtain the following formulae:
\ni n^ n^ Mj /
Having determined the positions of the Focal Points and of the
Principal Points, we can find at once the magnitudes of the Focal
Lengths/, e'; for
f= FA ^ FA, + A,A,
e' = EA' = E'A^ + A^A',
and, hence, by formulae (160) and (161):
/ = F/nl, 6' = - Fin,. (162)
If the given ray, to which the symbols u, h refer, is incident on the
first surface of the system in a direction parallel to the optical axis,
we must put Mi = 00 in the above formulae, whereby we obtain for
the Focal Lengths:
/
fl. U^ 'U^* * -U U, 'Urn' • *u
= -- • , c = ; U037
n^ U2'U^'"U^ u^u^'-u^
in agreement with the results as expressed by formulae (150) and (153).
LENSES AND LENS-SYSTEMS.
ART. 56. THICK LENSBS.
197. A centered system of two spherical refracting surfaces (m = 2)
is called a Lens. In the following discussion of Lenses it will be as-
sumed that the media of the incident and emergent rays are identical
in substance, and the symbol
n = n\/n, = njn^
19
274 Geometrical Optics, Chapter VIII. I § 197
will be employed to denote the relative index of refraction of the
medium of the lens-substance and the surrounding medium. A Lens
is usually described by assigning: (i) The magnitude of the relative
index of refraction (n); (2) The positions A^, A2 of the vertices of
the two spherical surfaces, whereby the optical axis of the lens may
be directly determined, both as to its position and as to its direction,
by drawing the straight line from Aito A^. The distance
A1A2 = d,
called the * 'thickness" of the Lens, is reckoned always in the positive
direction of the optical axis, so that d is essentially a positive magni-
tude; and (3) The magnitudes and signs of the radii r^ = -4^^,
fj = A2C2. Employing the same letters and notation as are used in
the preceding portion of this chapter, we shall regard a Lens as a com-
pound system consisting of two single spherical refracting surfaces,
whose focal lengths, in terms of the above data, will be expressed as
follows:
FA =^f = ^* E'A = fi' = ^^
n — I ' * * n — I
' ' •'* n — 1 ** * n — I
(164)
In order to determine the positions of the Focal Points F, E' and
the magnitudes of the Focal Lengths /, e' of the Lens, we shall employ
formulae (136), in which the determination-data are the Focal Lengths
/,, e[ and/2, 62 of the two partial systems and the * 'interval" A between
them. This latter magnitude is defined as follows:
A = E[F2 = E[A, + A.A^ + A^F^,
that is,
A = «;-/2 + d; (16s)
and, accordingly, expressed in terms of the original data, this magni-
tude is as follows:
A = »JfiJZlA±d(!LzJl J ^ , (,66)
n — I (« — i)
where
iV = (n - I) {nCfj - fj) + d{n - i)} (167)
denotes a constant of the Lens.
§ 198.] Lenses and Lens-Systems. 275
198. (i) The abscisscE of the Focal Points F, E' of the Lens with
respect to F^, £3, respectively:
These are obtained immediately from the first two of formulae (136)
as follows:
F^F^-"^, E',B:=-~\ (168)
(2) TTie Focal Lengths /, e' of the Lens:
Likewise, from the last two of formulae (136), we obtain:
F4=/=^*=-e' = 4'£'; (169)
where A, A' designate the positions on the optical axis of the two
Principal Points of the Lens. We see that in every Lens, surrounded
by the same medium on both sides, the Focal Lengths f, e' are equal in
magnitude, but opposite in sign. (This is true of an optical system
consisting of any number of spherical surfaces, provided »i = »1j
see § 193.)
Hence, also, the Nodal Points N, N' of the Lens coincide with the
Principal Points A, A', respectively; which is characteristic likewise
of any optical system for which the media of the incident and emergent
rays are identical (§ 193).
(3) Abscissce of the Focal Points F, £' of the Lens referred to the
vertices ilj, A^, respectively:
Since
A,F^A,F,+F,F^''f,+F,F,
a^e: = i4,£; +£;£' = - «;+£;£',
we derive, by means of formulae (164) and (168), the following form-
ulae for locating the positions of F and E' :
Nr, + n{n- i)rf ^ Nr^ ~ njn ~ i)rl
AtF^ TT. r , A2E = TT/ T • (170)
* N{n — i) ' N(n — i) ^ '
(4) Abscissce of the Principal Points A, A' of the Lens, reckoned
from the vertices A^, A^, respectively:
Since
A^A = A^F + FA, A^A' = A^E' +E'A\
formulae (169) and (170), together with formula (165), give the fol-
lowing formulae for determining the positions ol A, A':
276 Geometrical Optics, Chapter VIII. [ § 199.
whence we see that the abscissae of the Principal Points are in the same
ratio to each other as the radii of the surfaces of the Lens.
The distance A A' between the two Principal Points may be ex-
pressed as follows:
AA ^^^ AAi "pAj2T.2 I A^A I
whence we obtain:
199. Character of the Different Forms of Lenses.
An inspection of formulae (167) and (169) will show that the sign
of the Focal Length /, which determines the character of the Lens,
depends not only on the magnitudes and signs of the radii r^ r,, but
also on the thickness d of the Lens. It will depend also on whether n
is greater or less than unity, but in the following discussion U wiU be
assumed that the Lens is of the type of a glass lens in air, that is,
n — I > o.
Evidently, with this assumption, the magnitude denoted by N will be
greater than, equal to, or less than, zero, according as
d i ^_, ■ (173)
n — I
Now d itself is always positive; and hence in any form of Lens, for
which fi — fj is negative, the two lower signs in formula (173) cannot
possibly occur, so that for any Lens of such form, the sign of N will
necessarily be negative.
How the sign of the Focal Length / of the Lens depends on the
magnitude of the thickness (2, will be apparent in the following clas-
sification of the different forms of Lenses.
(i) Biconvex Lens (r^ > o, r, < o). In a Biconvex Lens the radii
fi, fa have opposite signs, and hence the Focal Length
^ N
of a Biconvex Lens is positive, so long as the thickness
n(ri - r,)
d<
n — I
so that a Biconvex Lens whose thickness d does not exceed this limiting
value is a convergent Lens. This is the usual character of a Biconvex
§199.] Lenses and Lens-Systems. 277
glass Lens in air, an example of which is shown in Fig. loi, where
fi = + lo, f 2 = — IS ^^d d = +3. For comparatively small
values of d, as here, the Principal Points A, A' are situated within the
Lens itself. If the Lens is made thicker, the two Principal Points
A\AA' 1*4 cT
Fio. 101.
CONVBRORMT BICONVEX GLA88 LBNS IN AlR.
fi"3/2: n-^iCi-« + 10; n--<4iCa = — 15; rf=^i^«-> + 3; >4i/^=— 11.66;
-^t£' = + U.25; ^i^ = + 0.833; A%A'^ — IJ25\ /=/^/^ =— *' --r^'-E* - + 12.5.
will approach nearer to each other, until when d attains the value
d = fi — r,, so that the two surfaces of the Lens have a common
centre, the Principal Points A^ A' coincide with each other at this
common centre. Fig. 102 shows a Biconvex Lens with concentric
surfaces; such a Lens, made of glass and surrounded by air, will be
convergent. Here, likewise, belongs the Spherical Lens, character-
Pio. 102.
Comnmosirr Bicohvbz Glass Lbns in Air: Special Case— Two Sttrfaces Concentric.
n— 3/2; ri=-+5; n — — 3; rf= + 8; ^i/^=— 0.625; yf i£* =+ 2.625 ;
^M--^l<ri--^lCi- + 5; ^t^'-^ia = ^«Ci =— 3; /=-/^^-—•=-^'-E'= +5.625.
ized by the value d ^ ti -- T2 ^ 2ri. The Spherical Lens is also to be
regarded as a particular case of the Equi-Biconvex Lens (r^ = — fj,
Ti > o). A Spherical Lens is shown in Fig. 103.
If we suppose the thickness of the Biconvex Lens to be greater
than (fj — fj), the Lens continues at first to be convergent, but the
Primary Principal Point A will lie now beyond (or to the right of)
the Secondary Principal Point A'; and as the thickness d is increased,
these points separate farther and farther from each other, so that at
length we shall find the Secondary Principal Point A' in front of the
278
Geometrical Optics, Chapter VIII.
[§W.
Lens and the Primary Principal Point A beyond the Lens, the Lens
still being convergent. And when d attains the value
i = n{jy^ - r^/(« - i),
PlO. 103.
CONVBROBNT BQUnUCONVSX GLASS I«BN8 IN AlR : SPBCXAL CaSB — SPBE&ICAX.
i«=3/2; n=^iCi = — n-C4i=-+3; «f = ^i^i-+6; -4i-P— iE'yIt- + 1.5;
AxA^ AxC'^ A' AfCAf ^l\ /=-P^ =-— /-yl'^- +4.5.
the Biconvex Lens becomes a Telescopic Optical System, with its
Focal Planes, and its Principal Planes also, at infinity. The case of
a Telescopic Biconvex glass Lens in air is shown in Fig. 104; for
Fig. 104.
Tblbscopic Biconvex Glass I«ens in Air.
i«»-3/2; n = ^iCi-+3: n-^«Ct-— 2: rf->4i^a- + 15: ^iO-+9; /-—•-•.
which the determination-constants have the following values:
fj = + 3, fa = — 2, d = + 15. The Optical Centre 0 of this Lens
coincides with the Focal Point £[ of the first surface of the Lens and
with the Focal Point F^ of the second surface of the Lens.
'' And, finally, in case d > n(ri — r2)/(« — i), a Biconvex glass Lens
in air will be divergent (/ < o). But no matter how great the thick-
ness d becomes, we shall find that the Focal Point F ol z. Biconvex
Lens lies always in front of the Lens.
(2) Biconcave Lens (fj > o > r,). Here also, as in the case of the
Biconvex Lens, the radii of the two surfaces have opposite signs, fi
being negative and r, being positive. Hence, assuming that n is
greater than unity, we find in the case of a Biconcave Lens that the
constant N is always positive, and, therefore, / is negative; so that
§ 199.[ Lenses and Lens-Systems. 279
a Biconcave glass Lens in air is always a divergent Lens. The Prin-
cipal Points A f A' of Si Biconcave Lens lie always in the interior of
the Lens, the Primary Principal Point A being situated in front of the
Secondary Principal Point A\ Fig. 105 shows a Biconcave glass Lens
c.
Pio. 105.
DxvBROBxrr Biconcave Glass I«bns nr Anu
iB-3/2; n--^iCi-— 10; n - -^tCt - + 15 ; <f-^i>4s- + 3: i4iF-+12Jj
ylt£'- — 12.7; AiA''-\-0,T7; ^t^'- — 1.154; /-/^^ -—/-^'^ - — 11.54.
in air, for which the constants have the following values : ^i = — lo,
r, = + IS» d = + 3-
In an Equi-Biconcave Lens we have Tj = — r^, fj > o.
(3) Lens with One Surface Plane. In this case, therefore, one of the
radii fi, r, is infinite.
If the first surface is the plane surface, then r^ = oo, and we find:
•^ n — i'
so that the character of the Lens depends on the sign of the curvature
of the curved surface. Thus, for example, in a Piano- Convex Lens
(fj = 00, r, < o), / is positive, and the Lens is a convergent Lens.
On the contrary, in a Piano- Concave Lens (r^ = oo , fj > o) , / is negative,
and the Lens is a divergent Lens. (In these statements it is assumed,
as always in this discussion, that n > i).
For fi = 00, we find also:
n(n — i) ' * n — I '
A^A = d/n^ A^A! = o.
When one surface of the Lens is plane, one of the Principal Points
will coincide with the vertex of the curved surface.
The diagrams (Figs. io6 and 107) show the cases of a Piano-Convex
Lens and of a Piano-Concave Lens (n = 3/2). The two Lenses aife
280
Geometrical Optics, Chapter VIII.
[§199.
represented as having the same thicknesses, and the absolute value
of the radius of the curved surface is the same for both Lenses. In
the figures the first surface is the plane surface (r^ = oo); but if the
light is supposed to go from right to left, so that the curved surface
Q
£'
Pio.106.
Plano-Convbz Glass Lens in Air. This I^ens is always Conversent.
Aui'^-k'2; AtA'^0; /-/V4 =— •=^'-fi'- + 12.
is the first surface, the figures, except for certain obvious changes in
the letters, will be correct.
(4) ConcavO' Convex, or Convexo- Concave, Lens. In a Lens of this
form the two radii r^, r, have the same sign, and, hence, the sign of the
Focal Length / = nr^rjN will be the same as the sign of the constant
N. If, therefore, N is positive, the Lens will be convergent; if N
is negative, the Lens will be divergent; and if N ^ o, the Lens will
e.
Pio. 107.
Plan o-CoNCAVS Glass I«bn8 in Air. This I^ens is alwasrs Divergent.
ii-3/2; n-i4iCi-»; n-^«Ca-+6; «f--4i^«-+3; ^i-P-+14; AtE' '^
AiA^+2i AtA'^0: /-/^<^-— •-^'JE'- — 12.
-12;
be telescopic. According to (173), the sign of N will depend on the
value of d.
Let us suppose that both radii are positive, so that the first surface
of the Lens is convex (r^ > 0) and the second surface is concave
(r, > o). It will be necessary to consider only this case, since we
have merely to suppose that the direction of the light is reversed in
order to obtain the opposite case.
Accordingly, assuming that both radii are positive, we have to
consider the following three cases of the Convexo-Concave Lens:
(a) Positive Meniscus (r, > r, > o).
Since in this case (fj — r^ is negative, and since d is always positive,
§ 199.1 Lenses and Lens-Systems. 281
it follows that d > n(ri — r^/(n — i), and, therefore, N is positive.
Hence, a Lens of this form, called a "Positive Meniscus'*, is always
a convergent Lens (Fig. io8). It will be remarked that the Primary
c. a E'
Pio. 108.
PosTTtVB Msmscus (Glass Lbns in Air). This I^ens is always Convergent.
II-3/2: n-i4iCi-+6; n--r4iCi- + 12; d^AiAt^-\-2; ^i/!*- — 22.8; ^i£'- + 19.2;
Aui"—!.!; AtA'^—lAi /-/^^/l -— •- i^'JE*- +21.6.
Principal Point A lies to the left of the vertex A^, and the Secondary
Principal Point A' lies to the left of the vertex ^l,; and that the line-
segment AA' \s always positive.
(6) The case when r^ > fj > o. In this case {r^ — r^ is positive,
so that the Lens may be divergent, convergent or telescopic, depend-
ing on the value of the ratio d/(ri — r^.
The most common case under this head is that for which
d < n{ry - r^lin - i).
When this is the case, we have a divergent Lens, called a " Negative
Meniscus'' (Fig. 109). The Principal Points A, A' lie beyond (that
c, C
Fio. 109.
Nboativb Meniscus (Glass I«bns in Am).
«-3/2; n-^iCi-+12; n->4iCi- + 6; d^A\At^-\-2; A\F^-\-2Q\ AtE^^ — 25.h',
^M- + 3; ^t^'-+1.5; /- /^^ -— •- ^'£^ - — 27.
is, to the right oQ the vertices A^, A^^ respectively. If d = fj — fj,
the two Principal Points coincide at a point which is also the com-
mon centre of the two surfaces of the Lens (Fig. no). An infinitely
thin Lens of this kind is not divergent, but telescopic.
Again, if when f i > f 2 > o» we have also
n{r, - fg)
a = ,
n — I
282
Geometrical Optics, Chapter VIII.
[§199.
the Lens will be of the kind represented in Fig. 1 1 1 , where the constants
have the following values: n = 3/2, fi = + 12, fj = + 6, d = + 18;
E'
Fig. 110.
CoNVBzo-CoNCAVB GLASS I«BNS IN AiR : SPBCiAL Casb OF Nboativb MsNiacus. Two surfsoef
of I^ens have the same centre : I^ens is Divergent (Focal Point F not shown in the diagram: it
lies far to the right.)
w-3/2; rx^AxCx^AxCt^AiA^AxA'^-^rl', rt^ A%Ct^ AtCx^ AtA^ A%A' ^ -^-Ix
rf-/ll/l«-n — n-+2: y<iF-+27.5: ^lE*-— 19.5; /-F/I-—/-^'JE'-— 22.5.
whence we find /==—«' = 00. This type of Lens may, therefore,
be called a * 'Telescopic Meniscus".
As d increases from the value d = fj — fj to the value
d = n(ri - r^\{n - i),
the Principal Points, which, as we saw above, were coincident, sepa-
rate farther and farther from each other, both moving along the optical
r^
Fio. 111.
CONVBZO-CONCAVB GLASS I«ENS IN AlRZ SPECIAL CASB — TBLBSCOPIC MBNiaCUS.
w-3/2: ri-^iCi-CiCt- + 12; n- -<4tCt- Ci-<4«- +6; /-— •- •.
axis in the positive direction of that axis, but A' keeping ahead of A
until they both arrive together at infinity.
And, finally, if, when fy^>f^> o, we have also
d > n(f^ - r^l{n - i),
as in the case of the Lens represented in Fig. 112, where the constants
V33r^^
Fio. 112.
Convergent Meniscus (n > ^i > 0) : Glass Lbns in Anu
«-3'2; n--4iCi=+3: >^=/lja-+2; rf-^Mt-+6; ^iF-— 24;
AzE'-'-VA', A\A^-\2; ^j.^'-— 8: /- /^^ - — / - -^'JE* - + 12.
§200.]
Lenses and Lens-Systems.
283
PlO. 113.
CONVBROBNT COZTVBZO-CONCAVB
Glass I«bns in Air: Special Case—
Two surfaces have equal curvature:
Called " I^ens of Zero Curvature.*'
d^ AiAt" CiCt.
have the following values: n = 3/2, fj = + 3, r, = + 2, d = + 6,
the Lens will again be a convergent
Lens, and now the Principal Points A,
A' will lie in front of the vertices Ai,
A 2, respectively, and A will lie in front
of .4'.
(c) The last case to be considered is
the case when r^ = r, > o. In this form
of Lens, sometimes called ''Lens of Zero-
Curvature", the curvatures of the two
surfaces are equal; and since r^ r, have
the same sign, and N is positive, the
Focal Length / is positive, so that this
Lens Is always convergent. The diagram
(Fig. 113) represents the case of a Lens
of Zero-Curvature, determined by the values: » = 3/2, / = — c'
= 6rJ/d. Obviously, in this Lens we have
In the limiting case when (I = o, this Lens will be an infinitely thin
"telescopic Lens.
ART. 57. THIN LBNSBS.
200* Practically speaking, the thickness of a Lens is almost always
^mall in comparison with the other linear constants of the Lens. And
Cexcept in the case of the so-called **Lens of Zero-Curvature", for
^^^hich fi = r,) the term (» — i)d which occurs in the expression of the
cronstant N is generally quite small in comparison with the other
"t^rm n{r^ — r^). The value of N may be written:
iyr = n(«-i)(r,-r.){,+^^};
d, hence, if we neglect terms involving powers of d higher than the first,
have:
I I I (n ~ i)d
N^nin- iXfj - fi) r «(^2 - O '
S^^bstituting this value of i/N in formulae (169) and (171), we obtain
following approximate formulce of Thin Lenses:
284 Geometrical Optics, Chapter VIII. [ § 202.
(174)
n{r^-ry nir^-r;)'
These formulae are useful for approximate determinations of the ma^i-
tudes of the Focal Lengths and of the positions of the Principal Points.
201. Infinitely Thin Lenses. If the Lens is infinitely thin, the
formulae above may be still further simplified by putting d = o. Thus,
we obtain :
d = AiA2 = o, /=—«' =
fif,
(n-i)(ra-r,r
AiF^-z ^ r = £'^. A,A=A,A'^o.
> (w-i)(r2-ri) *
• (i7S)
These formulae, which are identical with those formerly obtained in
Chapter VI, need no further remark here.
ART. 58. LBNS-STSTBMS.
202. Consider a compound system consisting of two Lenses with
their optical axes in the same straight line, and let A^, A[ and F^, Ej
designate the positions of the Principal Points and of the Focal Points,
respectively, of the first Lens; similarly, let A^, A'^ and F,, E\ designate
the positions of the Prii;icipal Points and of the Focal Points, respect-
ively, of the second Lens. Thus,
F,A, =/i = - «; = A[E\, F^, =/, = - 4 = i4;£;;
where /j, e\ and /a, «2 denote the Focal Lengths of the two Lenses.
Moreover, here let us put
A\A2 = d, E'lFj = A.
Then, since
A = E\A\ + A\A2 + A^F^,
we have:
A= -(/i+/2-d).
Finally, let F, £' and A, A' designate the positions of the Focal Points
and of the Principal Points, respectively, of the compound system of
the two Lenses, and let/, e' denote the Focal Lengths of the compound
system. Then, by processes entirely similar to those employed above
in Art. 56, we derive the following system of formulae for an Optical
§204.1
Lenses and Lens-Systems.
2S5
System composed of two Lenses:
/=-«' =
/1/2
fx+f,-d'
F,F =
f\
«' v^'
^,F=-
AiA
/,(/. - d)
£!£' = -
fl
t -r-i'
fxd
fi+ft-d'
fr+f,-d'
A'u- - /»(/i ~ d)
' ~fx+f,-d'
A' A' _ f^
' ~ A+f,-d-
(176)
Thus, being given the two Lenses and their positions relative to each
other, we can, by means of the above formulae, determine completely
the compound system.
203. If, instead of two Lenses, we had Two Systems of Lenses^ the
formulae (176) can be employed to determine the compound system,
provided the letters with the subscript i and the letters with the sub-
script 2 be understood as applying to the first and second systems of
Lenses, respectively.
204. A case of considerable interest is an Optical System composed
of Two Infinitely Thin Lenses. Since the two Principal Points of an
infinitely thin Lens coincide at the optical centre of the Lens, the
letters Ai and A^^ as employed in formulae (176), will designate for
this case the positions of the optical centres of the two Lenses, and,
therefore,
d = A^A^ '^ A1A2 ^ A1A2
denotes now the distance of the second Lens from the first. If the
two infinitely thin Lenses are in contact (d = o), we find:
l//=l//l+l//2,
in agreement with the general formula (106).
Assuming, for the sake of simplicity, that the optical system con-
sists of two infinitely thin Lenses, we may discuss formulae (176)
briefly, as follows:
(a) Suppose that both Lenses are convergent (Ji > o, /2 > o.) If the
two Lenses are in contact (d = o), we have:
/=
/,/,
and, cxHisequently, / > o. But this is the smallest positive value of/;
286 Geometrical Optics, Chapter VIII. [ § 204.
so that as we increase the distance d between the two Lenses, the
resulting system is less and less convergent; until, when d = /i +/j,
we have / = oo, in which case the compound system is telescopic.
If we continue to separate the Lenses still farther, we have at first a
feebly divergent system; but the divergence increases as d is made
greater and greater.
(b) In case both Lenses are divergent (/i < o,/2 < o), we have alwa3rs
/ < o, so that the compound system will be divergent. The diver-
gence will be greatest when the two Lenses are in contact, and will
decrease as the Lenses are separated farther and farther.
(c) Finally, suppose that one of the Lenses is convergent, and the
other divergent. For example, let us assume that fi>o and /, < o.
In this case the compound system will be divergent, if d < (fi+f^;
convergent, ii d > (Ji+ f^; and telescopic, if d = /j + /,. Since /i, /,
have opposite signs, there are two cases here to be considered, as
follows:
1st, The case when (/^ 4-/2) < o: that is, the absolute value of the
Focal Length of the convergent Lens is less than that of the divergent
Lens. Since d is essentially positive, the only possibility here is
d > (fi + f'^, and hence this system will also be convergent. The
greatest value of / is obtained by placing the two Lenses in contact
(d = o); and as the Lenses are separated farther and farther apart,
the convergence increases.
2nd, The case when (/i 4-/2) > o: that is, the absolute value of/,
is greater than that of /j. When the two Lenses are in contact (d = o),
the system is divergent and / has its least negative value. As d in-
creases, the absolute value of / increases, its sign remaining negative;
until, when d = /j + /g, / is infinite, and the system is a telescopic
system. For values of d greater than (/j + /J, the sign of / will be
positive, and the system will be convergent, the convergence increasing
with continued increase of d.
CHAPTER IX.
EXACT METHODS OF TRACING THE PATH OF A RAY REFRACTED AT
A SPHERICAL SURFACE.
ART. 59. INTRODUCTION.
205. In the preceding chapter we have seen how an ideal image is
produced by a centered system of spherical surfaces so long as the
rays concerned are the so-called ''Paraxial Rays" which are all con-
tained within the infinitely narrow cylindrical region immediately
surrounding the optical axis of the system. In this case to a homo-
centric bundle of incident rays corresponds a homocentric bundle of
emergent rays.
But, according to the Wave-Theory of Light, in order to have an
optical imagery, a mere homocentric convergence of the rays is not
sufficient. This theory requires not only that the wave-front after
the light has traversed the optical system shall be spherical, so that
the rays of light proceeding originally from a point shall meet again
in a point, but that the effective portion of the wave-surface shall be
as great as possible in comparison with its radius, which means that
the effective rays shall constitute a wide-angle bundle of rays (see § 45).
Only when this last condition is complied with will the resultant effect
of the spherical wave be reduced approximately to a point at the centre,
so that there will be point-to-point correspondence between object
and image.
Moreover, there is also still another practical reason why we find
it necessary to use wide-angle bundles of rays in the production of
an image. For if the wide-angle bundle of rays is a condition of a
distinct, clear-cut image, it is equally essential for the production of
a bright image, since the light-intensity will evidently be greater in
proportion as the effective portion of the wave-surface is larger.
Both theoretically and practically, therefore, we require to have an
optical system which will, if possible, converge to a point a wide-angle
homocentric bundle of incident rays, so that not merely those rays
which we call Paraxial Rays but those rays which have finite inclina-
tions to the optical axis will be converged again to one and the same
image-point. Generally speaking, this requirement is found to be
impossible of fulfilment. Indeed, there may be said to be only one
actual optical system which perfectly satisfies the condition of collinear
287
288 Geometrical Optics, Chapter IX. [ § 206.
correspondence, viz., the Plane Mirror; which, inasmuch as it pro-
duces only a virtual image without magnification, hardly deserves to
be ranked as an ''optical instrument" at all. The ^'Pin-Hole Camera"
is no exception to this statement, because only when the aperture
through which the rays enter the apparatus is a mathematical point
will there be strict point-to-point correspondence of object and image
— even then assuming that there were no exceptions to the Law of the
Rectilinear Propagation of Light such as we encounter in Physical
Optics.
Instead of the ideal case of coUinear correspondence of Object-
Space and Image-Space, the theory of optical instruments is compli-
cated by numerous practical and, for the most part, irreconcilable
difficulties, due chiefly to the so-called ** aberrations*' — some of which
are aberrations of sphericity^ while others are chromatic aberrations —
and due also, in a less degree, to the assumptions at the foundation of
Geometrical Optics, which, as we have pointed out, are not entirely
in accordance with the facts of Physical Optics. It is not our purpose,
however, to enter into a discussion of these questions here, as they
will be extensively treated in subsequent chapters of this treatise.
In this chapter we propose to investigate the path of a ray which
makes a finite angle with the axis.
ART. 60. OBOMBTRICAL METHOD OF INVESTIOATINO THE PATH OF A &AT
REFRACTED AT A SPHERICAL SURFACE.
206. Construction of the Refracted Ray.
In § 29, we showed how to construct the path of a ray refracted at
a surface of any form, and that method is, of course, applicable to
the refraction of a ray at a spherical surface. The following elegant
and useful construction of the path of a ray refracted at a spherical
surface was first given by Thomas Young in his lectures on Natural
Philosophy.^ Weierstrass,^ in 1858, and LiPPiCH,* in 1877, gave
the same construction, each entirely independently.
Let C (Figs. 114 and 115) designate the position of the centre, and
let r denote the radius, of the spherical refracting surface /«, and let
^ A course of lectures on Natural Philosophy and the Mechanical Arts, by Tbomas
Young, M.D., London, 1807 (two volumes); II, p. 73, Art. 425.
' See article by K. Schellbach entitled " Der Gang der Lichtstrahlen in dner G]a»>
kugel ": Zft. phys. chetn. Unt., 1889, II, 135.
'F. Lippich: Ueber Brechung und Reflexion unendlich duenner Strahlensysteme an
Kugelflaechen: Denkschriften der kaiserl. Akad. der Wissenschaflen flu Wiem^ xxzvn
(1878), pp. 163-192. See also a paper published by F. Kesslsr in Wied, Atm, Pkys^
XV (1882).
§206.1
Path of Ray Refracted at Spherical Surface.
289
QB represent the path of the incident ray meeting /* at the point B;
and let n, n' denote the absolute indices of refraction of the first and
second medium, respectively. Concentric with the spherical refracting
surface /i, and with radii equal to n'rfn and nr/n\ describe two spheri-
cal surfaces t, t', respectively. Let Z designate (as shown in the
diagram) the point where QB, produced if necessary, meets the auxil-
iary spherical surface t. Join Z by a straight line with the centre C,
and let Z' designate the point where this straight line intersects the
Pio. 114.
YouKO*8 Construction of thb Path of a
Rat Refracted at a Spherical Surface.
The figure shows the case when the surface is
conTez and the second medium more highly
lefracting than the first (n' > n). If the
letters O and ^. Z and Z' and r and r* are
interchanged, and if the arrow-heads are re-
versed, the same diagram will show Young's
Construction for the case when the ray is re-
fracted at a concave spherical surface into an
optically less dense medium.
Fig. 115.
Young's Construction of the Path of a
Ray Refracted at a Spherical. Surface.
The figure shows the case when the surface is
concave, and the second medium more highly
refracting than the first (»'>»). If the
letters O and J^, Z and jf and r and i' are
interchanged, and if the arrow-heads are re-
versed, the same diagram will show Young's
Construction for the case when the ray is re-
fracted at a convex surface into an optically
less dense medium.
Other auxiliary spherical surface t'. The path BR' of the refracted
ray is determined by the straight line which joins B and Z'. In
making this construction, we must be careful to select for the point
Z that one of the two possible points of intersection of the incident
ray QB with the spherical surface /i which will make the piece of the
incident ray which lies in the first medium and the piece of the re-
fracted ray which lies in the second medium fall on opposite sides of
the incidence-normal CB, in accordance with the Law of Refraction.
The proof of the construction is very simple. Since
CZ'.CB ^ CB\ Cr = n' :n,
20
290 Geometrical Optics, Chapter IX. [ § 207.
the triangles CJBZ, CBZ' are similar, and, hence, Z CBZ = Z BZ'C.
But in the triangle CBZ,
sin Z.CBZ CZ n'
sin jLBZC" CB'^ n'
and, since by the Law of Refraction
sin a/sin a' = n'/»>
where a = Z CJBZ, it follows that Z.BZC ^ Z C5Z' = a! and,
therefore, -BjR' is the path of the refracted ray.
In both diagrams (Figs. 114 and 115) the case represented is that
in which the first medium is less dense than the second (n' > n);
but by a suitable change of the letters and a reversal of the arrow-
heads, the same diagrams will suffice to exhibit the case when the ray
is refracted into the less dense medium (n' < n). In this latter case
the spherical surface t will be the inner, and the spherical surface t'
will be the outer, of the two auxiliary spherical surfaces; thus, in this
case, a ray may be incident on the spherical refracting surface /i with-
out meeting at all the auxiliary surface r; which means that such a
ray will be totally reflected.
207. ''Aplanatic" Pair of Points of a Spherical Refracting Surface.
The first point to be remarked in connection with Young's Con-
struction is the extraordinary property of every pair of such |x>ints
as Z, Z'. Any straight line drawn through the centre Cof the spher-
ical refracting surface will determine by its intersections with the
auxiliary spherical surfaces t, t' a pair of points Z, Z', at distances
from C equal to n'r/n, nr/n', respectively, characterized by the prop-
erty that to a homocentric bundle of incident rays Z corresponds a
homocentric bundle of refracted rays Z'. Moreover, this property
is entirely independent of the angular opening.of the bundle of inci-
dent rays, and is true, therefore, of a bundle of rays of finite aperture.
The pair of conjugate points Z, Z', which lie on the axis of the
spherical refracting surface (Fig. 116), and which are situated as above
described, are called the **aplanati€'^ pair of points of the spherical
refracting surface ; with respect to these points the spherical refracting
surface is an "aberrationless" surface.
Since rays which are directed towards the centre C enter the second
medium without being changed in their directions, the point Cmay also
be regarded as a pair of coincident conjugate points (§44) which possess
a property similar therefore to that of the aplanatic points. Moreover,
each point on the surface of the refracting sphere is a "self -correspond-
§ 207.1 Path of Ray Refracted at Spherical Surface. 291
ing", or "double", point. But the only pair of such points that are
separated is the pair Z, Z'.
Since jLBZ'C = a, ZJBZC = a', it follows that the angles of in-
clination to the axis of the incident and refracted rays are equal to
the angles of refraction and incidence, respectively; so that the apian-
Fio. 116.
SO-CALI.BD APLANATIC (oR ABBRRATIONLBSS) POINTS OF A RBFRACTINO SPHBRB.
atic points are likewise characterized by the fact that the sines of the
angles of inclination to the axis of any pair of conjugate rays crossing
the axis at Z and at Z' have a constant ratio. Another way of re-
marking this characteristic property of the aplanatic pair of points
of a spherical refracting surface is by the relation :
BZ/BZ' = n'fn.
Moreover, since
CZ'CZ' = r",
the geometer will recognize that Z, Z' are the so-called 'inverse'*
points with respect to the spherical surface of radius r, which are
harmonically separated by the end-points of the diameter on which
they lie.
The points Z, Z' lie always on the same side of the centre C of
the spherical refracting surface, so that whereas the rays will pass
**really" through one of these points, the corresponding rays will pass
"virtually" through the other point. Thus, one of the spherical sur-
faces r, t' is the virtual image of the other.
292
Geometrical Optics, Chapter IX.
[§208.
The circle of contact, in which the tangent-cone, drawn from that
one of the points Z, Z* which lies outside the spherical refracting
surface, touches this surface, divides it into two portions, and the
incidence-point B lies always on the greater of these two portions of
the surface.
In the case of Reflexion at a Spherical Mirror (n' = — n), Young's
Construction evidently fails. A Spherical Mirror has no pair of *'apla-
natic" points corresponding to Z, Z'; or, more correctly speaking, the
points Z, Z' coincide at the vertex of the mirror.
208. Spherical Aberration.
In general, however, a homocentric bundle of rays incident on a
spherical refracting surface will not be homocentric after refraction.
Consider, for example, a bundle of rays diverging from a point L
(Fig. 117), and incident directly on a Spherical Refracting Surface,
so that the chief ray of the bundle is directed, therefore, towards the
centre C. Since there is symmetry around LC as axis, it will be
sufficient to trace the paths of those rays which lie in a meridian section
of the bundle, for example, the section made by the plane of the dia-
Fio. 117.
Spherical Aberration. Whereas the incident rays all cross the axis of the spherical snrface
at one point L, the corresponding refracted rays cross the axis, in general, at different x>oints L'
Z", etc.
gram ; for it is obvious that the entire bundle of rays may be r^arded
as generated by the rotation of this meridian pencil around the chief
ray ^ C as axis.
If LB is an incident ray, and BU the corresponding refracted ray
meeting the straight line L C in L\ and if LBV is revolved around LC
as axis, to the incident rays lying on the surface of the right-circular
cone CLB will correspond a system of refracted rays lying on the
surface of the right-circular cone CL'B.
The position of the point L' can be seen to depend, in general, on
the slope of the incident ray LB, so that different rays of the pencil
§208.1
Path of Ray Refracted at Spherical Surface.
293
of incident rays L will determine different positions of the point L\
Accordingly, whereas all the rays of the bundle of incident rays L
will be grouped in cones, which have a common vertex at L, the
E'
Fig. 118.
Spherical Abbr&atioxt. Case when a Pencil of Parallel Incident Rays is Refracted at a
Spherical Surface.
corresponding refracted rays will be grouped in cones, which, while they
have all a common axis L C, will, in general, have different vertices L'.
This variation of the position of the point V corresponding to a fixed
Fio. lip.
Spbbxical ABXUiATioir. Case when a Pencil of Parallel Incident Rays is Refracted through
an Equi-biconvex I^ens (glass lens in air).
position of the point L is called Spherical Aberration (see § 260) : which
will be treated at length in a special chapter devoted to that subject.
294
Geometrical Optics, Chapter IX.
[§209.
The diagram (Fig. ii8) shows the case of a meridian pencil of in-
cident rays parallel to the axis of the spherical refracting surface.
The paths of the refracted rays have been traced by Young's Con-
struction. The outermost ray of the pencil is refracted so as to cross
the axis at a point marked L', whereas a Paraxial Ray will be refracted
to the Focal Point E' of the Image-Space. The line-segment E'L' is a
measure of the so-called Longitudinal Aberration along the axis.
Fig. 119 shows in the same way the Longitudinal Aberration along
the axis of an Equi-Biconvex Glass Lens in Air.
TRIGONOMETRIC COMPUTATION OF THE PATH OF A RAY OF FINITE
INCLINATION TO THE AXIS. REFRACTED AT A
SINGLE SPHERICAL SURFACE.
Casb I. When the Path of the Ray Lies in a Principal Section of the
Spherical Refracting Surface.
ART. 61. THE RAT-PARAMETERS, AND THE RELATIONS BETWEEN THEM.
209. Any section made by a plane containing the optical axis will
be called a Principal Section of the spherical refracting surface, and
under Case I we shall consider only such rays as lie in the plane of a
principal section.
In the diagram (Fig. 120) the plane of the paper represents a prin-
cipal section of the spherical refracting surface, the centre of which
gy-^^
Fio. 120.
TmOONOMBTRIC CALCULATION OP THB PATH OP A RAT RBPRACTBD AT A SPHERICAL SURFACE:
Cask when the rav-path libs in thb plane op a Principal Section. The stndirht line QB
shows the path of the incident ra>'. and the straisht line BH^ shows the path of the oorrespondinc
refracted ray,
AC'^r, AL^v. AL'^x/. CL^c, CL' " e , DB^h, Z^CBL'^^, ICBL'^^,
^BCA''^, ^ALB"; lAL'B^t^, BL'^l, BL'^t, CH'^b, Clf'^V.
is at the point designated by the letter C Through C draw a straight
line in the plane of the paper meeting the spherical surface in the
point .4. This straight line we shall take as the optical axis, so that
the point designateil by A will be the vertex of the surface. Let the
straight line QB^ intersecting the optical axis at a point L, represent
§ 209.1 Path of Ray Refracted at Spherical Surface. 295
the path of a ray of light incident on the spherical refracting surface
at the point B, and draw the radius B C.
We shall employ here pretty nearly the same letters and symbols
as were used in Chapter V, with such changes, however, as will be
necessary in order to distinguish the present case from that of a Parax-
ial Ray. Moreover, as we shall also have to introduce symbols for
several new magnitudes, and as the relations derived below will be
frequendy referred to in the course of this work, it will be well to
define clearly the precise meaning that is to be attached to each of the
symbols employed; which we therefore proceed to do.
I. Notation of the Linear Magnitudes,
The abscissa of the centre C, with respect to the vertex A, will be
denoted by r; thus, AC ^ r.
The abscissae, with respect to the vertex A , of the points designated
by L, L', where the ray crosses the axis, really or virtually, before and
after refraction, will be denoted by r, v\ respectively; thus, i4L = r,
AV = v'.
The abscissae, with respect to the centre, of the points L, L' will
be denoted by c, c', respectively; thus, CL = c, CL' = c^\ and,
consequendy,
c = » — r, c' = r' — r.
In regard to the signs of these abscissae, they are to be reckoned
positive or negative according as they are measured in the positive or
negative direction of the axis: the positive direction of the axis being
determined by the direction of the incident axial ray (§193).
The *^ ray-lengths'* are the segments of the incident and refracted
rays, measured from the point of incidence B to the points L, V where
the incident and refracted rays, respectively, cross the axis. These
magnitudes will be denoted by the symbols /, /'; thus, jBL = /, BL' = /'.
These magnitudes are to be reckoned positive or negative according as
they are measured along the ray in the same direction as the light
goes or in the opposite direction.
If D designates the foot of the perpendicular let fall from the inci-
dence-point B on the axis, the magnitude DB = A is called the "in-
cidence-height" of the ray, and is reckoned positive or negative accord-
ing as the point B lies above or below the axis.
The perpendicular to the optical axis erected at the centre C of
the spherical and refracting surface will be called the "central perpen-
dicular", and the intercepts CH, CH' of the incident and refracted
rays on the central perpendicular will be denoted by the symbols b, V;
296 Geometrical Optics, Chapter IX. [ § 210.
thus, CH = 6, CH' = b'. This intercept 6 is to be reckoned positive
or negative according as the point H lies above or below the optical
axis. And a perfectly similar rule obtains with regard to the sign of b'.
2. Notation of the Angular Magnitudes.
The angles of incidence and refraction are denoted by the symbols
a and a'; thus, in the diagram, Z CBL — a, L CBV = a'. These
angles are the acute angles through which the radius CB must be
rotated around the point B in order to come into coincidence with the
straight lines to which the incident and refracted rays belong.
The **Slope'* of the incident ray, or its inclination to the axis, is
denoted by the symbol 0; and, similarly, the symbol 6' is used to
denote the "slope" of the corresponding refracted ray. Thus,
/.ALB--e, /,AVB = e'.
These are the acute angles through which the axis must be turned
around the points L, L', in order to be brought into coincidence with
the straight lines to which the incident and refracted rays, respectively,
belong. Moreover, since
tan ^ = - h/DL, tan ^' = - h/DV,
the signs of 6 and 0' are the same as the signs of — h/v and — h/t/.
respectively.
The acute angle through which the radius CB drawn to the inci-
dence-point B has to be turned around C in order for it to coincide
with CA will be denoted by the symbol (p; thus, Z BCA = ip.
210. We proceed now to remark a number of useful relations be-
tween the magnitudes denoted by the symbols r, /, 6, A, a, $ and ip.
The position of a straight line is determined so soon as we know
the positions of two points on the line or the positions of one point
together with the direction of the line. The equation of a straight
line lying in a given plane — for example, the plane of a Principal
Section of the spherical surface — involves at most two arbitrary con-
stants or parameters; and to each set of values of any such pair of
parameters there corresponds a perfectly definite straight line of the
given plane.
Thus, for example, the position (but not the direction) of the in-
cident ray LB lying in the plane of the Principal Section will be
completely determined provided we know the values, say, of the param-
eters V, 6, which are called by some writers the "ray-co-ordinates".
Instead of using r, 0, we might also define the position of the ray by
§ 210.]
Path of Ray Refracted at Spherical Surface.
297
means of various other pairs of the magnitudes denoted by the symbols
V, If b, h, a, 6 and ^. L. Seidel, for example, uses in his system of
optical formulae ray-parameters that are equivalent to A, 0,
The relations between these magnitudes are obtained easily by an
inspection of the triangle LBC. Evidently,
a = 0 -j- <p.
(177)
(178)
This formula exhibits the connection between the angular magnitudes.
By the so-called Law of Sines, we derive from the triangle LBC
the following formulae:
/•sin ^ = — f -sin ^,
— f -sin a = (r — r) sin 6, "
(v — r) sin ^ = /sin a;
and by the so-called Law of Cosines:
/2 = (v - r)^ 4- r^ + 2r{v - r) cos ^, 1
(r — f)^ = r^ + f — 2r/cos a,
r* = (v - r)' + /" - 2l(v - r) cos 6. J
(179)
Finally, by projecting two of the sides of the triangle LB C on the
third side, we obtain:
r = /cos a — (v — f ) cos ^,
i; — r = /-cos 6 — r-cos ^,
/ = r-cos a+ (v — r) cos 6. -
Also, in the right triangles CBD, LBD, we have:
sin <p = A/r,
sin ^ = — A//.
(180)
and
(181)
(182)
Finally, if Y designates the foot of the perpendicular let fall from
the centre C on the straight line B H, we have evidently :
CF = f-sina = 6cos^.
(183)
By priming the magnitudes denoted by v, /, 6, a and 6 in the above
formulse (177), (178). (i79)» (180), (181), (182) and (183), we shall
obtain the corresponding relations for the refracted ray BL\
298 Geometrical Optics, Chapter IX. [ § 211.
ART. 62. TRIOONOMSTRIC COMPUTATION OF THE PATH OF
THE REFRACTED RAT.
211. The problem is as follows: Given the spherical refracting
surface and tiie values of the indices of refraction », n' of the two
media separated by it, and the position of the incident ray, to deter-
mine the position of the corresponding refracted ray. In other words,
being given the constants denoted by n, n' and r, and the coordinates
r, B of the incident ray, we are required to find the co-ordinates v\ ^
of the refracted ray.
By the Law of Refraction:
nsin a = n'^sin a'.
Moreover, since
a = ^ + ^, a' = d' + ^,
we have the invariant-relation:
a - ^ = a' - d'.
By means of these equations and the second of equations (178) above,
we obtain easily the following system of equations for calculating the
values of v\ Q' :
sin a = (i — vlr) sin Q, sin a* ^ n- sin aln\ 1
\ (184)
^' = ^ + a' - a, v' = r(i - sin a'/sin O. J
We may also remark here a number of other useful relations between
the parameters of the incident and refracted rays. For example,
since the incidence-height has the same value for both rays, we have
the following invariant relation:
/•sin^ = r-sin^'; (185)
and, since
sin a/sin ^ = — c/r, sin a' /sin ^' = — c'/r,
and, therefore,
sin O'/sin 0 = nc/n'c',
we obtain from (185):
nc/l = nV/r,
which may also be written:
V — r , v' — r
n — r- = n
This formula is, in fact, a mere transformation of the Optical Invari-
ant (§ 25)
K = n-sina = n'sina'
5211.1
Path of Ray Refracted at Spherical Surface.
299
for the special case of Refraction at a Spherical Surface. The magni-
tude
n{v - r) n'(v' - r)
/ = — — = — p;— . (186)
or
r-sin ^'
which remains unchanged as the ray is refracted from one medium
into the next, and which may be called the 'invariant of refraction
at a spherical surface", plays an important part in Abbe's Theory of
Spherical Aberration.
Note I. In the special case of Reflexion at a Spherical Mirror,
we have only to put n' = — n in the above formulae (see § 26). For
example, putting «' = — « in formulae (184), we obtain:
sma
a'
= sin 9(1 — v/r),
6 — 2a,
Reflexion at
Spherical Mirror.
Note 2. The following formula, adapted to logarithmic computation,
is convenient as a "check" formula in calculating the magnitude v':
f/ -- AL' =^ AC+ CD + DU
— r — r - cos if — h ' cot Q'
= 2r • sin* ffli — r • sin ^ • cos d'/sin B'
(. ip ip cose'\ . ip
sm cos - • - — T> I sm - ;
2 2 sm ^' / 2
so that, finally, we may write :
2r
1/ =
•sm- • COS! 6 + - ]
2 \ 2j ^ _
. iP a' + ^'
2r*sm - • COS
2 2
sin^'
and, similarly:
2r
t; as —
•sm- • cosi ^ + ~ I
2 \ 2/ ^ _
sin^'
2r'sm- • cos
2 2
sin^
sin ^
300 Geometrical Optics, Chapter IX. I § 211.
Dividing one of these formulae by the other, we obtain:
^, sin^.cos(^'+^j
sin^'.cos(^ + - j
In the special case of Refraction at a Plane Surface^ putting r = oo,
we have ^ = o, and the above formula becomes:
r'-tan 6' = v-tan ^, (Refraction at Plane Surface),
which may also be easily derived directly (§ 52).
Note 3. Spherical Aberration. The co-ordinates v\ 6' of the re-
fracted ray BR' can be found, as we have shown, in terms of the co-
ordinates », 6 of the incident ray. If in the formula
n{v -r)/l = n'(v' - r)/V
we substitute for /, /' their values as given by the first of formulae (179),
it is obvious that v' will thus be expressed as a function of n, »', r, v
and fp. The magnitudes n, n' and r are constants, so that v' is, in
fact, a function of the variables v and <p; and, therefore, if v is kept
constant, it is obvious that we shall, generally, obtain different values
of v' by merely changing the value of <p. This is the analytical state-
ment of the fact of Spherical Aberration mentioned in § 208.
The positions on the axis of the so-called **aplanatic'' pair of points
Z, Z' (§ 207) can be found easily by means of the formulae obtained
above. The condition that the abscissa v' corresponding to a certain
fixed value of v shall be independent of the angle (p must be imposed
upon the equations. Since
nc/i = «v/r,
and
/* = c* + r* + 2rC'Cos ^,
I' = c' + r* + 2f{:'cos ^,
we obtain:
(w'* - n^cV' + (n'V* - n^ey + 2rcc'(n'V - n\) cos ^ = o.
If, for a given value of c, the value of c' derived from this equation
is to be independent of ^, we must have:
n
§ 211.1 Path of Ray Refracted at Spherical Surface. 301
which shows that for this particular pair of values c, d must have the
same sign; that is, the points Z, Z* must lie on the same side of the
centre C If in the equation above we substitute this special value
of d^ we obtain
n
This equation gives two values of c, of which only the value
n
c=+-r
is admissible here where we have to do with optical rays as distin-
guished from mere geometrical rays. (The value c = — n'r/n cor-
responds to the other intersection of the ray with the auxiliary spheri-
cal surface t in Figs. 114 and 115.) Thus, we find:
r = r + n'rjn, v' = r + nrjn'
for the abscissae AZ, AZ' of the pair of aplanatic points of a spherical
refracting surface; in agreement with the results of § 207.
A characteristic property of the aplanatic points of a single spheri-
cal refracting surface, which was also remarked in § 207, may be stated
as follows: If ^, ^' denote the slopes of the incident and refracted
rays BZ, BZ\ then
sin^/sin^' = n/n';
that is, the ratio of the sines of the "slope"-angles is independent of
the magnitude of the angle of incidence, and constant, therefore, for
all pairs of corresponding incident and refracted rays. If F denotes
the value of the Lateral Magnification by means of Paraxial Rays for
the pair of conjugate points Z, Z', we shall find that:
Y = nyn\
and, hence, the relation obtained above may be written:
sin^/sin^' =^n'Y/n.
Expressed in this form, this relation, which we have obtained for the
aplanatic points of a single spherical refracting surface, represents
a very important general law of Optics known as the Sine- Condition
(Art. 86), which will be fully considered in a subsequent chapter.
Note 4. If the position of the ray is defined by means of its slope-
J«wiDBcncu opens,. Chapter IX. [ § 212.
«=- aLiscwc > Jii die central perpendicular, then by
^ , ■ a.^ . ^e - juowMHf iax-ariint-rdation:
^u^ 'r**^^ o*ctt :ae jonunecefs 4, B of the incident ray, we can find
t ^.x*^«i^«r»^ ' *' >*'' ^^ retracted ray by means of the following
« = — } — f
n
. . n cosB
n cosO'
>
(187)
.^^ -; - _ ,' :;ia ^* :iie coeaiection between the intercept AL = v
V ^ uco«. — ^»^2^ -***i ^^ atMtept CH = 6 on the central perpen-
^ «^^tw*.a ^ .« :orattjii:
> m ^f -. r) tan 6.
^, ^ M^m^"^ m ?^Hit nxum^ the point of intersection and
.>M;^A3^rM* W 1A£X OTHER OF A PAIR OF REFRACTED
<«^^ ,tMii^ :fil ?SB KANE OF A PRINCIPAL SECTION
.^ riHK SMHBHCAL REFRACTING SURFACE.
>^, xA. -Mill ."% v3ic iKSdecii rays, distinguished as the chief of the
..5^ . Ui . v^:<^ Jwc^ it the point L (Fig. 121); which, when the
. ;v . iiiic* * •< Nit^cle of incident rays, will coincide with the
. ^ .. . . iii .tik^H^v .>i vW *"$top'\ or circular diaphragm, which is
iiu., Hi v»«;i^;)c s>f object-rays that are permitted to pass
^ .^ ;v \.x-s\*»» x>>a».^n: and let the incidence- point of the chief
>, vx^^«*.<.s^ > .»V Ivcter B. The other ray (which we may
^ >v »->*^> ''^^ vf\^«55esi the optical axis at the point L, and
V V » *is» ^^""^ N**KfcvX dc the point B. The positions of both of
.X . , -^,>v»«fc\i ,\> t)c tttown, so that we may consider that we
• ^^ALB. 9^ZALB,
V. wv*^>->vx ^^a ihc v^ptioU axis,
§ 212.] Path of Ray Refracted at Spherical Surface. 303
so that we also know (or can find) the angles of incidence,
a = Z CBT, a = Z CBT,
eAe point of intersection of the secondary ray with the chief ray being
designated in the diagram by the letter T. The magnitudes
ay be regarded as the co-ordinates of the secondary ray with respect
^lie chief ray. This intercept / on the chief ray is measured always
froixi the incidence-point B as origin, and is to be reckoned positive
^^-^^f--
Fig. 121.
Figure represents a i»ir of rays, lying in the plane of a principal section of a spherical refracting
surface, and incident on this surface at the points designated by B and B, These rays cross the
optical axis at the points designated by £ and L, and intersect each other at the point designated
by 7! The refracted rays are not shown.
AL - r. AL' V, AC^ r, BT^ t, I BCA - ^, Z BCA - ♦,
IBCB^X, /.ALB^B, /ALB^B, IBTB'^K
or negative according as the light travels along the straight line BT
in the direction from B towards T or in the opposite direction. The
"aperture-angle" X is defined as the acute angle through which the
chief ray BT must be turned around the point T in order to bring it
into coincidence with the secondary ray BT,
Putting
LBCA^ip, ZBCA^^,
we have:
^ = <|) + X, (i88)
where x = ^ -B CB denotes the increase of the central angle <|).
From the figure we have evidently :
a-X=a + x. (189)
304 Geometrical Optics, Chapter IX. [ § 213.
Fn)m the centre Cdraw CY perpendicular to the straight line BT
at F. The orthogonal projection of the radius CB on the straight
line CY is equal to the sum of the orthogonal projections on CY of
the Hue-segments CB and BT\ and, sinqe these projections are equal
to r sin a, rsin(a + X) and — /sinX, respectively, we obtain the
relation:
rsin a = rsin (a + X) — /sin X,
or
rsinX .
= sm (a + X) — sm a. (190)
If iu formulie (189) and (190) we prime the symbols (, X, a and a,
we shall obtain the formulae for the corresponding pair of refracted rays.
Knowing* therefore, the positions of the pair of incident rays, and
U'ii^ giNvn the values of the magnitudes denoted by (, X, we can
fimi the values of the magnitudes denoted by (', X'. Thus, since
a — X — a = a' — X' — a' = x»
auvl
< ' sin X . , , XX . ^' • sin X' • / # . xa . /
«■ sin (a + X) — sm a, = sm (a + X') — sin a ,
f r
<ukK alsi>. since
a' + a' + y
sin (a' + X') — sin a' 2
sm (a + X) — sin a a + a + X
cos
2
we derive the following formulae:
X'=X+(a-a') - {a + a')r
a' + a' + X'
. . ^ cos
/' sin X 2
/ sin X' a + a + X
cos
(191)
Caue U. When the Path of the Ray Does Not Lie in a Principal
Section of the Spherical Refracting Surface.
ART. 64. PARAMETERS OF OBLIQUE RAT.
2L%. The e<iuation of a straight line in space involves as many as
liaii arbitrary constants, and the forms of the refraction-formulae which
\vi' ^hall obtain will depend on how these ray-parameters are chosen.
I ii u.^ take the centre C of the spherical refracting surface as the
§ 214.] Path of Ray Refracted at Spherical Surface. 305
origin of a system of rectangular co-ordinates. Naturally, also, we
shall take the optical axis itself as the :>:-axis. The plane of a prin-
cipal section of the spherical surface may be conveniently selected as
the xy-plane; nor will it at all affect the generality of the following
treatment if for this plane we take that meridian section of the spher-
ical surface which contains also the object-point. The plane of the
principal section, which is perpendicular to the xy-planet will then be
the x2-plane, and a transversal plane at right angles to the optical
axis will be the >^z-plane. For convenience, we may suppose that the
axis of y is vertical, and that the axes of x and z are horizontal.
The letters G, H and / will be used to designate the points where
the incident ray, prolonged if necessary, crosses the xy-, yz- and xz-
planes, respectively; and the rectangular co-ordinates of these points
will be denoted by
Xgf yg, o; o, y^, z^, and x<, o, z,.,
respectively.
In the following we shall explain the methods of A. Kerber and
L. Seidel of calculating the path of a ray refracted obliquely at a
spherical suriace.
214. Method of A. Kerber.
In the calculation-system of A. Kerber,^ the position of the ray
is determined by the co-ordinates of the points G and /, where the
ray crosses the vertical plane of the principal section (xy-plane) and
the horizontal meridian plane (jcz-plane). In the figure (Fig. 122)
the spherical triangle AA^Ai represents a piece of the spherical re-
fracting surface. The point Ay where the optical axis crosses this
surface, is the vertex of the surface; AA^Cis the plane of the principal
section, and AA^C is the meridian section perpendicular to the prin-
cipal section. Let
Z A CA^ = tpg, LA CAi = <Pi.
These angles are precisely defined by the following relations:
tan ^^ = - ~ , tan ^. = - -* . (192)
Xg Xi
Also, regarding the radius i4^C as a secondary axis of the spherical
surface, let us denote the abscissa of the point G, with respect to A^
as origin, by v^; and, similarly, regarding the radius A^C as another
secondary axis, we shall denote the abscissa of the point /, with respect
to Ai as origin, by v^; thus, v^ = A^G, v,. = AJ. From the figure,
* A. Kbrbbb: Beitratge wur Dioptrik, Heft II (Leipzig, Gustav Fock, 1896). pages 5-8.
21
306
Geometrical Optics, Chapter IX.
[§214.
we obtain:
and, since,
we have:
Xg = CG ' cos tpgt x< = CI * cos <Pit
CG ^Vg- r, CI = Vi- r,
v.-r^
r, - r =
^^ . (193)
cos^^ " cos^i ^ ^•'^
The projection of the incident ray in the plane of the principal
section (:x;jp-plane) makes with the optical axis an angle €, and with
Pio. 122.
Krrbrr'8 Method op Dbalino with thb Obliqub Rat. The plane of the paper {xyvHamt}
represents a principal section of a spherical refracting surface, centre at C, and optical axis
coinciding with the jr-azis of coordinates. A ray, whose path does not lie in the plane of the
principal section, is incident on the spherical surface at the point J?. This ray croeses the xyplaae
at the i>oint designated by G and the x^-plane at the point designated by /. The spherical triangle
AAffAi is formed by the intersections of the vertical jr,r-plane. the horisontal jrr-plane and the
plane of incidence with the spherical refracting surface.
Z ACAg - 4t„ Z ACAi - 4n, ^ A,GB - Bg, I AJB - •♦. Z GBC^ «. A^G - tv. Ait^vu
the ray itself an angle 5; these angles being exactly defined by the
following formulae:
y9
tan€ =
X: — X,
Z.'COSt
tan 5 =
(194)
§ 215.] Path of Ray Refracted at Spherical Surface. 307
Moreover, let B designate the point where the ray meets the spheri-
cal refracting surface, and let us put
The angle B^ may be determined from the following relation:
cos ^^ = cos (€ - ip^ • COS 5, (i9S)
which may easily be derived from the figure; and the angle B^ may be
determined in terms of B^ by means of the formula:
sin B^ = -^ sin B^, (196)
r,. — r
which may also be derived without difficulty.
The plane A^A^C contains the incident ray GI and the incidence-
normal -BC, so that this plane is the plane of incidence. The radii
il^C, A^C both He in this plane, as do also the line-segments denoted
by r^, r,. and the angles denoted by ^^, B^\ and, consequently, regarding
il^C and i4^C each as axes of the spherical surface, we have evidently
the following relations exactly similar to the relations expressed by
equation (177) and the second of equations (178):
a = ^y + fpj, = ^< + ^i, (197)
and
— r • sin a = (v^ — r) sin B^ = (v< — r) sin ^,., (198)
where a denotes the angle of incidence.
If in the figure the letters G and / are primed, the diagram will
answer to show the corresponding case of a ray refracted at a spherical
surface, and by priming all the symbols x, y, z, v, B, a, € and b in the
formulae (192) to (198) above, we shall obtain the corresponding rela-
tions between the parameters of the refracted ray.
215. Method of L. Seidel.
Instead of determining the position of the ray by its points of inter-
section with two selected planes, L Seidel^ makes use of only one
such point, and, in place of the co-ordinates of a second point, employs
two angular parameters to define the direction of the ray. The point
of the ray which he selects is the point designated by H (Fig. 123)
^L. Seidkl: Trigonometrische Formeln fUr den allgemeinsten Fall der Brechung des
Lichtes an ccntrirtcn sphaerischen Flaechen: SUzungsber, der math.-phys. CI. dcr kgl. bayr.
Akad. der Wissenschaften, vom 10. Nov. 1866. Reprinted in Beilage III of Steinhkil&
Vorr's Handbuch der angewandUn OpUk, Bd. I (Leipzig. B. G. Teubner. 1891), pager
257-270.
308
Geometrical Optics, Chapter IX.
[ § 215.
where the ray crosses the transversal (or yz-) plane. Moreover, in-
stead of using the rectangular co-ordinates (y^^, z^) of this point, he
introduces a system of polar co-ordinates (/>, t) in the ya;-plane. Em-
ploying other symbols than those used by Seidel himself, we shall
write:
p = CH, T = Z HCy,
which magnitudes are connected with the rectangular co-ordinates of
H by the following relations:
yh = P'cos T, z^ = p'sin T. (199)
Both the radius-vector p and the polar angle t are to be considered as
always positive in sign. The angle t, which may thus have any value
At
•^x
Method of t,. Sbidrl. The •traiffht line Blf represents a ray incident obliquely at the point B
on a spherical refractinir surface, whose centre is at the point designated by C. The optical axis
coincides with the jr-axis of co-ordinates, and the plane of the paper is the plane of a principal
section (jr^-plane) ; Ai*- beinff the section of the spherical surface made by this plane. CB is the
incidence-normal, and ACB is the plane of incidence. The ray B/f croaacB the x^^plane at the
I>oint desiiniated by G. and the:>'.r-plane at the point designated by Jf, The polar co-ordinates of
the point //sltc p * CH, v « Z HCy. The anffle at ^ is the anffle of incidence «. The acute ansle
made by the ray with the ;r-axis is the ansrle denoted by t ; and the anffle made by the pixiiectiQn
of the ray on the j'^-plane with the positive direction of the ^axis is the ansle denoted by ^.
comprised between 0° and 360®, may be defined as the angle through
which CH has to be turned about C, always in the sense of positive
rotation, in order that it may come into coincidence with the positive
direction of the y-axis.
§ 215.1 Path of Ray Refracted at Spherical Surface. 309
One of the two angular magnitudes that define the direction of the
ray is the acute angle (t) between the direction of the ray and the
positive direction of the x-axis; this angle being reckoned always as
positive.
The other angular magnitude selected for this purpose by L. Seidel
is the angle (^) made with the positive direction of the y-axis by the
projection of the ray on the transversal (or yz-) plane. This angle,
likewise, is always reckoned as positive, but it may have any value
comprised between o° and 360®.
If the direction-cosines of the straight line HI are denoted by a,
3, 7, then
- = — — = ^
and, since
tan \p = y/p
(as may be easily verified), we obtain:
tan^ = -\^\ (200)
whereby the angle ^ is precisely defined.
Moreover, since
a* + j8^ + 7^ = I , and a = cos t,
we find (taking the minus sign, which is in agreement with the defi-
nitions above) :
j8 = — sinr-cos^;
and, hence:
tan T = — ^^-^, (201)
Xi ' cos f
or
tan T = , (201 a;
which is, therefore, the definition-equation of the angle t.
An auxiliary angle On) is also employed in the calculation-scheme
of L. Seidel. Let B designate the point where the ray meets the
spherical surface; in the triangle BHC, the angle at H, but not
necessarily the interior angle, is the angle denoted by /x. This angle,
which is also reckoned as positive, may have any value comprised
between o** and 180®, and is defined exactly by the following formula:
cos/i = — sin T-cos (^ — t), (202)
310 Geometrical Optics. Chapter IX. [ § 216.
a relation which may easily be verified from the above definitions of
the angles denoted by t, t and ^.
From the triangle BHC v/e derive also the following formula con-
necting the angle of incidence a at B and the auxiliary angle a* at H:
r-sina = psinfi; (203)
wherein it should be noted that, according to this formula, since by
definition both p and sin /x are positive magnitudes, and the angle a
is an acute angle, the sign of the angle a must be reckoned always as the
same as the sign of the radius r.'
The point where the refracted ray crosses the transversal yz-plsme
is designated, similarly, by if'; and if the symbols x, y, 2, p, ^, t, ^, /i
and a in formulae (199) to (203) above are primed, we shall obtain
at once the relations between the corresponding magnitudes which
relate to the refracted ray.
ART. 65. TRIGONOMETRIC COMPUTATION OF PATH OF RAT REFRACTED
OBLIQUELT AT A SPHERICAL SURFACE.
216. The Refraction-Formulss of A. Eerber.
The problem is as follows: Being given the rectangular co-ordinates
(Xgf y^ and {x^, 2,) of the points G and / where the incident ray crosses
the xy- and jcg-planes, respectively, to determine the co-ordinates
{Xgy y^ and {x\^ z^ of the corresponding points C and /' where the
refracted ray crosses these same planes.
By the Law of Refraction, we have:
nsin a = n'-sin a';
and, moreover, since
« = ^^ + fpy = ^, + *f>i, a = ^i + fP^ = ^\ + ^o
we have:
a — ^^ = a' — ^^, a — B^^ cl — B\.
By means of these formulae and the formulae (192) to (198), we obtain
A. Kerber's^ System of Refraction- Formuke, as follows:
» This is practically equivalent to the method used by B. Wanach in a paper entitled
Ueber L. v. Seidel's Formeln zur Durchrechnung von StrdhUn durch tin seniriertes Lin-
sensystem, nebst Anwendung auf photographische Objective, published in Zeiischrifl fUr In-
strumentenkunde, xx. (1900). pp. 162-171. In Seidel's formulae, as originally published*
the symbol R is used to denote the absolute value of the radius of the refracting surface,
so that Sbidel has to employ the double sign in order to include the cases of both convex
and concave surfaces. Seidel adopted this method by preference, as being, in his opinion,
practically the most convenient.
'A. Kerber: Beilraege zur Dhplrik, Zweites Heft. (Leipzig, Gustav Fock. 1896),
pages 5-8.
S 217.]
Path of Ray Refracted at Spherical Surface.
311
tan ^^ = - yjxg, tan ^^ = - zjx^ ;
tanc ^
Vs
Xi-Xg
tan5 =
a;,-cos€
9
r - r, = -
cos^,
r — r, = —
Xi
9
COS ipi
r — V
cosSg = cos(€ — iPg)'COs6, sinSi = _ ^sinO^;
r — V.
sin a =
sin^ •
sin a = —7 sin a:
n
r '
, fsina
" sin u„
, r-sina
f — r, =
(204)
sin^; '
Xg= —{r- v^) -cos <pg, Xi = - (r - v'.) cos ip^;
y'g^ - x^-tan^^, z^ = - :>:'-tan ^,..
217. In the special case of a Plane Refracting Surface, the centre
C is the infinitely distant point of the optical axis, and, hence,
Jhe origin of co-ordinates will have to be shifted from C to the point A
^where the optical axis meets the refracting plane, which is effected
^ery simply by writing x — r in place of x. If we do this, and then
put r = 00, Kerber's Formulae for a Plane Refracting Surface will be
:found. as follows:
tanc =
fg
^—
fP< =
0;
*'9
, tan h =
^9
cose
"» =
*,
» ^i
= ^.;
costf.
=
COS6
•cos 3
•
^, = ^ , ^' = ^' :
^i ^gt ^g ^% »
n
sme'g = ^sin^^;
, tan Bg
*'» = "' s;i^:'
, tan ^i
V. = V,
'tan^;'
X, = V
»*
yii = y«; 2.=2.-
(205)
312 Geometrical Optics, Chapter IX. [ § 21&
218, In case the angle 6^ is very small, the determination of this
angle by means of the formula
cos 6g = cos (c — fPg) • cos 5
is not satisfactory, and a greater numerical accuracy will be possible
by determining, first, the value of the angle fi between the plane of
incidence and the vertical plane of the Principal Section by means of
the following formula:'
tan P = -:— 7 r ; (206)
sm (c -<pg)
whence we can find afterwards:
sm 6^^ r— 1 . (207)
^ smjS ^ '^
In connection with Kerber's Refraction-Formulae, the following
suggestion, also due to Messrs. Koenig and von Rohr,* is worthy of
remark:
By taking as the ray-parameters the co-ordinates x^, y^ and the
angular magnitudes denoted by d and c, the calculation of all of the
magnitudes denoted above by symbols with the subscript i can be
entirely avoided. Since, by (207), we have:
we obtain:
. sm5 sin 6
sm p = : — — = : — T? ,
sm 6g sin 6^
sm 6' = -:—r sm 5, (208)
sin^ ^ ^
whereby we can determine the angle 5'; and the value of the angle c'
may be found by the formula:
" cos 0
or by the formula:
. . , . tan 5'
^ This suggestion is found in Die Theorie der opiischen InstrumenU (Berlin, Julius
Springer, 1904), Bd. I, II Kapitel, "Die Durchrechnungsformeln": von A. Kobnig und
M. VON ROHR. p. 65.
' Same reference as preceding.
* .»tn of Ray Refracted at Spherical Surface. 313
219. The Refraction-Formulas of L. Seidel.^
Here the problem is as follows: Being given the angular magnitudes
(t, ^), which define the direction of the incident ray, and the polar
co-ordinates (/>, t) of the point H where this ray crosses the >^2-plane,
to find the corresponding parameters (r', ^') and (/>', v') of the re-
fracted ray.
Since the plane of the triangle BHC contains the incident ray BH
and the incidence-normal BC, this is the plane of incidence, which
likewise, therefore, contains the refracted ray BH\ That is, the two
j>lanes BHC and BH'C coincide, and, consequently, their lines of
Intersection with the ^^z-plane coincide also. Hence, the three points
dJ, H and H' all lie on one and the same straight line; accordingly,
he radii vectores CH, CH' have the same (or opposite) directions,
that the polar angles t, t' are either equal or differ by i8o°. In the
:^ise of a refracting surface, we shall have:
t' = t;
,:mid for a reflecting surface:
t' = iSo*' + T.
By formula (203), we have:
r-sina = />*sln/x, rsina' = />'-sin/i',
ere /*, /*' are the two auxiliary angles at the vertices J?, H' of the
ngles BHCf BH'C; and hence, by the Law of Refraction, we
ive the invariant relation :
n-/>-sin/i = «'•/>'• sin /*'. (209)
xeover, since the angle at C is common to these two triangles, we
in also another invariant relation as follows:
H + a = n' + a\ (210)
xneans of the above formulae, the position of the point H' may be
irmined.
Stall another invariant relation, depending on the fact that the
P*^*>c of incidence and the plane determined by the optical axis and
^^^ radius vector CH coincide with the plane of refraction and the
9^axie determined by the optical axis and the radius vector CH\ re-
l*. V. Sbidkl: Trigonometriache Formeln fUr den allgemeinsten Fall der p-
^ l^icfates an centrierten sphaerischen Flaechen : Sitzungsber. d^ -
^'^^ Akad. der Wissenschaften, vom 10. Nov. rfi^^
< .» -.
1 220.] Path of Ray Refracted at Spherical Surface. 315
(2) Determination of the Direction (t', ^') of the Refracted Ray*'
sin r'-sin (^' — ir) = -; sin r-sin (rp — t),
sin M
sin /-sin (^' - x)
tan (t - ^') = ——7 ,
cos ft
, sin (^ - 7r)
tan T = tan r -; — 777 r .
sin (^ — TT)
(214)
Note. — ^The second of these formulae is obtained by combining for-
mula (212) with the formula:
cos /i' = — sin t' • cos (^' — t) ;
and it enables us to find the magnitude of the angle ^^
220. In the special case of a Plane Refracting Surface, for which
the centre C is the infinitely distant point of the optical axis, the plane
surface must be taken for the yz-plsuie, and hence the three points
B, H and H' coincide. Accordingly, for this special case we have:
p' = p, t' = T.
And since the incidence-normal is parallel to the optical axis, we have
also a ^ T, a' = t'\ and, therefore,
sin T = -ism r
n
is the equation for determining the magnitude of the angle r'. More-
over, since both the incident and refracted rays lie in the plane of
incidence, containing the incidence-normal, which here is parallel to
the X-axis, the projections of these rays on the 3^z-plane must coincide
with each other; and, therefore.
By means of the above equations, we can find the four parameters
p\ t', r' and ^' of a ray refracted at a Plane Surface.
§ 221.1
Path of Ray through Centered Optical System.
317
The following system of formulae (see § 211) may now be written:
sma;^
-(■-r:)-'"
sin oLj^ =
w*-i
«j
sm a^,
^k = ^*-i + «*
a
ife*
»,
(sin al\
»
*+l
= »! - 'St-
eals)
In these formulae we must give k in succession all integral values
from ife = I to ife = f», where m denotes the total number of spherical
Fio. 124.
Path op a Rat in a Principal Section op a Centered System op Sphbrical Repractino
surpacbs.
AkU-x — Vk, AkLk' ■■ Vk', AkCk — Tk, DkBk — hk, Ak-iAk ■ dk-\, Bk-iBk — **-i,
BkLk'-i'lk, BkU'lk\ l.AfiLk'-iBk-i'^^k'-i, l.AkU'Bk^9,!, L BkCkAk^ ^k.
surfaces. If we know the values of the constants «i^_i, n]^ and r^
for the ifeth refracting surface, and if we have determined the ray-co-
ordinates tfjfc, ^j^_j of the ray incident on this surface, the first four
of the formulae (215) above enable us to find the ray-co-ordinates
v\^ B\ of the ray after refraction at the jfeth surface; whereas the last of
these formulae enables us to pass to the next surface, provided we
know the axial "thickness" d^ between the feth and the (fe + i)th
surfaces. Thus, having found the magnitude r^+j, we can proceed
to make the same calculation for the (fe + i)th surface, and so on,
until we obtain, finally, the co-ordinates of the emergent ray, viz.,
v^ = i4^L^, ^^ = /.A^L'^B^. An actual numerical example, illus-
trating the calculation-process by means of formulae (215), is given
in Art. 67.
318
Geometrical Optics, Chapter X.
I i 224.
222. We have also a number of other relations, which are often
very useful and convenient. Thus, if the symbols ^4, A^, f^, ll have the
following significations :
where the letters designate the points shown in the diagram (Fig. 124),
we have immediately, in connection with formulae (215):
h= - **/sin Ci. ^» = - **/sin $[,
^* = «» ~ ^k-i = «* "" ^k'
(216)
223. If the position of the ray is defined by its "slope" (^l_i)
and its intercept J^ (^C^-ffJ on the "central perpendicular", we
obtain (see § 211, Note 4) the following calculation-scheme:
sm ofj =
ft^cosCi
sm ttj = — 7- sm a^, ^4 = ^4_i + a^ - a^,
, «l_i cos gi^_i
* «4 cos ^4 *
(217)
together with the following "transformation-formula", for passing
from the feth to the (fe + i)th surface:
where
*»+i = *l + «ik- tan^i;
(218)
(219)
denotes the abscissa of the centre Q+i with respect to the centre C^;
that is, a^fc = (^k^k+v
The relation between the intercepts b/^ and Vj^ is given by the fol-
lowing formula:
h = (^ - Vk) tan ^;.i. (220)
ART. 67. NUMERICAL ILLUSTRATION.
224. By means of the formulae (151), we can find the position of
the image-point M*^ , which corresponds by Paraxial Rays with the
axial object-point M^ (or Li), and by means of formulae (215) above
we.can determine the position on the axis of the point L'^ where the
§ 224.] Path of Ray through Centered Optical System. 319
extreme outside ray, or so-called "edge-ray", of the bundle of rays
crosses the optical axis after emerging from the centered system of m
spherical refracting surfaces: and thus we can compute the longitudi-
nal aberration along the axis:
In practice this is found to be a very useful way of computing the
magnitude of this aberration, especially in the case of optical systems
of comparatively wide apertures, to which the theory of aberrations
of the first order does not apply very well. By repeated trials in this
fashion, it is possible, also, to discover how the thicknesses and radii
will have to be altered so that, for example, the edge-ray will emerge
8o as to cross the optical axis at a point L'^, which coincides, very
nearly at least, with the so-called "GAUSsian" image-point M*^; in
which case for this pair of rays (that is, for a paraxial ray and the edge-
ray), we shall have ^1 — wl = o, approximately. In the design of
optical instruments this calculation-process is found to be extremely
serviceable. In order to exhibit the use of the formulae, we shall give
here a rather simple numerical illustration.
For this purpose, we shall select an example given in Taylor's
System of Applied Optics (London, 1906), page loi, as follows:
The optical system is a large Telescope Object-Glass, of 12-in.
aperture (A^ = 6 in.), consisting of a biconvex crown-glass lens and
a biconcave flint-glass lens, with the following radii and thicknesses
(all measured in inches) :
^i = + 59-8; di = + i; ^2 = - 90-15; ^2 = 0.013;
fs = " 84.7; d^= + 1; and ^ = + 410.
"TTie values of the refractive indices, for rays corresponding to the
RAUNHOFER-Line C, are as follows:
n, =s «i = n^ = i; «j = 1-5146; n^ = 1.6121.
The incident rays are parallel to the optical axis, so that
Wi = V, = 00, and $1 = o.
^^ccording to the first of formulae (216), we have, therefore, in such
a. case as this:
h
sinofi =-^, (rj = 00), (221)
^hich is the formula we must employ here in order to determine the
^^ue of aj.
320
Geometrical Optics, Chapter X.
[ § 224.
The calculation will be divided into two parts, as follows:
(i) The calculation of the Path of a Paraxial Ray, by means of
formulae (151) J and
(2) The trigonometric calculation of the Path of the Edge-Ray by
means of formulae (215) above, together also with formula (221) above.
The sign + or — written after a logarithm indicates the sign of the
number to which the logarithm belongs. Elach vertical column contains
the calculation for one surface: accordingly, in the present example,
where there are four refracting surfaces, each table will contain four
such columns.
For the Edge-Ray: Ai = 6 inches, v^ = 00 and $1=0: hence, ac-
cording to formula (221) above, we have:
IgAi = 0.7781513 -f
cig fi = 8.2232988 +
Ig sin «! = 9.0014501 -f
This forms the starting point for the calculation of this ray.
The two parts of the calculation follow.
I. Paraxial Ray: u^ = 00.
Formulae:
Wi._i I tiff ~^ w*— 1
I
«1
I
I
u
*+l
I
n.
I - ^*/«I- *
ClgMt
Ig («*-i/ni)
Ig {nit-ijuknk)
dgr*
Ig («*— «fc_i)
clgnt
Ig
rkftk
ft-1
8.2232988-1-
9.71 14698 -|-
9.8I97020-|-
ft-2
7-7569451 +
0.1802980 -h
7.9372431 +
ni-i/ukHk
(«»— n*— i)/f*«»
i/u'k
Clgttft
Igdk
Ig dk/uk
I —dk/uk
ClgMJk
dg (i —dk/uk)
Igl/Uk^l
7.7544706 +
0.0000000
4-0.0056816
4-0.0056816
7.7544706-1-
0.0000000
7.7544706 +
+0.9943184
7.7544706 +
0.002474s -h
7-7569451 +
8.0450343-
9.71 14698 —
0.0000000
7.7565041 +
+0.0086545
-i-0.0057083
+0.0143628
8.1572385 +
8. 1 139434 -f
ft«3
8.1573196 +
9.7926080 +
7.9499276 +
8.0721166 —
9.7868224-f
9. 7926080 -|-
7.6SI5470—
+0.00891 10
—0.0044828
ft-4
7.6481546 +
0.2073920 -j-
7.8555466 +
7.3872161 +
9.7868224 —
0.0000000
7.1740385-
+0.0044282
7.6462272 +
0.0000000
6.2711819 +
4-0.9998133
8.1572385 +
0.000081 1 +
8.1573196 +
7.6462272 +
+ 0.9955718
7.6462272 +
0.0019274 -|-
7.6481546 +
+0.0071705
—0.0014929
+0.0056776
7.7541648 +
U4
+ 176.13077 in.
§ 224.]
Path of Ray through Centered Optical System.
321
Formula for the Focal Length e':
i/e' = - (i - dju[){i - dju^)(i - dju^){ifuj.
Ig (i - dju[) = 9-997S2SS +
Ig (i - d^/u^) = 9.9999189 +
Ig (i - dju^) = 9.9980726 +
clg wl = 7.7541648 +
clge' = 7.7496818 —
e' = — 177.9583 inches.
II. Elx?£-ifAK; See formulae (215) of this Chapter.
Ig (l —Vk/fk)
Igsin^*— 1
Igsinou
lg»*-l/H*
Igsinajfc
— a*
^i —a*
Ok
Igsmo*
<lg8m^
Ig (sin ol/sin ^)
ol/sin^
— smol/sind*
/ 8inai\
[fk
V'k
dk
S Vk+i/rk+i
*-+i/r»+i
ft-l
k^2
ft-3
ft-4
9.00I450I +
9.8197020+
0.4679039 +
8.5340672 —
0.2585662 +
8.9356090 —
9.654751 I +
8.4225418 —
9.OOI97II —
0.1802980 +
9.1941752-
9.7926080 +
8.0772929 —
0.2073920 +
8.8211521 +
9.I82269I —
8.9867832 —
8.2846849 —
*vO «' *v'
000
-5**45'3o'.3
-i'*57'36'.3
+5°45'55'.3
-4V46'.3
+8**59'48'.2
-iV57'.8
+0^41' 4'.5
-5''45'30^3
+3°47'S4'.o
+3°48'i9'.o
-8''45' 5^3
+4** 3' i'.9
-5°33'59^7
-o°49'53'.3
-I** 6'i3'.2
-i°57'36'.3
-4**56'46'.3
-iV57'-8
-i'*56' 6'.5
8.8211521 +
1.4659328 —
9.1822691 —
1. 0643910 —
8.9867832 —
1.5774582-
8.2846849 —
1.4715561-
0.2870849 —
0.2466601 +
0.5642414+
9.7562410 +
— 1.9368
+1.764655
+3.666414
+0.570481
+2.9368
-0.764655
—2.666414
+0.429519
0.4678744+
1.7767012 +
9.8834656 —
1.9549657-
0.4259276 —
1.9278834-
9.6330430+
2.6127839 +
2.2445756+
1.8384313 +
2.3538110 +
2.2458269 +
+175.6206
— I.OOOO
+68.93365
— 0.013
+225.8452
— 1 .0000
+ 176.1273
+174.6206
+68.92065
+224.8452
2.2420955+
8.0450343 —
1.8383494 +
8.0721 166 —
2.3518836 +
7.3872161 +
0.2871298 —
9.9104660 —
97390997 +
-1.9370
—0.813703
+0.548403
+2.9370
+ 1.813703
+0.451597
MaLa «s »i — «i =s — 0.0035 inches.
22
i
322 Geometrical Optics, Chapter X. [ § 2
Case II. When the Path of the Ray does not Lie in the Plane of a Punch
Section of the Centered System of Spherical Refracting Surfaces.
ART. 68. TRIGONOMETRIC FORMULiB OF A. KERBER FOR CALCULATi:
THE PATH OF AN OBUQUE RAT THROUGH A CENTERED
SYSTEM OF SPHERICAL REFRACTING SURFACES.
225. In the calculation-scheme of A. Kerber ^ (see §§214 and 21
the parameters of the ray before refraction at the Jfeth surface of 1
system of spherical refracting surfaces are the co-ordinates (x,, ^^ y^
and (Xi^ kf Zi, *) of the points G^ and Ij^ where the ray crosses the t
meridian CQ-ordinate planes, viz., the X3^- plane and the x^-plane,
spectively; and, similarly, the parameters of the ray after refract!
at this surface are the co-ordinates (jc^,*, y'g^jt) and (xi,*, «i, J of 1
points G4 (or G»+i) and /^ (or I^+i) where the refracted ray cros
the xy- and xz-planes, respectively. In order to obtain the refra
ion-formulae for the kth surface, we have merely to affix to the sy
bols in formulae (204) the fe-subscript to indicate that the formulae i
to be applied to the feth refracting surface.
It will also be necessary to obtain a system of ** Transformatii
Formul(B'\ whereby, having ascertained the values of the co-ordinai
(^i.ft. yi.*) and (x*.*, zi,*) of the points G^ (or G^+i) and fj, (or Z^+i), i
ferred to the centre C^ of the kth surface as origin, we can compi
the values of the co-ordinates (x^,*+i, yg,k+\) and (x<,*+i, «<,*+i) of the
same points referred to the centre C^^+i of the {k + i)th surface
origin. This shifting of the origin along the x-axis will affect on
the x-co-ordinates. Thus, evidently, we shall have:
where
«A = CkCk^x = ^A + ^ik+i - ''*. (22:
Accordingly, in the Calculation'Scheme of A. Kerber, we have tt
following system of formulae:
(i) Refraction- FormuUz for Finding the Values of the Parametei
Xg^kf yg,k, Xi^h and Zi^u of the Ray After Refraction at the kth surface:
tan ipg^k = - yg^/xg^k, tan ^<,* = - Zi.*/xi.»;
yg,k
tan €^.1 =
tan «;_, =
*i, k — Xg^k
Zi, k' COS €j,^l
(2^
' A. Kerber: Beitraege zur Dioptrik, Heft II (Leipzig, Gustav Fock, 1896), pages $
§ 226.]
Path of Ray through Centered Optical System.
323
cos ^^, ft COS^i.A
COS ^, *-i = COS (€i«i — <pg^ ft) • COS 6i-i ;
sin ^i, ft-, = sin Sg^ *-, ;
sina^h =
r* — Vg,k .
sin
e;
ft— 1»
/ ^ft— I •
sin ttjfc = — r- sin ofj ;
Wft
^,* = ^,*-i — a* + «*f ft,* = ft,ft-i — «* + «*;
^*-«'^,» =
r^-sina^
n- Vi,ft =
r^ • sin a^
(223, con-
tinued)
1
(224)
sine^,.ft' '* ''*•*" sina;.ft'
^^,ft = — (fft - v^.t) -cos <pg^k, xj.* = - (fft - Vi,ft) COS ^<,ft;
yg,k = — x^,ft-tan ^^,ft, 2i,ft = — xj,ft-tan ^i.j.
(2) Transformation' Formuks for Determining the Parameters x^.t+i,
%.ft+if ^i.ft+1 fl^ ^,*+i ^/ ^f^ R^y Before Refraction at the (fe + i)th
surface :
^9,k+l = ^^.ft + *** ~ ^k+l — rfjfci Xi^k+1 = ^<,A + y** — ''a+1 — ^ft
yg, *+l = y^, Jki 2i, ft+, = Zi, ft.
226. The Initial Values.
The position of the ray incident on the first surface of the centered
system of spherical refracting surfaces will be defined generally by
^ving the co-ordinates of the object-point Pj, whence the ray ema-
x:iates, and the co-ordinates of the point Pj, where the ray crosses
'^he plane of the so-called "Entrance-Pupil" (see § 257 and § 361).
Usually, it will be possible to select as the plane of the principal sec-
^ion (xy-plane) the meridian plane of the optical system which con-
ti^ns the object-point P„ so that this point will, therefore, coincide
ith the point designated by Gp If Afj designates the foot of the
irpendicular let fall from P, on the optical axis, and if we put
Ui = i4,Af„ 17, = JlfiP,,
tHe co-ordinates of the point P„ referred to a system of rectangular
^^es with origin at Cj, will be:
Xg^i = CiMi = wi - r„ yg,i = AfiP, = vu Zg^i = o.
324 Geometrical Optics, Chapter X. [ § 226.
In every actual optical instrument the angular opening of the
bundle of ''effective" rays, which, emanating from the object-point
Pp traverse the system of lenses, is limited in some way, usually by a
"stop**, consisting of a plane screen perpendicular to the optical aads
with a circular opening in it, whose centre (called the "stop-centre")
lies on the optical axis of the instrument. Even when no screen of
this description is employed, the cone of effective rays will be deter-
mined by the rim of one of the glasses — in some instances, also, by
the iris of the eye of the observer. The "stop" is not always situated
in front of the entire system of lenses; it may lie between one pair
of them, or it may even be placed beyond them all. Let us take the
most general case and assume that the "stop" is situated between^
say, the 6th and the (b + i)th surfaces of the system of m spherical
surfaces, and let us designate the position of the stop-centre by iH^*
This point ilf^ will be the image, formed by Paraxial Rays, after
having traversed the first b surfaces of the system, of a certain axial
object-point Afj; which latter point is the centre of the so-called
"Entrance-Pupil". The transversal plane Cj perpendicular to the
optical axis at Af j (which in any given optical system will always be a
perfectly definite plane) is the Plane of the Entrance-Pupil. And the
point where an object-ray, emanating from the object-point Pj, crosses
this plane will be designated here by Pp as has been stated above.
Moreover, we shall put A^Mi = u^ and shall denote the co-ordinates
of Pi, referred to rectangular axes with Ci as origin, as follows:
As has been remarked, the position of the object-ray is usually
given by assigning the values of the magnitudes denoted here by the
symbols u^, rji and t)i, |^p By drawing a simple diagram, the reader
will easily perceive that, if K designates the projection of the point Jj
on the xy-plane {CiK = x<,i, KIi = 2i,i), we have the following re-
lations:
t,i"AfiX' I, "M.M,^
whence, since
MiK = M^A^ + A,Q + CiK = x<.i + n - Uu
MiK = MtAi + A,Ci + CiK = x,., + n - u„
and
AfiAfi = M^Ay^ + AyM^ = tf 1 — «lt
§ 227.] Path of Ray through Centered Optical System. 325
we obtain:
2l ^ ^<, 1 + n - t<i 2<j ^ Xi,i + ri -ui
til JCi,i + ri-tti' 5, tti-tti
Thus, we obtain the initial values Xi,i, 2<,i as follows:
'yi - til
^ 1 + ^1 ~ ^1 » __ ^1 »
111 — tti lyi — Til
(225)
In case /A« object-paini Pi is infinitely distant, the object-rays will
constitute a bundle of parallel rays; and, since, in general, T/p as well
as tt„ will be infinite, the value of JCi,i, as given by the first of formulae
(225), will be illusory. Under these circumstances, we shall require
to know the direction of the object-ray, and, since all the object-rays
proceeding from one and the same point of the object are parallel, it
will be sufficient if we are given the slope-angle 81 of that one of the
bundle of object-rays which crosses the optical axis at the centre Af 1
of the Entrance-Pupil.^ Now, evidently,
tane.= "•-"•-
MiMi tt, — tti'
if, therefore, in the expression for x<,i given in (225), we substitute
the value of the ratio t<i/i7i, as obtained from this last equation, and
then put Wi = i7i = 00, we shall derive the first of the two following
formulae:
«<,i = tti -iircotOi - fi
J («i = «) (226)
^.1 = &•
1
The latter formula is obvious immediately from the second of formulae
(22s).
ART. 69. THE TRIGOHOMBTRIC FORMULiB OF L. SEIDEL FOR CALCULATIHG
THB PATH OF AN OBLIQUE RAT THROUGH A CENTERED
SYSTEM OP SPHERICAL REFRACTING SURFACES.
227. Employing here the same notation as was used in §§215
and 219, where the calculation-scheme of L. Seidel for the case of
the refraction of an oblique ray at a single spherical surface was given,
' This will not be the Chief Ray of the bundle, unless the stop-centre coincides with
the centre of the Entrance-Pupil; or unless, with respect to these two points, the spherical
a.berration of that part of the optical system which precedes the stop-centre has been
abolished.
326 Geometricad Optics, Chapter X. [ § 227.
we shall designate the points where the ray crosses the Jfeth transversal
(or yZ') plane, before and after refraction at the jfeth surface, by H^,
It^, respectively; and shall denote the rectangular co-ordinates of
these points by (o, ^a,*, 2*,*), (o, yi,*, zi,*), and their polar co-ordinates
by (Pkt ^a)» (p'kf ^ib)» respectively: the relations of these two sets of co-
ordinates being defined as follows:
y*.ft = Pk'^os ir^, Zk,k = pk'Sin ir», J
where
iri = Vj, (228)
(or, in case the kth surface is a reflecting surface, ir^ = ir^ -H 180®).
The directions of the ray, before and after refraction at the Ath
surface, are defined by two pairs of angular magnitudes denoted by
Tj, ^4 and Tf^, ^4, respectively. Since the direction of the ray after
refraction at the jfeth surface is identical with its direction before re-
fraction at the (ife + i)th surface, we have:
'Tk = ^*+i. ^k = ^»+i; (229)
which are, therefore, the "Transformation-Formulae" for Seidel's
Direction-Parameters. These Direction-Parameters are defined, pre-
cisely as in § 215, by the following formulae:
tan ^4 = — -r = tan ^j+j =
tan r^ = — 77 = tan rt+i = r
Xi, ft- cos ^4 *^' X<,jH.lCOS^
(230)
ft+i
It remains to obtain L. Seidel's Formulae for the transformation
from the parameters ir^, />i to the parameters x^+i, ^4+1; which we
proceed to do.
If the Direction-Cosines of the ray after refraction at the Ath surface
are denoted by a, j8, 7, then, precisely as in § 215, we have:
- = — tan Tj-cos ^jfc, - = — tan r^-sm ^4;
and since this ray goes through the two points Hi and Hj^i, whose
rectangular co-ordinates, referred to the centre of the fcth surface as
origin are:
(o, pl ' cos Ta, pI • sin ir^) and (a^, p^^+i • cos t^+i, Pk+i • an tjh-i),
§ 227.1
Path of Ray through Centered Optical System.
327
respectively, we have:
Pk+i'sin ir^+i - />l-sin ir^
Eliminating a, j8, y from these two sets of equations, we obtain:
pi,+i • cos T4+1 - />^ • cos T4 = - a^ • tan r^ • cos ^i,
/)4+iSin ir^H-i - />lsin ir* = - a^tan r^sin rp'^.
ombining these equations, we obtain easily the Transformation-
brmulae of L. Seidel,* as follows:
Piri^i'cos (^; - ith-i) = pt'cos (^i - tJ - a*tan r
, } (231)
Accordingly, in the Calculation-Scheme of L. Seidel for the re-
F^'"2tction of an oblique ray through a centered system of spherical
irfaceswe have the following formulae (see formulae (213) and (214)):
(i) Determination of the Position of the Point Htj^ by means of its
*d)lar Co-ordinates (/>4, irj :
cos Ma = - sin ri_i • cos (^'^.i - irj ; ^
sina» = />AsinMjkM; .
(232)
sm a^ = — r- sm aj^;
Mi = /** + «*-«!;
, _ singj _ «l_i sinjx^fe
^ sm ttj n^ sm /x^^
t) Determination of the Direction (t^, ^i) of the Refracted Ray:
Sin r^-sm (^^ - irj = -7-— sin t^^.! -sin (^^_i - irj ;
sm fi]^
tan (iTjk - ^1) =
sin ri • sin (^^ — ir;t)
COS/X4
(233)
^ ^ ' ^ ' sin (^;., - n)
tan Ti = tan Ti._t • — ; — 77? r- .
* ' sm (^;t - irj
1-. bEiDBL: Trigonometrische Formeln ftir den allgemeinsten Fall der Brechung des
*J/*^t.«8 an centrierten sphaerischen Flaechen : Sitzungsher. der math.-phys. CI. der kgl.
v^yr. Akad, der WissenschafUn, vom lo. Nov. 1866. Reprinted in Beilage III of Stein.
^»ti. & VoiT's Handbuch der angewandten Optik, Bd. I (Leipzig, B. G. Tkubner, 1891),
^8^ 257-270.
i
328 Geometrical Optics, Chapter X. [ § 228.
(3) Transformation- FormuUz for finding the parameters ^j+ii '*+i
of the Ray Before Refraction at the {k + i)th Surface:
\
(234) o
The equality of the two expressions on the left follows from the first
of formulae (232); and the equality between each of these and the
third expression can be deduced easily from the first of formulae (232)
and the first of formulae (233). Accordingly, this "control" formula
(235) serves to test only the accuracy of computations by these for-
mulae from which it is derived.
The values of the sines of the angles of incidence and refraction
are checked, along with the value of />', by the double calculation of
this latter magnitude by means of the two expressions for p' in formulae
(232). But as it is possible that, even though we have found the
correct value of the sine of an angle, an error may be introduced in
determining the value of the corresponding angle itself, or that a
mistake may be made in obtaining the difference a — a', thereby
involving also a mistake in the value obtained for the angle /i', and as
the "control" formula (235) would not enable us to detect an error of
any of these kinds, Seidel suggests also a second "control" formula,
as follows:
; ; 1 = ~~f T ; 1230)
which is a simple consequence of the Law of Refraction. The magni-
tude on the right is constant for all rays of the same wave-length
refracted between the same two media; so that in case the calculation
has to be made for a number of such rays (as usually happens in such
calculations), it will not be necessary to calculate at all the value of
228. Seidel's "Control" Formulae.
In order to check the numerical work from time to time, and thereby '^^^V
to avoid the disagreeable necessity, in case of arithmetical errors, of
having to repeat sometimes a very considerable portion of the calcu-
lation, L. Seidel has proposed, in connection with the above formulae,
several so-called '' Control'* FormuUz, the first of which is as follows r
sinjv^injx^ ^ sin /x]^ - sin r[_x ^ sin (a^^ — «!) . ^ j)
sin (^;_i - O ■" sin (^^ - irj "" sin {^/'^^ - ^J' ^^^5)
* a
tn of Ray through Centered Optical System.
the left-hand side of the equation, but it will be sufficient merely
see that the values of the expressions on the right are the same for i
the rays. Moreover, in the usual case of an optical system consistii
of a series of glass lenses, each surrounded by air, where, therefon
the ray proceeding from a medium (n) into a medium (n'), emerge
again into the medium (»), the values of the constant on the right-
hand side of (236) for two successive refracting surfaces will be equal
in magnitude, but opposite in sign ; and in such a case it will merely
be necessary to calculate the values of the expression on the left-
hand side for each surface, and see that the condition above-mentioned
is fulfilled.
Finally, a third "control " formula, deduced from the two trans-
fonnation-formulse (231), is as follows:
^*+i «*-tanr;fc p^
f
sin (^i - tJ sin (ir^ - t^+j) sin (^1 - t^h-i) '
(237)
In Steinheil & Voit's Handbuch der angewandien Optik, I. Bd.
^Leipzig, B. G. Teubner, 1891), the reader will find numerous complete
<2dculations by means of the trigonometric formulae of L. Seidel.
229. The Initial Values.
The position of the object-ray will usually be defined by the posi-
^on of the object-point Pi{u^ — r 1, lyi, o) and the position of the point
\(a, — fi, iji, 5i) where the ray crosses the plane of the Entrance-
^upil (see § 226). This ray crosses the first transversal (or yz-) plane
^t Hi(o, yh,u «A.i) and the horizontal xz-plane at I\{Xi^u o» ^,i)-
The positions on the x-axis of the points designated below by Q, Afj,
J^x ^^^ ^ ^^^ defined as follows:
fi = AiCu Ui ^ AiMu tti = AiMu Xi^i = CiK.
6y drawing a figure, the following relations will be immediately
obvious:
y*.i QK 2*., MtCt
Here
111 MiK' It M,Mi'
MyEi = Af,^, -f A,C, + C,X = X,., + r, - tt„
JlfiCj = Afii4i + i4,C, = fi - tti,
and
AfiAfx = M^A^ + iliAfi = tti — ttj;
and if for X|, 1 we substitute its value as given by the first of formulae
330 Geometrical Optics, Chapter X. [ § 229.
(225), we obtain:
H,(Ui - r,) - Ti,(«, - r,) «■ - ^
By means of formulae (238), together with (225), we can determine
now the magnitudes of the direction-parameters (tj, ^j) of the object-
ray; for according to the definition-formulae of these angles we have:
tan ^1 = , tan ti =
and, consequently:
tan ^1 = — ^- , tan Ti = — ^ . (239)
Til — lyi tti — «!
The initial values pu ti of the other two SEiDEL-parameters may be
determined by the equations:
\ (240)
/>iCos(^i - Ti) ==fciyj.cos^i - (r, - tti)-tanTi,l
wherein the upper sign must be used in case the object-point lies
above the optical axis, and the lower sign in the opposite case.
In the special case when the object-paint Pi is the infinitely distant
paint af the abject-ray, then, in general, both ^1 and Ui will be infinite.
In this case, instead of being given the co-ordinates ttj, iji, we shall be
given the direction of the ray — ^which will usually be done by assigning
the value of the slope-angle 81 of that one of the bundle of parallel
object-rays which crosses the optical axis at the centre M^ of the
Entrance-Pupil, and which, therefore, crosses the first central traasver-
sal plane at a point whose distance from the optical axis is:
(fi — tti) • tan 81.
If tji, 5i denote the co-ordinates of the point where the general object-
ray lying outside the plane of the principal section crosses the plane
of the Entrance-Pupil, we shall have in this case the following formulae
for determining the parameters pu iri:
y*,! = pi'CosTTi = Til + (fi — tti)-tan8i,
z*.i = />i-sinTi = 5i
|, (tti = i?i = «>). (
241)
Evidently, also, for the case of an infinitely distant object-point, we
have ^1 = 0° or 180° and ri = =*= 81.
CHAPTER XI.
GENERAL CASE OF THE REFRACTION OF AN INFINITELY NARROW
BUNDLE OF RAYS THROUGH AN OPTICAL SYSTEM.
ASTIGMATISM.
ART. 70. GBHSRAL CHARACTERISTICS OF A NARROW BUNDLE OP RATS
REFRACTED AT A SPHERICAL SURFACE.
230. Meridian and Sagittal Rays.
To an infinitely narrow homocentric bundle of incident rays re-
fracted (or reflected) at a spherical surface there corresponds, in
general, an astigmatic bundle of refracted (or reflected) rays, which,
provided we neglect magnitudes of the second order of smallness, is
characterized by the following properties:
The chief ray u' of the bundle of refracted rays is that one of the
refracted rays which corresponds to the chief ray u of the bundle of
incident rays. All the refracted rays meet two infinitely short straight
lines, the so-called Image- Lines (§ 47), which lie in two perpendicular
planes both containing the refracted chief ray «', and which are perpen-
dicular to u\ These two planes are the planes of Principal Curvature
of the element of the refracted wave-surface at any point P' of the re-
fracted chief ray w', which pierces the surface-element at P' normally,
a,nd their traces on the element of wave-surface at P' are two elements
of arc intersecting at right angles at P'. The two pencils of rays of
the bundle of refracted rays which lie in the planes of Principal Curva-
ture have their vertices on the refracted chief ray u' at the centres
of curvature S' and S'. Thus, to an object-point 5 lying on the
incident chief ray w, which is the vertex of an infinitely narrow homo-
centric bundle of incident rays, correspond two image-points S', 5'
lying on the refracted chief ray u', which we shall call the Primary
and Secondary Image-Points, respectively. The two Image-Lines are
perpendicular to the refracted chief ray u' at these Image-Points.
Thus, the I. Image-Line is perpendicular to the refracted chief ray
at S\ and lies in the plane of Principal Curvature of the refracted
wave-surface for which the 11. Image-Point S' is the centre of curva-
ture; and, similarly, the II. Image-Line is perpendicular at S' to the
chief refracted ray u\ and lies in the plane of Principal Curvature of
the refracted wave-surface for which the I. Image-Point 5' is the centre
of curvature.
331
N
332 Geometrical Optics, Chapter XI. [ § 230.
When the plane determined by the chief rays w, «', which we shall
call the Plane of Incidence, is at the same time a plane of Principal
Curvature of the refracted wave-surface, one of the image-lines will
lie in this plane, and the other will lie in a plane perpendicular to the
plane of incidence.
The special problem which we have to consider presents a compara-
tively simple case; for, since the refracting surface is spherical, the
two systems of incident and refracted rays are symmetrical about an
axis. Thus, if C designates the centre of the spherical refracting
surface, not only this surface but the incident and refracted wave-
surfaces as well are surfaces of revolution around the straight line
5C as axis. The plane of incidence wC, containing the common axis
of these three surfaces of revolution, is a meridian plane of each oner
of these surfaces, and is, therefore, also a plane of Principal Curvature^
so that one of the Image-Lines will lie in the plane of incidence,
and the other will He in the plane perpendicular to the plane of inci-
dence which contains the refracted chief ray u'. According to the
usage of most writers on Optics, we shall designate the latter as the
I. Image-Line and the former as the II. Image-Line/ The II. Image-
Line is perpendicular to the chief refracted ray u' at the point ?
where this ray crosses the axis of symmetry SC,
Thus, in the case of an infinitely narrow homocentric bundle of
incident rays refracted at a spherical surface, the directions of the
Image-Lines of the astigmatic bundle of refracted rays will depend
only on the position and direction of the chief refracted ray u'\ so
that to a range of object-points lying on a given incident chief ray u
there will correspond a series of parallel I. Image-Lines and a series
of parallel II. Image-Lines.
The planes of Principal Curvature of the wave-surface determine
two principal sections of the infinitely narrow bundle of rays. The
plane of incidence u C, which in the case of a spherical refracting sur-
face coincides with one of these planes, cuts the infinitely narrow
homocentric bundle of incident rays and the corresponding astigmatic
bundle of refracted rays in a pencil of incident rays with its vertex
at the Object-Point 5 and in a pencil of refracted rays with its vertex
at the I. Image-Point 5'. These are the so-called Meridian Rays;
since the plane of incidence u C is at the same time a meridian plane
of the spherical refracting surface.
If the incident chief ray u is supposed to be revolved about SC
1 Some writers, however, for example, Lippich, use the contrary method of designating
these lines.
x^l
IN arrow Bundle of Rays at Spherical Surface.
as axis through an infinitely small angle to one side and the othe^
its actual position, it will coincide in succession with all the n
which lie on the surface of a right circular cone of which 5C is 1
axis and the straight line SB (where B designates the point whe
the chief ray meets the refracting surface) is an element. The corr
sponding refracted rays will likewise He on the surface of a right cii
cular cone generated by the revolution of the chief refracted ray j55
about the same line as axis. Provided we neglect infinitely smaL
magnitudes of the second order, this group of incident rays may be
regarded as lying in a plane t which contains the incident chief ray
and is perpendicular to the plane of incidence uC (or ir); and, simi-
larly, the corresponding refracted rays may also be regarded as lying
in a plane t' which contains the chief refracted ray u' and is likewise
perpendicular to the plane uC. These planes are evidently tangent
to the conical surfaces generated by the revolution of «, w' around SC
as axis. Following the usage of most modern writers, we shall call
the incident and refracted rays lying in the planes ir, ir', respectively,
the SagiUal Rays.*
231. Different Degrees of Convergence of the Meridian and Sagit-
tal Rays.
The diagram (Fig. 125) shows a meridian section of the spherical
^refracting surface m containing the chief incident ray u and the chief
Fio. 125.
CoxrvB&OBircB op Meridian Rats aftbr Refraction at a Spherical Sxtrpacb. All the
3ies in the figure lie in the plane of a meridian section of the refracting sphere.
^"^fracted ray «', the plane of the diagram being, therefore, the Plane
'^^f Incidence. The numerals i and 2 in the figure are used to desig-
*^ate two points of the meridian section of the spherical surface both
"V-ery close to the incidence-point B of the chief incident ray and lying
^ " Sagittal " is a term borrowed from Anatomy. Many writers use the antonym
** tangential " instead of *' meridian **. On the other hand, some writers, who use the
term ** meridian **, prefer to be more consistent and use therefore the word " equatorial "
instead of " sagittal *'.
334 Geometrical Optics, Chapter XI. I § 232.
on opposite sides of this point. Thus, Si , 52 belong to the pencil of
meridian incident rays. After refraction, these rays will intersect the
chief refracted ray u' in the points designated in the figure by 5", S"\
which, while they are infinitely close together, cannot, in general, be
regarded as coincident unless we neglect infinitesimals of the first
order. In fact, the position of the I. Image-Point S' depends on the
arc B\ , so that for different rays of the meridian pencil we shall obtain
values of BS' which differ from each other by magnitudes of the same
order of smallness as the arc JBi. Hence, the convergence of the re-
fracted rays in the meridian section is said to be a ''convergence of the
first order".
The convergence of the refracted rays in the sagittal section is of
a higher order than the first. Thus, in Fig. 126, which is the corre-
Pio. 126.
CONVBROBNCB OP SAGITTAL RAYS AFTER RBFRACTIOIT AT A SPRBRICAL SURFACB. The plRIW
of the paper represents a meridian section of the ref ractinn: sphere. The points designated in the
diairram by the letters 5. ^, B and C lie in this plane. The points designated by the Roman
Numerals I and II are both infinitely near to the point B: these points lie on the line of inter-
section of the two planes which are perpendicular to the plane of the paper and which '^n*******
the incident ray SB (or u) and the corresponding refracted ray B^ (or ai'). respectively.
sponding diagram for the case of the sagittal rays, if the triangle
SBS' is supposed to be revolved about the central line SCS* as axis
through an infinitely small angle above and below the plane of the
paper, the chief incident ray and the chief refracted ray will coincide
in succession with the rays of the bundles of incident and refracted
rays, respectively, which lie on the conical surfaces generated by this
revolution. These refracted rays all intersect exactly at the point i?
and these rays are very nearly identical with the sagittal rays them-
selves. In fact, it is easy to see that the convergence of the sagittal
rays at S' is a "convergence of the second order", and is optically
more effective than that of the meridian rays.
232. The Image-Lines.
The astigmatic bundle of rays may Le regarded as composed either
of pencils of meridian rays whose chief rays all meet in the II. Image-
Point 5', or as pencils of sagittal rays whose chief rays all meet in the
§ 232.] Refraction of Narrow Bundle of Rays at Spherical Surface. 335
I. Image-Point 5'. The vertices of the meridian pencils form the
I. Image-Line, and the vertices of the sagittal pencils form the II.
Image-Line.
An incident ray proceeding from 5, lying in the plane of the meridian
section, and meeting the spherical refracting surface at a point i a
little above B (Fig. 125) will be refracted so as to intersect the central
line 5C at a point slightly to the left of 3' (Fig. 126); and this
point will be the point of convergence of all the refracted rays which
correspond to incident rays lying on the conical surface generated by
the revolution about 5C as axis of the incident ray Si. And, simi-
larly, if the point of incidence lies in the meridian section at a point
2 slightly below S, the rays incident on the spherical surface at points
in the arc of the circle described by 2 when the figure is revolved about
SC as axis will be refracted so as to cross the central line at a point
a little to the right of S\ Thus, all the rays of the infinitely narrow
astigmatic bundle of refracted rays will cross the central line 5 C within
an infinitely short piece of it lying on either side of the II. Image-
Point S\ This line-element may be regarded, and, indeed, from a
purely geometridal point of view, should be regarded, as in reality
the II. Image-Line.^ However, this line is not perpendicular to the
chief refracted ray w', and it is more convenient and quite permissible
to consider both of the Image-Lines, according to Sturm's definition,
as perpendicular to the chief ray of the astigmatic bundle (see § 49).
In fact, as Czapski* and others have pointed out, a section of the
lundle of rays made by a plane through S' perpendicular to the chief
xay u' differs very little from a straight line; the actual shape of the
section is a curve with two loops, not unlike a slender figure 8. It is
^^asy to see that this is so; for whereas the rays of the sagittal section
p>roper all intersect in S\ the rays of the other so-called sagittal sec-
tions intersect in points which lie on the axis to one side and the other
^:>f S', and, hence, the rays of each of these latter pencils will meet the
plane, which is drawn perpendicular to w' at S\ either before or after
they meet each other at the vertex of the pencil on the central line
SCf according as this vertex lies to the one side or the other of the
IL Image-Point 5'. Thus, we see that the section of the bundle
made by this plane opens out on each side of S\ Moreover, it can
very easily be shown that the width of this section is a magnitude of
* See, particularly. L. Matthibssbn: Ueber die Form der unendlich duennen astig-
matischen Strahlenbuendel und ueber die KuMMER'schen Modelle: Silzungher, der math.-
pkys. CI. der hoenigL bayer. Akad. der Wissenschaften zu Muenchen, xiii. (1883). 83.
*S. CzAPSKi: Zur Frage nach der Richtung der Brennlinien in unendlich duennen
optiscben Buescheln: Wied. Ann., xliii. (1891), 332-337-
336 Geometrical Optics, Chapter XI. [ § 233.
the second order of smallness, and hence the section itself may be
regarded as a straight line, since we are neglecting infinitesinials of
the second order. As Czapski says, the two 8-shaped sections, which
we have at both the I. Image-Point 5' and the II. Image-Point S\
with the axes of the 8's at right angles to each other, are as near an
approach to what may be called the Image-Lines of the astigmatic
bundle of rays as any other pair of lines.
ART. 71. THE MERIDIAN RATS.
233 . Relation between the Object-Point S and the I. Image-Point S'.
Let the chief ray u of an infinitely narrow homocentric bundle
of incident rays proceeding from an Object-Point S meet the spherical
refracting surface m in the point B (Fig. 127), and let the refracted
chief ray u' corresponding to w be constructed as in Young's Con-
struction (§ 206) by means of the concentric spherical surfaces t, t'
described around C as centre with radii equal to n'r/riy nr/n', respect-
ively, where C designates the centre of the spherical refracting surface,
and r denotes its radius, and n, n' denote the absolute indices of refrac-
tion of the first and second medium, respectively. Let G designate a
point of the spherical refracting surface in the plane of incidence u C
and infinitely near to B, so that SG will represent a secondary ray of
the pencil of incident meridian rays. This ray will meet the auxiliary
spherical surface t in a point N infinitely near to the point Z where
the chief incident ray u meets this surface, and the refracted ray cor-
responding to the incident ray SG will meet the spherical surface t'
in a point N' infinitely near to the point Z' where the chief refracted
ray u' meets this surface. The point of intersection of this refracted
ray with the chief refracted ray will determine the I. Image-Point 5',
which is the vertex of the pencil of meridian refracted rays.
The relation between the Object-Point S and its I. Image-Point y
may be found in various ways, either analytically or geometrically.
A very elegant geometrical method, involving however certain kine-
matica! notions which appear to be a little foreign in a treatise on
Optics, is given by L. Burmester in his interesting paper, "Homo-
centrische Brechung des Lichtes durch die Linse" {Zs. /. Math. u.
Phys., xL, 1895, 321). The method which is given below is in some
ways very similar to that used by F. Kessler in a paper entitled
" Beitraege zur graphischen Dioptrik" {Zs. f. Math. u. Phys., xxix.,
1884, 65-74).
On BC (Fig. 127) as diameter, describe a semi-circle meeting the
chief incident ray u in a point Y and the chief refracted ray «' in a
S 233.] Refraction of Narrow Bundle of Rays at Sptierical Surface.
point ¥', and let us imagine ttiat straiglit lines aie drawn connecting
tile points Y, Y' with each of the points G, Z and Z'. Since the
^^les aie inhnitely small, we can write the following proportions:
ZCY'B
ZGYB
/.gsb'
ZGSB
i.NYZ'
BS
BY'
YZ
SZ'
ZGS'B
iGS'B
BS
'BY"
Y'Z'
' S'Z' ■
iN'Y'Z'
"ieiefore, multiplying each of the two upper equations by the c
338 Geometrical Optics, Chapter XI. [ § ^ -^•
directly below it, we obtain :
ZGYB _ BS'YZ AGTB _ BS'Y'Z'
LNYZ " BY'SZ' /.N'Y'Z' " BTS'Z''
Since we are neglecting here infinitesimals of the second order, we o^
regard the point G as lying on the circumference of the circle BYlT '
and therefore we can write:
LGYB^LGY'B.
Moreover, since
LBYC^LBY'C^ 90^
the semi-circles described on CZ and CZ' as diameters will go throim
Y and Y\ respectively. These semi-circles may also be regarded
going through the points N and N' which are infinitely near to
and Z', respectively. Accordingly,
LliYZ =/,NCZ = Lli'Y'Z'\
and, thus, we obtain the following relation :
BS'YZ _ BS'Y'Z'
BY'SZ' BY'S'Z"
or
{BZSY) = {BZ'S'Y').
In this equation the points designated by the letters 5, Z, Z\ Y an<
Y' are all fixed points lying on the given incident chief ray u or on th^^
corresponding refracted chief ray u'; whereas 5' is the I. Image-Poin
on w' corresponding to an Object-Point S lying on u. Interpretini
the equation, we can say:
To a range of Object- Points P, Q, R, S, • • • lying an the chief incu
dent ray «, there corresponds a projective range of I. Image-Points P',
Q\ R\ S\ • • lying on the chief refracted ray u'. And, moreover,
since the two ranges have the incidence-point B in common, they are also
in perspective.
That the points F, F' are in the relation to each other of Object-
Point and I. Image- Point is evident not only from the above equation,
but geometrically also; for if we imagine an infinitely narrow pencil
of meridian incident rays with its vertex at F, these rays will meet
the spherical refracting surface at points infinitely near to B which
may all be regarded as lying on the circumference of the semi-circle
5 FC; so that for all rays converging to F, the angles of incidence,
being all subtended by the arc CY, will all be equal, and hence, the
as
Z
5 234.) Refraction of Narrow Bundle of Rays at Spherical Surface. 339
angles of refraction also must all be equal, and be angles in the circum-
ference standing on the arc CY\
234. The Centre of Perspective K is determined by the inter-
section of the straight lines YY\ ZZ\ The existence of this point
seems to have been recognized first by Thomas Young/ The point
K was afterwards found again, independently, by Cornu^ in 1863
and by Lippich^ in 1878.
Since Z CBZ' = Z CYY', both being inscribed angles standing
on the same arc CY', it follows that YY' is perpendicular to CZZ'
at K. Thus, we have the following simple Construction of the
/. Image- Point S' corresponding to an Object-Point S on the chief
incident ray u:
Having constructed the refracted chief ray w' corresponding to the
chief incident ray u, draw CY perpendicular to w at F and YK
perpendicular to CZ at K\ the straight line connecting S with K
will intersect the chief refracted ray u' in the I. Image-Point 5'.
The position of the Centre of Perspective K may also be computed
as follows:
Since
CY = r-sin a, CK = CF-sin a' = r-sin a-sin a\
"^ve obtain:
^„ nr- sin* a nV-sin*a' ,
CK -7— = . (242)
n n
If we draw the straight line BK (Fig. 127), then
Z CBK = Z CBZ' - /.KBZ' = a' - Z.KBZ'
--a' "{/.BKC-ABZ'C) = a -{- a' - Z BKC;
us, in the triangle BKCwe obtain:
BC _ sin ZBKC sin j^BKC
CK " sin Z CBK " sin (« + «'- j^BKC) '
'Tbomas Young: On the Mechanism of the Human Esre: Phil. Trans., 1801, xcii.*
33. This paper is reprinted in The Works of Thomas Young, in three volumes, edited
do. Pbacock, D.D. (London, John Murray, 1855); Vol. I. pages 12-63. See " Prop.
-• on p. i6.
■A. CoRNu: Caustiques — Centre de Jonction: Nouv. Ann. de Math., 1863, (2), li..
^"* *— 317. See also A. Cornu: Construction g4om4trique des deux images d'un point
^Uxsijiieux produit par refraction oblique sur une surface spherique; Journ. de physique,
^^r. III. X. (1901), 607.
*F. Lippxch: Ueber Brechung und Reflexion unendlich duenner Strahlensysteme an
^^xefflaechen: Wiener Denksch., 1878, xxxviii., 163-192.
340
Geometrical Optics, Chapter XL
[§2=1^:35.
And since
YC
BC
= sin a,
we have also:
BC
CK
YC
I
= sin a ,
so that
CK sin a • sin a' '
sin Z.BKC
whence we find :
sin a ' sin a' sin (a + a' — Z BKC) '
tan Z.BKC ^ tan a + tan a\
Commenting on this result, we observe that /.BKC, and, hence,
Z CB K = a + a' — /.BKC, is independent of the radius of €^
spherical refracting surface, so that the values of these angles
depend only on the angle of incidence a and on the indices of refi
tion n, n'.
Indeed, it is obvious also from the geometrical relations in the di
gram, that if, keeping the incidence-angle a unchanged, we sup
the radius of the refracting sphere to be variable, although the actu
distances from B of both C and K will vary, the directions of
straight lines B C and B K will remain unaltered. Perhaps, the easi
way of seeing this is by drawing through any point on 5 C a straigh
line parallel to CZ, and constructing a point on this line exactly i
the same way as the point K was constructed on CZ. It will be
that the point thus determined will lie always on the straight line BK
235. The Focal Points / and /' of the Meridian Rays.
If the Object-Point S is the infinitely distant point / of the chi
incident ray w, the meridian incident rays will be a pencil of
rays to which will correspond a pencil of meridian refracted rays meet-
ing the chief refracted ray u' in the " Flucht " Point V of the range
of I. Image-Points. And, on the other hand, if the I. Ims^ne-Point
5' is the infinitely distant point /' of the chief refracted ray u\ the
meridian incident rays will intersect in the " Flucht " Point / of the
range of I. Object- Points lying on the chief incident ray u. The
" Flucht " Points / and /', or, as we shall now call them, the Primary
and Secondary Focal Points of the Meridian Rays, may be located by
drawing through K (Fig. 128) straight lines parallel to u' and u meet-
ing u and u' in the points / and /', respectively.
According to a well-known law of projective ranges of points, we
have evidently :
JBTB = JSrS' = (JB + BS) {VB + BS^)\
jzr
§ 235.] Refraction of Narrow Bundle of Rays at Spherical Surface.
341
and if we put
BS = 5, BS' = 5',
we obtain the following equation :
BJ . Br
(244)
which, as the reader will remark, is completely analogous to the
relation
AF/u + AE'/u' = I, or f/u + e'/u' = - i,
which we found for the case of an infinitely narrow bundle of normally
incident rays refracted at a spherical surface; see formulae (148).
Fio. 128.
JtBFSACnON AT A SPHBRICAL SURPACB OF AN INFINITELY NARROW BUNDLE OF RAYS. Per-
ReUtions of the Rasffe of Object-Points lyin^ on the chief incident ray u and the Ranses
I. and n. Imase-Points lyin^ on the correspondinfl: refracted ray vf.
The positions of the Focal Points /, /' may be calculated as follows:
From the figure (Fig. 128), we obtain:
BJ = rK= -KZ'-
sin a
sin (a — a')
, sin a ' cos a
sin (a — a') '
Br ^ JK^KZ
^^d since
sin a
in (a — a!)
sin
7n = CZ . -
sin a' • cos' a!
sin (a — a') *
CZ = n'rin, CZ' = nrln\
342 Geometrical Optics, Chapter XI. [ § 2 ^^ \
the above formulae may be written :
„^ r-sin a'-cos' a nr-cos*a
5/= -
sin (a — a') »'-cos a! — n*cos a
_ r-sin a cos' a' n'r • cos' a!
sin (a — a') n' • cos a' — n • cos a *
236. Formula for Calculating the Position of the I. Image-Point ^^ u.
S corresponding to an Object-Point 5 on a given incident chief ray u. ^ ^^'
If in formula (244) above we substitute the values of BJ and Bl\ •
as given by formulae (245), we obtain the following relation:
fi! • cos' a' n • cos' a n' • cos a' — n • cos a
5'
(246)
Thus, if the chief incident ray u is given, and if the corresponding
chief refracted ray v! has been calculated trigonometrically, so that
the values of both a and a' are known, this useful formula enables us
to calculate the value of s' in terms of that of 5. It should not be
forgotten, however, that this formula has been obtained by neglecting
magnitudes of the second order of smallness, and is correct, therefore,
only to that degree of approximation. The formula may be written
in Abbe's differential system of notation (§ 126) as follows:
A f J— ^ ) ^ 7 ^(^cos a). (2460)
This formula may be derived also without much difficulty fronrft
formulae (191) of Chap. IX by regarding X, X' and the differenced^
(a — a), {a! — a') all as small magnitudes whose second powers may
neglected. Rayleigh^ has obtained the formula also in a very simpl'
way by the use of the Principle of the Shortest Light-Path (§ 38)
237. Convergence-Ratio of Meridian Rays.
^ If (Fig. 127) we put
Z BSG = dX, Z BSG = rfX',
then
^-"rfX
will denote the convergence-ratio of the pencil of meridian rays. Now
» J. W. Strutt, Lord Rayleigh: Investigations in Optics with special reference to
the Spectroscope: Phil. Mag., (5). ix. (1880). 40-55-
c=
i
i 239.] Refraction of Narrow Bundle of Rays at Spherical Surface. 343
we saw above that
BY BY'
ZBSG ^ ^'ZBYG, ZBS'G^^^'ZBY'G, ZBYG^ ZBY'G;
^Liid, accordingly:
Hence, since
obtain:
ZBS'G BY' BS
ZBSG "" BY ' BS'*
BY = 2r-cosa, BY' = 2r-cosa',
„ d\' S'cosa'
^M = 3r = 1 • (247)
" a\ 5 -cosa ^ ^'^
ART. 72. THE SAGITTAL RATS.
238._ Relation between the Object-Point S and the n. Image-
Point S\
Let S designate the vertex of the pencil of sagittal incident rays.
If the bundle of incident rays is homocentric, S will coincide with 5.
We have seen that the vertex 5' of the pencil of sagittal refracted rays
is the point of intersection of the chief refracted ray u' with the central
line 5C; and, hence, without further study, we may make the fol-
lowing statement:
The range of Object- Paints P, Q, R, S, • • • lying on the incident
chief ray u is in perspective with the corresponding range of II. Image-
Points P', Q\ R', S', • • • lying on the chief refracted ray u' ; the centre
C of the spherical refracting surface being the Centre of Perspective of
these two corresponding ranges, since the straight lines PP'^ QQ'% RR\
iSS', ' * ' all pass through the centre C.
239. The Focal Points J, T of the Sagittal Rays.
If the Object-Point S is the infinitely distant point 7 (or I) of the
^ief incident ray u, the sagittal incident rays will be a pencil of
parallel rays to which will correspond a pencil of sagittal refracted
xays with its vertex at the "Flucht" Point T of the range of II. Image-
Points lying on the chief refracted ray u'\ and, similarly, if the II.
Image-Point 5' coincides with the infinitely distant point J' (or J)
of the chief refracted ray «', the sagittal refracted rays will be a pencil
of parallel rays to which will correspond a pencil of sagittal incident
rays with its vertex at the **Flucht" Point 7 of the range of II.
Object-Points lying on the chief incident ray u.
The positions of the "Flucht" Points 7 and /', or, as we shall now
call them, the Primary and Secondary Focal Points of the Sagittal Rays,
344
Geometrical Optics, Cnapict -..
may be found by drawing through C straight lines parallel to v! a
u which will meet u and «' in the points J and 7', respectively.
Since the ranges P, Q, R, S^ • • • and P', Q', R\ S', • • • are prc^'
jective, and since the point B is a double-point of the two ranges,
have the following relation:
JB'TB = J5P5' = (JB + BS)(TB + 550;
and, hence, if we put _ _
BS = 5, BS' = ?,
we derive an equation exactly analogous to formula (244) which was
obtained for the meridian rays, viz. :
LV»-
B7 , BT
^- +^r = I.
s s
(248)
The positions of the Focal Points J, /' may be calculated as follows:
Since
BJ = I'C, BT = JC,
we obtain directly (Fig. 128):
BJ = - -.
BV =
r-sin a'
nr
sin (a — a')
r-sina
n'-cosa' — n-cosa *
nV
sin (a — a!)
n''COsa' — n*cosa '
(249)
240. Formula for Calculating the Position of the n. Image-Pdnt
S' corresponding to an Object-Point 5 on a given chief incident ray u.
Substituting in formula (248) the values of B7 and Bl\ as given
by formula (249), we obtain the following formula for determining ?
in terms of s:
1'
n n''C08 a' — fi'cos a
(250)
or, in Abbe's method of writing:
A f __ I _ -A(n-cosa).
(2Soa)
-«7as
%)
. 0-S-.
Ce
^n
zr
)
5 242.] Refraction of Narrow Bundle of Rays at Spherical Surface. 345
infinitesimals of the second order. If we retained infinitesimals of the
second order, the formula which would be obtained for the meridian
rays would give values of s' which depend on the inclinations of the
secondary rays to the chief ray of the pencil of meridian rays: that
is, the values of 5' would differ from each other by infinitesimals of
the first order. Thus, as has been stated above (§ 231), the conver-
gence of the meridian rays at the I. Image-Point is a "convergence of
the first order", whereas the convergence of the sagittal rays at the
II. Image-Point is a ''convergence of the second order".
241. Convergence-Ratio of the Sagittal Rays.
Let dX, dX' denote the angular apertures of the pencils of sagittal
incident and refracted rays. Obviously, we have the following re-
lation:
2. = -^ =--7; (251)
where Z. denotes the Convergence-Ratio of the Sagittal Rays.
ART. 73. THE ASTIGMATIC DIFFERENCE, AND THE MEASURE OF THE
ASTIGMATISM.
242. If the bundle of incident rays is homocentric, the Object-
Points S and 5 on the chief incident ray u are coincident, and in this
<:ase, therefore, we shall have 5 = 5. Thus,
To a range of Object- Points P, Q, R, S, • • • lying on the chief inci-
^dent ray u there corresponds a projective range of I. Image- Points P', Q\
5', • • • and a projective range of II. Image-Points P', Q\ R\ S\
, both lying on the chief refracted ray u' ; and, hence, also, the two
'^nges of Image- Points are projective with each other.
The points designated in the figures by the letters B and Z' are
Mie double-points of these two projective ranges of Image-Points. At
e incidence-point B the Object-Point and its two Image-Points coin-
de. The point Z\ as we saw, is the vertex of the bundle of refracted
ys corresponding to a homocentric bundle of incident rays — which
not be an infinitely narrow bundle — ^with its vertex at the Object-
int Z where the chief incident ray meets the auxiliary spherical
jface T (see § 207).
In the case of an infinitely narrow homocentric bundle of Object-
ays proceeding from the Object-Point 5 and undergoing refraction
a spherical surface, the astigmatic difference is the segment
S'S' = 5' - ?
^5 the chief refracted ray u' comprised between the II. Image- Point
1
\ 243.] Refraction of Narrow Bundle of Rays at Spherical Surface. 347
rhe points Z, Z' are the so-called aplanatic points of the spherical
^fracdng surface.
ART. 74. HISTORICAL NOTE, CONCERNING ASTIGBIATISM.
243. The theory of Astigmatism, at least in its beginnings and
early development, is due almost entirely to British men of science.
The earliest investigations along this line, of which we have any
record, are to be found in the optical writings of the distinguished
mathematician Isaac Barrow, professor of Geometry in the Univer-
sity of Cambridge (1663-1669) and the preceptor of Newton, who
succeeded to his chair in the university, and who aided Barrow in
preparing for publication his Lectiones OpticcR (London, 1674). In
this excellent and interesting work, Barrow investigates very skil-
fully the paths of rays lying in the meridian plane of a spherical
refracting surface, and shows how to construct the I. Image-Point.
But the real discoverer of Astigmatism was Sir Isaac Newton
bimself, who in his Lectiones Optica Annis 1669, 1670, 1671 (London,
1728) deals with the problem of the refraction of a narrow bundle of
rays at both plane and spherical surfaces, and who not only recognizes
the existence of the two Image-Points, but seeks also to determine what
intermediate point is selected by the eye as the place of the image.
The next most important advances in this study were made by
Robert Smith, who investigated very thoroughly the properties of
::austics both by reflexion and by refraction at spherical surfaces, and
who showed clearly the relations between the Object-Point and the
[. Image-Point, not merely for the case of refraction or reflexion at a
single spihercal surface, but for the general case of refraction through
I centered system of spherical surfaces. See especially Chapter I X
>f Book 2 of Smith's Compleat System of Opticks (Cambridge, 1738).
rhus, in Sec. 423 (Vol. i., p. 165) Smith finds that
JSrS' = JB'VB,
vhere the letters here used refer to Fig. 128. This result is a direct
x>nsequence of the fact that if P\ Q\ R\ S\ etc., lying on the chief
lefracted ray w', are the I. Image-Points of P, Q, -R, 5, etc., respect-
vely, lying on the chief incident ray «, these two ranges of points,
IS we found in § 233, are projective with each other, so that we have :
(PQRS) = {FQ'KS'),
?*or example, according to this relation, we have (Fig. 128) :
JS'I'S' = JYTY' = JZ-rZ' = JB'VB = a constant.
348 Geometrical Optics, Chapter XI. [ § 2.-^-
For the Construction of the Focal Points /, /', Smith gives (i
419, Vol. 1., p. 164) the following convenient method: From t_ ^i.\\e
centre C (Fig. 128) of the spherical refracting surface draw CY^ C. '^!IZY*
perpendicular at F, F' to the chief incident and refracted rays u, u^^ j^'
respectively; and draw the radius CB to the point of incidence i^ ^
From F, F' drop perpendiculars on CB, and through the foot of thB" ^i^he*
perpendicular let fall from F draw a straight line parallel to «', ar^:^ .
through the foot of the perpendicular let fall from F' draw a straigf-^^L.
line parallel to u. These straight lines will intersect u and «' in tt ^::^-
required points / and /', respectively.
Among the most important contributions to this subject are thcnrzjise
of Thomas Young, who recognized clearly and distinctly the va^^£/e
of Newton's discovery of Astigmatism. In Young's celebrated pav^jer
**0n the mechanism of the eye" {Phil, Trans., 1801, cii., 23-88; '^
printed in the Miscellaneous Works of the late Thomas Young, Lorid^^'
1855), he gives the formula, obtained first by l'Hospital {**AnC^'f^
des infiniment petits'\ Second Edition, Paris, 1716), for calcul^^^ ^
the intercept s' on the chief refracted ray of the meridian rays, ^^
shows how to find the positions of the Focal Points /', I' of both^^^ ^,
meridian and the sagittal rays. Moreover, Young perceived the ^.
spective centres K and Cof the rays of the meridian and sagittal sp^^ uv
ions of the narrow bundle of rays, and also discussed very thoroug^^-^^t^
the astigmatism of the eye. In his Lectures on Natural Philosoj^^^^^^^
(London, 1807), Young gives, likewise, the formula for the iviter&^^ 1^
5' of the sagittal refracted rays and applies all these various formic ^^m^pen
to a number of important special cases. He seems also to have b^^"^^^
the first to recognize the existence of "image-lines". Moreov*-^ *
Young was cognizant of the so-called "aplanatic" points of a ref ra» ^'^
ing sphere. ^{
The contributions of Airy ^ and of Coddington * to the theory "'^^^-t
astigmatism deserve also to be ranked among the most importar^ '^^na-
For a complete and very learned account of the theory of Astigm-^^^^ ^^
tism from the earliest times to the present, the reader is referred r -. p
the historical note, "Ueber den Astigmatismus", at the end of %- ^^t^
Culmann's article entitled "Die Realisierung der optischen Abbil^:^ _ i^
ung", which is Chapter IV of Die Theorie der optischen Instruments^
edited by M. von Rohr (Berlin, 1904).
* G. B. Airy: On a peculiar defect in the eye and mode of correcting it: Comb. PWfc^^^id©'
Trans. (1827), ii., 227-252. Also: On the spherical aberration of the eye-pieces of tek'i^*^^
scopes (Cambridge, 1827); and in Camb. Phil. Trans., iii. (1830), 1-64. ^, j jg.
' H. Coddington: A Treatise on the Reflexion and Refraction of Ligjkt: London, il ^
\ 244.] Refraction of Narrow Bundle of Rays at Spherical Surface. 349
ART. 75. INQUntT AS TO THE NATURE AND POSITION OF THE IMAGE OF
AN EXTENDED OBJECT FORMED BT NARROW ASTIG-
MATIC BUNDLES OF RATS.
244, It appears, therefore, that, in general, when an infinitely
narrow homocentric bundle of rays is refracted at a spherical surface,
the bundle of refracted rays will not be homocentric, but will be astig-
matic; so that to an Object- Point there corresponds, not a single
Image-Point, such as we 'have in the case of ideal imagery, but a pair
of infinitely short Image-Lines at right angles to each other and lying
in diflferent planes. An eye placed on the chief refracted ray may
accommodate itself to regard either of these two Image-Lines as the
image of the Object-Point whence the rays emanate.
If, instead of one single Object-Point, we have an aggregation of
such points forming an extended object, each of these points being the
vertex of an infinitely narrow bundle of incident rays whose chief rays
(we may suppose) all meet the spherical refracting surface at the same
point 5, the image of the object will be more or less blurred and
distorted. Thus, if the eye is accommodated to view the primary
Image-Lines, the dimensions of the object parallel to these lines will
be exaggerated in the image, whereas when the eye is focussed on
the other set of Image-Lines, there will be a similar exaggeration
[>arallel to these lines, so that in either case the quality of the image
mil be defective. Thus, as a rule, we do not obtain either faithful or
distinct images by means of astigmatic bundles of rays. It is assumed
^y most writers that on the whole the best image in such a case will
•ye obtained by accommodating the eye to view neither of the two sets
rf Image-Lines of the astigmatic bundle of rays, but a place lying
omewhere between these, the place of the so-called '* Circle of Least
Oonfusion*\ In fact, it is said, the eye unconsciously selects these
ections of the astigmatic bundles of rays.* Corresponding to each
»oint of the object, the eye will thus see a small area, so that according
tD this view of the matter, the image of an object, as Heath expresses
t, "is taken to be the aggregation of the overlapping 'Circles of least
onfusion'." In general, this is no doubt a correct explanation, but
TM, some special cases a more perfect and satisfactory image may be
btained by viewing the Image-Lines directly.
CzAPSKi* considers, for example, the case of an infinitely short
> See. for example. Heath's Geometrical Optics (Cambridge. 1887), Art. 145. Also.
>. I^ummer's work on Optics, published as Vol. II of the Ninth Edition of Mueller-
^ouillet's Lehrbuch der Physikt Art. 183. — The designation of this section of the
>tiiidle of rays as the place of "least confusion" is rather misleading, as the definition is
^^t.ter at either of the two Image-Lines.
'S. CzAPSKi : Theorit der optischen Instrumente nach Abbe (Breslau, 1893), S. 76. Seet
i 245.] Refraction of Narrow Bundle of Rays at Spherical Surface. 351
spread over a small area in this plane, so that if a screen were placed
It right angles to the "mean" refracted chief ray at 5', it would inter-
sect the bundles of refracted rays in a line. Thus, an eye placed on
the "mean" refracted chief ray and accommodated to the II. Image-
Point S' would view there a linear image 6' (Fig. 129) of the linear
object b.
An entirely analogous case is presented when the Object-Line is
a small line c lying in the plane containing the "mean" incident chief
ray and perpendicular to the plane of incidence. An eye placed on the
"mean" refracted chief ray and accommodated for the I. Image-Point 5'
would see there an image of c in the fcfrm of a straight line c' (Fig. 129)
parallel to c itself; whereas if the eye were focussed on the II. Image-
Point 5', the image of c will be found to be a rectangular figure c'
(Fig. 129) perpendicular at 5' to the plane of incidence.*
kSLT. 76. COLUKEAR RBLATIONS IN THS CASB OF THE RBFRACTION OF
A NARROW BUNDLE OF RATS AT A SPHERICAL SURFACE.
245. The Principal Axes of the Two Pairs of Collinear Plane
Systems.
To the chief ray u of an infinitely narrow homocentric bundle of
hcident rays which meets the spherical refracting surface at the inci-
lence-point B corresponds the chief refracted ray u' of the astigmatic
bundle of refracted rays. Both the incident meridian rays and the
•efracted meridian rays proceed in the plane uu', which may, therefore,
oe designated as the plane t or t'. Similarly, the planes of the inci-
ient sagittal rays and the refracted sagittal rays may be designated by
he symbols ir, 5r', respectively.
Consider, first, a point V lying in the plane t of the meridian rays
Liid very near to the chief incident ray u; and let us suppose that V
tself, regarded as an Object-Point, is the vertex of a narrow bundle
rf incident rays all meeting the spherical surface at points nearly
adjacent to the incidence-point B. The incident ray VB (or v)
ydng in the plane t may be treated as the chief ray of this bundle.
!^he angle at B between the rays u, v being an infinitesimal angle of
he first order, so likewise is the angle between the corresponding re-
nacted rays w', v'; and, since v lies in the plane uC or t, v* will lie in
hie plane u' C or t', which is coincident with tt; and to the pencil of
"mcident rays proceeding from V and lying in the plane tt will corre-
^ For a very clear and interesting treatment of the images formed by astigmatic bundles
r rays see L. Matthibssen: Ueber die Form der unendlich duennen astigmatischen
t.rahlenbuendel und ueber die KuMMER'schen Modelle: Sitzungber. der math.-phys, CL
k. bayer. Akad, der Wiss. zu Muenchen, xiii. (1883), 35-51-
352 Geometrical Optics, Chapter XI. [§245.
spond a pencil of refracted rays lying in the plane ir' and converging to
the I. Image-Point V on v\ Thus, if we utilize only such rays as
before and after refraction proceed infinitely near to u and «', respect-
ively, the plane-fields ir, ir' will be characterized by the property th^^
to a homocentric pencil of rays of ir there corresponds by refraction*
homocentric pencil of rays of tt'.
In the next place, let us consider a point TT lying in the plane of '^^
sagittal section and also infinitely near to the chief ray u of the bun^^
of incident rays. Regarding W as an Object-Point, we shall supj
that it is the vertex of a narrow bundle of incident rays whose chief
w meets the spherical refracting surface also at the point S so tl
the angles between the two incident chief rays u and w and betwe^^^
the corresponding refracted chief rays u' and w' are both infinitesii
angles of the first order. If we use Young's Construction (§ 206) fc
drawing the refracted ray w\ it will be obvious that, if we neglec^^
infinitesimals of the second order, w' will lie in the plane t'; anc§^
with the same degree of exactness, all incident rays proceeding fron^
TT and lying in the plane t will, after refraction, lie in the plane x'
and will converge to the II. Image-Point TT' on w' corresponding tc^
the Object- Point W on w.
Thus, within the infinitely narrow region surrounding the so-called
'*mean** incident chief ray u in the Object-Space and the corresponding
refracted chief ray u' in the Image-Space, we have a collinear relation
between the plane-fields ir, ir' and also between the plane-fields t, ir';
because to every incident ray in ir (or tF) there corresponds a refracted
ray in ir' (or tF'), and to every Object-Point of ir (or 5r) there corre-
sponds a I. (or II.) Image-Point of ir' (or ir').
It may be remarked also that the plane-fields x, x' have in common
the range of points which lie in the plane of incidence x along the
tangent to the spherical refracting surface at the incidence-point B;
whereas the plane-fields x, x' have in common the range of points
which lie in the line of intersection of these planes.
These results, which appear to have been first obtained by Lippich,*
may, accordingly, be stated as follows:
(i) The plane-fields x, x' lying in the plane of incidence are in
perspective with each other; and
(2) The plane-fields x, x', which are both perpendicular to the
plane uu\ and which contain w, «', respectively, are likewise in per-
spective with each other.
^F. Lippich: Ueber Brechung und Reflexion unendlich duenner Strahlensysteme
an Kugelflaechen: Denkschriften der kaiserl. Akad. der WissenschafUn wm Wien, xncviiL
(1878). 163-192.
§ 245.] Refraction of Narrow Bundle of Rays at Spherical Surface. 353
In order to get a proper idea of the imagery which we obtain by means
of the meridian rays, suppose we consider an infinitely short object-
line SV lying in the plane of incidence SBC and perpendicular at
5 to the "mean" incident chief ray SB (or u). To the narrow pencil
of meridian object-rays with its vertex at V will correspond a pencil
of meridian image-rays with its vertex at the I. Image-Point V" on
the refracted ray BV^' corresponding to the incident chief ray VB;
and if 5' is the I. Image-Point on «' corresponding to the Object-Point
S on tt, the infinitely short line 5'V", which is the image of the object-
line 5F will, in general, not be perpendicular to the **mean" refracted
chief ray u' (or -B5'). Accordingly, let us draw 5'F' perpendicular
to BS' at 5' and meeting SF" in the point V. Now the distance
between the two points F' and V" and also the angular aperture
of the pencil of image-rays F" are infinitesimals of the first order,
and, therefore, the piece of S'V intercepted between the two extreme
rays of this pencil will be an infinitesimal of the second order, and,
consequently, may be treated as a mere point, since here we neglect
infinitesimals of order higher than the first. Thus, according to Abbe,
we may regard the point F' as the vertex of the pencil of image-rays
corresponding to object-rays proceeding from F and S'V\ therefore,
as the image of SV. In brief, provided we neglect infinitesimals of
the second order, we have a right to say that the image, by means of
meridian rays, of an infinitely short object-line perpendicular to the
"mean" incident chief ray is an infinitely short line perpendicular to
the "mean" refracted chief ray.*
If dX denotes the inclination to the chief ray w of a secondary ray
of the pencil of meridian object-rays whose vertex is at 5, and if d\'
denotes the inclination to the chief refracted ray u' of the corresponding
refracted secondary ray, it is a very simple matter to show that
^always neglecting infinitesimals of the second order) we have for the
sneridian rays the following relation:
n'SV'd\ = n'S'V''dk';
^^irhich will be recognized as perfectly analogous to the Law of Robert
Smith for the refraction of paraxial rays, the so-called Lagrange-
Helmholtz Formula, § 194.
And, finally, if we consider in the same way the imagery in the
* See CzAPSKi's TheorU der optischen InstrumenU nock Abbe (Breslau, 1893). S. 78.
Also. P. Cuucann's *' Die Realisierung der optischen Abbildung ", which forms Chapter
IV of Die Theorie der optischen InstrumenU, edited by M. von Rohr (Bd. I. Berlin. 1904).
S. 171.
24
r *■'
354
Geometrical Optics, Chapter XI.
[§246.
)X
^r^t
planes t, ir', it will be obvious, on mere grounds of symmetry, that tb^
image, by means of the sagittal rays, of an infinitely short object-Ii^^
5TF lying in the plane t and perpendicular at 5 to the **mean" incid^^^
chief ray u will be an infinitely short line 5'TF' in the plane r' ^-^^^
perpendicular to the refracted chief ray w' at the II. Image-Point "^
corresponding to the object-point 5; provided that here also we negl^^^
infinitesimals of the second order.
Thus, according to Abbe, the '*mean" incident chief ray u and tt-
corresponding refracted ray u' are to be regarded as the Princi
Axes of the narrow coUinear plane-fields x, x' and also of the
coUinear plane-fields ir, x', since in both cases to an object-line perpe
dicular to u there corresponds, as we have seen, an image-line perpe
dicular to u\ This was not the case in Lippich's mode of treati
this matter, but it will be found to simplify the problem very greatlf
to be able to consider the chief rays u, v! as the Principal Axes of
two pairs of coUinear plane systems.
n-
fl-
ing
tly
e
Pzo. 130.
PlOURB FOR FXlfDXNO TBB SBCONDA&T POCAI. I«BN0TH (Ai') OP THE
Rats.
8T8TBM OP
246, Having determined the Principal Axes, we can now p:
to obtain the formulae for calculating The Focal Lengths of the
plane systems of rays; the Focal Lengths being defined as in § i
For example, Fig. 130 represents the case of a narrow pencil
parallel meridian incident rays to which corresponds a pencil of
fracted meridian rays with its vertex at the Focal Point /'.
incidence-points of the chief ray u and a secondary ray of the pen
of incident rays are designated in the diagram by the letters B and
respectively; the corresponding refracted rays are BV (or «') and Gi
i
246.] Refraction of Narrow Bundle of Rays at Spherical Surface. 355
it B erect BU, BU' perpendicular to w, w' and meeting the secondary
icident ray and the secondary refracted ray in the points Z7, U\
sspectively. If, then, we put d\' = Z BVG, and if e^ denotes the
econdary Focal Length of the system of meridian rays for which
and u' are the chief incident and refracted rays, according to the
efinition referred -to above, we shall have:
^- " dW •
Similarly, in the case of a pencil of parallel meridian refracted rays
manating before refraction from the Focal Point / on the chief ind-
ent ray «, the Primary Focal Length /^ will be given by the formula:
BU'
^^^ d\ '
yhere dk =^ Z BJG.
If a, a' denote the angles of incidence and refraction of the chief
•ay, we have evidently the following relations:
BU = BG'Cos a, BU' = 5Gcos a',
_ _ 5C7 _ _ BGcosa ^, _ _ BIT _ _ BG- cos a'
^^ BJ " BJ" ' ^ ^ EI' ' BI' '
irhence, therefore, we obtain:
cos a' , .,_ cos a
f, = JB——, e'^^I'B-
COS a ' cos a'
Phus, we see that the Focal Lengths /^ and el are not equal to the
egments JB and I'B onu and u' comprised between the Focal Points
'' and /', respectively, and the incidence-point B, as, having in mind
he special case of normally incident rays, where we have / = FA ,
' = E'A, we might have expected.
Substituting for JB and I'B their values as derived from formulae
245), we obtain finally:
wr-cosa-cosa' , »'r-cosa-cosa' , .
» -cosa — ncosa » -cosa — »-cosa •'^
By a process exactly similar to the above, we shall obtain, even
ore simply, for the Focal Lengths /„, e^ of the system of sagittal
Lys, for which u and u' are the chief incident and refracted rays,
^pressions as follows:
^ that in the case of the sagittal rays the distances of the incidence-
356 Geometrical Optics, Chapter XI. [§247.
1 ^
point B from the Focal Points / and T are equal to the Focal Lengths. ^ ^
Thus, employing formulae (249), we obtain:
7= "E g' r^ . (^.ss')
" n'-cosa' — n-cosa* " w'cosa' — »-cosa
For given values of the constants », »' and r, the Focal Lengths f^^ ^*
and 7„, e[,t as we see from formulae (254) and (255), depend only^ ^ .
the angle of incidence (that is, therefore, on the slope and positi ^^ !
of the chief incident ray u. In the special case when the chief ii
dent ray meets the spherical surface normally at the vertex A,
putting a = o, in the formulae (254) and (255), and writing Fin ph
of / or J and £' in place of /' or /' and A in place of 5, we obta;- ^.^i^'
For a = o: /. =/„ =f=FA= nr/(n' — »),
e' = el ^e^^E'A = -n'r/{n' - »);
as in formulae (147) of Chapter VIII.
Moreover, we find also:
/./«: = IK = - n/n'. (2S(^ ^6)
which corresponds with the relation already found in Chapter VIIL ^^'»
viz.,//e' = — n/n\
The magnification-ratios for the meridian and sagittal rays may bd^ be
derived without difficulty by means of the formulae given in Chag[^sp.
VII. § 179.
ART. 77. REFRACTION OF NARROW BUNDLE OF RATS THROUGH A CBC^^SF-
TERED STSTEM OF SPHERICAL REFRACTING SURFACES.
.^ur-
n-
247. Formulas for Calculating the Astigmatism of the Bundle of
Emergent Rays.
We shall consider here only the simple case when the chief incid^^si^t
ray Wi lies in a plane which contains the optical axis of the cent^*^^^
system of spherical surfaces. Thus, all the meridian sections of ^•^^
astigmatic bundles of rays arising by refraction at the successive
faces will lie in this plane.
Let Bf^ designate the point where the chief ray meets the kth spl^-
cal refracting surface, and let
denote the length of the path of the chief ray comprised between
incidence-point 5^ at the feth surface and the incidence-point ^^ "^^^
at the {Jk + i)th surface. Moreover, let 5^, 5^ designate the positi* ^^^
on the chief ray of the I. and II. Image-Points, respectively, af
I
r
7.] Refraction of Narrow Bundle of Rays at Spherical Surface.
357
refraction of the ray at the kth surface. We shall employ also
following symbols:
Tie relations between these intercepts on the chief ray before and
r refraction at the *th surface are given, for the meridian rays,
formula (246) and, for the sagittal rays, by formula (250) of this
ipter. Thus if r* (= -^^CJ denotes the radius of the feth spherical
ace, and if a^, a^ denote the angles of incidence and refraction of
chief ray at this surface, we shall have:
»j^*cos a
«i_, • COS^ ttfc
n
w*-i
"" «' T ^kf
s
US7)
re, by way of abbreviation, we have put:
»4 • cos a^ — tif^^i • cos a^fc
= V '
^ki
(258)
magnitude being called sometimes the ''astigmatic constant*' of
Jfeth spherical surface for the ray incident on it at the angle a^.
or the Logarithmic Computation of the positions on the emergent
;f ray of the I. and II. Image-Points Sl^ and 5^ corresponding to
Object-Point Si on the chief incident ray, it will be necessary, in
first place, to determine, by means of the system of formulae (215)
"hapter X, the path of the chief ray through the centered system
n spherical refracting surfaces, whereby we shall obtain the values
he angles of incidence a, a' at each surface in succession. We may
1 proceed to employ the following system of formulae, which are
tten in a form adapted to logarithmic work :
K =
<«, • sin (a^fc - al)
I. Meridian Rays:
r^sina^
T
Sk
V,
n;^_,'COs af, i_
»4 • cos a^ s^ fij, ' COS aj,
II.
^*+l — -^A ■" ^*-
Sagittal Rays :
»^_l I
Si,
n
- +
n,
^k^i "~
St, — 5f..
(259)
358 Geometrical Optics, Chapter XL [ §
In these formulae k must receive in succession all integral values frc^
k=i to k = m {8^=0). Accordingly, if we are given the values of t^^^
constants of the optical system, that is, the magnitudes denoted by r^ -»■ ^
and d, and if we are also given the ray-co-ordinates (vp ^,) of the ch. i^^
ray incident on the first spherical surface, so that we have the data
determining, by means of formulae (215) of Chapter X, the magnitw
denoted by a, a' and 5; and, if finally, we are given the positions on t.
chief incident ray of the I. Object-Point S^ and of the II. Obj(
Point Si, that is, if we are given the values of the intercepts 5i(= Bj.
and $1 ( = B^Si); we can, by successive substitutions in formulae (25^
obtain the values of the magnitudes 5^ ( = B^S'^) and s'^ (^ Sm^m)f ^^
thus determine the positions on the emergent chief ray of the I.
II. Image-Points 5^ and S^, and the magnitude of the AstigmatS:
Difference 5^5|„ = ^1 — ^l- The calculation, to be sure, is quite loi
and tedious, especially if the system consists of as many as four or fi^
refracting surfaces; but there is no shorter process of solving thf
required problem.^
The condition that the Astigmatic Difference of the bundle of emer-
gent rays shall vanish is 5^5^ = o, or s'^ = 5^. If the Optical Systei
consists of a single Lens (w = 2), it is not difficult to show that this
condition leads to a quadratic equation for determining 5|(« ^|).
The problem of the Homocentric Refraction of Light-Rays througi
a Lens has been beautifully and completely investigated by L. BuR-
MESTER.* By a simple process of geometrical reasoning, he shoi
that when an infinitely narrow bundle of rays is refracted through
Lens, there are two object-points (which may be real or imaginary
and which may be coincident) lying on the chief object-ray, to eac
of which there corresponds on the chief image-ray a "Homocentric'
Image-Point. Moreover, the same reasoning can be extended immc
diately to show that the same thing is true also in the case of a
system of any number of spherical refracting surfaces. BurmesT^^^r
shows also how to construct the two object-points and the cof f ^
sponding ** Homocentric** Image-Points in the case of a Lens,
discusses a number of interesting special cases.
248. Collinear Relations.
Within the infinitely narrow region surrounding the chief ray bef^
and after refraction at the feth spherical surface, we have a col
* See A. Gleichen: Lehrhuch der geomeirischen Optik (Leipzig und Berlin. B. G.
NER, 1902), pages 441-467, for the complete calculation of the ''Astigmatlsche BUdpunl
of P. GoERz's Double Anastigmatic Photographic Objective.
' L. Burmester: Homocentrische Brechung des Lichtes duich die Linse: Zfi, /. Jf<
u. Phys., xl. (189s), 321.
248.] Refraction of Narrow Bundle of Rays at Spherical Surface. 359
elation between the plane-systems x^^i and x^, which He in the
)lane of the meridian section of the centered system of spherical re-
racting surfaces; and, likewise, a collinear relation between the plane-
ystems fi^i and t^, which lie in the planes of the sagittal sections
rf the astigmatic bundles of rays before and after refraction at the
rth surface. In Art. 76 we saw that the chief rays before and after
efraction at this surface were to be regarded as the Principal Axes
rf each of these two pairs of collinear plane systems. And since the
hief ray after refraction at the *th surface is identical with the chief
ay before refraction at the (k + i)th surface, the following is the
tate of things which we have here:
The Principal Axis of the Image-Space of the feth surface is at the
ame time the Principal Axis of the Object-Space of the (Jfe + i)th
urface; and it will be recalled that this is precisely the one condition
hat was assumed in Chapter VII, Art. 52, in deriving the formulae
or finding the determining-constants of a compound system due to
he combination of any number of given simpler systems. Thus, if
ire know the positions of the Focal Points Jj,, /^ and J/,, 7[ and the
lagnitudes of the Focal Lengths/^,*, el,* and /«,*, el,» for the Meridian
nd Sagittal Rays, respectively, for each one of the m spherical surfaces
f the centered system, we can employ straightway the formulae re-
erred-to above, in order to determine the positions of the Focal Points
\ r and /, 7' and the magnitudes of the Focal Lengths/^, e^ and /«» ^1
f the entire compound system.
Obviously, we may also employ here exactly the same method as
/as used in Chapter VIII, Art. 54, for finding the Focal Lengths of a
entered system of spherical refracting surfaces for the case of Paraxial
Cays. Thus, for the Sagittal Rays we should find without difficulty:
e: = tb^ . 'i:'!."V^ . (260)
For the case of the Meridian Rays, since (Fig. 130)
BuVk ^ cosal.
B 1,17 I, cos a/
wre shouki find, in the same way, the following formula:
r/D cos g, cos a^'" cosa^ ^i ' 4 ' ' ' -^1-1 /^z-^x
** " cos «! • cos ttj • ' • cos a^ h'^i' ' '^m
Thus, having found by means of formulae (260) and (261) the magni-
360 Geometrical Optics, Chapter XI. [ § 249.
tudes of the two Secondary Focal Lengths e^ and e^, the magnitudes of
the Primary Focal Lengths /^ and /« can be calculated from the fol-
lowing relations:
ART. 78. SPECIAL CASES.
(il
249. The Special Case of the Refraction of a Narrow Bundle
Rays at a Plane Surface.
When we are given a chief ray u incident at a certain point B ^
a spherical refracting surface, we have seen how we can construct
corresponding refracted ray u' (Chapter I X, § 206) and determine
position of a certain fixed point if (§ 234), which is the centre of per^'^
spective of the range of Object- Points lying on u and the corresponding^
range of I. Image-Points lying on u\ just as the centre C of the sphe
is also the centre of perspective of the range of Object-Points lyin:
on u and the range of II. Image-Points lying on u\ We saw also^^
that when the radius of the spherical surface varies, these points
and K do not remain fixed, but move along two fixed straight lines. In
particular, if the radius of the refracting surface becomes infinite, so
that this surface is, therefore, a Plane Surface, the points C and K
will be the infinitely distant points of the two fixed straight lines.
And, hence, in the case of a Plane Refracting Surface, as was shown
in Chapter III, Art. 20, the straight lines joining the Object- Points
lying on the chief incident ray u with their corresponding II. Image-
Points lying on u' will all be parallel to the fixed straight line BC
normal to the refracting plane; and, similarly, the straight lines join-
ing the Object-Points lying on u with their corresponding I. Image-
Points lying on u' will all be parallel to the other fixed straight line B K.
In this special case, therefore, the range of Object-Points on u and the
two ranges of I. and II. Image-Points on u' are three similar ranges of
points}
The refracted ray u' corresponding to a given ray u incident on
a plane refracting surface fifi at the point B (Fig. 131) may be con-
structed by using Young's Construction, as follows:
On the incidence-normal take a point 0, and with this point as
centre and with radii equal to n'OBJn and n-OB/n' describe in
^ See F. Lippich: Ueber Brechung und Reflexion unendlich duenner Strahlensysteme
an Kugelflaechen: Denksckr. der kaiserl. Akad. der Wissenschaften su Wien, xzzviiL (1878),
163-192.
Also, F. Kesslsr: Bdtraege zur graphischen Dioptrik: Zfl. f. Maih. «. Pkys., xxbu
(1884). 65-74.
§ 250.] Refraction of Narrow Bundle of Rays at Spherical Surface.
361
the plane of incidence the arcs of two concentric circles c^, Cj. If Z^
designates the point of intersection of the incident ray u with the
arc Cj, and if Z, designates the point of intersection of the straight
line OZi with the arc c,, the straight line BZ^ will be the path of
tJie refracted ray u'.
The normal to the plane refracting surface gives the direction of
<Jie infinitely distant point C The direction of the infinitely distant
s'^'j:^' '
PlO. 131.
RBFAACnoif OF Iif Fiif ITELT NARROW BuNDLB OP Rays AT A Planb SURFACE. Construction
^tf Chief Refracted Ray ar' Correspondinir to Chief Incident Ray u ; and Construction of I. and II.
^msse-Points S^ and "S^ corresponding to a sriven Object Point .ST on u. Centres of Perspective Cand
"^both at infinity. Plane Surface is icffaxded as a Spherical Surface with Infinite Radius.
^x>int K is found by drawing OY perpendicular to BZ^ and YH per-
^)endicular to OZ^. Then the point K will be the infinitely distant
:(X)int of the straight line BH.
The I. Image-Point 5' and the II. Image-Point 5' corresponding to
^^n Object-Point S on the chief incident ray u are found by drawing
'^Jirough S straight lines parallel to BJ^ and BC, which will meet the
^^hief refracted ray u' in the required points 5' and 5' respectively,
le Focal Points of the Meridian and Sagittal Rays coincide with
infinitely distant points of the chief incident and refracted rays.
By putting r » oo in the formulae of Arts. 71 and 72 of this chapter,
shall derive immediately the same formulae as were obtained in
Chapter III, Art. 19.
250. Reflexion at a Spherical Mirror Treated as a Special Case
^ Refraction at a Spherical Surface.
In the case of Reflexion {n' jn = — i), we cannot use Young's
\
362
Geometrical Optics, Chapter XI.
(§250.
4D{
Construction for constructing the reflected ray w' corresponding to a
ray u incident on a spherical mirror, for the obvious reason that the
auxiiiar>' spherical surfaces t and t', and with them the Aplanatic
Points Z, Z\ used in this construction (§§ 206, 207), have here i^^
meaning. Except, however, such properties as depend on these p^'
ticular features, we have in the case of Reflexion at a Spherical Mit^^^
relations corresponding precisely to those which we found in the i^'
N'estigation of Refraction at a Spherical Surface. It is very easy ^^
obtain these relations independently, but it is also instructive to aO^'
sider the problem as a special case of refraction (§ 26).
If in Fig. 132, where C designates the position of the centra
the Spherical Mirror mm. the chief incident ray u meets the mi
at the point 5, the co
sponding reflected ray u'
have a direction such
ZCBtt= Zu'BC. On
as diameter describe a cii
cutting w, It' in the points -
F', respectively. O b v
ously, exactly as was th"^*^
case in refraction, the poir^^ ^^
F' on tt' is the I. Imag^rS^
Y
be
K
he
Pio. 152.
Kkvukxion op Inpinitbly Narrow Bundlb op
KvYH AT A Sphkrical MiRROR. M, yf Chief Incident
HuU KvtIwtcU K«y». respectively.
4.CHS»^^ IS^BC, BS'^s, BS'^/, bS^'^Y,
Point of the Object-Point
on u (§ 233); and, hence,
centre of perspective
(§ 234) of the range of
ject-Points on u and t
range of corresponding
Image-Points on «' will
I.
Ue
on the straight line 3^ ^^ •
The actual position of K is found by drawing CK perpendicular ^^
YY*\ thus, K is seen to be the point of intersection of the straight l^
rr and CB.
The 1. and II. Image-Points 5' and 5' on the chief reflected ra;
corrt^>J|Kmding to an Object-Point 5 on the chief incident ray u
determined by drawing from 5 straight lines through K and C;
iuiersei'tions oi SK and SC with w' will determine the points S* and
iVJiiHH'tively. Straight lines drawn through K and C parallel to
tay 14 will determine by their intersections with the reflected raj ^^
the KvKal Points /' and 7', respectively. Similarly, the Focal Poi^^
7 .ukI J im H are found by drawing through K and C, respecti
«U«u^ht line^ |>arallel to u\
the
3^.
S 251.] Refraction of Narrow Bundle of Rays at Spherical Surface.
363
The Metric Relations which we have for the case of the Reflexion
of a narrow bundle of rays at a Spherical Mirror can be derived from
the corresponding Refraction-Formulae, which have been obtained in
this chapter, by merely putting n' ^ ^n and a' = -- a. However,
In the formulae derived in this way, the reader should bear in mind,
that, according to the convention we made in § 26, the positive direction
Df any straight line is the direction along that line which light would
pursue if the line were the path of an incident ray^ and, accordingly,
the positive direction along a reflected ray is the direction exactly
opposite to that in which the reflected light is propagated along it.
Failure to note this point has been a source of frequent confusion
with writers on Optics.
We derive, therefore, the following set of FormukB for the Reflexion
>/ a Narrow Bundle of Rays at a Spherical Mirror:
I. Meridian Rays:
Cif^f sin* a, ZBKC = o;
f^^e'^^JB^ tB = -
r-cosa
s s rcosa
• 5
II. Sagittal Rays:
J^^e:^jB=^TB
2 cos a
112 cos a ^ s
s s r s
(263)
251. Astigmatism of an Infinitely Thin Lens.
Provided we assume that the length of the path of the chief ray
within the Lens is negligible (which may sometimes be a rather big
assumption, even though the Lens is infinitely thin), and accordingly
put BiB^ =5 o, we shall have:
5, = s,
29
Si — ^2,
and since here there is no possibility of confusion, we shall find it
convenient to write: 5 = Si, s' = ^2 and J = J^ T = Jj. Moreover,
since the Lens is supposed to be surrounded by the same medium on
both sides, we may also write : ni = n, = », n[ = n'. Thus, for the
case of an Infinitely Thin Lens (w = 2), formulae (259) give the fol-
1
364
lowing relations:
Geometrical Optics, Chapter XI.
[i
(*
)
_ n'-cosaj — n-costti w-cosa^ — n^-cosgg
I. Meridian Rays:
£ _ cos* «! -cos' ttj I cos* «2 { ^\ i ^2 \ .
5 cos «! • COS a2 s n- COS a, \ cos aj cos Oj /
II. Sagittal Rays:
The conditions that to an Object-Point S Ijang on the chief obj
ray u there shall correspond a '' Homocentric'* Image-Point S' lying "^j^"
the chief image-ray u' are s = s = 5S, 5' = 5' = 5S', whence we fir^^
52 =
n(cos* «! • cos* ttj — cos* a[ • cos* a2)
F/2' 2' 2\ Tr 2'»2'»
i(cos a^cos a, — cos aj — Fj-cos a|-sm a.
(2
)
lA
J
In general, therefore, on every incident chief ray u there is one sui
Object-Point 2 to which on the corresponding emergent chief ray
there corresponds a "Homocentric** Image-Point S'.
A case of both theoretical and practical interest occurs when I
chief ray goes through the Optical Centre of the Infinitely Thin
(which is easily contrived by placing a screen with a small drculai^ ^m
opening right in front of the Lens). In this case the paths of the^
incident and emergent chief rays are along the same straight line, and,^
accordingly, we have:
ctx = ci2\ and also a| = aj!
and, therefore,
rxVx + r^V^ = o.
Introducing these values in formulae (264) above, we obtain for this^^
special case:
FormukB for Calculating the Astigmatism of an Infinitely Thin
for the cctse when the Chief Ray goes through the Optical Centre :
I. Meridian Rays :
II V,
f, — ft
II. Sagittal Rays :
s
n-
cos*
«i
U
I
I
s "
n
r^
ri
•
(2
S 251.] Refraction of Narrow Bundle of Rays at Spherical Surface. 365
The positions of the Secondary Focal Points /' and T of the systems
of Meridian and Sagittal Rays of the astigmatic bundle of emergent
rays may be found by putting 5 = 5 = oo in the above formulae.
Thus, if A designates the position on the axis of the Optical Axis of
the Thin Lens, we have
AV = iTTT — - — c cos' ttp AT = TTT — - — ^ . (267)
If in formulae (261) and (262), we introduce the special conditions
ivhich we have in the present case, viz. : m = 2, s\ = $2, a^ = a^, a\ = a^
and fly = ni, we find for the Focal Lengths of the system of Meridian
Rays:
/„ = - el = AV.
In the same way, formulae (260) and (262), give for the Focal Lengths
Df the system of Sagittal Rays:
/,= -?: = AT.
Accordingly, in the special case which we have here the Focal Lengths
Df both systems of rays are equal to the distances of the Focal Points
from the incidence-point A, Thus, we have:
fit
/« =fu'COs^cti = - el = - il-cos^tti = yr7--^—\ cos^a^; (268)
'IV 2 M/
SO that now formulae (266) may be put in the following forms:
I. Meridian Rays : i/s' — i/s = i//^;
IL Sagittal Rays: i/J' — i/5 = i//„. ^
(269)
These equations, as will be immediately recognized, have precisely the
same form as the formula for the Refraction of Paraxial Rays through
an Infinitely Thin Lens, formula (99) of Chap. VL The Focal Lengths
^, and 7« are both functions of the slope-angle a^ of the chief ray, and
for the value ai = o we obtain:
(«! = 0), f^=J^=f = nr^rjin' - n){r2 - fi).
"When the chief ray goes through the Optical Centre of the Infinitely
Thin Lens, the Astigmatic Difference vanishes, in general, only for
366 Geometrical Optics, Chapter XI. I § ^^*
the case when the Object-Point is in contact with the Lens; but wr
«! = o it vanishes for all positions of the Object-Point.*
Another interesting special case which has been investigated by "•
Harting* is the case when the chief ray crosses the optical axis at
the common vertex of a System of Thin Lenses in Contact.
» See Die Theorie der optischen Instrumenie, edited by M. voN Rohr (Bd. I, B^^
X904); IV. Kapitel, " Die Realisierung der optischen Abbildung". von P. Culmann, S- ^'^^'
• H. Hartinc: Einige Bemerkungen zu dem Aufsatzedes Hrn. B. Wanach: Ueb^^^*
V. Sbidrls Formeln zur Durchrechnung von Strahlen u. 8. w.: Zfl. f. Instr.^ xx. (i^^^'*
234-237. See also Die Theorie der optischen InstrumerUe, edited by M. von Rohr (HP ^- ^'
Berlin, 1904); V. Kapitel, " Die Theorie der sphaerischen Aberrationen*'. S. 254.
CHAPTER XII.
THE THEORY OF SPHERICAL ABERRATIONS.
I. Introduction.
ART. 79. PRACTICAL IMAGES.
252. The requirements of a good image are (i) that it shall be
sharp or distinct, corresponding, therefore, to the object point by point,
C^) that it shall be accurate, that is, completely similar to the object,
and thus faithfully reproducing it, and (3) that it shall be bright.
This last condition necessarily implies the use of wide-angle bundles
of rays, because obviously the light-intensity at any point will be
greater in proportion to the number of rays that unite at that point.
On the other hand, the first two requirements, which are both purely
geometrical, will, in general, be fulfilled by an optical system only in
the special and unrealizable case when the bundles of rays concerned
in the production of the image are infinitely narrow. Thus, in the
theory of the Imagery by means of Paraxial Rays, which was developed
according to general laws first by Gauss, and which has, therefore,
been appropriately called "GAUSsian Imagery" (§ 188), we have seen
that for an optical system of centered refracting (or reflecting) spher-
ical surfaces a distinct and accurate image was formed only when the
rays concerned were all comprised within an indefinitely narrow cylin-
drical space immediately surrounding the optical axis; this region be-
ing more explicitly defined by the condition that a "paraxial** ray is one
for which both the angle of incidence a and the central angle tp were
so small that all powers of these angles higher than the first could be
neglected (§ 109).
In general, even with infinitely narrow bundles of rays, stigmatic
imagery, except in the case of normally incident rays just mentioned,
is possible only for certain special positions of the object-point.
It goes without saying that from the standpoint of the optician the
Cormation of images under such impracticable restrictions is almost
without interest. Without dwelling on the obvious objection that
such images would be of infinitesimal dimensions (as would be like-
wise true of the objects to be depicted), we encounter a still greater
difficulty in the fact that Physical Optics — which in all optical ques-
tions IS the court of last resort — pronounces that these images are
not true images at all. For according to the Wave-Theory of Light,
367
368 Geometrical Optics. Chapter XII. [ § 252.
a mere homocentric convergence of the image-rays is not of itself suf-
ficient for the formation of a distinct optical image. If the wave-surf*
ace in the Image-Space is spherical — so that the image-rays all meet
in one point, viz., at the centre of the spherical wave-surface — instead
of an image-point, we shall obtain a resultant effect (in the pl^^^
perpendicular to the optical axis through the centre of the sphet^^
wave-surface) consisting of a central luminous disc surrounded W
alternate dark and diminishingly bright rings. The brightness of *^
disc fades from the centre out towards the circumference. The gr^^^
the extent of the effective portion of the spherical wave-surfac^
compared with its radius, that is, the wider the angle of the hc^^ ^
centric bundle of image-rays, the smaller will be the diameter of
diffraction-disc, which therefore tends to be reduced more and
nearly to a point as the angular aperture of the bundle of image-i
is increased. From the point of view of Physical Optics, as lAiiAXC^^^^
remarks, this is the only sense in which the term "point-image"
have any meaning. Thus, both for a clear and distinct image as
as for a bright image, theory insists that wide-angle bundles of
must be employed (see §45).
On the other hand, from the geometrical standpoint the fundament^
requirement of optical imagery is the convergence of the rays to 01^^
point; and, in general, this requirement in the case of bundles of finil^ ^
aperture is impossible.
Consequently, actual optical images, which are necessarily formec^
by bundles of rays of finite aperture, are, in general, more or les^
faulty. These faults — ^which are called aberrations — may sometime^^
escape unnoticed merely because the eye which views the image cannot^
or does not distinguish the defects which it contains. But to the *
practical optician who strives to obtain an image as nearly perfect as
possible it is of the highest importance to ascertain the nature of these
various so-called aberrations, to distinguish them the one from the
other, and to perceive clearly what factors contribute to produce them
in each instance, so that in the design of an optical instrument he may
contrive to reduce, perhaps to abolish entirely, at any rate those
aberrations which for the particular tyf)e of instrument are to be re-
garded as the most objectionable. Along these lines, and especially
since the rise of Photography, wonderful progress has been achieved
in the design and construction of optical instruments.
The plan that is employed is to combine optical systems in such a
way that, although each single refracting or reflecting surface gives
^ See Muellbr-Pouillbt's Lehrbuch der Physik (neunte Auflage). Bd. II, 447.
§ 253.] Theory of Spherical Aberrations. 369
by Itself a point-to-point imagery only within the narrow region to
which the paraxial rays are confined, in the compound system these
limitations are very considerably extended in one direction or another
or perhaps in several directions simultaneously. The duty of refract-
ing the rays so that they will emerge finally in suitable directions is
not assigned to a single surface, but is distributed over a number of
separate surfaces. By suitable combinations, it has been found pos-
sible in this way to construct systems which by means of wide-angle
bundles of rays will give a true image of an axial object-point or of
a small surface-element placed at right angles to the optical axis.
The objective of a microscope, for example, is a system of this kind.
In the eye-piece, or ocular, on the other hand, we have an illustration
of a system which by means of relatively narrow bundles of rays pro-
duces the image of a large object. Thus, in the compound microscope
the duty of the objective is to produce an image of a small object by
means of wide-angle bundles of rays, whereas the duty of the ocular
is, by means of narrow bundles, to spread over the large field of vision
the image produced by the objective. In the case of the photographic
objective, we must have both wide af)erture and extensive field of
vision, and in order to meet both of these requirements at once, some-
thing else has to be sacrificed, and, accordingly, we are obliged to be
content with a less distinct image than we require in the case of the
objective of a microscof)e.
Of course, it would be idle for the optician to seek to produce an
image which is free from faults that could not be detected by the eye
if they were present. The resolving-power of the human eye is com-
paratively poor ((/. § 377). Thus, for example, details in the object
which are separated by an angular distance, say, of one minute will not
be recognized by the eye as separate and distinct. Accordingly, the
practical image need be perfect only to the degree that in it those
elements of the object which are to be preceived as separate must be
presented to the eye at a visual angle of not less than one minute of arc.
ART. 80. THE SO-CALLED SEIDEL IMAGERY.
253. The theory developed by Gauss* in his Diopirischen Unter-
^suchungen proceeds on the assumption that the central angle ^ is so
small that the second and higher powers thereof are negligible. The
theory is applicable, therefore, only to optical systems of narrow af)er-
ture. and of small visual field, since both the incidence-points of the
rays and the object-points whence they emanate must all lie very close
' C. F. Gauss: Dioptrische Untersuchungen (Goettingen, 1841).
25
370 Geometrical Optics, Chapter XII. [ § 254.
to the optical axis of the centered system of spherical surfaces. The
investigations of Euler,* Schleiermacher,* Seidel* and others
wea* first directed to^-ards taking account of the aberrations due to
increase of the aperture of the system; but, later, with the rise of
Photography and the de\'elopment of the Photographic Objective,
it became necessary to take into consideration not only a wider aper-
ture but a greater field of vision, in order to portray objects which
Heiv at s*.>nie distance from the optical axis. The complete theory of
spherical aberrations was worked out by J. Petzval* and L. Seidel,*
aiKl in the following sections of this article the methods of these two
investigators form the basis of the mode of treatment.
^54« Order of tiie Image, according to J. Petzval. Taking the
v,>pucal axis of the centered system of spherical surfaces as the 5:-axis
v>t a s>ste«i of rectangular co-ordinates, let us denote the co-ordinates
\A au obKVt-p*.>int P by {, iy, f . The transversal plane <r, which passes
ihiv>u^h P sXvuA is perpendicular to the optical axis, will be called the
I >^Ki i- FluH^. Let P (|, tj, 5) designate the position of the point where
ih\* i\viitiiK\ir |xith of an object-ray, proceeding from the object-point
t^, a\»ss*.^ a sec\>nd fixed transversal plane o* parallel to the object-
V^aiK^ ^ . ITk' ix^tion of this object-ray will be completely determined
b> itK^ tvHir v\iranH>ters iy, f , ij, J. In the image-space let <r', a' designate
a |Mir v^ fi\t\i transN-ersal planes perpendicular to the optical axis; we
jJkUI call the |>Iane 9' the Image-Plane. Let P', P' designate the posi-
livHix^ v4 iht* jxnnts where the rectilinear path of the image-ray, corre-
x^sMKlin^ 10 the object-ray PP, crosses the planes <r', o*', respectively,
aiKl let the rectangular co-ordinates of P', P' be denoted by ({', iy', f')
xUkI Ia vt*» ^'» J'), respectively. The position of the image-ray will,
ihcivUMW U^ defined by the four parameters 17', f', ij', J'.
Siuvv to eN*ery object-ray there corresponds one, and only one,
^ ) . Ki'lkr: IHo^iricm pars prima (Petersburg, Akad. Wiss., 1769); pars secunda
,i>ia\. i::o>; p*»rs ttrtia {ibid,, I77i)-
^ I . SvitLKiKKMACUBR : U^fcr den Gebrauch der analyiischen Optik bei ConstnuUon
.^JSl.^»^-r WfrkMHjut (PoGG. Ann,, 1828. xiv.); also, Analytische Optik (Baumgartnbrs
uuil \vKN KiriNiiSHAvnutNs lift, f. Phys, u. Math,, 1831. ix.. 1-35; 161-178; 454-474:
i\ij. V. 171 Joo; 32^357)-
^ I SkiuHL: Zur Theorie der Femrohr Objective: Astr, Nachr., 1853, xxxv. No. 836,
xv»i ,110.
* .K»s»kwi'u Pktzval : Bcricki u^fcr die Ergebnisse einiger dioptriscker Untersuchuniem
,\\M\. ii^^), See also: Bertcht ueber optische Untersuchungen. SUMungsber. der maik.-
s,j...f .«i.>A. a. der kaiserl, Akad. der Wissenschaften, Wien, xxvi. (1857), 50-75. 93-xo5,
I \> I L^. <.Sec eapecially page 95. in regard to the '* order '* of the image.)
"^I.. SkiDKL : Zur Dioptrik. Ueber die Entwicklung der Glieder 3ter Ordnung.
w^lvlu' lea Weg eineit ausaerhalb der Ebene der Axe gelegenen Lichtstrahles durch ein
v iMii hivxhciider Medien be»timmen: Astr. Nachr,, 1856, xliii.. No. 1037, 289-304;
\o. loJvS, SOS 3^0; No. 1019. 321-332.
§ 255.] Theory of Spherical Aberrations. 371
image-ray, it is obvious that each of the four parameters of the image-
ray must be a definite function of the four parameters of the object-
ray, so that we may write:
where the functions /p/jj/a,/^ can be deduced by the laws of refraction.
Moreover, taking account of the symmetry with respect to the
optical axis, we observe that if the signs of the parameters ?;, f , t), J
are all reversed, the signs of the parameters ly', f', t)', J' will all like-
wise be reversed; and, consequently, if each of the functions above is
developed in a series of ascending powers and products of ly, f , t), J,
each of these series can contain only the terms of the odd degrees.
And, hence, if the parameters of the ray are regarded as magnitudes
of the first order of smallness, these series-developments will contain
only terms of the odd orders of smallness.
Now, if for aU rays proceeding from the object-point P we obtain
exactly the same values of the co-ordinates rj', f', we shall obtain at
-P' a perfect image of the object-point P. In general, however, this
^11 not be the case, and for a second object-ray coming from P, whose
parameters are, say, 17, f , t) -f 6% J -f 5^, we shall obtain a new set
of values 17' + W, f ' + 5f ', ^' + ^^'f V + ^' for all four of the par-
^mieters of the corresponding image-ray. Obviously, in the series-
developments the differences 617', 5f ' will contain also only the terms
of the odd degrees. If, as compared with the magnitudes 77, f , t), J,
t:hese differences 617', 5f ' are, say, of the (2k -f i)th order of smallness,
then, according to J. Petzval, the spot of light formed around P'
by the totality of all such points as P' is to be considered as an **ifnaf^e'*
of the (2k + i)th order in the image-plane <r' corresponding to the
object-point P. The higher the order of the imager the more nearly
perfect it will be. An image of the 3rd order is one in which there
are uncorrected faults of the 3rd order.
255. Parameters of Object-Ray and Image-Ray, according to L.
Seidel. A complete development of the theory of Spherical Aberra-
tions was first published by L. Seidel, who extended Gauss's theory
90 as to take account of magnitudes of the 3rd order of smallness,
neglecting therefore the terms of the 5th and higher orders. Thus,
in the so-called Seidel Imagery, the image is of the fifth order.
The comparative simplicity and elegance of Seidel's methods are
due to his choice of the four parameters which define the rectilinear
path of the ray, viz., the two pairs of rectangular co-ordinates (17, f)
372 Geometrical Optics, Chapter XII. [ § 255^
and (t), 5) of the points P, P where the ray crosses the two fixed trans-
versal planes <r, <r. In order to make this clear, let us suppose now
that PP represents the path, not of the object-ray itself, as formerly,
but of this ray before refraction at, say, the kth surface of the optical
system, and, in the same way, let P'P' represent the path of the ray
after refraction at this surface. The actual locations of the four trans-
versal planes <r, <r' and <r, <r' have not been specified; and, accordingly,
we may establish an arbitrary connection between, say, <r and <r', on
the one hand, and between <r and <r', on the other hand. If, for
example, Jlf , M' designate the points where the optical axis meets the
planes <r, <r', resf)ectively, these points may be selected with reference
to each other so that, in the sense of Gauss's Theory, Jlf, M' are a pair
of conjugate axial points with respect to the spherical refracting sur-
face which is here under consideration. And the same relation can
be established between the pair of points Af , Af' where the optical
axis crosses the transversal planes <r, <r', respectively. Thus, by
Gauss's Theory, the transversal planes <r, <r' and <r, <r' will be two pairs
of conjugate planes with respect to the spherical surface in question.
If A designates the vertex and C the centre of this spherical surface,
and if we put:
.4C = r, AM^u, AM'^u\ 4Af = a, AM'^u',
the relations between M and M' and between M and M' will b^
expressed as follows (see § 126):
"0-i)-"'0-i--)-^'
(270)
The co-ordinates of the four points P, P, P', P' may now be ex-
pressed as follows:
wherein the first term on the right-hand side of each of these equations
denotes the approximate (or *'GAUSsian") value of the parameter ob-
tained by neglecting the terms of the 3rd order, and the second term
denotes the correction of the "^rd order, which, being added to the prin-
cipal, or approximate, value, gives a value which will be exact except
S 256.) Theory of Spherical Aberrations. 373
for residual errors of the sth and higher orders. Evidently, the points
Q(y, z) and Q'(y\ z'), lying in the planes <r, a' and not far from the
points P, P\ respectively, are a pair of conjugate points according
to Gauss's Theory; and the same thing is true also of the pair of
points Q{y,z) and Q'{y\z'),vihxc\i lie in the transversal planes a, a'
and not far from the points P{r\, J), P'{r\\ JOt resf)ectively.
256. The Correction-Terms or Aberrations of the 3rd Order.
Thus, Seidel employs two independent systems of transversal planes
perpendicular to the optical axis of the centered system of spherical
surfaces, so that for each medium traversed by the ray there is one
plane of each system. The position of the object-ray before refrac-
tion at the first spherical surface is given by assigning the co-ordinates
(^if f i)» ("HuSi) ^^ ^^^ points Pp Pi where the ray crosses two arbitrary
transversal planes (Tp 0*1.
For the plane c^ naturally we shall select the transversal plane
^which contains the object-point Pi(i7i, fi); this is the so-called Object-
Plane mentioned above (§ 254). Moreover, without affecting at all
tiie generality of the discussion, we may select the jcy-plane of the
system of rectangular co-ordinates so that the object-point P, lies in
tihis plane, in which case we shall have fi = o. Since the bundle of
object-rays is homocentric, the point ftCVi* ^i) will coincide with Pp
that is, 5^1 = o, hz^ = o.
The plane cr^ is the transversal plane, which, according to Gauss's
theory, is conjugate, with resf)ect to the first k spherical surfaces of
the optical system, to the Object-Plane a^. After refraction at the
ith surface, the ray (prolonged either forwards or backwards, if neces-
sary) will cross the plane c^ at the point P^iyi, f*). If m denotes the
total number of spherical surfaces, the corresponding image-ray,
emerging from the optical system, will cross the Image-Plane c^ at
the point P^, whose co-ordinates are:
where y^, z^ denote the co-ordinates of the point Q'^, which, by Gauss's
theory, is the image of the object-point P, (or Q^. The magnitudes
y^, z^ can be determined by the approximate formulae of Gauss.
Obviously, the point Q^ will lie in the meridian plane through the
point d, and since Oi is coincident with P„ if the meridian plane
containing the object-point is taken as the xy-plane, we must have:
fi = 2i = 2I = o;
and, hence, f^ = hz^.
374 Geometrical Optics, Chapter XII. [ § 25'
The magnitudes denoted by by^^ bz^ are the correction-terms, ot
aberrations of the yd order, which measure the errors of the image-
By some writers dy^, Bz^ are called the ''TangentiaV and *' Sagittal' "^ ^»
abtrratioHs. rtfspectively, in the GAUSsian Image-Plane <r^. We ma]
^y
alj<> call them the y-aberration and the z-aberration in this plane.
^57. Planes of tiie Pupils of the Optical System. So far as th
MivMiiin^s of the magnitudes 5y^, bz^ are concerned, it is a matter
iu» ^.xuiseviueace what plane Ci is selected for the initial plane of
v>ihci* system ^,or o'-sN-stem) of transversal planes. In all optical i
xii umcuts the aperture of the cone of effective rays is limited by
lain vJiaphragmii or circular openings, called **stops", which are plac:^
vkilh their plaiie* perpendicular to the optical axis and with their c:^^'
iivs ou thiii x\is. E\*en in case there is no such artificial diaphra^^^'
ih\' ruu«i \.>r fastenings of the lenses themselves will act as such \ i
I hat of all the rax-s emitted from an object-point only a certain limi \^,
number succeed in making their way through the entire apparat^::^^^ ^
Whv'u there are se\-eral diaphragms, the effective stop is that (^^^ ^.
v^ hich tvrtuits the fewest rays to pass. This stop may be situated, ^ ^
s\>idiikl^ to circumstances, in front of the entire system or somewhc^^ u^
\^ iihia the s>-stem or e\-en beyond the entire system. It is found to • i^j^
!U\»«it ^.\»aveuient to select the plane Vy so that one of the planes of th^**"^^^.
.'^Hii^ of transversal planes shall coincide with the plane of the eflecr^ ^--#iin
i\c *iop. lu the most general case, when the stop is situated withi:^ ^_ral
ihc optical system, say, between the *th and the {k + i)th spheri< m^ma
nui Ukvs, the plane of the stop will be the transversal plane a^, and ^^^^^^
^\y>\^sx^\\XXK^ will be at the point Af]^ where the optical axis crosses thi^^^^^j
plane. The axial point Af,, whose image produced by the refraction o%^^ _-,
) Ml axial rays through the first k spherical surfaces of the optical system^^^^^^
is Mi,, will determine, therefore, the position of the initial transversal •--^ri,
plane Qx' ^ ^*^ object-rays cross the plane Ci at points lying within
the siKUv which would be covered by a thin circular disc placed lyith
it?» cvata^ on the optical axis at Afj and perpendicular to the optical
axis aiul of such dimensions that, with respect to the first 4 surfaces
\>\ I lie optical system, its GAUSsian image at Af]^ exactly coincided
wiUi I he crtective stop there. This circle around Af, in the plane Ci
luirt Uvn well called by Abbe the Entrance- Pufnl (see § 361) of the
opiical s>stenr. and we shall, therefore, speak of the initial plane 0*1 as
lUc Plane of the Entrance-Pupil". Similarly, all the image-rays will
X iv'o. ihc last transversal plane a^ in points contained within a circle
uvuuvl A#„,. which, with respect to the entire system, is the image,
!»> vivi v^'s theory, of the Entrance-Pupil. This circle is called the
{ 259.) Theory of Spherical Aberrations. 375
Exit-Pupil, and the plane <ri^ is called the "Plane of the Exit-Pupil".
Obviously, by this method of selecting the initial plane Ci we have
the advantage of knowing the greatest possible values which the co-
ordinates 3^1, 2| can have in the case of a given optical system, and,
since the values of 8y^, 8z„^ depend also on the values of y^, «j, this
knowledge will be of service in considering the relative importance of
the various terms in the series-developments.
258. Chief Ray of Bundle. Of all the rays proceeding from the
object-point Pi there is one, which, lying in the meridian plane through
Pi, will, in traversing the medium in which the stop is situated, go
through the centre of the stop. This ray, distinguished as the chief
ray of the bundle, will, in general, cross the optical axis for the first
time at a point Li not very far from the centre Afi of the Entrance-
Pupil. The slope of the chief ray of the bundle of object-rays emanat-
ing from Pi is
ZJlfAPi = 61,
more exactly defined by the following equation :
- AfiPi til
where i>, = AiLi denotes the abscissa, with respect to the vertex Ai
of the first surface, of the point Lj. Of course, under certain circum-
stances the point Li may coincide with Afj, as is often the case.
259. Relative Importance of the Terms of the Series- Develop-
jnents of the Aberrations of the 3rd Order. The maximum value
of yi will be fixed by the limits of the required field of vision, and in
the same way the maximum values of 3^1, «i will depend on the size
of the aperture of the optical system. Thus, for example, in the case
of an optical system of relatively small field of vision, and, on the
other hand, of relatively large aperture, the most important terms in
the series-developments of the aberrations dy^, dz'^ will be the terms
which do not contain y^ at all, that is, the terms 3^1, y^z^, yizl and
ij. Next in importance will be the terms which contain the first
power of 3^1, viz., 3^13^^, yiyiZi and y^sf ; and then the terms which con-
tain the second power of y^y viz., yly^ and y^z^; and, finally, least
important of all for this particular case, the term which contains
yj. In the developments of the aberrations of the 3rd order, where
y„ «! = o, 3^1 and Zi denote the approximate values of the parame-
ters of the object-ray, the ten terms above-mentioned are all that can
occur.
376 Geometrical Optics, Chapter XII. [ § 2
The expressions for the aberrations hy^, hz^, which are developo^^^
by Seidel ^,see Art. 102), enable us to compute the resultant defects
oi the ja\I order of the image of an object-point, and by specializing
the^e gienenil formulae las Seidel himself does), we can ascertain the
nature of the various component defects which go to make up this
resuIrjLttt. Howe\-er. in order to obtain a clear comprehension of these
errors k b be« to follow the plan adopted by Koenig and von Rohr* in
their o^mirjible »ind exhausti\'e treatise on the Theory of Spherical Aber-
r-dtioihk JLnd thus* turst,to develop separately the formulae for each one
ot these st^rvTol Aberrations, and afterwards to give, at the end of the
chancer. SfciUEL's general theory (Arts. 102, foil.). Accordingly, this
tucttK\t will be pursued here also.
tl. r^JI :!^n»3UC.%2. AlXKKATlOS IN THS CaSB WHEN THE ObJBCT-PoINT LiBS ON TBfi
Optical Axis.
4ltt. «l« CmUtACTlK OF A BUHDLB OP REFRACTED RATS EMANATIHG
OiOfiOBALLT ntOM A POIHT ON THE OPTICAL AXIS.
MK LoQfitiidiMl Abemtian, or Aberration along the Optical
Axis^ The Amplest case of all is the case when the object-point lies
^Mi the o(.>ticJil xxis of the centered system of spherical surfaces, so
that the vx>iut designated by P| coincides with Afj, that is, y^ = o*
When the bundle of image-ra>'s is symmetrical about an axis, as is the::
v.\usc when the rdN-^ emanate originally from a point on the optical axis,
vUK* ^.^" the caustic surfaces (§ 46) is a surface of revolution around the
axis uikI u* touched by each ray of the bundle; whereas the other caustic
«urUicx\ in this particular instance, collapses into the segment of the
u.\is wmpriscil between the point M' (Fig. 133) where the paraxial
r<^\s cn.v» the axis and the point V where the outermost rays of the
buiKile meet the axis.* All the rays of the bundle will intersect the
axis at ^KHuts which are comprised between the two extreme points
M aiKi L\ This axial line-segment M'V is called the Longitudinal
.l\7fa/*i>i« of the outermost ray. Let A designate the vertex of the
spherical surface, and let us put AM' = u\ AV ^ xf. If the Spheri-
^ V Kv^knk: uud M. \x>N RoBR: Die Theorie der sphaerischen Aberrationen; bdng
V t^^uv( \ vI>«m;^ ''^ \U^) of ^^ TheorU der optischen Instrununie, Bd. I. edited by
M. \v>\ Kv^iH vBefUn. igo4). This treatise of Messrs. KoBNicand von Robr has been
.>• >us>aumabl« skeivic^ to the author in the preparation of the present chapter of this
* VU iho lvtt\H« in the figure should, as a matter of fact, be written with the subaoript
•A.. iv> auluau' th«it the letters relate to the rasrs after refraction at the last, or mth, surface
i!kx- .> .u-4ik. Hut in «U »uch cases as the one here considered the surface-numerals
s\ .v.Mi 4* r.ulkkUiiiH ii»a,v be conveniently omitted where only one of the surfaces of the
■ s ..s .It t.^ tK'iu^ ucatvsl. ^uce there is no risk of confusion.
^ ..cory ot Spherical Aberrations.
377
cal Aberration along the axis, or the Longitudinal Aberration M'V
is denoted by hu\ we shall have:
hv! = v' - u\
How if ^ =» LAVB denotes the slope of the ray which, in the plane
of the diagram, crosses the optical axis at the point designated by L',
it is evident that hu' is a function of this angle ^'; and, moreover, it
£s also evident that if the function hu' is developed in a series of as-
Pio. 133.
OP BUMDLB OF Rays Stmmbt&ically Situated with rbspbct to the Optical
Axis.
AM*^^, AL'^x/, M'L'^l^, Ary^iy, DB^h, IAUB^V,
ing powers of %\ only the even powers will occur, because for a
r lying in the same meridian plane and symmetrically situated on
^ other side of the optical axis, so that its slop)e-angle is equal to
6^, we shall obtain the same value of the function hv! . If, there-
in, ^ may be regarded as a magnitude of the ist order of smallness,
can write:
hu' = r' - u' = a'^'*, (273)
^*^Bce all the succeeding terms of the series, involving magnitudes of
^^^ orders of smallness higher than the 3rd, are, by the limitations
^f this investigation, to be neglected.
The co-efficient a' is entirely characteristic both of the magnitude
^nd of the nature, or sign, of the aberration hu\ since, for a given
'Value of ^, we can determine hu' , so soon as we have ascertained
also the value of a'. Thus, if a' — o, we have v' = u', in which case
^ say that the system is ^'spherically corrected'' for this ray. Ac-
■ ■.■■>• -.r.': fffect of the other can
- rx ■jcreen is advanced still
s* -i: the centre of the circuli
,-. -ivc. A plane pcrpendicula
■ Tt! rigurc will meet the outs
- 1 'xn these rays cR^iy the cau
■? -iar-tf we shall evidently havi
■ ; 1 :he bundle of rays. By somi
* Tich appears on the screen w
. -;. Circle of Aberration.
r: •.•cer.tv-height at the last spheri
t "uiKile, whose slope-angle is
■ = .AL'B:
•X iifurel mark the position of tl
- -t. ;!(«.• .-thcr ray of the pencil of im
vire oxwses the optical axis. TI;
t<!v«d by 9". Finally, let » dci
rr*vf:-on of this general ray with tl:
..t. 1 .'E I will be the radius of tt;
.vtS" - cottf'):
■-rx'wd aberration-co-cfficient (§
5 262.) Theory of Spherical Aberrations. 379
we obtain :
2^".sin»^"(cot^" - cot^O = ^'' - ^"';
and expanding the trigonometric functions in series/ and neglecting
terms involving powers and products of $', B" higher than the 3rd,
we find :
which IS satisfied by the values B" = B' and B" = - B'\z, The first
of these values corresponds to the maximum value of i represented in
the diagram by the ordinate G'H'; whereas the second value
gives the slope of the ray for which i = i^ is a minimum. The inci-
<lence-height of this ray is, therefore, approximately half that of the
outside ray, but opposite in sign. If this value of B'^ is substituted in
the above equation connecting i and B", we shall find (neglecting, as
Ijefore, powers of 6' above the 3rd) for the radius of the Least Circle
of Aberration:
to = - a'0^/4.
The position on the axis of the point N' can be determined from
tJie fact that N'L' must be equal to — io-cot $'\ hence, to the same
<legree of approximation, we find :
N'L' = a'//4 = M'V/4.
Accordingly, the distance of the Least Circle of Aberration from the
GAUSsian Image-Point M' is equal, approximately, to three-fourths
of the Longitudinal Aberration of the extreme outside ray.
262. The so-called Lateral Aberration. Exactly what point on
the axis is to be regarded as the image of the axial object-point M in
such a case as that which we are here discussing is a question that
cannot be decided by merely theoretical considerations; especially,
too, as there is some diversity of opinion on the subject. In order to
be answered, the matter, as Czapski observes, needs to be considered
rather from the point of view of Physical Optics than from that of
Geometrical Optics. Most optical writers are agreed, however, that
the place probably selected by the eye as most nearly reproducing the
axial object-point is the place of the least circle of aberration. This
* The development of the cotangent in series is as follows:
cot « = I /« -— «/3 — «*/45 — 2X*/945 * '•
I
i
:^ = -a'e'.
r -t Longitudinal Aberration 8u' is
-.:r^4 Aberration By' is of the 3rd
, , a«^ - "^£S ?CaMT7LA FOR THE SPHERICAL
i .. . r « ?iRScr buhdle of rats.
r
: > imanating originally from a po
..■:*: >y:em of spherical surfaces is
j . - -Mis it will be sufficient to inves
-. x Consider, therefore, any ray
'ione containing this ray be tl
•^.■^- -.^itfs. Hence, for this ray not o
> - t jJLse for all the rays of the bt
^ :ii> lerm in the series-developm
%::1 >! the3'J-term (see § 259).
i^ 4- -iw subscript-notation, let us des
ic r* ,\i:h of this ray crosses the optic
_>.^ ^a .. :he tth spherical surface. Emp
.^ A* -? :itix*is OS in § 209, viz. :
* .11'. ZJ5Ci4=^, ZALB =
^AL'B = e\
.^.<^ » iic!^ence and refraction by or, a', n
^^ , ^.raiccon by n, n', we may write the ;
S263.]
Theory of Spherical Aberrations.
381
If, also, My M' designate the points where the path of a paraxial ray
crosses the optical axis, before and after refraction, respectively, at
the *th spherical surface, and if
then (§ 126)
Vloreover,
AM = tt, AM' = u\
"(-;-^)=«'0-.^)---
ML = «« = »-«, M'V = bu' = v' " u'
viU denote the magnitudes of the Longitudinal Aberration of the ray
jefore and after refraction at the spherical surface.
If we neglect all magnitudes higher than those of the second order,
lie approximate values of the slope-angles 6, B\ expressed in terms
)f the central angle ^, are ^ = — f ^/w, B' — — tiplu'. But if, as we
>ropose to do here, we take account also of the terms of the 3rd order,
lie expressions for ^, B' must evidently have the following forms:
^= -
-ip + Aip^,
^'= --,^ + i4V%
u
(276)
'herein the co-efficients A^ A' are undetermined. Moreover, since
a = d + ^, a' =B' + tp,
may expand or, a' likewise in a series of odd powers of <p, as follows:
Jr
a = —<p + Afp^,
ft
Jr
a'- -T<p + A'(p^,
ft
(277)
''here the co-efficients A^ A' have the same meanings as in formulae
^^76).
If X denotes a small magnitude of the ist order, and if we take
^uxount of terms as far as x^^ then
sin jc = jc — x^/6;
and, hence, employing formulae (276) and (277), we have here the fol-
.vumrtntnti t.^pucs. Chapter XII.
[§
cfi^pMUJciiii^ :or ':he sanes of the angles a, a\ 6^
. -J,
-11 ;
-i:i J
(2:
>. -^ ■
^ . » !ui^iiiciiih:s of orders higher than the 3rd,
%•»
,-.A).j'^{y^).
. ..^ - ..:t :r>*. rhree of equations (275) these values of t!^Kie
- 4 -" in. v" =- u' + 8u',
.WH>v. .^^ .irsc of these equations by nfu and the sec
.s^jinjict die first from the second, we obtain:
* ^ V-tii nuj Vtt tt/ Jr^
.. .^ > . > iwotfe^ of the third of the equations above,
.. ^ . , . j< cs4»uK :«iie both of the unknown co-efficients A,
^^ ;^.vs«^ ^vVttv^aient Difference-Notation, whereby t
.x^>*«^^ '^Hf values g, q' of a magnitude before an^
,1^ <^ >^ .,iM»^«J by Af/, we derive the following equation
> V «« « » /
«d
Theory of Spherical Aberrations. 383
since (§126)
n r n nu
u \ nu r nj
e still further simplified as follows:
A^=-Jf»^'J«.A — . (279)
provided we know the Longitudinal Aberration 8u of the given
fore refraction at the spherical surface in question, we may, by
of formula (279), compute the magnitude 8u' of the Longitudi-
jerration after refraction.
From the incidence-point B draw BD perpendicular to the
I axis at -B, and put DB = A, so that h denotes the ordinate of
:idence-point B, that is, the incidence-height of the ray of slope
len, since
h = f-sin <p,
degree of approximation required in this investigation, we may
h = r(p — r -jr; (280)
onsequently, in formula (279) we can put r*^* = A*. If we do
nd if now at the same time we attach to the symbols the surface-
;r in the form of a subscript, noting also that the point L^^ where
y crosses the axis before refraction at the ifeth surface is identical
he point Li_, where the ray crosses the axis after refraction at
— i)th surface, so that
ly write the formula for the ifeth surface as follows:
"^-'^P^--mJl(-'-^--r^). (281)
\bb£'s abbreviated notation:
>4 Oeometrical Optics, Chapter XII. (§265.
-rjSsi '?y TWiixx* of this recurrent formula (281), we can obtain
•!iaii> *tf oiue f :he Longitudinal Aberration 6u^ of the ray after
*r*;a*..«ii ..L lie *.a*t surface of the centered system of m spherical
-^;a*.t^ -k»* :his is done, we proceed now to show.
^y , aieuiux^ :ormulae (276) and (280), we find:
- -«? + (^i*-^)^'= -ttr + (^V-0^»,
(282)
..Ai» >»iict». again introducing the subscripts, and remarking that the
..it^io ceiLuceii by S^ and d'j^^ are identical, we may write, taking
.icv^xHuii oi the terms of the 3 id order:
^J=%^ (283)
't, :h«retore. we multiply both sides of equation (281) by *J, and
li. ii)^ ;^UK time use the relation (283), we shall derive the following
--"«•''(»-.).•
'r, isjw, '\a chis^ formula we give k in succession the values 1,2, • • - ^'
taJ lUu •A>^ther the equations thus obtained, and note also t^^^'
>i,iivv -ih; iHjiiKlle of object-rays is supposed to be homocentric, we t^^
>u.. J*»i - >.\ >*e obtain finally:
^-BxmMa-
{r^^^
u .i>iA AH inula we need to know, in addition to the const^^^. j^.
^.uv I .;vvVi»«i»ie the optical system (refractive indices, radii, th^^ ^
v *^.v xhv \ '.Hilv the position on the axis of the object-point Jf, ^^^.^e
\ .iv<viKx^lvi^ht ik| of the object-ray; for then we can comp^^^^
>v > .*.uv',x .s alt the other magnitudes that occur on the right-ha-^^|^^
^vn -^ 'K vxtuatkm. Practically, the formula is very convenie^^^j^jj
\x *v.>^ ; o\I>iSts the effect on the Longitudinal Aberration bu^ whir ^^jj]
. v>\ xaw Ai vNU^i refraction. For a given axial object-point, it ^^ q[
X s^ ^ V , XNHvttcuUy |x>ssible, by employing a sufficient number ^ ^^^
,iv\,x A^ sXHurtNe s^> that the aberration hu^ = o; the conditic
V x>v\^ ^^
|>i-^-4n^)r°- ^"^"^
Theory of Spherical Aberrations. 385
lust be remembered, however, that the accuracy of this formula
I abolition of the spherical aberration along the axis depends on
agnitude of the aperture of the bundle of rays; for it has been
ed throughout that we can safely afford to neglect the powers
slope-angle 6 higher than the 3rd. Thus, for example, in the
f the objective of a telescope, the aperture of which, although
means negligible, is relatively small, the formula will usually
very high approximation. On the other hand, in the calcula-
f a photographic objective the formula would generally not be
iccurate. In the objective of a microscope the magnitude of
gle 6 is often equal to nearly 90®, and the approximate formulae
erived are not applicable to wide-angle systems at all.
. Abbe's Measure of the "Indistinctness" of the Image. By
of formulae (274) and (282), we find for the Lateral Aberration:
mce:
i.» denotes the length of the object-line perpendicular to the
I axis at JIfi, whose GAUSsian image at M]^ is equal to the value
Lateral Aberration 6y,^ after refraction at the Jfeth surface, it is
it that details in an object at M^ which are separated by an inter-
eater than ei^k will, on account of the spherical aberration, not
' separated in the image formed after refraction at the ifeth
». Thus, according to Abbe, the magnitude denoted by «!,»,
red at the object, affords a convenient measure of the lack of
or *Hndistinctness'\ of the image.
approximate value of the slope-angle ^^ is:
tf^ =^ r = ;
«* «*+i
ence by the Law of Robert Smith (§ 194) :
nihiei,k ^ Khf^'Syl
hf, n^. Ui , / « X
386 Geometrical Optics, Chapter XII. [ § 267.
Thus, from formula (286) we obtain:
'- ' - k'^wy H^X'
(288)
which shows that the ''indistinctness" is proportional to the cube of
the aperture A| of the bundle of object-rays.
In case the object-point Mi is very far away, it will be convenient
to determine the angle €i,« subtended at the vertex i4, of the first
surface by the linear magnitude ^i.^; thus, since
we have:
■—mik)M^A'
and, hence, the angular value of the lack of detail in the image, on
account of spherical aberration, is proportional to the cube of the
linear aperture ftj of the bundle of object-rays. For example, in ^^
case of the objective of a telescope, it is proportional to the cube of ^^
diameter of the objective.
ART. 83. SPHERICAL ABERRATION OF DIRECT BUNDLE OF RATS t^
SPECIAL CASES.
267. Case of a Single Spherical Refracting Surface.
If the optical system consists of a single spherical surface (m =^
we have for the Longitudinal Aberration of the bundle of image-
corresponding to a bundle of object-rays proceeding from the
point M (u ^ AM):
2» \nu nuj
and if we substitute for / its value, viz.:
n'{u' - r)
J-
ru
we obtain:
iu = 3 — -, • (28
znruu
In each of the following three cases the Longitudinal Aberration wi^
be equal to zero:
(i) When tt = w' = o, in which case object-point M and image^^
point M' coincide at the vertex A of the sphere;
§ 268.] Theory of Spherical Aberrations. 387
(2) When u =^ u' ^ r, in which case object-point M and image-
point M' coincide at the centre C of the sphere; and
(3) When nu = nV, in which case:
tt = (w + n')r/nf u' = (n + n')r/n\
and the points Af, M' coincide with the aplanatic points Z, Z\ re-
spectively (§ 207, § 211, Note 3). This latter case is the only one that
may be said to have any practical importance.
For all other positions of the object-point M the point V will not
coincide with M\ The sign of du' will depend on the sign of the factor
I I _n— w/i n +n i\
nu nu n^ Xi" w « / '
^nd for any given spherical surface may be positive or negative de-
pending on the sign of u. If the object-point M is at an infinite
distance, the sign of Su' will depend on that of («' — «)/r. If this
expression is positive, the refracting surface will be a convergent sur-
face, the sign of Su' will be negative, and the spherical surface will be
* 'spherically under-corrected" (§ 260).
268. Case of an Infinitely Thin Lens.
When the optical system consists of two spherical surfaces, we must
ut m = 2 in formula (284). Assuming that the Lens is surrounded
the same medium on both sides, we may conveniently write:
n = nj/nj = njn^,
that in the following discussion n will be used to denote the rela-
index of refraction for the two media concerned. Moreover, in
case of an Infinitely Thin Lens, we have:
u[ = «,,
d we may therefore afford to dispense with the subscripts in the
xTibols Wi and 1*^, and write these u and u\ respectively. Likewise,
shall write: fc = Aj = A,. Under these circumstances, we obtain
the general formula (284) the following expression for the Longitu-
Aberration of an Infinitely Thin Lens:
2 l\ri uj \nui uj Xr^ u J \nu^ uj\ ^ ^'
F'or the case of an Infinitely Thin Lens we shall employ a special
^'^Uuion^ as follows: Thus, let x = i/w, x' = iju' denote the recip-
388 Geometrical Optics, Chapter XII. [ $ 268.
rocals of the intercepts on the axis of the paraxial object-rays and
image-rays, respectively, and let c = i/fj, c' = i/r, denote the curv-
atures of the bounding surfaces of the Lens. Finally, let ^ = i//
denote here the reciprocal of the primary focal length of the Lens.
With this system of symbols the formulae of Chapter VI, Art. 41,
for the Refraction of Paraxial Rays through an Infinitdy Thin Lens
will have the following forms:
^
= (n - i){c
-C),
X
^X + ip,
I
x+(n-
i)c
«i
n
•
(291)
Employing these relations, we can eliminate from formulae (290) the
magnitudes denoted by u|, x' and c\ and thus we shall obtain:
»«•- --f1(,-^)v+(,-^){— <-+-)(-4
+ ^ (c - «) j (n + 2)(C - *) - 2I» j J'
or
,qi)
2 \\n — ij 'Vw— I n — I /^
n n n \
If the object-rays are parallel to the axis {x = o, x' = ^), the im^^^
point M' coincides with the secondary focal point E\ and for ^
special case we obtain :
2(n— i) 1 n <p ^<p n— I \ ^
In the case of a Thin Lens of semi-diameter h and focal length/, whc^
thickness is greatest along the optical axis, one can easily see iiC^^
the geometrical properties of the circle that the thickness of the
is very nearly equal to
2
\r, rj-z^' ^>^-2(n-i)'
and thus for a I^ns of this character the expression within the larg^
§ 269.] Theory of Spherical Aberrations. 389
brackets of formula (293) is the factor by which the thickness of the
Lens has to be multiplied in order to obtain the spherical aberration
along the axis for an infinitely distant axial object-point. If the Lens
is a convergent glass Lens in air (n = 3/2), its thickness is very nearly
equal to **//.
By way of illustration, let us compute by formula (293) the Longi-
tudinal Aberration for Lenses of special forms; thus, we shall find:
(i) In case the first surface of the Lens is plane (c = o) :
E'U^^(-^A'^^' for n = 3/2, E'V^-^-^.
\n - 1/ 2/' ^' ' 2 /
(2) In case the second surface of the Lens is plane (c = ^/(n — i)) :
E'V = -f -Ta y, for « = 3/2, E'V = - ^ — .
n{n— ly 2/ ^' ' 6 /
(3) In case of an Equi-Biconvex Lens (c = — d — ^/2(w — i)) :
r^,jf 4^^ - 4n' - w + 2 h^ 5 A'
^^ 8n(n - i)^ 7' ^^' ""^^^ £'L'=--J.
Assuming, therefore, that the focal length / of each of these Lenses has
the same numerical value, we see that the Longitudinal Aberration is
greatest in the Lens with its plane side turned towards, and least in
the Lens with its plane side turned away from, the object-rays.
260. The next question to be investigated is. What are the conditions
that the Longitudinal Aberration of a Thin Lens shall vanish ?
If, for brevity, the expression within the large brackets in formula
(292) is put equal to Z, we may write the formula for the Longitudinal
Aberration of a Thin Lens as follows :
.2. '2
hu' = -Z. (294)
If ^ s o (that is,/ = 00), we shall have hu' = o. In this case u = u\
r, = r,, so that the two surfaces of the Infinitely Thin Lens are parallel.
This case has evidently no practical interest. It remains, therefore,
to investigate the cases when the function Z vanishes.
We shall assume that we have given a Lens of a definite focal
length, and that the position on the axis of the object-point M is also
given; and, under these circumstances, we are required to determine
the form of the Lens in order that the Longitudinal Aberration shall
be zero; that is, we must ascertain the curvatures c, c' of the two surf-
aces of the Lens. Since c' = c + ^, and since the value of ^ is sup-
posed to be prescribed, the problem, in reality, consists merely in
390 Geometrical Optics, Chapter XII. [ § 269.
finding the curvature c of the first surface. This process of varying
the curvatures of the surfaces without altering the focal length is
called ** bending'' the Lens.
Accordingly, regarding c as the independent variable, and treating
both X and tp as constants, we shall write the function Z in the follow-
ing form :
2 = !H:_V_f4(M2)^^2«.+I^^
n \ n n — I J
3^ + 2 , . 3^+1 , f n_y s , X
For Z = o, we obtain two values of c, as follows:
4(w' — i)x + njzn + i)^ ifc n V^{n — i) Vx + ^) — (4n - iV
^ "" 2(n - i)(n + 2)
and if these values of c are to be real, the expression under the radical
must be positive; that is, for real values of c, we must have:
4«(« + ^) - /^ _ ^y^ > o;
or, since jc + ^ = ^',
Accordingly, we see that a necessary condition that the aberra*^^
shall vanish is that x and x' shall have the same sign; which m^^^
that the object-point and image-point must lie both on the same ^^^
of the Lens. In the practical and more important case when the im^ .
is a real image, it is impossible to abolish the Longitudinal AberraC^^
of an Infinitely Thin Lens.
The condition above may also be put in the following form:
{n- I +Vn{n + 2)]^ x' (n - i -Vn{n + 2)Y
4w — I X 4w — I
Thus, with a glass Lens in air (n == 3/2), it is possible to abolish
Longitudinal Aberration only in case_the ratio x' jxy or u' ju, is co
prised between the values (11 — Vzi)! 10 and (11 + V^2i)/io, th
is, between the values 0.642 and 1.558.
In exactly the same way, by considering Z as a function of x,
treating c and tp as constants, we shall find that in order for Z
§ 270.] Theory of Spherical Aberrations. 391
vanish for real values of x, the Infinitely Thin Lens must have a form
such that
^ _i_£ (n + i)(3w - i) ^
ip n — I <p 4(n — i)
that is, the ratio cf(p must be comprised between the values
gm^ ^
2(n — i) 2(w — i)
For example, for « = 3 fz, the value of c/<p must lie between (2 — V^39)/2
and (2 + '^39) f 2 that is, between — 2.1225 and + 4.1225.
Practically speaking, these results are without value.
270. Since, therefore, it is practically not feasible to abolish en-
tirely the Longitudinal Aberration in the case of an Infinitely Thin
Lens, let us seek now to find the condition that the Aberration shall
be a fninimum.
Equation (295), in which c and Z are to be considered as the vari-
ables, evidently represents a Parabola with its axis parallel to the
Z-axis of co-ordinates and with its vertex at the point:
nUn - i) , n
" 4(w — i) (n + 2) ^ n + 2 ^ ^''
^•" n + 2 * + 2(n-i)(n + 2)^'
(296)
and it is obvious that for a Lens of given "power" (<p) and for a given
position (x) of the object-point M on the axis, the minimum value of
the function Z will be Z = Zq.
So long as xx' = x(x + <p) is not positive, the value of Z^ cannot
be equal to zero; if xx' •< o, then Zq > o. That is, for a real image-
point M' on the other side of the Lens from the object-point M, the
minimum value of Z is positive. In case xx' > o, Zq will, in general,
be negative, and in special cases it may be equal to zero, in agreement
with the results found in the preceding discussion. We need consider
only the case when Zq > o. The minimum value of the Longitudinal
Aberration is:
^0 = r— Zq.
For an infinitely distant object-point (x = o, x' = ^), the curvatures
392 Geometrical Optics, Chapter XII. [ f 271.
of the Lens-surfaces for minimum aberration are:
^; (3f = o);
^0 =
«(2« + l)
4 = ;
4
2W* -
- » -
-4
2{n - i)(n +
2)^'
+ 2)
and the
minimum aberration
is:
(E'L% = -
n(4n-
I)
*v
8(»
- 1)*0
* + 2)
For w = 3/2, we find: Cq = 12^/7, {E'L% = — is*V/i4; and for
w = 2 (diamond), Cq = S^/4, (E'L% = — 7AV/16. The minimum
value of the Longitudinal Aberration of a Diamond Lens is very much
less than that of a Glass Lens of equal focal length. And, generally,
for values of n greater than unity, it is easy to show that the minimum
value of the aberration decreases with increase of n.
When jc = o, we have:
, _ njin + i)
^••^•""2n*-n-4*
which, for w = 3/2, gives cJcq = — 6. Thus, with an infinitely dis-
tant object-point a biconvex glass Lens has the least Longitudinal
Aberration, viz., — I5AV/i4» when the curvature of its first surface
is six times as great as the curvature of its farther surface.
271. Case of a System of Two or More Thin Lenses.
If the optical system consists of a system of m Infinitely Thm
Lenses, with the centres of their surfaces ranged along one and the
same straight line, we can determine the Longitudinal Aberration by
means of the formula (284). We shall employ here a notation entirely
similar to that used above in the case of a single Lens (§ 268); but it
should be noted also that in the following formulae the subscript attached
to a symbol will indicate, not, as usually, the ordinal number of the
spherical refracting surface, but the ordinal number of the Lens to
which the symbol has reference. The bundle of object-rays is sup-
posed to emanate from an object-point Jlfi on the optical axis, and the
point where the paraxial image-rays cross the axis will be designated
here by ilf^^, and, similarly, the point where the outermost ray of the
bundle of image-rays crosses the axis will be designated by L'^. ^^
the Longitudinal Aberration of the system of m Lenses, we obtain*
KL: = -^'fki<p^„ (297)
2«« *=1
§ 272.1 Theory of Spherical Aberrations. 393
where
\Wa-i/ \ n^fe- I w^fe- I V
In this formula w^ denotes the relative index of refraction from air
into the medium of the ftth Lens; Cj^ and ^^ denote the reciprocals
of the radius of the first surface and the primary focal length, re-
spectively, of this lens; x^ denotes the reciprocal of the intercept
Ajjdj^f where Mf, designates the point where paraxial rays cross the
axis before entering the ifeth Lens; A^ denotes the incidence-height of
the outermost ray at the feth Lens; and, finally, u^ = il^ilf^ is the
intercept of the paraxial image-rays.
If the distances that separate the Lenses are all negligible, so that
we have 2l System of m Thin Lenses in Contact^ the formula becomes:
Ki: = -^*i:%^.. (299)
Here the relation x^^^ = ^a + ^a will also be of service.
272. We may consider somewhat more in detail the special case of
^in optical system consisting of Two Infinitely Thin Lenses in Contact,
The condition that the Longitudinal Aberration of a combination of
^his kind shall vanish is:
If the focal lengths of the two Lenses, or their reciprocals ^j, ^2» ^ire
assigned, and if also we know the reciprocal jc^ of the distance u^ of
t:he axial object-point from the first Lens, then, since ^2 = ^1+ ^n
the analytical condition for the abolition of the spherical aberration
^11 be an equation of the 2nd degree in c, and c^. We may, therefore,
choose arbitrarily the value of one of these two magnitudes; in which
case there will always be two values of the other, real or imaginary,
which will fulfil the above requirement.
Since, therefore, we have here two arbitrary variables c, and c,
and only one equation to determine them, we may impose one other
condition. For example, a very natural idea would be to make the
curvatures of the second surface of the front Lens and the first
surface of the following Lens identical {c\ = c^ , so that the two Lenses
could be cemented together. However, if the two Lenses are made of
different kinds of glass, with unequal co-efficients of dilatation, a com-
396
Geometrical Optics, Chapter XII.
[ § 273.
Ig hjh, = 9-997S2SS +
Ig ^s = 1.8426804 +
clg ttj = 8.1572385 +
Ig A3/A1 = 9-9974444 +
Igw^ = 2.3518454 +
clg ttj = 7.6462272 +
IgV*! = 9-9955170 +
The following scheme exhibits the process of the calculation:
k = i
nk-~\Uk
clg tift
clg Hk-l
clg n'k^iUk
Clgtijk
ClgMJk
clg n'kUk
i/niui
— I lni~iUk
_i i_
nM
clgn
.In
— i/m
i/rk—i/ut
Ig(i/r» — i/uk)
Igni-i
IgJk
lgA(l/»|M)»
Ig ihk/hi)^
IgPk
7.7544706+
9.8197020+
7.5741726+
+0.0037512
0.0000000
+0.0037512
8.2232988+
8.2232988+
0.0000000
8.2232988+
k = 2
7.7569451 +
9.8197020 +
7.5766471 +
8.1572385 +
0.0000000
8.1572385 +
+0.0143628
— 0.0037727
+O.OIO59OI
8.0450343 -
—o.oi 10926
+0.0057I4I
—0.0168067
8.2254825 —
0.1802980+
6.4465976+
7.5741726+
0.0000000
4.0207702 +
8.4057805 —
6.8II56IO+
8.0249001 +
9.9901020+
4.8265631 +
* = 3
8.1573196+
0.0000000
8.1573196+
7.6462272+
9.7926080+
7.4388352 +
+0.0027469
—0.0143655
— 0.0116186
8.0721 166 —
— 0.01 18064
+0.0143655
—0.0261 719
8.4178352-
0.0000000
8.417835a-
6.8356704+
8.0651538 —
9.9897776+
4.8906018 —
* = 4
7.6481546'
9.7926080-
7.4407626-
7.7541648-
0.0000000
7.7541648-
+0.0056776
—0.0027591
+0.0029185
7.3872161 +
+0.0024390
+0.0044479
—0.0020089
7.3029583-
0.2073920+
7.5103503-
5.0207006+
7.4651597+
9.9820680+
a.4679283 +
2P*
+ 104.899
+ 670.754
+ 2.937
+ 778.590
- 777.324
= + 1.266
10"
10"
lo-
10
10
-8
10
Ig 2Pt = 2.1024337 +
Ig {hjh^y = 0.0089660 +
lg*J= 1.5563025 +
h^\ =4.4916704 +
clg 2»4 = 9.6989700 +
7.8583426 +
Accordingly, we find :
M\L\ =: — 0.0072 inches.*
^ Taylor, computing the Spherical Aberration by a formula equivalent to the one
employed by us, obtains a different value and one which agrees very closely with the
exact value. But there appears to be a numerical error in his calculation of what he caOs
the ** first parallel plate correction".
§ 274.] Theory of Spherical Aberrations. 397
It will be perceived that the value of the longitudinal aberration
Af^L^ thus obtained is in fact rather more than twice as great as the
exact value obtained in Art. 67 by the rigourous process of trigonomet-
ric computation, and at first sight it might appear, therefore, that the
approximate value was utterly unreliable. However, the two values
are of the same order of magnitude, and a little reflection will convince
anyone that in this particular example, at least, we have no right to
expect an agreement between the two values beyond the second place
3f decimals. At least one of the values of $1 is very nearly equal to
5**, and if we bear in mind that when we use the formula of the first
approximation we are neglecting all terms involving the powers of
this angle above the second, we can easily see that the agreement
ibove in the first two figures to the right of the decimal-point is all
Jiat we could look for here.
In order to find Abbe's measure of the Angular Value of the Lack
>f Detail in the Image on account of the Spherical Aberration, we
Droceed as follows:
IgSP* = 2.1024337 +
3 Ig *i = 2.3344538 +
clg 2ni = 9.6989700 +
lg«i.« = 4-1358575 -
Phis angle is expressed here in radians. It will be found to be less
than o'.3.
ART. 85. CONCERNING THE TERMS OF THE HIGHER ORDERS IN THE
SERIES-DEVELOPlftENT OF THE LONGITXTDINAL ABERRATION.
274. The formulae derived in Art. 82 were based on the assumption
that we could put » — « = a^; thereby in the series-development of
the expression for the Longitudinal Aberration neglecting all the terms
after the first. So long as the slope-angle d is relatively small, this
procedure is fairly justified, and even though the formulae thus ob-
tained cannot claim to be entirely accurate, they will often enable us
to compute very approximately the magnitude of the Spherical Aber-
ration. Applied to optical systems of relatively narrow aperture, the
formulae will be found to be extremely serviceable in so far as they
exhibit clearly the effect that will be produced by a variation of any
>ne of the factors (radii, intervals, etc.). that are involved in the
problem: so that the optical designer, instead of having to grope his
Biray by means of tedious trial-calculations, can proceed methodically
to make such alterations as he sees will tend to diminish the Spherical
398 Geometrical Optics, Chapter XII. [ § 275.
Aberration. Especially, in the design of the Objectives of Telescopes
— a problem which ever since the time of Galileo has engaged the
attention of some of the greatest mathematicians of the world—
these approximate formulae have proved to be of the greatest value.
If the Longitudinal Aberration 8u is developed in a series of ascend-
ing powers of one of the variables a, 6, tp or A, it is obvious that the
greater the relative magnitude of this variable, the more terms of the
series will it be necessary to take account of. Thus, provided the
slope-angle d is not too great, it may suffice to take account of only
the first two terms of the development, and then we may write:
5tt = o^ + be*.
The development of the formulae for the co-efficients a and J, by
Abbe's Method of Invariants, is given by Koenig and von Rohr in
their treatise on Die Theorie der sphaerischen Aberrationen} The re-
current formula obtained in this way for the aberration-co-eflScient of
the second term, viz. 6^, is not too complex to be often very service-
able in the practical design of optical instruments; but the co-eflScients
of the succeeding terms of the series lead to exceedingly complicated
algebraic expressions, and are not usually of much value on this ^'
count, especially also as we begin to encounter well-nigh insurmoi^^^'
able numerical difficulties in trying to evaluate by means of tl>^
expressions the radii of the spherically corrected system. In ca^^ .
is necessary to take account of these higher terms, the only s^
factory procedure is to resort to the laborious method of trigonon<^ ,^
rical calculation of the ray-paths. After a number of trials it is neai^
always possible by suitable alterations of the radii, thicknesses, e
to contrive so that some selected ray shall emerge from the system
as to cross the optical axis approximately at the same point as
paraxial image-rays; and although this by no means implies th-^
any other ray of the same meridian section will also intersect th^
axis at this point, it is usually a first step in the direction of dimir^
ishing the Longitudinal Aberration. The method is very fully
plained, with a great number of actual numerical illustrations, i
Steinheil & VoiT*s Handbuch der angewandten Optik (Leipzig, 1891)^
275. The Aberration Curve. If the Longitudinal Aberration hu o^*
a ray of incidence-height h is developed in a series of ascending powers
of hy and if we take account of only the first two terms, we may writer^
bu = ah^ + bh*;
»This is Chapter V of von Rohr's Die Theorie der optischen InsUrumtnU (Berlin, 1904);
see pages 217-219 and pages 235-239.
5 275.]
Theory of Spherical Aberrations.
399
where a and b are co-efficients independent of the variable h.^ If a
and b both vanish, the Longitudinal Aberration will be zero for all
values of i, and in such a case (which never actually occurs) the optical
system would be entirely free from aberration for the axial object-point
in question.
If we suppose that the co-efficients a, b have opposite signs, we
shall find that the above equation represents a curve, of the general
.X
PlO. 134.
ASBRRATIOlf-CURVB : Casb OF Undbr-Cor-
RBCTION.
Fio. 135.
Aberration-Curvb : Casb of Ovbr-Cor-
RBCTION.
form shown in Figs. 134 and 135, which is symmetrical with respect to
the jc-axis, and which is tangent to the i-axis at the origin. This
curve is called the Aberration Curve. For the value du = o, we obtain:
i = g = zfc V - a/b;
consequently, the ray whose incidence-height is equal to g will cross
the optical axis at the point M where the paraxial rays converge, and
the system is, therefore, said to be spherically corrected for this ray.
For all values of h comprised between A = o and h == g, the sign of
Bu remains unchanged, so that all the intermediate rays will be either
spherically under-corrected {du < o), as in Fig. 134, or spherically
Dver-corrected (du > o), as in Fig. 135.
Moreover, we have a maximum (or minimum) value of the Longi-
rudtnal Aberration du at the origin and also at the points whose ordi-
lates are:
i = J = d: ;/— a /lb,
* These co-effidents a and b are, of course, not the same as the co-efficients denoted
^ these same letters in the development of m in a series of ascending powers of ^.
400 Geometrical Optics, Chapter XII. [ § 276.
The absolute value of the Longitudinal Aberration will be greatest,
therefore, for the ray whose incidence-height is j = g/Vi, and this
value is nearly equal to — a^l\b. The smaller this greatest value is,
the more nearly will the system be spherically corrected.
Without knowing the values of the co-efficients a and 6, the Aberra-
tion Curve can be plotted by qalculating by the trigonometrical for-
mulae the values of u corresponding to given values of the incidence-
height h, and by practical opticians this method is used to exhibit
graphically the performance in respect to spherical aberration of the
optical system as finally completed.
Concerning the Choice of a Suitable Aperture for the Objective, the
question arises. Which ray of the bundle shall be "corrected " so as
to cross the optical axis at the point where the paraxial rays converge?
According to Gauss,* if H denotes the radius of the aperture of the
objective, we should choose for this purpose the ray for which
A = g = H 1/6/s.
The value
A = g = HV1I2
has also been recommended as a suitable value of the inddeiice-
height of the corrected ray; in this case the working part of ^
spherical refracting surface will be divided by the circle of radium I
into two equal zones, so that half of the refracted rays will be und^^*
corrected and half will be over-corrected.
III. The Sinb-Condition. (Optical Systems of Wide Aperture and Small ¥t^^
OF Vision.)
ART. 86. DERIVATION AND liBANINO OF THE SINE-CO NDITIOH.
276. We have seen that it is possible to design an optical syst^^
of centered spherical surfaces which for a pair of conjugate axial poii^^
is free, or practically free, from spherical aberration; so that to
homocentric bundle of object-rays proceeding from a point M on xf^
optical axis there will correspond a homocentric bundle of image-ra]
with its vertex at the GAUSsian image-point M'. If the optical S3
tem consists of a single spherical refracting surface, it will be recalled
that it was the pair of so-called Aplanatic Points Z, Z' that were thu ^
characterized by the property that to an incident chief ray crossing
the axis at Z at any angle B corresponded a refracted ray crossing th^
1 See Gauss's Letter to Brandes. given in Gehlers Physik, Woerterbuchd^p^z, i83i)«^
Bd. vi., I. Abt.. S. 437* This letter is quoted at length in Czapski's Tkeorit dtr apUscken^
InstrumenU (Breslau, 1893), p. 96.
S 276.] Theory of Spherical Aberrations. 401
axis at the conjugate point Z' (§207). But this was not the only
characteristic of this remarkable pair of points, for we found, also
(§211, Note 3), that the slope-angles ^, %' of the incident and re-
fracted rays were connected by the relation:
ny sin B = n'y* sin B\
xrhere y' {y = Y denoted the Lateral Magnification of the imagery by
neans of paraxial rays with respect to the pair of conjugate axial
x>ints Z, Z'. If the relation between the Object-Space and the Image-
>pace were a coUinear relation (as it would be if all the rays concerned
irere paraxial rays), the slope-angles ^, B' would be connected by the
^w of Robert Smith (§ 194), viz.:
ny tan % = n'y' tan B'\
)ut, since for finite values of 0, B' these two equations cannot both
DC true at the same time, it is manifest that the correspondence by
means of wide-angle bundles of rays between the Aplanatic Points
of a single spherical refracting surface is not the same kind of corre-
spondence as we have in the ideal case of optical imagery. Here is
a matter, therefore, that requires to be investigated.
The mere fact that an optical system has been so contrived that
for a pair of conjugate axial points M, M' the Spherical Aberration is
sensibly negligible, by no means implies also that the system will be
free from aberration for any other object-point, for example, for a
point Q very near to M, If the aperture of the system is so narrow
that the rays which are concerned in producing the image may be
regarded as altogether paraxial rays, we know that to an infinitely
imall object-line MQ perpendicular to the optical axis at M there will
x>rrespond, point by point, an infinitely small image-line M'Q' per-
pendicular to the optical axis at Af'; but, in general, if the incident
ays which come from the object-point Q constitute a wide-angle
lundle of rays, only those rays which proceed very close to the axis
nil emerge from the system so as to meet in the corresponding Gauss-
dSi image-point Q^, Even in those cases where the Spherical Aber-
ation with respect to the axial points Af , M' has been most completely
bolished, the points of the image which are not on the axis will appear
o blurred and indistinct that the diameters of their aberration-circles
le actually comparable in magnitude with their distances from the
ads. According to Abbe,' the explanation of this indistinctness is to
^ E. Abbb: Ueber die Bedingungen des Aplanatismus der Linsensysteme: Sitzungsber,
er Jenaischen GeseUschafi /fir Med. u. Naiwrw., 1879, 129-142; also, GesammelU Alh
^ndlungen, Bd. I, 213-226.
27
402
Geometrical Optics, Chapter XII.
[§277.
Fig. 136.
SlNE-CONDITION.
be found in the fact that the images of the object-line MQ (Fig. 136)
produced by the different zones of the spherically corrected system
have different magnifications; and, thus, although all these images
will lie along the same line perpendicular to the optical axis at M\
being of unequal lengths, they will overlap each other and produce
therefore a confused image.
Q ^^ Xz^' ^f th^ angular aperture of
the objective is pretty
large, the differences in
these magnification-ratios
may amount to as much as
50 per cent, or more of the
lateral magnification pro-
duced by the central or pa-
raxial rays. Evidently, under such circumstances there will be no
imagery at all in any practical sense. The problem consists, there-
fore, in finding the condition that the magnifications of all the differ-
ent zones of the objective shall be equal to each other, that is, equal
to the magnification
F = M'QUMQ
of the imagery by means of paraxial rays.
277. Consider an object-ray u proceeding from the axial object-
point M to which corresponds an image-ray v! crossing the opti^^^
axis at the point M' conjugate to M\ and let B, 6' denote the do^'
angles of this pair of corresponding rays. Since the optical syst^'J
is supposed to be spherically corrected with respect to the points ^'
Af', to an infinitely narrow bundle of object-rays whose chief ^^r
is u will correspond an infinitely narrow homocentric bundle
image-rays whose chief ray is «'; so that the I. and II. ima-^
points coincide with each other at the axial point Af '. We saw (§ 2
that within the infinitely narrow region of space surrounding
**mean" incident chief ray in the object-space and the correspond^"
emergent chief ray in the image-space, there was a coUinear coi
spondence between the plane-fields x, x' of the Meridian Rays a
also between the plane-fields 5r, x' of the Sagittal Rays; of such
character that to an infinitely small object-line MV lying in the plai:^
of the meridian section and perpendicular at M to the "mean" inc^^
dent chief ray u there corresponds an infinitely small image-line M'V^
in the same plane and perpendicular at M' to the emergent chief ra]^
w'; and, similarly, to an infinitely small object-line MW lying in th^
277.] Theory of Spherical Aberrations. 403
lane x of the pencil of Sagittal object-rays and perpendicular at M
-) the "mean" incident chief ray u there corresponds an infinitely
nail image-line M'W' in the plane w' of the pencil of Sagittal image-
lys and perpendicular at M' to the chief image-ray «'.
We shall use the symbols
F, = M'V'/MV, F, = M'W'/MW,
) denote the lateral magnifications of the Meridian and Sagittal Rays,
jspectively. The line-elements MW and M'W are perpendicular to
le optical axis at M and M\ respectively; but the same thing is not
•ue with respect to the line-elements MV and MV. If in the
leridian plane we draw VR^ V'R' perpendicular at F, F' to MV,
W and meeting in i?, R' the axis-ordinates erected at M, M\
ispectively, so that
MV = AfiJ-cos B, M'V = JIf' iJ'-cos ^;
len
M'R' .. cos B
= F
MR " cos B
/»
ad in order that the image at M' of a plane element perpendicular to
le optical axis at M shall be identical with the GAUSsian image, or
le image produced by means of the central (paraxial) rays, we must
M'^ _ M'W' _ M'Q'
^^ " 'mw ~ ^Q *
lat is,
'*cosr "
r all values of the slope-angle 0.
If
^-"dx' ^-"dx'
-note the angular magnifications, or ''convergence-ratios", of the
^cident and emergent pencils of Meridian and Sagittal Rays, respect-
''^ly, then, since the formulae which were deduced in the case of
•ollinear Imagery are applicable here, we have (see Chap. VII, § 179,
•^ Chap XI, § 246) the following relations:
404 Geometrical Optics, Chapter XII. [ § 277.
where n and n' denote the refractive indices of the media of the inci-
dent and emergent rays, respectively.
Let us consider, first, the Imagery in the Plane of the Meridian
Section of the infinitely narrow bundle of incident rays whose chief
ray is u. Obviously,
and from the above relations we obtain:
M'R' _ ncosOde _ ''n-disinS)
MR "n^cose'dS'^n'-disind'y
This equation shows that the lateral magnification perpendicular to
the optical axis at the points ilf , M' produced by the Meridian Rays
depends on the slope-angle 0 of the chief incident ray u; and, hence,
the condition that this magnification shall have the same value for
all values of the slope-angle 6, between the value ^ = o and the value
of 6 for the edge-ray is:
n-d (sin 6) _ _^
and, since this equation must be satisfied by all values of 9, ^f ^^'
eluding very small values, it may be written:
sm 0 n
In the next place, we proceed to consider the Imagery of the 5ag**7^
Rays of the same infinitely narrow bundle of rays. The value of
angular magnification in the Sagittal Section may easily be found
imagining the figure to be rotated about the optical axis throug"^^
very small angle, in which case the angles between the initial ^^
final positions of the chief incident and emergent rays u, u' will be
angles dX, d\' whose ratio d\'/d\ is equal to Z^. According to form_
(251) of Chap. XI and formula (185) of Chap. I X, we have for the 1^
spherical surface:
- _ ^* _ ^* _ sin^;
and, since
n
2. = n z,. »,
§ 278.] Theory of Spherical Aberrations. 405
we shall find :
— ^ sin B'^ sin 6'
* Sin 0| sin B
since here we write B and B* in place of ^i and B'^^ respectively.
The lateral magnification Y^ of the the Imagery by means of the
Sagittal Rays must be equal to the lateral magnification Y of the
imagery by means of the Paraxial Rays; and, hence, since
F^.Z, = Yl^ = nln\
we obtain here also:
sin ^ _ ^ V
sin ff n
as the condition that the magnification of the Imagery by means of
the Sagittal Rays shall be constant and equal to that by means of the
Paraxial Rays; and this condition is seen to be precisely the same as
was found above for the Imagery by means of the Meridian Rays.
It will be observed also that it is likewise identical with the character-
istic relation which we found to be true always in regard to the pair
of aplanatic points of a single spherical refracting suriace (§211,
Note 3).
The law here derived, known as the Sine-CondUian, is one of the
most important of the valuable contributions of Abbe^ to the theory
of Optical Instruments. It may be stated as follows:
The necessary and sufficient condition that all the zones of the spheri-
cally corrected optical system shall produce equaUsized images at the
axis-point M\ conjugate to the axial object-point M, is that, for all rays
traversing the system, the ratio of the sines of the slope-angles of each pair
of corresponding incident and emergent rays shall be constant; that is,
sin 6 /sin 0' = constant.
The value of this constant, as we see from formula (300), is n'Y/n.
278. Other Proofs of the Sme-Law. The so-called Sine-Condition
^is enunciated by Abbe, in 1873, for the special case of a centered
sBystem of spherical refracting surfaces might have been seen to be
^E. Abbb: Bdtraege zur Theorie des Mikroskops und der mikroskopischen Wahr-
»%ghmung: M. Schultzbs ArchivfUr mikroskopische Anatomie, IX (1873), 413-468. Also,
^I^esammeUe AbhatuUungen, Bd. 1. 45-100. See also paper entitled: Ueber die Bedingungen
^Ics ApEanatismus der Linsensysteme: SUzungsber, der Jenaischen GeseUschaft f&r Med. li.
-^aiurm., 1879. 129-142; reprinted in Carls Repertorium der Exper.'Phys,, XVI (i88o),
393-316, and in Gesammelte Abhandlungen, Bd. I, 213-226.
406 Geometrical Optics, Chapter XII. [ § 278.
contained in a far more general law of Clausius's^ based on the
Second Fundamental Principle of Thermodynamics; which may be
stated thus:
If the energy radiated by an element of surface Ar, in a medium of
refractive index w, by a bundle of rays of solid angle dw, is transmitted
entirely to an element of surface d<r', in a medium of refractive index
w', by a bundle of rays of solid angle rfw', then we must have the fol-
lowing equation:
n^'cosd'dw dtr'
n' •cos^'^Ai)' da '
where B, B' denote the angles between the chief rays and the corre-
sponding surface-normals.
Applied to the case of an optical system of centered spherical surf-
aces, to the axis of which the surface-elements (2<r, dc' are supposed
to be perpendicular, this equation is easily reducible to the form given
by formula (300). For in this special case the magnitudes ^, ^
evidently denote the slope-angles of the incident and refracted rays,
and
d(a sin B'dB
Ay'^sin^'-eW"
so that Clausius's equation becomes:
n^'djsiri'B) da'
n'''d{sm'B')'' da '
and, since d<r'/d<r = F', we obtain by integration:
sing _^Y
sin B n
Applying the Law of the Conservation of Energy to the Radiat^^
of Light, Helmholtz* has given also another mode of deducing Abb^ ^
Sine-Condition, which is interesting, inasmuch as this important ^
suit is thus obtained from still another point of view.
Finally, let us mention here the extremely simple and elegant pi
' See Browne's EngUsh Translation of Clausius's Mechanical Theory of Heat (Londo^^
1879), p. 321. The law of Clausius's here referred to was first published in the cd^*
brated paper, Die Concentration von Waerme und Lichtstrahlen und die Grenzen ihrc^
Wirkung: Pocg. Ann,, cxxi. (1864), S. i.
' H. Helmholtz : Die theoretische Grenze fUr die Leistungsfaehigkeit dcr Mikro-^
skope: Pogg. Ann,, Jubelband, 1874. 557-584. See also WissenschafUUhe Abhandlungem^
II, p. 185.
\ 279.] Theory of Spherical Aberrations. 407
of the Sine-Law published by Mr. Hockin/ which is based on the
general law of the equality of the optical lengths (§ 38) of all the ray-
paths between the pair of conjugate axial points Af, M' for which the
system is assumed to be spherically corrected.
ART. 87. APLANATISM.
279. We must explain here the meaning that is to be attached
to the term ''aplanatic'\ as it is employed by Abbe and modern
writers on Optics. Formerly, this word was applied to an optical
system merely to mean that it was free from spherical aberration,
and this is the sense in which the term is used by Coddington,
Herschel, etc. But, according to Abbe, in order for an optical
system to be aplanatic, it must fulfil each of two requirements,
viz.: (i) It must be free from spherical aberration for a pair of
conjugate axis-points M, M'\ and (2) The sine-condition must also
be satisfied for this pair of points My M'. Thus, the aplanatic pair
of points Z, Z' of a spherical refracting surface are rightly so-
called, because not merely are these points free from aberration,
but, as we have seen, they fulfil the Sine-Condition also. On the
other hand, the focal points of a reflecting ellipsoidal surface are
not aplanatic, because they do not satisfy the Sine-Condition, and the
same observation applies also with respect to the infinitely distant
ixial point and the focal point of a parabolic reflector.
Accordingly, the Aplanatic Points of an optical system are the points
m the axis for which the spherical aberration is abolished^ and which
it the same time satisfy the Sine- Condition.
Abbe* has described a very ingenious and simple mode of testing
lie aplanatism of a lens-system; consisting in viewing through the
jystem a certain sheaf of concentric hyperbolae, the plane of the object-
figure being placed perpendicularly to the axis with the common centre
It the proper distance from the aplanatic point; which should yield as
image two sheaves of mutually perpendicular, equidistant parallel lines
(see'§ 291). By means of this device. Abbe has investigated the older
typesof microscopes, and hehasshown that, long before the publication,
* Charles Hockin: On the estimation of aperture in the microscope: Joum, Royal
Uic. Soc., (2), IV (1884I. 337-346. See also J. D. Everett's note on Hockin's proof of
the Sine Condition. Phil, Mag., (6), IV (1902). p. 170. Hockin's Proof of the Sine-Con-
dition will be found given also in the 9th edition of Mueller-Pouillet's Lehrlmch der
Pkysik, Bd. II, Optik, and in Drudb's Lehrbuch der Opiik.
' E. Abbe: Ueber die Bedingungen des Aplanatismus der Linsensysteme: SUzungsher,
ier Jenaischen Cesellschafl fUr Med. u. Naiurw., 1879, 129-142; also. GesammeUe Ab^
handlungen, I, 213-226; also, reprinted in Carls Rep. der Exper.'Phys., XVI (i88o).
303-316.
.^vfumcmcal Optics, Chapter XII. [ § 280.
-.. . .It :?uie-<-\)mlition, microscope-designers, without knowing
.-. ... ::t..ic ^z Icsb perfectly fulfilled this essential requirement
.^ :ic ^L>oucioa of the spherical aberration. As Lummer'
-s.. ..^^, > ^ jiiiy another of the many instances in which correct
.s.- .;- .^^^ ^ie\:ciieii theory.
^r. A«. ras suix-coirDrnoN in the focal planes.
N<* ■ u:» jeeu pointed out (§276) that the imagery which we
. ..« .witii wiie 5iae-Condition is fulfilled is not governed by the same
.»o ..> vc .:ave ia the case of Collinear Imagery. This diflference is
.,. .. >^ iAin^L> iiiciuifest, for example, if the aplanatic pair of points are
V. ...LiLivci> aisuiac point of the optical axis and one of the Focal Points
i*. . i-'»-^»>^^' system; as we shall proceed to show. Let MB be an
. V..UW .A> i.>rucetding from the axial point M and meeting the first
.,. o.v.(. ^i :iie »v6tiem at By and let us put:
BM --h ^AMB = By
.-x»c i ^.icsi^ndtes the vertex of the first spherical surface. H *
. .^ w;.*^ .ik: incidence-height of this ray at this surface, then
*
'^iv^*.N.vv:4* t A - FAf denotes the abscissa of the object-point Af V«^ .
.,>^xv . .u '.ihj Primary Focal Point F, then (see Chap. VII, § r T.
iv UV4^ tuo^uilicatioa of the imagery by means of paraxial ray^ ^
X
^% Is. V . UC4WW* the Primary Focal Length of the optical system. A^^"
'' n
.. s,.v a<?u*H«* the Secondary Focal Length of the system, evidently^
n X
• I, -4 ,uvvhci.Mirof aplanatic points of the system, the Sine-Condi- ^
s> Vw«w1'AkJI\^u-i*st*9 Ltkrbuch der Physik, Bd. II, Optik, neunte Auflage, Art.
§ 281.] Theory of Spherical Aberrations. 409
tion expressed by formula (300) may be put in the following form:
h le'
sin B' X '
And if we suppose now that the object-point M is the infinitely distant
point E of the optical axis, and, consequently, the image-point M'
coincides with the secondary focal point E\ then / = x = 00, in which
case we find :
Similarly, for the case of an infinitely distant image-point f cor-
responding to an object-point at the Primary Focal Point F, we should
obtain:
sinO •''
If, therefore, supposing that the aplanatic pair of points is the pair
JS, £', to which the first of these two equations applies, we describe
around the Secondary Focal Point E' as centre a sphere of radius e\
ill the points of intersection of the parallel object-rays with their cor-
"esponding image-rays will lie on the surface of this sphere, whereas
n the case of Collinear Imagery, these points of intersection of the
ncident and emergent rays all lie in the Secondary Principal Plane
vrhich touches the above-mentioned sphere at its vertex.
ART. 89. ONLT CHE PAIR OF APLANATIC POINTS POSSIBLB.
281. When an optical system is so contrived that for a certain
lair of points ilf , M' on the optical axis not only is the spherical aber-
a.'tion abolished but at the same time the Sine-Condition is fulfilled,
£at element of luminous surface placed normally to the axis at M
rill be distinctly delineated as a flat surface-element at M' by bundles
f rays of any angular width (not exceeding the angular aperture of
t»^ system) : but it by no means follows that the system will give at
^' a distinct image of a plane area at M of finite dimensions; nor,
wiced, that it will produce such an image even of an element of surface
It is situated at any other place on the axis. In fact, an optical
ystem cannot have even two pairs of adjacent aplanatic points; for
f this were possible, the system would have to be spherically corrected
^r both pairs of points, and this requirement, as we shall show, is in-
j^^nipatible with the condition that either of the two pairs of points
^ aplanatic.
410 Geometrical Optics, Chapter XII. [ § 281.
In the diagram (Fig. 137) Af, Af' are supposed to be a pair of apla-
natic points of the optical system. A ray MB emanating from the
object-point M and inclined to the axis at an angle 0 will, after trav-
ersing the system, emerge so as to cross the axis at the image-point
JIf ', the slope of the image-ray being denoted by B'. This pair of
corresponding rays may be regarded as the chief rays of two infinitely
FIO. 137.
An Optical System can bavb only one pair of Aplanatic Points Af, At*
narrow pencils of corresponding Meridian Rays; let /' designate the
position on the emergent chief ray of the Secondary Focal Point of
this pencil of Meridian Rays (see §§ 235, 246). The image Jlf' -R' of
an infinitely small object-line MR perpendicular to the optical axis at
M will be determined by constructing the path through the system of
a ray proceeding from R parallel to MB which will emerge in adired?-
ion very nearly the same as that of the emergent chief ray, and whid^
will intersect this ray at /', and which, by its intersection with ti^^^
normal to the optical axis at Af ' will determine the image-point ^
corresponding to R, Let P, P' designate the points where this r^
crosses the axis before and after refraction through the optical syste
The pair of axial points P, P' are adjacent to the aplanatic pair
points M, M'\ and, therefore, let us write:
MP = dx, M'P' = dx'.
Let us now assume also that the optical system is spherically correctec^
for the points P, P', so that they also are a pair of conjugate points; '
in which case the ratio dx'/dx will be the value of the axial magnifica-
tion, at the points Af, Af', of the imagery by means of paraxial rays,
Hence (see Chap. VII, § 179), we find:
dx n *
where Y denotes the lateral magnification, at the conjugate points
Af , A/', of the imager>' by means of paraxial rays.
S 281-1 Theory of Spherical Aberrations. 411
Now from the figure we obtain:
MR = - dxtsLTie, M'R' = - rfx'-tand',
since /LM'P'R' differs from the angle Q' by only an infinitesimal
magnitude; and, hence,
ix' M'R' tan 6
dx " MR tSLTiS''
Here, we may recall that in § 277 we found :
M'R' n^cosO'de
/ »
MR n'-cosS'-de
and therefore equating the two expressions above for dx'/dx, and at
the same time introducing this last relation, we obtain the following
equation:
»'*
sine-de^ -^Y* -sine' -de";
which, being integrated, gives:
j^
'#9
COSd= -TF*'COSd' + C,
where C denotes the integration-constant. The value of C can be
found by putting d = d' = o; thus, we obtain:
n
Substituting this value of C in the above result, we find :
«'^
I - cos ^ = -2 F*(i - cos flOi
ti
which can be written finally as follows:
. 0
sin- ,
?-!LV
sin —
2
Evidently, this equation cannot be satisfied at the same time with the
Sine-Condition expressed by equation (300) . Consequently, an optical
system can have only one pair of aplanatic points.
This result might have been established immediately by merely re-
marking again (§ 276) that the Sine-Condition Imagery is essentially
Jtfumecicai Opcks» Chapter XII. [ § 283.
ii«^rtii- -.nn xf liixKar tmacer>*. Now we know that if as many as t^o
«rfiRr;i*.> ^-inate ^ifpendiciilar to the axis are portrayed by similat
.•*!i'-r-«4!crita.2^ .aao [^erpemiiciilar to the optical axis, the Imagery
■ui* T . .iiiurar, JLxiii hence it follows that tha Sine-Condition cannot
T ^.^-s*:t:u .T wo i>ain* of axial points. Thus, for example, the oto-
.'^.. ic . . ritcnjsicope must always be computed for that pair d>V
..iv^iwi.x vuii.< :or which it is to be used; and in order to obtain ^
s*^ 1^ . Utfjje •-i :^he object, the latter must be placed at the aplanatic
^.i.««i4>^ rnereiore, that with all the means at his disposal, tl^^e
.•.iit^<>^ Ml :m practical optician, employing wide-angle bundles i^-^/
v^« •a •-ot.'e ;.u achieve is the approximate realization of one or oth( r
i >%w :woniHc±i uus«ibilities: To produce a perfectly sharp ima^^e
^i«f / :n indijimiely small element of surface perpendicular
i« «VK>, z: y. :istr jf an indefinitely small element of the aocis ilseP^
: > . ■i.M.^.tcaily iLtipus»;$ibie to obtain a sharp image of even an indef
»*.&•> Naitt&l uwidi etement of volume; for the conditions which ai
tsu.«c\« V .\i :u&blled in order to portray distinctly its dimensic^n
vM..»^tfi o ihi o|HiciI axis are at variance with the conditions th^3it
iwc*< X >4«.t:»aed in order to produce a distinct image of its lateral
<^. ^ 'JftYMOMEIIT or THE FORMULA FOR THE SIHB-CONDITl
s/» n&B ASSQMFnON THAT TH£ SLOPE-ANOLES
AftE COMPARATIVELT SMALL.
."^^ '-CC uti^ js^uuie now that the effective bundles of rays
Ufc^^^^ >> A saicabie stop so that the slope-angles 6 are all compa^
• v,.N ^^aail-^ 5*^ s^itaiU that we may neglect powers of 6 above
'.^v. *.1i« ruUo^iaijL method of development is practically the sa
.V i^. x^^va !>> IvoJiXiu ;iiid voN RoHR.*
XxNv* .i>,xv&ain^ cu fi^rmula (185),
sin el h
r' »
-je
a-
sin di_i Ik
%y siiit^ * Hi iJfi sin e'i,_^ »i ii, I'j, *
>5^ '. IS. ;iii. tk*ii obtain the development of the ray-length in ^
^ ,iHiVUfcim^ i>owvrs v>f the central angle tp. Since, by the secom
■-. "V
V -i-Niv .^iw M. VON lii^^iiJi: Die Theorie der sphaerischen Abenationen: Chaptci^^
X vv\%t^> ■ *•< •'*•«#«< d» i^pHschen Instrumente (Berlin, 1904), Bd. I, 302-304.^-^
i 284.] Theory of Spherical Aberrations. 413
of formulae (i8o),
r — r = /-cos 0 — r«cos ^,
we obtain, neglecting powers of 0 and tp above the third,
or
,.(.-,|■)(..^)-.(..^r|).
Now
U — 2 f
U
and, hence, finally, we obtain:
where J denotes the so-called zero-invariant (§ 126).
Since
V ^ u + du, t^i^ = A^
we may also write this formula as follows:
/ = «fi+--A' — Y (302)
\ u 2nu) ^^ '
^^nd for the ray-length /' of the refracted ray we have merely to prime
^he letters n and u in this formula.
To the same degree of approximation, we obtain, therefore, for the
tio l\V the following formula:
^ = ^,(i + A'^A-i--A*-^Y (303)
284. Thus, re-introducing the subscripts, we obtain:
If this expression is to be constant for all values of ^i (or i,), then
must have:
i'f»y.A(i)-24^")-o.
2ibsi \nu)^ k^\ \u Jt,
414 Geometrical Optics, Chapter XII. [ § 284.
Now, since (see formulae (270))
\u u) \u! u )
we have :
and
^ hu I ( n-bu , fi'biA
A — = 7 \ I A A — 2~ If
A = (7- J)A —
u ^ u — u
and, hence.
ife V"A"lfe \tt-uA S/fc-*/* \ tt' A
But, since
«*A - %-i = d*-i = w* - "1-1.
we have evidently:
Z A ( ) = '— + . " . = o,
since in the present case, in which the system is supposed to be sphen-
cally corrected for the two axial points Jlfp Jlf^, we must have:
5tt, = o = 8u^.
According to formula (281a), we have:
A -^- =- iA*/*-A— ;
u ^ nu
and, hence, we find:
Accordingly, the Sine-Condition may be expressed as follows:
or
nt ^Jk-Jk \nuJk
Let Q, designate the end -point of the infinitely short object-liD^
Jlf,Ci perpendicular to the optical axis at M^, and let fc, denote the
incidence-height of the paraxial object-ray which, proceeding from Oi
§ 2S5.] Theory of Spherical Aberrations. 415
IS directed towards the centre Afi of the Entrance-Pupil (§257), and
let h^ denote the incidence-height of this ray at the feth spherical surf-
ace. Introducing the relation given by formula (155) of Chapter
VIII viz •
hji,(J, - J,) = hMA - Ji).
svhich, in the way it is employed here, is admissible, since we neglect
:xiagnitudes above the third order, we obtain finally the formula for
the Sine-Condition in the following form:
z/ja*/.j*a(;^)^=o. (304)
Seidel^ notes the fact that Fraunhofer in his characteristic con-
struction of the telescope-objective, appears to have satisfied this
condition, and he, therefore, calls formula (304) the Fraunhofer
Condition.
If this condition is fulfilled, along with the condition of the aboli-
tion of the spherical aberration for the conjugate axial points Jlfj, M^^
we shall have (cf. Chapter VI, § 138) :
«i sin ^1 Wj thul 7 '
IV. Orthoscopy. Condition that the Image Shall Be Free from Distortion.
JkRT. 91. DISTORTION OF THE IMAGE OF AN EXTENSIVE OBJECT FORMED
BY NARROW BUNDLES OF RAYS.
285. In case the object to be depicted is, say, a plane surface of
iinite dimensions placed perpendicular to the optical axis of the Lens-
System, our only chance of obtaining an approximately correct image
•nil be by introducing a small circular stop, or diaphragm, whose duty
ivill be to limit the angular widths of the operative bundles of rays
emanating from the various points of the object. It is obvious that
:his mode of producing an image will be attended also by a number of
liflficulties of one kind and another, which may be described in a
^neral way as aberrations due to the obliquity of the rays proceeding
rom the lateral parts of the object. In general, a plane object will
^ot be reproduced by a plane image, but on account of the astigmatism
>f the narrow bundles of rays, the image will be resolved into a double
mage, symmetrically situated with respect to the optical axis on two
* L. Seidel: Zur Dioptrik. Ueber die Entwicklung der Glieder 3ter. Ordnung. welche
den Weg eines ausserhalb der Ebene der Axe gelegenen Lichtstrahles durch ein System
brecbenden Medien. bestimmen: Astr. Nach., No. 1029, xliii. (1856). See Section 9 of
SnDKL's paper.
Theory of Spherical Aberrations.
417
n where the stop is, go through the centre 0 of the stop. The path
ray will lie in the meridian plane containing the object-point
ich is here the plane of the diagram (Fig. 138). If the image-
x,'>r'
w —
Q'
Fio. 138.
TiON OP THK IMAOB. OTX represents the Optical Axis of a centered system of spherical
The position of the stop-centre is marked by O. The Object-Plane and the Imase-Plane
lated by v, o*. The straight lines with arrow-heads show the directions of portions of the
le chief ray which has its origin at the Object-Point P (or Q) in the Object-Plane v,
r' is supposed to be occupied by a screen, what actually appears
3 screen may now be called the practical image of the plane
perpendicular to the optical axis at M. Immediately around
ial image-point M' there will be other sharp image-points, but
:tle distance from the axis we shall have image-spots instead of
points, and these images will be more and more indistinct, the
• they are from the axis. If the diameter of the stop is reduced,
ect will be to diminish the dimensions of the image-spots, and
miting case we may even suppose that the stop contracts into
: point or pinhole-opening at 0, so that only the chief rays etna-
from the points of the object-plane a succeed in getting past the
These chief rays, which constitute a sort of skeleton of the
« of effective rays, will determine by their intersections with
lage-plane a' the positions in this plane of the image-points
x>nding to the points of the object-plane <r. Thus, in this view
matter, the point P', where the chief ray emanating from P
crosses the image-plane <r', is to be considered as the image of
ict'point P.
Measure of the Distortion, The position in the image-plane
he point Q', which, by Gauss's Theory, is conjugate to a point
[le object-plane <r, is defined by the equation:
Y denotes the magnitude of the Lateral Magnification of the
system for the pair of conjugate axial points Jlf, M', If we
S 288.] Theory of Spherical Aberrations. 419
then, since,
LM = LM + MA+ AM =- u - u - 8u,
LM' = LM' + M'A' + A'M' ^ u' - u! - hu\
we obtain:
i\' v! - u! — hu! tan 6'
17 tt — tt — 5u tan 6
Now if the image is to be free from distortion, the point P' must coin-
dde with the ideal image-point Q'\ that is, rf = M' P' must be iden-
tical with y = M'Q\ which means that we must have:
n y '
and, hence, the Condition of Orthoscopy ^ which requires that all pairs
of conjugate chief rays shall trace similar figures on the object-plane
and image-plane, may be expressed by the following formula:
tan 6' _ tt — u — 5u .
tan e " w' - u' - 5u' ^' ^^^5)
"which involves, therefore, not merely the ratio of the tangents of the
slope-angles 6, 6^ of the chief ray, but also the Longitudinal Aberrations
SUf bu' at the centres of the Pupils.
If the Lateral Magnification of the system with respect to the Pupil-
CTentres Af , Af' is denoted by F, it may readily be shown, by the aid
of formulae (127) and (153), that we have always the following re-
Isition between Y and Y:
^ n' ' MM Y ' ^3^^
mrliere n, n' denote the indices of refraction of the first and last media.
H«noe, we may also write formula (305) above in the following form:
tan 6' n v! — u! u ^ u — 8u i
tan e " n' ' u - u ' u' - u' - du'' Y '
In case the object-point P is infinitely distant, the image-plane <r'
WTi 1 1 coincide with the secondary focal plane e' and the point M' will
com xidde with the secondary focal point E\ Under these circum-
xices, we find :
tane;_n M'E' i
tane " n'' M'E' - M'U' Y'
SP "that in this instance the Longitudinal Aberration of the ray at the
i
420 Geometrical Optics, Chapter XII. [ § 289.
Entrance-Pupil does not matter. And if, as in the case of the Tele-
scope, both object and image are infinitely distant, so that £' is also
the infinitely distant point of the optical axis, then:
tan 6' n I
tenT = n''Y = '=°""*^"*'
and the Condition of Orthoscopy in this special case is independent
of the aberrations at both Pupil-Centres.
It is sometimes stated that the constancy of the tangent-ratio
tan 6'/tan 6, known as Airy's Tangent- Condition,^ is the necessary
and sufficient condition of freedom from distortion; but, as M. VON
RoHR* has pointed out, this is evidently by no means the case except
under special circumstances. For example, if, as is the case with a
certain class of Photographic Objectives, the stop coincides with the
Exit-Pupil, so that the three points designated by O, M\ I! are all
coincident, then for an infinitely distant object-point, just as also in
the case of the Astronomical Telescope, the constancy of the Tangent-
Ratio is the condition of orthoscopy.
289. Case when the Pupil-Centres are without Aberration. If the
stop is placed, say, between the jfeth and the (jfe+ i)th spherical sur-
faces, the optical system will be divided into two parts, an anterior
part (I) composed of the first jfe spherical surfaces and a posterior part
(II) composed of all the spherical surf aces after the ifeth. If the part
(I) is spherically corrected for the centre of the Entrance-Pupil an^
the centre of the stop, and if, similarly, the posterior part is spherically
corrected for the centre of the stop and the centre of the Exit-Pupl
the chief rays which are obliged to go through 0 will also go through
the points Af , M\ In this case the Longitudinal Aberrations atAfi^
will vanish, that is, hu — du' = o; and now the condition of orthos-
copy is:
tan 6' n i
This is Airy's Tangent-Condition above-mentioned, viz., that the
ratio of the tangents of the slope-angles of every pair of conjugate
' G. B. Airy: On the spherical aberration of the eye-pieces of telescopes: Camb* ^'*'*
Trans., Ill (1830), 1-64. This paper was published separately in Cambridge thr« J^
before it appeared in the Phil. Trans.
•M. von Rohr: Beitrag zur Kenntniss der geschichtlichen Entwicklung, ^^
sichten ueber die Verzeichnungsfreiheit photographischer Objektive: Zfi. f. Imtr»l^
(1898), 4-12. Sec also A. Koenig und M. von Rohr: Die Theorie der sphacrisd*"
Aberrationen: Chapter V of Die Theorie der optischen InsirumenU, Bd. I (Berlin, I9W^»
edited by M. von Rohr; see page 241.
(291.1
Theory of Spherical Aberrations.
421
I
Jiief rays must be the same. The requirement of spherical correction
jf the stop-centre for the two parts (I) and (II) of the optical system
is called by von Rohr the Bow-Sutton Condition.^ When both condi-
tions are satisfied, the points Af, Af' are called the Orihoscopic Points
of the system.
290. The Two Typical Kinds of Distortion. I f when the two partial
sjrstems are spherically corrected with respect to the stop-centre the
ratio tan 6' : tan 6 is not constant, the magnification of the image
dose to the optical axis will be constant, but out towards the edges
there will be distortion. For example, if with increasing values of 6,
the ratio tan 6' : tan 6 also increases, the magnification ri' jy will in-
crease towards the margin of the field, so that spaces of equal area in
the object-plane will appear distorted in the image into spaces of
gradually increasing size as we go out from the axis. If the object
consists of a network of two mutually perpendicular systems of equi-
distant parallel lines, as in Fig.
139 (a), the image will appear
as in Fig. 139 (6). This case
is known as '' Cushion-Shaped
Distortion", sometimes called
ilso Positive Distortion. On
he other hand, if tan 6' : tan 6
lecreases as the slope-angle
ncreases, the magnification 17^3^ will diminish out from the centre of
:he image; and then we have the case known as ** Barrel-Shaped Dis-
:ortion", or Negative Distortion (Fig. 139 (c)).
291. Distortion when the Pupil-Centres are the Pair of Aplanatic
Joints of the System. If the points Af , Af' are the pair of Aplanatic
'oints of the system, they must satisfy the Sine-Condition, viz.,
in O'/sin 6 = constant; and since this condition is necessarily opposed
o the Tangent-Condition, the image in thiscase will be distorted in such
ashion that 17'will be less than the ideal value y\ Moreover, since the
angent of an angle increases faster than its sine, the difference y' — ij'
rill increase as y increases, and therefore the distortion will be *'barrel-
haped" (Fig. 139 (c)). If the object consists of two sheaves of hyper-
bolae resembling Fig. 139 (6), and if Af, Af' are the pair of Aplanatic
Points, the image in this case will be the two systems of parallel
straight lines (Fig. 139 (a)). This is the test which Abbe invented to
' R. H. Bow: On Photographic Distortion: Brit. Journ. of Photography, VIII (1861),
paces 417-419 and 440-442.
T. Sutton: Distortion Produced by Lenses: Phot, Notes, VII (1862), No. 138, 3-5.
b
Fig. 139.
Showing thb Typical Kinds of Distortion.
l.\
Theory of Spherical Aberrations.
423
>btaiii
?5 - , I ^
/ — I -f 'J
nk-\nk-\^'k
proceed, therefore, to develop an expression for the quotient
nrtu'
\ expression relates to the Jfcth spherical surface, but for the present
ill be convenient to drop the subscripts. The subscripts are like-
omitted from the letters in the diagram (Fig. 140), which repre-
Fio. 140.
DRB USED IN THE DERIVATION OP THE DISTORTION-ABERRATION FORMULA. The fiffure
tents the path of a chief ray incident on the k\h surface of a centered system of spherical
tinsT surfaces.
AC'-r, AAf-'u, AM-'u. AL'^v, MP^ri, IBCA"^, JtB-'h,
s the path of the chief ray before refraction at the jfeth surface,
vertex and centre of which are designated accordingly by the
rs A and C, respectively. This ray crosses the axis at the point
^;nated in the figure by L and is incident on the Jfcth surface at the
t designated by B, The place where the transversal plane of the
stem belonging to the medium immediately in front of the jfeth
ice is cut by the optical axis is marked by the letter Af, and the
t where the ray crosses this plane is designated by P. Finally,
Foot of the perpendicular let fall on the optical axis from the inci-
:e-point B is designated by D, Let us also use the following
bols:
i4 C = r, ilJlf = tt, i4L = i;, MP ^ ri, LMLP = 8,
Z,BCA = ^.
§ 292.] Theory of Spherical Aberrations. 425
and since this equation must be true for all the chief rays, that is,
for all values of the central angle ^ (as far as the extreme value per-
mitted by this approximation), we may equate to zero the co-efficients
of ^ and ^^; whereby the magnitudes of the co-efficients /, m are
determined as follows:
- tt — tt
/ = r,
u
tt — tf.fi u e I 1
~ u L 2ru tt(u — tt) r* 6r* J
Substituting these expressions for the co-efficients /, m in the series-
development of the function ij, and at the same time using the in-
variant-relation obtained by combining formulae (270) :
f 7 — ^(" — ^) n'{u' — u')
uu uu
we derive the following formula:
Similarly, for the ray refracted at this surface, we obtain the corre-
sponding formula for n'ti' ju' by merely priming the letters u and c on
the right-hand side of equation (308). Doing this, and dividing the
latter equation by the former, we obtain :
wiytt' ^ 1 2r u r (j — J) u J
2«low
^ U U 2 WU
we found in § 263. Moreover, according to formula (77),
u n
us, we obtain :
wiytt' 2\f n J — J nu y
If fc = Dfi denotes the incidence-height of the chief ray corre-
l^onding to the central angle ^, we may, neglecting magnitudes above
e 3rd order, put
§ 293.] Theory of Spherical Aberrations. 427
Finally, according to the Law of Robert Smith (§194), the rela-
tion between the conjugate ordinates 3^1, y^ may be expressed evi-
dently as follows:
«i «« '
and, moreover, we have also:
«iyi
fc. = -
and
I u,a,
/, -J, «,(«!- Ui)"
And, hence the formula above may be written:
whence it is seen how the Distortion-Aberration hy^ is proportional
to the cube of the ordinate y^.
ART. 94. THE DISTORTION-ABERRATION IN SPECIAL CASES.
293. Case of Single Spherical Refracting Surface.
When the optical system is composed of a single spherical surface,
formula (309) gives for the Distortion-Aberration
where, for the sake of brevity, we write:
r n nu
Dr
n' ^ n' + w 2n n' n
./•*"" —2 ^„ ^^. "t"
n " n u ru ru
f^-
Tf the image is to be free from distortion, we must have 6y' = o; which
mplies here one of two things: Either J = o, or else, 7" = o. If
^ = o, then u = tt' = r; which means in this case that the stop-centre
roinddes with its image at the centre C of the spherical surface, and
mder these circumstances the image will be free from distortion for
ill object-distances.
§ 295.] Theory of Spherical Aberrations. 429
The condition that X shall be a minimum for given values of ^, x
and X will be found to be:
_ 3(w+ 0 , w + I n(2n + i)
^ " 2(n + 2) ' "*■ 2(n + 2) ^ "*■ 2(n - i)(n + 2) ^-
V. Astigmatism and Curvature of The Image.
ART. 95. THE PRIMARY AND SECONDARY IMAGE-SURFACES.
295. In the imagery of extended objects by means of narrow
bundles of rays whose chief rays all meet at a prescribed point on the
optical axis of the centered system of spherical surfaces, there will,
in general, be astigmatic deformation of the bundles of image-rays;
in consequence whereof to an object-point P lying outside the axis
there will correspond, not a sharp image-point, but two short image-
lines perpendicular to the chief ray of the bundle at the so-called I.
and II. Image-Points S' and 3' (see Chapter XI). Thus, in case the
in^age-rays are received on a focussing-screen, the image of the object-
point as seen on the screen will generally be a small patch of light
corresponding to the cross-section of the bundle of image-rays at that
place, the dimensions of which, in one direction at least, will always
be comparable with the diameter of the narrow stop; so that such an
image formed by an astigmatic bundle will always be more or less
blurred and indistinct, and not to be compared in this respect with
the sharp image which is obtained when the object-point is on the
axis. The farther the object-point is frftm the axis, the more pro-
nounced this defect will be. In two special positions of the focussing-
screen the image will be deformed into a short line, which is vertical,
say, for one of the positions, and horizontal for the other position
— corresponding to the places of the two image-lines of the astigmatic
bundle (§ 230). Somewhere between these two positions the bundle
of rays will have its narrowest cross-section, which, in the case of a
centered system of spherical surfaces, will be approximately circular
in form. This is the place of the so-called * 'Circle of Least Confu-
sion" (§ 244) — a somewhat misleading phrase, inasmuch as the con-
vergence of the rays in either of the two image-lines is of a higher
order. However, we do obtain here perhaps the nearest approach to
a true image of the object-point.
If on every chief image-ray corresponding to such points of the
object as are contained in a meridian plane of the optical system, we
mark the I. and II. image-points S' and 5', the loci of these two sets
of image-points will be two curved lines which touch each other at
98.] Theory of Spherical Aberrations. 433
d, hence, to the required degree of exactness, we obtain:
cordingly, we derive the following approximate expressions for the
ignitudes of the aberration-lines, in the GAUSsian image-plane cr',
the meridian and sagittal rays:
P't7' = -^,ix', P'r=-^,dx'. (314)
298. Moreover, let tr" be any plane parallel to the GAUSsian image-
me a', and at a distance from it M'M" = e (say), and let P'\ t/"
d V" designate the points where the rays L'5'5', H'S' and G'3',
jpectively, cross the plane cr"; so that P"U" and P"V" will be the
lear aberrations in this transversal plane of the meridian and sagit-
I rays of the astigmatic bundle of image-rays. Evidently, if we
gleet the second powers of the aperture-angles dX', dV and the
iwers of the slope-angle 6' above the second, we shall have:
■f, therefore, supposing that we have d\' = dX', we wish to determine
^ position of the focussing-plane a" somewhere between the I. and
image-points 5' and 5' for which the linear aberrations P"U" and
T" are of equal magnitudes but of opposite signs, the two equations
j) give the following formula for this particular value of the
B<jssa e:
4 \R'^R'J'
i- > under these circumstances, we obtain:
the bundle of rays is received on a plane screen coinciding with
s position of the plane <r", we shall obtain on the screen, as was
^ted above (§ 295), perhaps the nearest approach to a true image
the object-point.
29
434 Geometrical Optics, Chapter XII. [ f 299.
In case the astigmatism was entirely abolished, so that
by placing the plane screen in the position for which e = y jtR'r
we should obtain on it an actual point-image of the object-point P.
But it will be remarked that the value of e depends on that of y, and
in order to obtain point-images of the different points of the object,
we should have to "focus" the screen so that its intersection with the
curved stigmatic image-surface would contain the point to be observed.
ART. 97. DEVELOPMENT OF THE FORMULiB FOR THE CURVATURES
ilR\ i/l'.
299. The Invariants of Astigmatic Refraction. The curvatures at M'
of the two image-surfaces have now to be expressed in terms of the
curvature of the object-surface at M and of the given constants of the
centered system of spherical surfaces. In the development of these
expressions we shall use Abbe's Invariant-Method, as given by Koenig
and VON Rohr in their treatise on Die Theorie der sphaerischen Aber-
rationen.
In Chapter XI, §§236 and 240, we derived two* formulae (24^)
and (250), which may be written as follows:
^ /cos a cos'a\ ./cosa' cos'a'\
5-'(T-|)-«'(^--r.>
(316)
where the functions denoted here by Q and Q, which have the same
values before and after refraction at a given spherical surface, are
called the Invariant- Functions of the Chief Ray of the Infinitely Narrow
Bundle of Rays. Each of these functions may evidently be developed
in a series of ascending powers of the central angle 4> of the following
forms:
Q = J + B^ =J + B'^,^
^ 2 2
Q = J + B^=^J + B'^,
2 2
(317)
wherein the coefficients B, 5, etc. are as yet undetermined, and
where, as usual, the terms involving powers of ^ higher than tb^
I 300.] Theory of Spherical Aberrations. 435
second are neglected. The relations, which we wish to find, will then
be given by writing:
5' - 5 = o, B' - B = o.
The easiest method of obtaining the expansions of Q and Q will be
:o develop the functions i /s, i /s and cos a each in a series of ascending
DOwers of 4>» and to introduce these expressions in the formulae (316)
ibove.
300. Developments of i /s, i /s and cos a in a series of powers of ^.
In the diagram (Fig. 143) the straight line SB represents the path
Fio. 143.
Path op Chief Rat op Pencil op Meridian Rays Incident on ^^h sxtrfacs of centered
SYSTEM OF spherical REFRACTING SURFACES.
AC'^r, MK'^R, BS^s, AM^u, AL'-v, BL-'l, ZBCA"^, ZSATAf'^^, ZALB'»9,
of the chief ray before its refraction at (say) the jfeth spherical surface.
In its progress through this medium the ray crosses the axis at the
point designated by L and is incident on the spherical surface at the
point B. The point designated by 5 is the I. Image-Point of the
astigmatic bundle of rays in the medium between the (k — i)th and
feth spherical surfaces, and the curved line MS represents the section
In the meridian plane (or plane of the figure) of the I. image-surface
9^hich is the locus of the I. image-points 5. The primes and subscripts
vhich naturally belong to these letters are suppressed for the present;
hey will re-appear, as usual, at the end of the investigation. For
he purpose of these developments, we shall employ, therefore, the
allowing symbols:
i4 C = r, MK = R, AM = w, AL =- v, BS = 5,
ZBCA = 4), ZSKM = ^, ZALB = 6.
tTie letter K is used here to designate the centre of curvature at M
rf the meridian section of the I. image-surface.
436 Geometrical Optics, Chapter XII. [§300.
From the figure we obtain easily the following relation:
I cos8
u -f- 2/v'Sin 2r • sm —
2 2
which, provided we neglect the powers of the angles 6, if and ^ above
the second, may be written:
I —
I 2
2 2
Moreover, when the angles 4^ and ^ are infinitely small, we have:
R^ ML ^ V — u u — u
^ " ^'*-* AL " ^^-^ V ^ "iT"-
This relation, which is strictly true in case 4^ » ^ « o, is also true
provided we may put sin 4> = 4^ and sin ^ = ^, that is, provided we
neglect the powers of these angles above the first; and even when we
retain, as here, the second powers of these angles, we may write:
In the same way, also:
U
Hence, eliminating 6 and ^ from these equations, we obtain:
I 2 U'
5
or, finally:
'b^^iiU-^H}]'
5 u u [r u R \u u/j2
The development of the reciprocal of 5 = BS will obviously ^^
precisely the same form as that obtained here for 1/5; the only^"*
ference being that we shall have R in place of R in formula (31*)*
Again, since
a*
cosa = 1
2
§ 302 J Theory of Spherical Aberrations. 437
and since
we obtain :
cosa
= 1 — 1 1-^ = 1 f • -- . (319)
\ u J 2 n 2 ^ ^^
301. The expressions for the co-efficients B, B and B', B'.
If now we substitute in the formulae (316) the series-developments
for 1/5, 1/5 and cosa, as found above in formulae (318) and (319),
we obtain the following expressions for the co-efficients B and B in
formulae (317):
nR n u uu nu
tiR n u uu
These expressions can be obtained in a more convenient form. Thus,
by simple transformations:
n u^ ^ u u^ n^ ' r u \r nj
n nu
and hence:
(320)
The expressions for the co-efficients 5', B' will evidently have the
^me forms as the expressions found above for 5, B, and can be ob-
:ained directly from formulae (320) by merely priming the symbols
^, R, R and u.
302. Imposing now the conditions 5' — 5 = o and B' — B = o,
aind at the same time introducing the subscripts and employing Abbe's
difference-notation, we derive the following formulae for the relations
between the curvatures of the image-surfaces before and after refract-
§ 303.] Theory of Spherical Aberrations. 439
two equations (322), we obtain:
and, hence, the condition of the abolition of the astigmatism of the bundles
of image-rays, viz., R'^ = i?^, becomes:
k=m J2 / I \
and, exactly, as in §292, we may employ here also formula (155) of
Chap. VIII, viz.:
hA(Ji - Ji) = hMJ, - A).
whereby formula (324) may evidently be put in the following form:
a formula of great simplicity and convenience, since, exactly as in the
case of the formula for the Longitudinal Aberration along the axis, it
enables us to see distinctly the effect of each single refraction, and
thereby to ascertain the factors which have the most influence on the
astigmatism.
303. Curvature of the Stigmatic Image. If the astigmatism is
abolished, we obtain for the curvature of the image :
whence it is seen that the curvature of the stigmatic image is independent
of the position of the stop.
This is the so-called **Petzval Formula**, which was published, un-
fortunately without proof, by Joseph Petzval, in his celebrated paper,
Bericht ueber die Ergebnisse einiger dioptrischer Untersuchungen (Pesth,
1843. Verlag von C. A. Hartleben).* The formula is applicable
only in case the image is stigmatic, and although Petzval does not
expressly even allude to this pre-requisite condition, it is hardly to
be supposed that he was ignorant of it.^
' See alflo J. Petzval: Bericht ueber optische Untersuchungen. Siizungsbfr. der math.'
MOiunriss. CI, der kaiserl. Akad. der Wissenschaften, Wien. xxvi (1857). 50-75, 92-105,
139-145. The PsTZVAL-formuIa is given here also without proof, on p. 95. but the re-
mainder of this contribution is chiefly devoted to a discussion of this equation, which is
shown to hold for a number of simple special cases.
' In regard to this question, see especially M. von Rohr's Theorie und Geschichte des
fholograpkischen Objektivs (Berlin. 1899). p. 270. L. Sbidel, in his paper, "Zur Dioptrik.
304.J
Theory of Spherical Aberrations.
441
»rmulae (92) of Chap. V:
^m^m^m ^l^l^l
u
m
U,
u
m
U,
inally, also, by formula (155) of Chapter VIII, we have:
ad, thus, we obtain:
If, therefore, employing the relation :
A.A,(A-J,)=AA(/*-/t).
€ take from under the two summation-signs in each of the formulae
; 28) the term
I U'tU
2 2
(/, - J,r - n\{u, - u,)
21
m! if we multiply both sides of these equations by n^/u^, at the
»iie time eliminating d\'^ and dX^ on the right-hand sides of the two
vjations by means of the formulae (330), and also expressing y'^ in
r^ns of yi by means of Smith's Formula:
n„,h„y^ n,h,yi
u
m
U,
obtain, finally, the formulae (328) in the following forms:
w,uf
5«!5 p' rr' _ ^ *i •"i**i ^,2-. Q
M,uJ
5y!«i-3,
(33O
, for brevity, we put:
(332)
S 306.] Theory of Spherical Aberrations. 443
306. Case of an Infinitely Thin Lens.
For the case of an Infinitely Thin Lens, we can write :
wherein, employing the same special Lens-Notation as in § 268, we
may put:
/-/--*. J-/. * y (n~ i){c-x) -mp
n — I
Introducing these symbols, we shall find :
where the symbol U is used as an abbreviation for the following
function :
n' , a . 2(n + i) . 2n , n+i
Thus, the curvatures of the images produced by a Thin Lens will be
for the case of a plane object:
(i) When the centre of the stop coincides with the centre of the
Infinitely Thin Lens (x = 00), we find U/(x ~ x)' = i, and hence:
I 3n + 1 I n + i
W^^ n ^' i?' " " n ^'
whence it appears, that under such circumstances, the curvatures of
the image-surfaces are independent of the distance of the object from
the Lens, and the chief rays proceed in straight lines from the points
of the object to the conjugate points of the image. The curvatures,
in fact, depend only on the focal length of the Lens and the value of
the relative index of refraction («), but not on the form of the Lens.
If » = 3/2, we find i2' = — 3//11 and 5' = — 3//$.
(2) The condition of the stigmatic image is
f7 = o,
444 Geometrical Optics, Chapter XII. [§307.
in which case the curvature of the image is:
I _ I ip
(3) In the special case of a System of Infinitely Thin Lenses k
Contact, with the centre of the stop situated at the common vertex of (he
Lenses (x = 00 for each Lens), the function U^(x — x)' is equal to
unity for each Lens, and, hence, the curvatures of the image-surfaces
will be:
I w + 3 I ^w + i
»
Accordingly, the condition of a fiat stigmatic image in the neighbour-
hood of the axis (i2' = 5' = 00) requires that we shall have in this
case:
S^ = o,
which means that the combination of Lenses must act like a slab with
plane parallel faces.
VI. Aberrations in the Case of Imagery by Bundles op Rays of Fimitb Slopis
AND OF Small Finite Apertures.
ART. 99. COMA.
307. The Coma-Aberrations in General. Heretofore, in the in-
vestigations of the aberrations in the case of object-points not on the
optical axis, it has been assumed always that the rays were limited by
a stop of infinitely narrow dimensions. In actual optical construction
this condition can never, of course, be absolutely realized ; nor, indeed,
in the case of certain optical instruments is it necessary that it should
be, so long as the diameter of the stop is relatively very small. On
the other hand, when it is required to produce the image of a fairfy
extensive object by means of somewhat wide-angled bundles of rays.
as, for example, is often the case with photographic objectives, the
diameter of the stop will enter as a chief factor in the study of the aber-
rations of the rays. Thus, whereas we saw (§ 304) that the aberra-
tion-lines in the case of infinitely narrow bundles of astigmatic rays
were proportional to the first powers of the aperture-co-ordinates yi»
*i (§ ^59) » we must now advance a step farther, and assume here that
the aperture is so wide that we will not be justified in leaving ^^
of account the second powers and products of these co-ordinates.
A bundle of rays of finite aperture, emanating from a point outside
§ 307.J
Theory of Spherical Aberrations.
445
the optical axis, may show aberrations of a character similar to the
spherical aberration along the axis of a direct bundle of rays (see § 208
and § 260). These aberrations will be manifest in both the meridian
and sagittal sections of the bundles of rays, but here a very impor-
tant difference is to be remarked, as will now be explained.
The rays of the sagittal section are symmetrically situated on op-
posite sides of the meridian plane, so that the point of intersection of
every pair of symmetrical rays in this section will lie in the plane of
the meridian section, for example, as shown in Fig. 144. But in the
Pio. 144.
Stmmbtrxcal Character op thb Aber-
rations OF THE RaY9 op THB SAGITTAL
Section of an Inclined Bundle of Rays
/>p FiNiTB aperture. The chief ray of Uie
handle is the ray marked u. The plane of
the meridian section is the plane containing
u which is perpendicular to the plane of the
Fig. 145.
Unsymmetrical Character op the
Aberrations of the Rays op the Me-
ridian Section op an Inclined Bundle of
FINITE aperture. The chief ray of the
bundle is the ray marked m. This is the ray
which at some sta^re of its proflrress ffoes
throuflfh the centre of the stop. The rays of
the meridian section are in general not
symmetrical with respect to the chief ray.
meridian section (Fig. 145) it is obvious that, in general, there will be
no symmetry at all. The chief ray of the bundle will depend on the
position on the optical axis of the centre of the stop. If the rays are
received on a screen placed perpendicularly to the optical axis, and
if a straight radial line is drawn in the plane of the screen through
the point where the screen meets the optical axis and intersecting the
light-pattern on the screen, there will be no symmetry in the pencil
of rays which meet the screen at points lying along this line: whereas
in the case of a pencil of rays which meet the screen at points lying
along a line at right angles to this radial line there will be symmetry.
The light-pattern on the screen sometimes presents the appearance of
a comet, with its tail turned either towards or away from the optical
axis; which accounts for the origin of the name **coina'\^
So far as the meridian rays are concerned, we have to ascertain only
the y-aberrations (§ 256), because, by the Laws of Refraction, the paths
* Some excellent drawings exhibiting these appearances are to be found in H. Dennis
Taylor's A System of Applied Optics (London, 1906). This work contains several chap-
ters in regard to Coma. Especially interesting in the diagrams are the drawings by Prof.
S. P. Thompson, Plate XVI.
§508.]
Theory of Spherical Aberrations.
447
308. The Lack of Symmetry of a Pencil of Meridian Rays of
Aperture. In the special case when the chief ray of the bundle
coincides with the optical axis, there will be symmetry in the pencil
oi meridian rays, as is exhibited in the diagram (Fig. 146), which
represents the meridian section of an optical system consisting of a
»ingle spherical surface. The centre of the stop is supposed here to
Fig. 146.
X«ACK OF Symmetry op a Pencil of Meridian Rays of Finite Aperture.
be situated at the vertex A of the spherical surface, and the object
is infinitely distant, so that the object-rays emanating from any point
of the object are parallel.
If the object-point is not on the optical axis, the chief ray of the
bundle of object-rays will be inclihed to the optical axis at some
angle, say 8; and it is evident by an inspection of the figure that the
meridian rays of this bundle produce an eflfect quite different from that
which we perceived in the case of a bundle of rays emanating from an
axial object-point. In the first place, the chief ray is no longer the
ray which meets the spherical refracting surface normally; and,
generally, this will always be a distinguishing peculiarity of such a
pencil of meridian rays, so that the chief ray will not (except for certain
special positions of the stop) go through the centre C of the spherical
surface; and even in case it did happen to pass through the centre of
Dne surface, it would not pass through the centre of the next following
lurface of a centered system of spherical surfaces. The straight line
Irawn through C parallel to the incident rays (which may, or may not,
^e the path of an actual ray of the pencil), is in a certain sense, an axis
h{ symmetry for the refracted rays in the same way as the optical
ixis is an axis of symmetry for the direct pencil of refracted meridian
"atys: but, since the stop cuts off more rays on one side of this line than
t does on the other, the actual pencil of refracted rays is not symmet-
rical with respect to this straight line of slope-angle 8 drawn through
i 309.] Theory of Spherical Aberrations. 449
^oints corresponding to the points 5 and R on the incident rays SB
ind RI, respectively; the actual positions of S' and R' being, of course,
lependent on the positions of 5 and R, respectively. The angles of
ncidence at B and / are supposed to differ from each other by an infi-
itely small magnitude of the ist order; and, consequently, the points
lesignated by 5' and R' are two infinitely near points on the caustic
urve of the meridian rays. The point of intersection of the refracted
ays BS' and IR' is designated in the figure by T'; and we may con-
ider S'T' as the longitudinal aberration along 55' of the infinitely
arrow pencil of meridian rays which are refracted at the points lying
a the arc BI.
The following symbols may be conveniently employed :
Z CBS' = a', Z Cir = a' + da\ Z ICB = d^, ABTI ^ d\\
BS' = s\ IR'--s' + ds\
With V as centre and with radii equal to VI and T'R\ describe
wo circular arcs meeting BS' in the points designated in the figure
y Y' and Z', respectively. The variation ds' = IK — BS' may be
^nsidered as consisting of a displacement S' Z' together with a dis-
lacentent Z'R'. The latter may be said, in a certain sense, to be
ue to the variation of the point of incidence from B to I\ whereas
le former is the displacement depending on the angle rfX' between
le refracted rays leaving B and /. We shall try now to obtain an
q>ression for the magnitude of the component
S'Z' = dg!
I the total variation; because, since BV and IT' are tangents to the
austic curve at the two infinitely near points 5' and R', and since,
herefore, the lengths S'T' and T'R' can differ from each other only
)y an infinitesimal magnitude of an order higher than either of them,
o that we can put
S'T = T'R' = T'Z',
he magnitude denoted by dq' is equal to twice the aberration 5' 7^.
Incidentally, also, we may observe that since (neglecting infinitesi-
lals of the 2nd order) S'T' + T'R' = dq' = the length of the element
f the caustic, the radius of curvature of the caustic at S' is equal to
^Idk'.
Throughout this present investigation we shall retain magnitudes
f the Ojrfer dtp. Hence, provided we neglect only small magnitudes of
n ord^r higher than the ist, we shall obtain from the figure the
30/
§ 310.] Theory of Spherical Aberrations. 451
The above formula has been derived for the rays after refraction
at the spherical surface here considered; but it is obvious that we shall
obtain in the same way a precisely similar relation connecting the
corresponding magnitudes before refraction, viz.:
idQ _K SK'Q ncos^ a dq
r d<p "^ f^ ns 5* dX'
Combining, therefore, these two formulae, and using Abbe's difference-
notation, we obtain:
. /ncos^ adq\ ^ ^ . / i \ . v
Thus, knowing the values of the magnitudes denoted by a, 5, dq and
rfX, which relate to the narrow pencil of meridian rays before refraction
at the spherical surface, we can calculate the magnitudes denoted by
a' and s\ and determine, by means of the formula just obtained, the
magnitude of the ratio dq'fd\\ which relates to the pencil of rays
after refraction.
310. Instead of a single spherical surface, let us suppose now that
the optical system consists of m spherical surfaces with their centres
ranged all along one straight line. Introducing in our notation the
surface-subscripts, we must write:
and, hence, for a centered system of m spherical surfaces, we obtain
by formula (333) the following recurrent formula:
s'J dXi *dX|\ Si'S^'S^ J Vcosa'icosai-cosai^^i/
+3 gf '*"^^' ' '^:>-i)Ycosa^rcosa^,. . ''''''.^)\.Q,.Jl) .
61 \^*+r^*+a- • s^J Vcosa^cosa^+i- • coso^.J * ^* VW*
If we write
then
and, hence:
arc BJf, = ij,
•'* cos a* cos al *
jk+i ^*+i cosal *
L2.] Theory of Spherical Aberrations. 453
1 if here we substitute:
can write finally:
s^ 2 Jm cos' a
-^ ri:(^A'Q,'K,'4-) . (336)
(12. Let us now impose the condition that the slope-angles 0, 0' of
chief rays are small magnitudes of the first order — of the same order
the aperture-angles X, X', as we shall now denote these latter angles,
tead of denoting them, as above, by the symbols dX, d\'. Without
fleeting ultimately the magnitudes of the 3rd order of smallness,
may obviously introduce in the above formula (336) the approxi-
te values of the magnitudes denoted by the symbols 5, j, Q and IC,
us, we may employ here the approximate relations:
cos a = I, sin a = a, 8 = — h/u and ^ = h/r;
ere h denotes the incidence-height of the chief ray and u = AM.
d, hence, since
a = e + 4),
can put:
hJ
a = — ,
n
i, therefore:
K = n-sin a = na = hJ,
>reover, approximately, also:
I, hence, if h denotes the incidence-height of a paraxial object-ray
anating from the axial object-point Afp we may use here also the
owing relation:
lally, we may put here Q = J. Accordingly, introducing these
ues in formula (336), and at the same time writing now bw' in place
iw\ we obtain:
Theory of Spherical Aberrations. 455
ly write formula (337) in the following form:
is, on the assumption that the slope-angles of the chief rays are
magnitudes, the condition of the abolition of the so-called
latic" Aberration of the meridian rays is:
|:"*JA*/*/*a(;^)^=o. (339)
>ver, if the reader will investigate also the y-aberration and the
ration of a ray of the sagittal section, as is done, for example,
essrs. KoENiG and von Rohr,^ he will discover that equation
is likewise the condition of the abolition of both aberrations of
Lgittal rays.
i¥ill be recalled that precisely this same equation was obtained
8 the expression of the Sine-Condition (formula 304).
ART. 101. SPECIAL CASES.
r. Case of Single Spherical Surface. The condition that the
tic aberration, in the case of a single spherical refracting surface,
vanish is evidently:
7J(i/nV — i/nu) = o;
I will be satisfied in each of the three following cases:
/ = o, or tt = w' = r: that is, when the object and image co-
; at the centre of the spherical surface — a case possessing no
leal interest;
y = o, or u =^ r: that is, when the stop-centre is situated at
intre of the spherical surface; and
nu = n'u': that is, when the pair of conjugate axial points
''' are the aplanatic pair of points of the spherical surface.
• Case of Infinitely Thin Lens. Employing the usual special
Notation (see §268), we may write the expression on the left-
side of formula (339) as follows:
JiJi(ifnu[ " x) + Jrfai^' — i/nu[) = <pV;
KoENiG und M. VON Rohr: Die Theorie der sphaerischen Aberrationen: Chapter
dI. I of Die Theorie der optischen Instrumente (Berlin, 1904); edited by M. voN
See pages 275-289.
456 Geometrical Optics, Chapter XII. [§316.
where
J^ — c — X, Ji == c — X,
(w- i)(c-x) -mp _ (n - i){c - x) - tup
^^^ n-i ' •^»"" n-i
I x + {n- i)c
2 , X ^x + tp.
nui n
Thus, we find :
n I n — I n n j
n' 2 , 2n + I . n 2n + i , w+i j
(n — i) n — I n — i'^ n n
The value of V will be a minimum when :
n(2n + 1) 3(^+0 , ^ + 1 3
^"■2(«-i)(n + 2)^"^2(n+2)*'^2(n+2)*' j
For real values of c, we must have:
{n + i)(5n + i) ^ 4^-1 a . (^ + 0*^
. 2(2n + i) 2 2(n^ + 4» + i) ^^^^o.
+ z ^--^px -i x^^
fi n ft
VII. Sbidel's Theory of the Spherical Aberrations of the Third Orot
ART. 102. DEVELOPMENT OF SEIDEL'S FORMULA FOR THE v- kJBCC^ *'
ABERRATIONS.
316. Gaussian Parameters of Incident and Refracted Rays.^-^ .
we take the vertex A of the spherical refracting surface as the o^^V,
of a system of rectangular axes, and choose the positive direction o^^
optical axis as the positive direction of the a:-axis, then, adopting^
method of Gauss,* we can write the equations of the incident ra^ ^
follows:
5x . „ Cx ^ ^
where the two pairs of constants 5, P and C, Q are the four par^^'
eters which are used here to determine the position of the incicf^^^
ray. And, similarly, the equations of the corresponding refract^
' C. F. Gauss: Diopirische Untersuchungen (Goettingen, 1841). page 3.
S 316.] Theory of Spherical Aberrations. 457
lay may be written as follows:
B'x . „, C'x , ^
^jvhere B\ P' and C\ Q' denote the corresponding parameters of the
refracted ray. In these equations «, n' denote the absolute indices of
refraction of the first and second medium, respectively. The relations
between the parameters of the incident ray and those of the refracted
ray, whereby, knowing the former, we can determine the latter, are
obtained by Gauss very simply as follows:
The abscissa of the incidence-point B is:
AD = r(i — cos ip) = 2r'Sin -,
where D designates the foot of the perpendicular let fall from B on
the optical axis, and where r = AC denotes the abscissa of the centre
C of the spherical surface, and <p = Z B CA denotes the central angle.
Since the point B is common to both the incident and refracted rays,
the value a: = r(i — cos tp) must satisfy both sets of equations; and,
consequently, we obtain:
2 — r • sin* — +P = 2^ r . sin' — +P',
n 2 n' 2
2 — r • sin — + 0 = 2 — 7 r • sin — + Q'.
n 2 ^ n 2 ^
(340)
Moreover, let H, H' designate the points where the incident and
refracted rays, produced if necessary, cross the transversal plane per-
IDendicular to the optical axis at the centre C of the spherical surface.
Since, according to the Laws of Refraction, BH^ lies in the plane
cx>ntaining BH and BC, the three points C, H and H' must lie all
in a straight line: and if in the triangles BHC, BH' C the angles at
tly H' are denoted by Mi m'» the following relation can easily be deduced
Csee Chap. IX, formula (209)) from the law connecting the angles of
incidence and refraction:
n- CH'sin /i = »'• Cil'«sin /x'.
Accordingly, if the co-ordinates of H, H' are (r, y^^, Zj)^ (f, y\^ 2^),
respectively, we shall have :
n n
458 Geometrical Optics, Chapter XII. [§317.
and
and since
we obtain :
. B'r ^, , C'r ^,
y'f^ Zf^ CH' n-sin/x
y^ "" Z;^ " CH ""n'-sin/i"
(5
y + nP) sin M = (5 V + n'P') sin m', 1
r (34O
r + nC) sin m = (CV + w^) sin m'. J
By means of these formulae (340) and (341), we can obtain the values
of the parameters B\ P' and C\ Q' of the refracted ray in terms of
those of the incident ray/
317. Approximate Values of the Gaussian Parameters, and the
Correction-Terms of tiie 3rd Order. In the following investigation it
is assumed that the aperture of the optical system is relatively smaUi
so that none of the effective rays are very far from the optical axis.
This being the case, we may regard the parameters denoted here by
5, P, C, Q and B\ P\ C, Q' as being all small magnitudes of the first
order. For the same reason, the magnitudes sin ^, cos /i, cos /*' ^
likewise to be considered as small magnitudes of the ist order. We
propose, according to L. Seidel,* to neglect here all terms of orders
higher than the 3rd; and, hence, if A denotes a small magnitude of
the first order, we may write this as follows:
A = a + 5a;
where the small letter a denotes the part of A which is of the ist
order, and ba denotes the correction-term of the 3rd order; for, as
was explained in § 254, if the parameters of the ray are regarded as
magnitudes of the ist order, the series-developments will contain only
terms of the odd orders.
If, therefore, in the exact formulae (340) and (341) we substitute
for 5, P, etc., ft + 5ft, p + bp, etc., respectively, we shall obtain a
set of approximate formulae which are accurate except for residual
errors of the sth and higher orders. Moreover, each of the new equa-
tions thus obtained will break up at once into two others, since, evi-
dently, the terms of the ist order on one side of the equation must be
^ See also Oscar Roethig: Die ProbUme der Brechung und Refiexion (Ldpiig, i^T^)*
pages 15-26.
«L. Seidel: Zur Dioptrik. Ueber die Entwicklung der Glieder 3ter Ordnung, vdche
den Weg eines ausserhalb der Ebene der Axe gelegenen Lichtstrahles durch cin Syston
brechenden Medien. bestimmen: Asironomische NackrichUn, xliii. (1856). Nos. ioa7»
1028. 1029.
§ 318.]
Theory of Spherical Aberrations.
459
equal to the terms of the same order on the other side; and since the
same is true also in respect to the terms of the 3rd order. Thus between
the approximate values 6, />, etc., and b\ p\ etc., of the parameters of
the ray before and after refraction we obtain the following set of
relations:
.fj.f
r r r r
(342)
and between the correction-terms of the 3rd order the following re-
lations:
«^-«^=f U-«}' «2'-«2=f U-»)'
[ib'+'^) - (56 +^^) = -^(j + f ) (cosV-cosVO; [ (343)
Obviously, in the further development, it will be sufficient to obtain
the formulae for the magnitudes 6, p, 6', p' which relate to the xy-
plane; then all we shall have to do to find the corresponding formulae
for the magnitudes c, g, c', q[ which relate to the a:2-plane will be to
substitute in the first formulae the latter magnitudes in place of the
former.
318. Relations between the Ray-Parameters of Gauss and Seidel.
Instead of the GAUSsian parameters
•
B = 6 + aft, P = p + 5p and C = c + «c, Q = ff + «2,
we have now to introduce the parameters
1? = y + 5y» f = 2 + 52 and T| = y + 5y, J = j + 5j,
which are employed by Seidel(§ 255), and which are the co-ordinates
of the points P, P where the ray crosses the two fixed transversal
planes a, <r, respectively. The abscissae of the points Af , Af where the
optical axis meets the transversal planes a, a will be denoted by 1^, u,
respectively; thus,
AM = tt, AM = u;
and, similarly, for the pair of axial points M\ M' conjugate to JIf, Af,
respectively, let us put:
AM' ^ u\ AM' = u'.
S 319.] Theory of Spherical Aberrations. 461
Robert Smith for a single spherical refracting surface (Chap. VIII,
§ 194)-
Moreover, we find :
ib
r J — J\ a u J
r J — J\ u u J
and, hence, substituting these values in the first and third of formulae
(343), we obtain, after some obvious reductions:
("''^-»?)-("''^-»f)='T■"•^-(^^'.-)■
Combining these two equations so as to eliminate the difference
A(n«3y/a), we find:
'"-?-iC7^(^;-'i)(^"'-^S+-V-o^.). (347)
319. It only remains now to obtain expressions for the small magni-
tudes ipj cos/x, cos/x'; wherein, however, we need consider only the
terms of the ist order, since these alone will have any influence of the
3rd order on the value of the expression for A(n'dy/u).
In order to obtain the approximate expression for the central angle
^, we shall proceed as follows: The distance from the vertex A of the
spherical surface of the point where the incident ray meets the yz-
plane of co-ordinates is approximately equal to Vp^ -f g^, and since
the length of the arc i4 5 is equal to rtp, we may, if we neglect the mag-
nitudes of the 3rd order, put:
and, hence, we obtain:
We must now derive an expression for cos* fx' — cos' fx.
462 Geometrical Optics, Chapter XII. [§319.
The approximate equations of the incident ray BH are:
X y — p z — q
n b c '
and the equations of the straight line CH are:
y z
jc = r, — = - ,
yk «A
and hence for the angle /x between these two straight lines, we have:
cos fi = — j-j — i — .
yl + ^l
Now since the point HCr, y^t sj is a point on the incident ray, we have:
if, for the sake of brevity, we write temporarily:
u u u u
Hence, since by formula (346) :
n n
we obtain:
, n* {(y - y)Y + (z - z)Z}*
COS M - ^ _ y) v„» • Y' + Z"
2 n'* {(y - y')Y + jz' - z^Z}*
"*''' "(7-y)VV Y* + Z'
Now evidently :
u'u'
(^_y)y+(^_rtz) =,{(^,-^)r+(^-i)2
and hence we find :
n" AB
cos* !>■' ~ COS* /!= 77 fTi ' rrj-
(J-jy Y* + Z"
§ 320.] Theory of Spherical Aberrations. 463
where for brevity we write:
n
Now
u u' f n n
I.I 2 J.I ^-^
u u r n n
and thus we can write:
Lw \r n n / u \r n n )
— 2
Accordingly, we obtain finally:
n^
cosV - co8*/x = Tj3^,A--il, (3So)
where -4 is defined by (349).
320. If now we substitute in formula (347) the expressions (348)
and (350), we shall obtain on the right-hand side of the equation:
where
w \ r n n ) n
\u uu )\r n njn
"which latter expression may also be written as follows:
u \J nu J r n /
H 2 — J'^ 2 J'A— . (351)
a «w uu nu ^•'■^
Thus, we obtain finally:
464
Geometrical Optics, Chapter XII.
[ { 321.
and, similarly :
(353)
where R is defined by (351).
321. Thus far the directions of the axes of y and z are entirely
arbitrary, except that it has been assumed they are both perpendicular
to the optical axis. We may select as the xy-plane the meridian plane
which contains the point Q, and which, according to Gauss's Theory,
will contain also the conjugate point Q'.- This evidentiy will not
affect at all the generality of the treatment, and it will lead to some
simplification, inasmuch as we shall have then 2 = 2' = o. Thus
if we put 2 = o in the formulae (351), (352) and (353), we obtain the
following set of formulae:
-t(i^
u \J nu J
r n J
, y + ^ y . I ^yy t K ^
u
nu
uu
nu
The y-aberration:
<'^)-^r^'{^i-'ih
The z-aberration:
(n-hz\
n
J-R.
u
(354)
2(7 - jy
These formulae give the variations of n'dyfu, n'hzju which result in
consequence of the refraction of the ray at a single spherical surface.
In case we have a centered system of m spherical surfaces, we must
introduce the subscript k to indicate that the formulae apply to the ith
surface, and then the formulae will be written:
where
« A"
zUu-JuY ' «
-.j'-zr'Ru,
^-^l^M4).-^^'r/(0J
+
J
Theory of Spherical Aberrations.
465
ow if A, h denote the incidence-heights of a pair of paraxial rays
nating from the axial object-points Jlfj, Afj, respectively, we have,
Robert Smith's Law (§ 194) :
u.
Ut
u.
Ui
u,
eover, we have also Seidel's Formula (Chapter VIII, § 195):
J _ J ^ *i^ (J ^ J) ^ ^^^^ ^'^^* "" ^^^
***»
****
«itti
we introduce these relations in the above equations, we shall
in the following formuls:
2 (u, - «,)* » M *r *i \««/* ]
(U, -Ml)' 1*1 rtl \»«/»J
yf»i „ ..1 /** ''it rt A /' ^ ^
(355)
31
466
Geometrical Optics, Chapter XII.
[§32L
Now
nU-«y;;_i nl'dyl K^^ni^^^dy,
k-i
Ui
Ui
K
u
k-l
and, hence, if we suppose that the object, situated in the first medium,
is free from aberration, so that the object-point Pj coincides with ftt
and therefore
5^1 = «2i = o.
we find:
that is,
and, similarly:
«1 *«*=! *i V « A'
Let us now employ the following abbreviations:
' (356)
322.]
Theory of Spherical Aberrations.
467
o that we may write finally :
+!
yiyf
2(u, -«,r ' '*« ^ ' -
y?
_„S*L cv.
2 (Ui - u,y "' A.
5^;
? = ;? xF^iI"*^ — 7 TsWiUi i — o
A.
^2 (u, -«,)»"! 'A.^
(357)
322. Conditions of the Abolition of the Spherical Aberations of
he 3rd Order. The expressions denoted here by 5', 5", 5"', 5'^,
S^ are practically equivalent to the famous five sums of Seidel,
dthough Seidel's expressions in their final form are different from
The equation 5' = o will be recognized as the condition of the
abolition of the spherical aberration at the centre of the visual field;
that is, the condition that the axial points ilfp M*^ shall be a pair of
'aberrationless" points (§ 265).
The equation 5" = o is at the same time the condition of the fulfil-
ment of Abbe's Sine-Condition (§ 284) and of the abolition of Coma
;§3i3)-
The condition of the abolition of the astigmatism of narrow oblique
Dundles of rays is 5"' = o (§ 302), and the conditions necessary for
I plane, stigmatic image are 5"' = o and S^^ = o; see formulae (332),
} 304.
Finally, the condition that the image shall be without Distortion
s 5^ = o; see formula (311) or formula (312), § 292.
The image will be perfectly faultless (except for residual errors of
he sth order) provided all five sums 5', 5", 5'", 5^^, and 5^
ranish together, and these five conditions are necessary if the image
s to have this degree of perfection in every respect.
Seidel's Formulae (357), which give the magnitudes of the y- and
:-aberrations of the 3rd order in the image-plane a'„, are derived by
V. Kerber* by the employment of Kerber's Formulae given in Chap-
;er IX, §§ 214, 216 for the refraction of a ray at a spherical surface;
prherein the trigonometrical functions are replaced by their series-
' A. Ksrber: Beitraege zur Dioptrik. Zweites Heft (Leipzig. 1896); pages 9-15.
468 Geometrical Optics, Chapter XII. (§323.
developments. Kerber's process is also given by Koenig and ^^"^
RoHR^ in their treatise on the Theory of Spherical Aberrations.
ART. 103. EUMINATION OP THE MAGNITUDES DENOTED BT h, u^
323. The natural determination-data of an optical system arer *^
radii (r) of the spherical surfaces, the thicknesses (d) of the interve^^^^^
media and the refractive indices (n). If in addition to these ma^-^P^'
tudes we know also the positions of the object and of the stop, wl
is equivalent to knowing the values of Wj and Up we can compute
values of the two systems of magnitudes A, u and h, u which occu:
Seidel's Aberration- Formulae (357). So long as these formulae
to be employed to investigate the defects of an image produced b;
given optical system, they answer their purpose excellently, ^xit:::^^ ^^
case the problem is to design an optical instrument which is to iv^^^
certain prescribed conditions, the fact that the equations contain Xr ^^^
sets of magnitudes which are not independent of each other is a c^^^*^'
advantage which must be got rid of by eliminating one of these s^^*^^
of magnitudes by means of the other set. In Seidel's final forms ' °
the aberration-formulae the magnitudes denoted here by A, u do n^c^^^^
appear.
This elimination is performed with the aid of the two formulae (15 *^SS'
and (156) of Chapter VIII, which are also due to Seidel, and whic=^^^'
by the introduction of the convenient abbreviating symbol T, m^^^^
be written here as follows:
^_^^j,g d^r y (3S
The magnitude denoted here by T depends only on the initial value--
of the magnitudes A, u and h, u. If we introduce, also by way o'^
abbreviation, another symbol and write:
formulae (358) may be put in the following forms convenient for
direct application to the expressions contained in the aberration-
' A. Koenig und M. von Rohr: Die Theorie der sphaerischen Aberrationen: Chapter
V of M. VON Rohr's Die Theorie der optischen InstrumenU, Bd. I (Berlin. 1904). page
317-323.
8)
>f
§ 323.] Theory of Spherical Aberrations. 469
formulse (357):
Proceeding now to eliminate the magnitudes /ij^, u^ from the express-
ions under the summation-signs in the formulae (356), we remark,
in the first place, that the sum 5^ which is the expression of the Co-
efficient of the Spherical Aberration along the axis, does not contain
these magnitudes at all. Passing, therefore, to the Coma-Co-efficient,
we obtain from the second of equations (360) :
Eind hence:
5" = |:44A(-).-(.+Jrj. (36.)
The first of the two terms on the right-hand side of this equation is
the co-efficient S^ which is concerned with the spherical aberration
along the axis. If the optical system satisfies Abbe's Sine-Condition,
it must be spherically corrected for the object-point Mi (§277
and § 279); that is, 5^ = o; consequently, the formula for Abbe's Sine-
Condition, which is identical with what Seidel has called the Fraun-
BOFER' Condition (§ 284), is:
Again, we find :
hence, for the Astigmatic-Co-efficient:
5"' = |rfi^(.+^J'A(i\; (363)
and, since
we find also:
5'^ = .|^/f '-a(^). (364)
470 Geometrical Optics, Chapter XII. I § 324.
The cx)-efficients of the expressions for the curvatures of the two
image-surfaces formed by the infinitely narrow pencils of meridian and
sagittal rays can be obtained by combining the two equations (363)
and (364).
Finally, since
and
we have the following expression for the Distortion-Co-efHcient:
ART. 104. REMARKS ON SEIDEL*S PORMULiB: AND RSFBRSNCES
OTHER GENERAL METHODS.
324. In a masterly discussion of his formulae, Seidel draws al^^^
number of important conclusions of a general kind, which, howc^^ *
can only be referred to here very briefly. Thus, for example, he pocJ^
out that it is impossible (except in certain special cases that h^^^
comparatively little practical interest) to construct an optical appa — ^'
tus which will produce a correct image of the 3rd order for all distanc
of the object. If it is required to form such images of objects at .^
distances, in addition to Seidel's five equations we shall have otli^^f^
conditions also, one of which, known as Herschel's Equation, is, ^^
general, in curious contradiction to the so-called Fraunhofer- ^^ ^^
Sine-Condition expressed by formula (362) : so that the two conditio^^^-^
can be satisfied at the same time only in particular cases, one of whic::^^
is that the image shall be of the same size as the object.
An image of this degree of perfection even in the case of one sped^^^
object-distance can only be attained by combining in the system c^'
lenses a sufficient number of separated surfaces. If the distances h^'
tween the spherical surfaces are all so small as to be negligible (^^
that in the formulae we may put d^ =0), it is easy to show that the
conditions of the abolitions of all the errors of the 3rd order are a5
follows:
2)7* -A — = o, (abolition of aberration along axis);
S/. A =0, (abolition of comatic aberration) ;
nu
§ 326.1 Theory of Spherical Aberrations. 471
njA^ = ni^i, S - A - =0, (condition of plane, stigmatic image) ;
n^ — nj = o, (abolition of distortion).
This last condition is compatible with the condition n^u^ = n^Uy^ only
in case the optical system is a plane mirror or an infinitely thin plate
of glass: and, hence, for an optical system which shall produce images
of the 3rd order it is necessary that some of the rf's at least shall be
different from zero.
325. In connection with the excellent exposition of Seidel's theo-
ries which is given by Professor Silvanus P. Thompson in an
appendix to his English Translation of Dr. O. Lummer's Beiiraege zur
photographischen Optik^ he directs attention to a remarkable memoir
published by Finsterwalder' in 1892, wherein the author, employing
Seidel's Formulae, derives the equation of the Focal Surface^ which
is the envelope of the bundle of emergent rays which have their origin
at a point outside the optical axis of a centered system of spherical
surfaces, and proceeds then to show in a very simple and elegant
manner how the definition of the image and the distribution of the
light in it depends on the extent of the visual field and on the aperture
of the system and also, in the case when the image is real, on the
position of the focussing screen.* Finsterwalder not only obtains
by his method results which are in complete accord with those of
Seidel, but, as Professor Thompson states, he has "also investigated
the distribution of the light in the coma, and its changes of shape
when the position and size of the stop are changed".
326. With regard to other general methods of investigation in
Optics, the following paragraphs, also quoted from Professor Thomp-
son's chapter on "Seidel's Theory of the Five Aberrations", may be
appropriately inserted at this place :
> O. Lummer: Beiiraege sur photographischen Optik: Zft. f. Instr,, xvii (1897). 208-
219; 225-239; 264-271.
Silvanus P. Thompson: Translation of Otto Lummer's Contributions to Photographic
Optics (London, 1900).
' S. Finsterwalder: Die von optischen Systemen groesserer Geffnung und groesseren
Gesicbtsfeldes erzeugten Bilder: Muench, Abhand. der k. bayer. Akademie der Wiss, II
O.. XVII Bd., Ill Abth., 519-587. Published also separately in Muenchen in 1891 by
G. Franz.
' Seidel himself had already determined the equation of the Focal Surface, without.
however, showing how the equation was obtained. See Seidel's paper entitled: Ueber
die Theorie der caustlschen Flaechen, welche in Folge der Spiegelung oder Brechung von
Strahlenbuescheln an den Flaechen eines optischen Apparates erzeugt werden: Gelehrte
Anteigen k, bayr. Akad, d. Wiss., xliv (1857). 241-251. See also a letter written by
Seidel to Kummer. and published, so Finsterwalder states, in Sitzungber. der k. Akad.
d. Wiss, SM Berlin, 1867.
472 Geometrical Optics, Chapter XII. [ § 326.
"Remarkable as these researches of von Seidel are, it is of interest
to note that an even more general method of investigation into Vet^
aberrations had been previously propounded. This is the fragm^^^'
ary paper of Sir W. Rowan Hamilton,^ introducing into op^^
the idea of a 'characteristic function* [see §39], namely the timet^^^..
by the light to pass from one point to another of its path. True.h^" ^^ .
not work out the relations between the constants of his formulae ^^ .
the data of the optical system. Yet the method, as a mathemaC^^
method of investigation, is unquestionably more powerful. It
recently, and independently, been revived by Thiesen,* whose eq^^^
tions include those of von Seidel.
"The latest development of advanced geometrical optics is due^^
Professor H. Bruns,' who has shown that in general the formi-^ -^
that govern the formation of images can be deduced from an originatS^ tr^i^S
function of the co-ordinates of the rays — a function termed by h -^^^*,
the eikonal — by diflferentiating the same, just as in theoretical mec:^^^
anics the components of the forces can be deduced by differentiati^ -^ "^
from the potential function. Bruns's work is based upon the theoi:^^^'^!^
of contact-transformations of Sophus Lie. But as yet neither tl^ ^^ ,
formulae of Bruns nor those of Thiesen have been reduced to suc^ ^^ .
shape as to be available for service in the numerical computation ~
t»
optical systems.'
In this connection it may be stated that the applications of Seidel*^ ^^
aberration-formulae to the calculation and design of optical system:,
are attended with much difficulty, and on this account practical
opticians seem still to prefer to resort to the methods of trigono--^^'^
metrical calculations of the paths of the rays, whereby with relatively"*-^ .
less trouble they arrive at safer results and are also able to keep track
more easily of the effects of each single surface. The complete solu-
tion of the Seidel formulae is indeed only possible in the case of sys-
tems of comparatively simple structure. The greatest practical value
of these general 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 particular
> On some Results of the View of a Characteristic Function in Optics, B. A. Report
for 1833. p. 360.
« M. Thiesen: Beitraege zur Dioptrik: Berl Ber.f 1890; 799-813. See also: Uebcr
voUkommene Diopter: Wied. Ann. (2) xlv (1892), 821-823; Ueber die Construction
von Dioptern mit gegebenen Eigenschaften: Wied. Ann, (2) xlv (1892), 823-824.
Also, J. Classen: Mathematische Optik (ScHUBERTsche Sammlung 40), Leipzig, 1901,
Chapter XI entitled "Thiesens Theorie der Abbildungsfehler."
* H. Bruns: Das Eikonal: Abhandlungen der tnath.-phys. CI. der k, saechsischcn Akad,
d. Wiss., xxi (1895). 321-436. Also published by S. Hirzel. Leipzig, 1895.
i 326.] Theory of Spherical Aberrations, x 473
lystem which he aims to achieve. Concerning the use of these formulae
he reader is referred to a valuable and interesting article by A.
CoENiG, entitled Die Berechnung opUscher Systeme auf Grund der
rheorie der Aberrationen}
In a series of learned papers C. V. L. Charlier^ has given also a
lethod of investigating the spherical aberrations of a centered system
f spherical surfaces, which is said to be especially adapted to the
ractical design of optical instruments. But it is impossible here to
o more than merely refer to this work.
' See Chapter VII (pages 373-408) of Die Theorie der optischen Jnstrumente, Bd. I
Berlin, 1904), edited by M. von Rohr. See also A. Kerber's Beiiraege zur Dioptrik,
iblished in Leipzig from 1895 to 1899.
• C. V. L. Charlier: Ueber den Gang des Lichtes durch ein System von sphaerischen
nsen: Upsala, Nova Acta, xvi (1893), 1-20; Zur Theorie der optischen Aberrations-
rven: Astr. Nachr., cxxxvii (1895). No. 3265, 1-6; Entwurf einer analytischen Theorie
r ConFtruction von astronomischen u. photographischen Objectiven: VierUljahrsschrift
r astronomischen Gesellschaft, 31. Jahrgang (1896), Leipzig, pages 266-278. See also a
ipcr by R. Steinheil: Ueber die Berechnung zweilinsiger Objektive: Zft. f. Instr., xvii
897). 338-344. in which the writer says that '* Die Arbeit des Hrn. Charlier bedeute
nen Schritt vorwaerts."
CHAPTER XIII.
COLOUR-PHENOMENA.
I. Dispersion and Prism-Spectra.
ART. 105. INTRODUCTORY AND HISTORICAL.
327. Relation between the Refractive Index and the Wave-
Length. In the preceding chapters it has been tacitly assumed that
the index of refraction (») of an isotropic optical medium was a con-
stant magnitude; which assumption was permissible so long as we
were concerned only with light of some definite kind or colour. The
length (X) of a light-wave depends on two factors, the speed of propa-
gation (v) and the vibration-number or frequency (N), according to
the familiar formula:
X = vjN.
Light of a definite colour is characterized by a definite value of the
frequency iV, which is not altered when the light is refracted from
one medium into another. On the other hand, the speed (v) with
which the light is propagated is different in different media, and,
consequently, the wave-length (X) must vary also. However, if we
select some standard medium (§ 24), as, for example, the free ether of
empty space (wherein also light of all colours is propagated with the
same speed), the wave-length of the light in this medium may he
employed also to characterize the colour of the light. In this chapteTi
therefore, the symbol X will be used to denote always the toave-lenifk
of the light in vacuo.
The refractive index of a given medium is a function of the wave-
length X; so that we may write:
n = /(X).
The exact character of this relation has never been definitely ascer-
tained, although a number of formulae have been proposed. The
earliest and best known of such formulae is the one suggested by
Cauchy,* as follows:
B C
n^ A +^ + ^4 + ---»
where A, B, C, etc., denote constants depending on the nature of the
medium and diminishing rapidly in magnitude as we proceed to the
^ A. L. Cauchy: Mhnoire sur la dispersion de la lumihre; published in Prague in 1836.
474
328.] Colour-Phenomena. 475
ligher terms of the series. The formula shows that the waves of the
horter wave-lengths are the more highly refracted. In media which
xhibit the so-called phenomenon of ** anomalous dispersion'* it is, how-
ver, not true that the shorter waves have the higher indices of re-
faction, so that the formula is by no means general; but within
ertain limits it is found to represent fairly well the results of experi-
lents. An investigation of the experimental data in regard to this
latter shows that, in general, as many as three coefficients A, B, C
^ill be required in order to express completely the relation between
and X for all optical media; although, as Schmidt^ has shown, in
le case of a number of substances, the relation may be right well
(pressed by a series with only two constants.
We see, therefore, that until we specify the kind of light that is
eing used, the refractive index of a medium is a phrase without mean-
ig; for a medium has just as many indices of refraction as there are
ifferent kinds of light. If, for example, a given straight line is the
3mmon path of rays of two or more kinds of light, these rays will,
I general, be separated by refraction and made to take different
outes when they enter a new medium. This phenomenon is called
dispersion of the Light, sometimes called also the * 'chromatic dis-
lersion".
328. Newton's Prism-Experiments and the Fraunhofer Lmes of the
Solar Spectrum. The discovery and explanation of the fact that the
ight of the sun is composite and consists of light of a great variety of
olours is unquestionably the greatest of Newton's contributions to
optical science. Admitting the rays of the sun through a small circular
opening in the window-shutter, Newton caused these rays to pass
hrough a glass prism, and was surprised to find that the image on the
opposite wall, instead of being a circular spot of white light (as was
►roduced before the interposition of the prism in the path of the beam)
fas an elongated spectrum^ with vivid colours, and about five times
s long as it was broad. Newton's remarkable series of prism-
xperiments was begun in the year i666: a complete description of
hem was afterwards published in his treatise on Optics.' He was
id to conclude that sun-light is not homogeneous, but is composed
f rays of different colours, some of which are more refrangible than
thers, the red rays being the least refracted and the violet rays the
lost refracted ; so that the coloured spectrum varied by impercept-
)le gradations of colour from red at one end to violet at the other;
* W. Schmidt: Die Brechung des Lichts in Glaesern (Leipzig, 1874).
• Isaac Newton: Opticks: or a treatise of the reflexions, refractions, inflexions and
4ours of light (London, 1704). The ditscovery of Dispersion and the explanation of the
>lour8 of the Spectrum was communicated to the Royal Society in 1672.
476 Geometrical Optics, Chapter XIII. [ § 328.
the order of the colours (as they were distinguished by Newton)
being red, orange, yellow, green, blue, indigo and violet.
The important practical problem of abolishing, if possible, the chro-
matic aberrations of optical instruments, espedally in the case of the
telescope, raised the question as to whether the dispersions of dif-
ferent substances were such as to allow of combinations which neutral-
ized the dispersion without at the same time neutralizing the refraction.
Newton himself conceived that he had proved by experiment (Opticks,
Book i. Part ii. Prop. 3) that achromatism involved necessarily the
abolition of ray-deviation also; so that in an achromatic combination
the emergent rays must needs be parallel to the corresponding inci-
dent rays. Newton concluded, therefore, that it was impossible to
produce an achromatic image by refraction, and it was this error
that **made him despair of improving refracting telescopes and ^^
him to turn his attention to the application of mirrors to these inst^'
ments".* Newton's authority on such questions was so great tt^
for a long time his view was accepted as settling the matter.
EuLER,* approaching the subject from a theoretical stand-pa^^*
and basing his argument on the erroneous assumption that the hur^^^
eye is an achromatic combination of lenses, deduced the correct c^^
elusion that such combinations were possible, and calculated the
ditions that were necessary therefor, although he lacked suffici^^
experimental data. In 1754 Klingenstierna,* in Sweden, succeec::^'
in showing by a combination of two prisms not only the deviation
the rays without dispersion, but also the dispersion of the rays wi
out deviation.
Heath* states that the mistake in Newton's experiment (abo
referred to) **was first discovered by a gentleman of Worcestersh
named Hall, who made the first achromatic telescope"; but th^
"this discovery was allowed to fall into oblivion, until the experime
was again tried by Dollond, an optician in London, who found th
the dispersion could be corrected without destroying the refractio-^
and therefore that Newton's conclusion was not correct". In 175
Dollond was able to construct an achromatic telescope by the use
two kinds of glass called **crown glass" and **flint glass", of whicf^
the former is the weaker in respect to both refraction and dispersion -
* See Heath's Geometrical Optics (Cambridge, 1887). Art. 179.
*L. Euler: Sur la perfection des verres obiccUfs dea lunettes: Mhn. de Berlin, iii
(1747). 274-296.
' S. Klingenstierna: Anmerkung ueber das Gesetz der Brechung bei Lichtstrablen
von verschiedener Art, wenn sie durch ein durchsichtiges Mittel in verschiedene andere
gehen: Svensk. VH. Acad, Handle xv (1754), 300-306.
* Heath's Geometrical Optics (Cambridge. 1887). Art. I79-
i 328.1 Colour-Phenomena. 477
[n this combination the convergent lens was made of crown glass and
the divergent lens of flint glass.
Dollond's success revived interest in the question, and a number
>f mathematicians, for example, Euler, Clairaut and D'Alembert,
proceeded to investigate formulae for calculating optical systems; but
» long as the numerical constants of the different kinds of glass were
lot available, these labours were necessarily unproductive; and no
arther progress worth recording was achieved until the era of Fraun-
lOFER (1814), whose brilliant researches marked the dawn of a new
lay in optical science. By looking through a prism at a very narrow
lit, formed by the window-shutters of a darkened room, Wollaston^
lad detected in 1802 that the solar spectrum was crossed by dark
lands; but it was not until these so-called Fraunhofer Lines were
adependently re-discovered by Fraunhofer' in a far more thorough
nd scientific manner that their real significance and value were recog-
lized.
In the Prism-Spectroscope, such as was afterwards used by Kirch-
[OFF and BuNSEN, the source of the light is an illuminated slit placed
parallel to the edge of the prism in the focal plane of a collimating
ens; whereby the rays incident on the first face of the prism are
endered parallel. If, after emerging from the prism, the rays are
made to pass through a second convergent lens, there will be formed in
the focal plane of this lens a series of images of the slit, each image
corresponding to light of a definite colour or wave-length (§ 327).
If the slit is illuminated by monochromatic light, there will be only
one image, but if the incident rays are composed of light, say, of two
kinds, of wave-lengths Xi and X,, we shall have two slit-images side
by side and more or less separated from each other depending, among
other things, on the magnitude of the interval Xj — X,. If
X2 = Xi -f dXi,
the two slit-images will be immediately adjacent to each other, and
they may partly overlap and blur each other. If the slit is illumi-
nated by white light emitted originally by an incandescent solid.
For example, the light of an electric arc, there will be formed in the
*W. H. Wollaston: A method of examining refractive and dispersive powers, by
^matic reflection: Phil. Trans., ii (1802), 365-380.
' A preliminary report of Fraunhofer's work was communicated to the academy of
idences in Munich in the years 18 14 and 181 5. See also: Joseph Fraunhofer: Bes-
immung des Brechungs- und Farbenzerstreuungsvermoegens verschiedener Glassorten,
n Bezugauf die Vervollkommnung achromatischer Fernroehre: Gilberts Ann., Ivi (1817),
^64-313-
478 Geometrical Optics, Chapter XIIL [ § 329.
focal plane of the receiving lens a continuous spectrum, consisting of an
innumerable series of coloured images of the slit of every gradation
of shade from red to violet, one image for each of the infinite varieties
of the light that is emitted by the source. A definite wave-length (X)
is associated with each colour, and to each wave-length there corre-
sponds also a definite value of the refractive index («), which increases
continuously from its greatest value for the extreme red light to its
least value for the extreme violet light.
However, the solar spectrum obtained when the slit is illuminated
by sun-light is not continuous, as Newton supposed, but is crossed
by a vast number of dark bands parallel to the slit, corresponding,
as we know now, to those radiations which are absent from the light
that comes to us from the sun. It would be more correct to say that
these dark places indicate a relative deficiency of intensity of certain
definite kinds of light in what we call sun-light. These Fraunhofer
Lines are irregularly distributed over the entire extent of the solar
spectrum, and although their actual positions will be altered if we
replace the prism of the spectroscope by another one of different
material, the order of the lines and of the coloured intervals between
them is always the same, so that any line can be readily recognized.
The great importance of these lines for optical science consists, as
Fraunhofer was quick to perceive, in the fact that each line corre-
sponds to a definite wave-length of light, and hence we can employ
them in the determinations of the refractive indices of a substance.
The more conspicuous of the lines in the different parts of the spectrum
were designated by Fraunhofer by the capital letters of the Latin
alphabet from A to H\ the violet end of the spectrum, as nearly as
he could locate it, being designated by the letter /. The indices of
refraction of a given substance for rays of light of wave-lengths corre-
sponding to the Fraunhofer Lines i4, B, C, • • • are usually denoted
by the symbols w^, Wj, Wcr, • • • .
329. The Jena Glass. Now that it was possible to determine accu-
rately the optical properties of different media, the great obstacle in the
way of perfecting optical instruments so as to fulfil as far as possible the
theoretical requirements was found to be the lack of suitable kinds of
glass. This deficiency, which Fraunhofer and others had tried to
supply by the manufacture of new kinds of optical glass, began to be
realized more and more with the development of the microscope and in
the construction of the photographic objective. Finally, in i88i, Pro-
fessor E. Abbe, who has been rightly called the ''Galileo of the
Microscope", undertook, in conjuction with Dr. O. SCHOTT, a s>'S-
§329.] Colour-Phenomena. 479
tematic investigation of the * 'optical properties of all known substances
which undergo vitreous fusion and solidify in non-crystalline trans-
parent masses".* The success of these ingenious and exhaustive ex-
periments, in which entirely new and remarkable compositions of
glass were obtained by using a far greater number of chemical elements
than had ever been essayed before and, especially, by employing in
the manufacture both boric and phosphoric acids as well as the usual
silicic acid, was almost immediate and beyond all expectations, and
I few years later (1886) the **Glastechnisches Laboratorium** of
\fessrs. ScHOTT und Gen., in Jena, was established, where the now
j^orld-famous "Jena Glass" is manufactured.
The important practical problem, suggested first by Fraunhofer, of
>roducing pairs of crown glass and flint glass such that the dispersions
rf the different parts of the spectrum should be as nearly as possible
«iual for both kinds of glass, with the object of abolishing or diminish-
ng the so-called secondary spectrum (Art. 112), was successfully solved
>y the labours of Abbe and Schott. Another problem of not less
mportance consisted in producing a large variety of kinds of optical
^lass of graduated properties, so that in the design of an optical system
he optician might be able to find a combination more or less exactly
idapted to his particular requirements. This result was likewise
ichieved.
The optical properties of the different varieties of glass are de-
scribed in the Jena-Glass Catalogue with reference to five bright
lines of the spectrum which are all easily obtained by artificial sources
of light, viz.: The red potassium line, which is very close to the
Fraunhofer Line A, and which may be designated, therefore, by
A'; the yellow sodium line which coincides with the Fraunhofer
Line D; and, finally, the bright lines of the spectrum of hydrogen, the
first two of which are identical with the Fraunhofer Lines C and F,
while the third, designated by G', is very near the Fraunhofer Line
G. The wave-lengths of the light corresponding to these lines are as
follows:
' Sec E. Abbe und O. Schott: Productionsverzeichniss des Klastechnischen Labora-
toriums von Schott und Genopsen in Jena: published as a " prospectus " in July, 1886,
ind re-printed in Gesammelie Abkandlungen von Ernst Abbe. Bd. II (Jena, 1906), 194-
201. See also: E. Abbe: Ueber neue Mikroskope: SUT,.-Ber. Jen. Ges. Med, u. Natw., 1886.
107-138; reprinted in Gesammelie Abkandlungen, Bd. I (Jena. 1904). 450-472.
Especially, see S. Czapski: Mittheilungen ueber das glastechniache Laboratorium
in Jena und die von ihm hergestellten neuen optischen Glaeser: Zft. f. Inst., vi (1886),
293-299 and 335-348. See also the very complete history of optical gla?s-manufacture
given in M. Von Rohr's valuable and learned work. Theorie und Geschichle des pho-
lographischen Ohjektivs (Berlin. 1899). 325-341.
§329.] Colour-Phenomena. ' 481
The index of refraction of each kind of glass for the D-hine is given
in the first column of the table. Since this line is about at the bright-
est part of the spectrum, and since also this radiation is especially
convenient to obtain, the value of itj) is usually employed to charact-
erize the refrangibility of an optical medium.
The next column of the table gives the value of the so-called mean
iispersion, that is, the difference (np — nc) of the indices of refraction
"or the light corresponding to the lines C and F. This difference is
ibout proportional to the length of the spectrum, since the greater
>art of the visible spectrum is included between the lines C and F.
The third column gives the value of the magnitude
.= -^^^^^. (366)
rhe numerator of this fraction is the difference between the mean
ndex of refraction («2>) of the material and the index of refraction of
lir (n = i); which difference occurs so frequently, for example, in the
bnnuls of Thin Lenses. The reciprocal of this fraction, viz., i/v^
s called the relative dispersion; and, hence, the greater the value of v^
the smaller will be the relative dispersion. It will be remarked that
the series of glasses are arranged in the table with respect to the
magnitude of this constant v from the greatest value of v to its least
value in descending order. This is due to the fact that the optical
character of a given specimen of glass is seen most clearly by a con-
»deration of its i^-value.
The values of the partial dispersions for the three intervals A'-D,
D-F and F-G't which appear in the next three columns of the
table, enable us to perceive also the behaviour of the glass as regards
dispersion; so that we can compare the dispersions of two different
kinds of glass for the various parts of the spectrum with a view to
iscertaining the degree of achromatism that is possible by a combina-
don of the pair. For this same purpose also the value obtained by
lividing the partial dispersion of one of these intervals by the
iralue of the mean dispersion nr — «c is entered in the same col-
unn immediately under the value of the partial dispersion to which
t belongs. It will be seen from the table that the partial dispersions
)f different kinds of glass are, in general, quite different. Moreover,
x>mparing the spectra produced by two different optical media, we
nay find that the dispersion of the red region is relatively greater,
ind at the same time the dispersion of the blue region is relatively
ess, for the first substance than the corresponding partial dispersions
32
§ 330.] Colour-Phenomena. 483
index and a greater dispersive power than his crown glass. But a
high refractive index does not necessarily imply also a great dispersive
power, as was formerly supposed, as will be seen by comparing the
following pair of products of the Jena-Glass Laboratory:
«D
*J7—»C
0.1309.
0.7260.
Densest Baryta Crown
Extra Light Flint
I.0II3
I.S39S
0.01068
0.01 143
Here it will be remarked that the more highly refracting of these two
specimens is at the same time the less strongly dispersive one of the
pair. It is easy to understand how the production of different kinds
of glass with such properties as we have noted marked an epoch in
optical engineering and made possible the extraordinary perfections
of modern optical instruments.
330. Combinations of Thin Prisms.
In connection with this subject it will be of service to consider here
briefly two combinations which have been mentioned above and which
have great practical importance, viz., the case of deviation without dis-
persion and the case of dispersion without deviation. Suppose that we
have two prisms made of substances whose indices of refraction for
light of a given wave-length X may be denoted by n and «'; and, for
the sake of simplicity, let us assume, for the present, that the refracting
angles fi and ff are exceedingly small, and also that the rays which we
employ meet the surfaces of the prisms at very nearly normal incidence.
Of course, these assumptions are widely different from the conditions
that we have in an actual case; but that need not affect the object
which we have here in view.
If € denotes the total deviation of the ray of wave-length X that is
produced by the pair of prisms in combination, then, by formula (28)
of § 72, we can write:
6 = (n - i)/3 -f (n' - i)/3'. (367)
The variation dt of the deviation in consequence of a variation of
the wave-length of light from the value X to the value X + ^ will be a
measure of the dispersion. Thus, by differentiation, we obtain:
d€ = P'dn + P'-dn'. (368)
(i) If the combination of the two thin prisms is to be achromatic
with respect to light of wave-lengths X and X + dX, then we must
put d€ = o, and, hence, the condition of achromatism requires that
331.] Colour-Phenomena. 485
)ersion for the interval comprised between the values X and X + ^
vill be expressed analytically by the following formula:
dx" dn[ dk ■•'aw; d\ "^ ■" ■*'a»; dk ' ^^^^
i¥herein it is assumed that there is no dispersion of the light in the
irst medium (dt/dn^ = o). In this formula m denotes the number of
^fracting surfaces. The partial differential co-efficients de/dn are
lot only functions of the refractive indices «i, n[, n\^ etc., but these
nagnitudes depend also on the forms and position-relations of the
lefracting surfaces; whereas the magnitudes dnjdk depend only on
he form of the function connecting the variables n and X (§ 327) and
>n the values of the numerical constants of the medium in question;
ind, hence, it has been suggested that the differential co-efficient dn/d\
night properly be called the * 'characteristic dispersion" of the medium.
Accordingly, the problem of finding the dispiersion in the case of a
:iven optical system consists in determining the values of the magni-
udes defdn for each medium. We propose now to investigate this
>roblem in the case of a system of prisms with their refracting edges
.11 parallel.*
According to formulae (43) of § 93, we have, for the refraction at
he ifeth plane refracting surface of a ray lying in a principal section
)f the prism-system, the following equations:
«;-sina; = «;_!• sin a^, 1
r (373)
€* = «*-«*; J
•
vhere n^ denotes the index of refraction of the {k + i)th medium for
ight of the given wave-length X ; a^^ a]^ denote the angles of incidence
ind refraction at the *th surface; and tf, denotes the angular deviation
rf the ray produced by this refraction. Moreover, if fij^ denotes the
efracting angle of the *th prism (that is, the dihedral angle between
he ith and the (k + i)th refracting planes, as in § 93), we have
Iso:
«t+i = "1-/3*; (374)
' See S. CzAPSKi: Theorie der optischen Inslrumente nach Abbe (Breslau, 1893). pages
45. foil.; H. Kaysbr: Handhuch der Spectroscopic, Bd. I (Leipzig. 1900). Arts. 297,
>I1.; and F. Lobwe*s " Die Prismen und die Prismensysteme *' which is Chapter VIII
f Die Theorie der optischen Instrutnente, Bd. I (Berlin. 1904). edited by M. Von Rohr;
ages 455-457.
§ 332.] Colour-Phenomena. 487
// the first and last media are both air^ we can put:
and, if, moreover, there is no initial dispersion, we can put also:
dai = o, 5€ = — da^.
Accordingly, under these circumstances, we have:
^'=S^*S"^«;'[ (377)
cosa^+i = i. J
332. Dispersion of a Single Prism in Air.
Assuming that there is no initial dispersion {da^ = o) and that the
prism is surrounded by air, so that we may write:
ni ^ n^ ^^ If W| ^ w,
and putting m » 2 in formulae (377), we obtain for the dispersion of a
single prism:
dt sinj3 dn
ex""cosa;-cosa;dx' ^378)
where
0 = a'l — a,
denotes the refracting angle of the prism. According to this formula,
the dispersion of a single prism for light of wave-lengths X and X + dX
. depends not only on the value of the refractive index n but also on
the refracting angle /3 and on the angle of incidence a,. When the
angle of emergence a^ = 90®, the dispersion 56/5X = 00 has its maximum
value. As the angle a^ decreases (in consequence of a corresponding
variation of the incidence-angle ofj), the dispersion Sc/^X diminishes
until it reaches a minimum value, after which farther decrease of the
angle ai is accompanied by increase of the dispersion. The fact that
for a certain value of the incidence-angle a^ the dispersion dt/dX is a
minimum was first remarked by J. F. W. Herschel,^ who found also
that this position was different from that of minimum deviation.
The dispersion will be a minimum for that value of the incidence-
angle ai for which cos a[ * cos a, is a maximum; but the solution of this
* J. F. W. Herschbl: Article " On Light *' in the Encyc. MetropolUana (London,
Z828).
§ 334.] Colour-Phenomena. 489
we shall have:
«i = - «i = «s = ' • • = «2.-i = - «i.- ='•• = - «!•»
«i = — «2 = «s = • • • = «2.-i = - «2i =••• = "" ««;
and evidently now each of the m terms within the brackets on the
right-hand side of the above equation will be equal to
sin a'l
cosai
the signs of the terms being all positive. Since, moreover, n«sin a[ =
sin ttp we obtain for the magnitude of the dispersion under these cir-
cumstances:
where w/2 denotes the number of glass prisms.
Comparing this result with formula (380), wie see that the dispersion
of a train of glass prisms adjusted as above described is equal to the
sum of the dispersions of the prisms taken separately. This formula
(381) is a useful one, because in actual practice the prisms of a prism-
spectroscope are usually adjusted in this way.
334. Achromatic Prism-Systems.
The condition that the rays of wave-lengths X and X -f rfX shall
emerge from the optical system along the same identical path is
Be/dX = o; in which case the deviations of the two rays will have the
same value €. This is the case of Deviation Without Dispersion.
Assuming, as is usually the case, that the incident rays are them-
selves without dispersion, we find by formula (377) the following
€umdition of achromatism of a system of prisms for light of wave-
lengths X and X + rfX:
4-1 ^_;k cos a^
cos««+i == I-
(382)
If, for example, the system is composed of three prisms (w = 4),
and if the first, third and last media are air («i = Wj ~ ^1 = ^)» so
that the system consists, let us say, of two glass prisms separated by
air, the combination will be achromatic for light of wave-lengths X
and X -f dX, provided we have:
sin j3|*cos Oj-cos a^'dn\ -f- sin jSj-cos a\ -cos a^'dn^ = o, (383)
490 Geometrical Optics, Chapter XIII. [ § 354.
where
denote the refracting angles of the two glass prisms. In this equati^^^
the magnitudes a'p aj, a, and a^ are connected by the relations:
sin a J = n[ -sin (aj - /SJ, sin a, = n^ -sin (a^ + jSj) ;
and, hence, if the first prism is supposed to be known, that is, if tl
magnitudes denoted by n[ and fi^ are given, and if also the angle
incidence ai and the index of refraction n^ of the second prism ai
given, there will still remain two arbitrary magnitudes, viz., jSj and
Under these circumstances, therefore, the condition expressed by equs
tion (383) may be satisfied in either of two ways, as follows: (i) An::
arbitrary value may be assigned to the refracting angle (fi^ of th»
second glass prism, and we shall have then to determine the correspond^&zd-
ing value of the angle a,, that is, we shall have to find the orientation ok — m>{
the second glass prism with respect to the first in order that the com^rrja-
bination may be achromatic; or (2) Assuming an arbitrary value <g ^f
the angle a^, we may then employ equation (383) to determine wha^^^
value the refracting angle of the second glass prism must have,
two glass prisms may even be made of the same kind of glass (n[ = n,"
As a concrete illustration, let us assume that the ray of wave-lengtzrij
X traverses each of the glass prisms symmetrically, that is, with min^^
mum deviation; in which case we have the following relations (§ 71) r
«l = "" «2» «l = ~ «2 = ^ » «S = "" «4 = ~ •
Introducing these values in equation (383), we obtain the condition of
achromatism for this special case in the following form:
\^
H
5ir
dn^ dn^
tan a. — 7- + tan ou -^-7- = o,
where
sin aj = »! 'Sin — , sin a, = nj-sin — .
The simplest case is that in which the system is composed of ^
glass prisms (usually cemented together along their common face), the
first and last media being air, so that n^ = «i = I. For this case
m = 3,andby formula (37s) we find:
X, = sin a\ 'dn[, X^ = sin d^-dfi^ — sin oc^'dni, X, = — sin aj-(IV»
Colour-Phenomena.
491
ce, employing formulae (377), we obtain after several obvious
tions:
sin j3| • cos aj • dn\ + sin /Jj • dn^
56 =
(384)
COS aj'cos ofj
J 01 = a\ — a2» ft = «a — «s
:e the refracting angles of the prisms.
therefore, this combination is to be achromatic for light of wave-
lis X and X + rfX, we must have:
sin /3, -cos a^'dn\ + sin ft-dwa = o- (S^S)
leans of this formula, the angle ft of the second prism can be
lated, so soon as we assign the value of the angle of incidence {a^
he value of the deviation-angle (c).
I. Direct-Vision Prism-System. We may consider briefly also
nportant practical case of a system which is constructed so that,
ugh the rays of wave-
hs X and X + dX are dis-
d, the standard ray of
-length X traverses the
tn without being devi-
(€ = o) — prism-system
on directe. If, as in the
U case considered in
, the system is com-
I of two cemented glass
IS (w = 3) surrounded
r, the dispersion is given
rmula (384) above. If
specialize the problem
arther by supposing that the ray of wave-length X emerges from
y^stem in a direction perpendicular to the third plane refracting
ce (Fig. 149), we have evidently the following system of equa-
for this Direct-Vision Combination:
Fig. 149.
DiRECT-Visiow Combination of Two Cbbcbittbd
Glass Prisms. The portion DEF of the Crown-Glass
Prism can be cut away, as no rays traverse this part.
See diagram of AMici-Prism. Fiff. 153.
n,
= ni = I, a^ — ci^- o, aj = ^j, '
n,
sin 02 = "^ sin /S,,
w,
€i = - €2 = «! - ttp
nj'sin €(
f
€• = € = O,
tan a| -
^2 = «i — /3i;
»l«COS€i — I
(386)
§ 336.] Colour-Phenomena. 493
is not merely to increase the dispersion dtjd^ but rather to obtain
as nearly as possible a pure spectrum, wherein the light to be ana-
lyzed is resolved into its simplest components, so that at any given
part of the spectrum the difference d\ of the wave-lengths that are
superposed shall be as small as possible. The spectrum is composed
of a series of images of the slit, each of which corresponds to light
of a definite kind or colour; and if the apparent width of the slit-
image for light of wave-length X is greater than the angular dis-
persion dt of the rays of wave-lengths X and X + dX, the slit-images
corresponding to these two radiations will partly overlap each other,
and, accordingly, the spectrum in this region will be more or less
impure. If the slit itself were a mathematical line of light, and
if there were perfect coUinear correspondence between object and
image, the spectrum would be absolutely pure, and the image of the
line-source for a given wave-length of light would be itself a line
occupying a perfectly definite and distinct position in this ideal spec-
trum.
Evidently, the purity of the spectrum will depend on the width of
the slit-image and on the length of the spectrum. Let ha denote the
apparent size of the slit as viewed from the first face of the prism, and,
similarly, let ha' denote the apparent size of the slit-image for light
of wave-length X as viewed from the last refracting plane. The greater
the dispersion Sc/^X of the light of wave-lengths X and X + dX, and
the smaller the magnitude of the angular width ha' of the slit-image,
the greater will be the purity of the spectrum at this place in it; and,
hence, as a measure of the purity of the spectrum, Helmholtz^ proposed
that we employ the following expression:
P = 1^ : ««'. (389)
What is here meant by the image of the slit is not the actual or "dif-
fraction" image, but the image as determined on the assumption of the
rectilinear propagation of light according to the laws of Geometrical
Optics.
*H. VON Helmholtz: Handbuch der physiologischen OpUk (zweite umgearbeitete
Auflage. Hamburg u. Leipzig, 1886), p. 394. In regard to this subject see also:
H. Kaysbr: Handbuch der Spectroscopies Bd. I (Leipzig, 1900), pages 305 & foil, and
pages 548 & foil.;
S. CzAPSKi: Theorie der optischen Instrumente nach Abbe (Breslau, 1893). pages 148
& foU.; and
F. Lobwb's " Die Prismen und die Prismensysteme *' in Die Theorie der optischen
InslrumenU, Bd. i (Berlin, 1904). edited by M. von Rohr, pages 448 & foil.
494 Geometrical Optics, Chapter XIII. [ § 337.
If, therefore, we leave out of account the diflfraction-effects, then,
according to formula (49) of § 97, the angular width of the slit-image
formed by a system of prisms, which is composed of m plane refract-
ing surfaces and in which the first and last media are both air («i =
n^ = i), is given by the following formula:
where ^aj denotes the angular width of the slit itself. Accordingly,
assuming that there is no initial dispersion, and employing therefore
formulae (377), we find for the purity of the spectrum produced by a
system of prisms, as it is defined by equation (389), the following
expression:
wherein the term cos Oq must be put equal to unity always. Thus, we see
that the magnitude P depends not merely on the properties of the
prism-system but on the width of the slit itself; and, hence, the purity
of the spectrum, as defined by Helmholtz, is not by itself a sufficient
criterion for the comparison of the spectra produced by different
prism-systems.
337. Purity of Spectrum in Case of a Single Prism. Consider the
spectrum of a single prism surrounded by air. In this special case let
us write according to our custom:
lt| = Wj *^ If W| = It.
According to formula (375), we have here:
X^ = sin «! 'dn, Xg = — sin a^'dn;
and, hence, putting m = 2 in formula (391), we obtain for the purity
of the spectrum of a single prism in the region corresponding to the
light of wave-length X:
^ sin/3 dn I /^x
cos a| • cos cKj aX oai
where /3 = a'l — a^ denotes the refracting angle of the prism. We see,
therefore, that the purity of the spectrum of a single prism surrounded
by air is proportional to the so-called ''characteristic dispersion
(§ 331) of the prism-medium and is inversely proportional to the width
of the slit. The purity of the spectrum varies also with the angle of
i 338.] Colour-Phenomena. 495
ncidence (ai), and when the prism is adjusted so that the incident
ay "grazes'* the first face {a^ = 90°), we find P = 00. The ad van t-
.ge of using the prism in this position is enormously discounted,
lowever, on account of the great loss of light by reflexion. As the
.ngle «! decreases from the value aj = 90°, the purity P diminishes
Jso until it attains a minimum value determined by that value of the
ingle «! for which the function cos a^-cos aj is a maximum. This
^alue of «! may be found by a process entirely analogous to that
employed by Thollon in ascertaining the position of minimum dis-
persion, which was alluded to in § 332; in fact, by merely interchang-
ing the symbols a[ and ofj ^^ formula (379), we obtain immediately
the following relation :
aj = - «Vi; (393)
nrhich gives approximately the position of the prism for minimum
purity of the spectrum in the region corresponding to the light of
Rrave-length X.
If the prism is adjusted in the position of minimum deviation for
the rays of wave-length X (which, in addition to other advantages,
is also the position in which the loss of light by reflexion is least),
nre must introduce in formula (392} the following relations:
/3 = 2a[ = — 202;
jvhereby we obtain:
_ 2 dn I
P = - tan «! TT r— ;
n * d\ Sui
i result which may be derived also directly from formula (389) by
nerely remarking that for the position of minimum deviation we have
see § 86) 5a' = Sag = Saj, whereas the value of defdX is given here
)y formula (380). The purity of this part of the spectrum depends
)nly on the refractive index n, the width of the slit and the form of
he prism. It is called the '^normal purity'',
338. Diffraction-Image of the Slit. The methods of Geometrical
i)ptics alone are not sufficient to enable us to ascertain the character of
Jie slit-image; this problem involves not merely the theory of refraction
Dut the theory of diffraction also. According to this latter theory, the
image of a luminous line (or very narrow rectangular aperture) parallel
to the edge of the prism is never itself a line, but a far more complicated
effect which we have not space to investigate here, especially too as a
X)mplete exposition of the matter can be found in almost any standard
Rrork on Physical Optics. In the accompanying diagram (Fig. 150) the
496
Geometrical Optics, Chapter XIII.
[§338.
plane of the paper represents a principal section of the prism-system, of
which the traces in this plane of the first and last surfaces, MiMi and/t,^,
are shown in the figure. The source of the light is supposed to be a
small luminous line perpendicular at S to the plane of the paper. The
rays emanating from this line-source are made parallel by a "colli-
mating" lens; so that the straight line PQ = bi is the trace in the
plane of the paper of the portion of the plane-wave which is due to
Priim S>jit%m
Cetitrwl Bamg
Cefffrol Bofid
fori
Fio. 150.
Resolution op I«inb9 in Prismatic Spbctrum. The plane of the i>aper n.preaenti the ^xot
of a principal section of the prism-system. The source of liff ht is a small luminous line perpeodic*
ular to plane of paper at the point marked S, The straight lines f&xMi and mw^ are the traces in
the plane of the paper of the first and last refractinff planes, respectively. PQ ^bx^ width of betfs
of parallel incident rays ; P^(/ » bw! "■ width of beam of parallel emergent rays of wave-lenctb K
I, S^OS^* a <c » angular distance of slit-images corresponding to light of wave-lengths A and A +^
arrive later at the first surface /*,/*i of the prism-system. The straight
line P'Q' = 6^ shows the trace in this plane also of the corresponding
emergent plane-wave for light of wave-length X. A convex lens inter-
posed in the path of the beam of emergent rays will produce on a screen
situated in the focal plane of the lens an image of the slit. If this
image is investigated by the methods of Physical Optics, we find that
the image of a vertical line at 5 consists mainly of a so-called "central
band" of light of a certain finite horizontal width (which depends on
the focal length of the lens, for one thing) and of maximum brightness
along a vertical line perpendicular to the plane of the paper at the
point designated in the diagram by 5'. On either side of this vertical
median line the brightness of the central band diminishes very rapidly
to absolute darkness. There is also a series of much fainter bands
situated symmetrically on both sides of the central band, but for all
practical purposes the central band alone may be considered as the
actual and effective image of the very narrow rectangular aperture
at 5.
If the slit is illuminated by light of wave-lengths X and X + iX,
Colour-Phenomena. 497
angular beam of parallel incident rays will be resolved by the
>rstem into two such beams, one for each of the two colours,
'e shall have at the points 5' and 5" on the screen the maxima
tness of the two slit-images corresponding to the light of wave-
X and X + d\. Now, in order that these images whose central
5 are at 5' and 5" may be far enough apart to be distinguished
sye as separate and distinct images, Lord Rayleigh^ has shown
/.S'OS" = «€ must beat least equal to X/6;^, where fti], = P'Q'
ndth of the beam of emergent rays of wave-length X. The
f the beam of emergent rays will depend on the orientation of
m-system, as is evident from formula (387), and the angular
bt of the centres of the two images will depend on this also,
der, therefore, to resolve a **double line** of wave-lengths X and
it is necessary that the angular interval de shall have the
ig value at least :
X cos ot
5€ = r n ^- (394)
*i ft= I COS a*
special case when the rays corresponding to the light of wave-
i traverse the prism-system with minimum deviation^ we have,
ig to the formula at the end of § 94:
kJi cos a^
ice the condition (394) becomes in this special case:
8€ = r .
^i
case of a single prism (w = 2), formula (394) is as follows:
X COS«t'COS«,
06 = r , ? . (395)
^i cosa^-cosaj
Ideal Purity of Spectrum. According to Rayleigh's investi-
, the least value of the angular interval 5c necessary in order to
a "double line" is equal to half the width of the central band
lifTraction-image of the slit; and, hence, on the supposition that
Krt is a luminous line, the methods of Physical Optics show that
ular width of the image = 2d€ = 2X/6^. Helmholtz defines the
Raylbich: Investigations in Optics: PhiL Mag, (5) viii (1879). pages 261-274.
477-486; and (5) ix (1880), pages 40-SS- See also article " Wave Theory ",
; Encyclopedia Britannica, xxiv, 430-434.
1 340.] Colour-Phenomena. 499
The straight line PQ (Fig. 151) represents the trace in the plane of
a principal section (plane of the paper) of the plane wave-front of the
light at some instant prior to its arrival at the first face of the prism;
and the straight line P'Q' represents in the same way the position of
the wave-front of the light of wave-length X at some subsequent instant
after the waves have traversed the
prism. Similarly, also, the straight line
P"Q^' represents at this same later
instant the position of the plane wave-
front for the light of wave-length X -}-
iX. As a matter of fact, the two rays
Df wave-lengths X and X + dX, which
meet the first face of the prism at the
same point, will thereafter pursue fxo. 151.
slightly different geometrical paths; resolvino power of a prism.
but by virtue of Fermat's Minimum
Principle (Art. 11), this difference will be entirely negligible in com-
parison with the actual distances traversed by the rays; and, hence,
i¥e may consider that the rays pursue the same routes both within
ind without the prism, as represented in the diagram. Thus, for
example, if s and / denote the lengths of the ray-paths within the
Drism of the rays of wave-length X which are nearest to the refracting
xige and farthest from it, respectively, these same magnitudes will
lenote also the lengths of the ray-paths within the prism of the corre-
sponding pair of rays for light of wave-length X + dX. The refractive
ndices of the prism for light of wave-lengths X and X + (fX will be
lenoted by n and n + dn, respectively. Finally, the prism is supposed
to be surrounded on both sides by air whose dispersion is so slight as
to be negligible.
According to Art. 11, the optical lengths of the paths from P to P'
and from Q to Q' for light of wave-length X are equal ; as is true, like-
wise, with respect to the optical lengths of the paths from Q to Q"
and from P to P" for light of wave-length X -H dX. Evidently, the
optical length of the path from Q to Q" for the light of wave-length
I -f rfX is longer than that from Q to Q' for the light of wave-length
, by the amount s*dn 4- Q'Q"\ and, in the same way, the optical
mgth of the path from P to P" for the light of wave-length X 4- dX
cceeds that from P to P' for the light of wave-length X by the amount
Jfg — P"P'\ and since these excesses must be equal, we find:
(/ - s)'dn = P"P' -f- QQ" = zP"P\
§ 341.1 Colour-Phenomena. 501
3 = 0, and formula (400) reduces to the following:
dn
p = t^- (402)
If the prism is adjusted in the position of minimum deviation, and if
the entire extent of the prism is traversed by the
rays, the Resolving Power of the prism depends only
an the thickness of the prism at the ba^e.
The formulae obtained above may be extended at
once to a system of glass prisms separated from
each other by air, provided the glass prisms are
all made of the same kind of glass. For such a
system, (/ — s) in formula (400) will denote the
difference in aggregate thickness of the dispersive
material through which the extreme rays of the F10.152.
pencil have passed. If the prisms are all adjusted
so that the rays traverse them symmetrically, and if the upper extreme
ray passes through the edges of all the prisms, then / in formula
(402) denotes the sum of the "bases" of the prisms.
Schuster* remarks that "the resolving power of prisms depends on
the total thickness of glass, and not on the number of prisms, one
large prism being as good as several small ones". Thus, all the prisms
shown in Fig. 152 "would have the same resolving power, though they
would show very considerable differences in dispersion".
341. According to Cauchy's Dispersion-Formula (see § 327), we
may write approximately:
n = i4 + B\-^;
and, hence, by formula (402) the resolving power of a prism of base
/ is:
/>=-25~s. (403)
We may say, therefore, roughly speaking, that the Resolving Power of
a prism is inversely proportional to the cube of the wave-length ; and
hence the Resolving Power is much greater for light of short wave-
lengths. The Resolving Power of a grating is the same for all wave-
lengths, and hence a grating-spectroscope is not so good as a prism-
spectroscope for resolving the ultra-violet lines of the spectrum.
The value of the co-efficient B in formula (403) depends on the
* A. SCHUsrBRi An Introduction to the Theory of Optics (London. 1904). p. 144.
502 Geometrical Optics, Chapter XIII. [ § 341
material of the prism. Lord Rayleigh gives the following calculation
of the thickness / of a prism made of the "extra dense flint" glass of
Messrs. Chance Bros, that is necessary in order to resolve the
Fraunhofer double Z>-line. The indices of refraction of this glass
for light corresponding to the Fraunhofer lines C and D are:
«c= 1.644866, «D = 1.650388;
and the wave-lengths in centimetres are:
\c = 6.562-10"*, Xd = 5.889-10"*.
Thus, we find:
Now
tip — no _j^
B = r=2 — 7^ = 0.984-10 .
X* X X* X*
zB d\ zB zB'dK
For the D-line: X = 5.889- 10"*, d\ = o.oo6-io"* (difference between
Di and D^.
Accordingly, we find / = 1.02 cm., which is, therefore, the necessary
thickness of a prism of this material in order to resolve the double
Z>-line. Moreover, Lord Rayleigh, testing this result by experiment,
found that, as a matter of fact, a prism-thickness of between 1.2 and
1.4 cm. was needed for this purpose.
342. The Resolving Power of a system of prisms of diflferent
materials is given by the following formula:
where s,, and /^ denote the lengths of the ray-paths of the extreme rays
between the ifeth and the (* + i)th plane refracting surfaces.
For example, in an Amici Direct- Vision Prism (§335), consisting
of two prisms of crown glass cemented to a prism of flint g^, as
represented in Fig. 153, we have:
5, = 5, = o, 52 = 5, /i = /s = ^. <a = 0, w[ « ni;
and hence by formula (404) :
In this combination the dispersion of the crown glass b opposed to
§ 343.] Colour-Phenomena. 503
Pig. 153.
Resolving power of Amicx " Direct-Vision ** Prism.
that of the flint glass, and the Resolving Power of the system is not
great.
II. The Chromatic Aberrations.
ART. 108. THB DIFFERBNT KINDS OF ACHROMATISM.
343. When a ray of white light is incident on a refracting surface,
it will, in general, be resolved at the point of incidence into a pencil of
coloured rays, since, as we have seen, the index of refraction depends
on the colour of the light. Thus, for example, if P designates the
position of a radiant point emitting, say, red and blue light, and if Bi
designates the position of a point on the first surface of a centered sys-
tem of spherical refracting surfaces, and, finally if t designates a trans-
versal plane perpendicular to the optical axis, then corresponding to
an incident ray proceeding along the straight line PB^t there will be
a red image-ray which will cross the plane t (really or virtually) at a
point P'and likewisealso a blue image-ray which will cross the plane ir at
a point P\ which, in general, will be different from the point P'.^ Since
the positions of the focal points and the magnitudes of the focal lengths
of an optical system depend also on the indices of refraction of the
media traversed by the rays, and since the values of these indices
depend on the colour of the light, it is evident that the same optical
system will produce as many coloured images of a given object as
there are colours in the light emitted by the object; and, in general,
also, these images will be formed at different places and will be of
different sizes. The entire series of images may be described as an
image affected with chromatic aberrations. Even if the image were
' In the following pages of this chapter, whenever we have to deal with two colours,
the letters and symbols which relate to the second colour will be distinguished from the
corresponding letters and symbols which relate to the first colour by means of a dash
written immediately above the character. It is true this same method of notation was
used in the theory of astigmatism to distinguish between the meridian and sagittal rays;
but no confusion is likely to occur on this account, and for various reasons it is convenient
to use this same device here in a new sense.
504 Geometrical Optics, Chapter XIII. [§343.
otherwise perfect and free from all the so-called spherical aberrations,
the definition of the image will generally be seriously impaired on
account of colour-dispersion alone, and hence one of the most important
problems of practical optics is to correct, as far as possible, the chro-
matic aberrations and to produce an optical system that is more or
less achromatic.
The problem here mentioned is still further complicated by the fact
that not only are the fundamental characteristics of the optical system
(viz., the positions of the focal points and the magnitudes of the focal
lengths) dependent on the indices of refraction of each medium, but
the various spherical aberrations, which are encountered when the
rays are not infinitely near to the optical axis, are likewise functions
of the indices of refraction; so that we may have also chromatic varia-
tions of the spherical aberrations, even though the optical system has
been corrected so that the focal points and the focal lengths are the
same for all wave-lengths of light. As a matter of fact, all the prop-
erties of a centered system of spherical refracting surfaces are depend-
ent in some way or other on the indices of refraction, and hence they
are all variable with the colour of the light. The term "achromatism"
by itself is, therefore, entirely indefinite, for the system may be achro-
matic in one sense and not at all so in other senses. For example*
the images corresponding to the different colours may all be formed
at the same place, and yet be of different sizes; or the system may be
achromatic with respect to Distortion or with respect to the Sine-
Condition, etc., and at the same time affected with colour-dispersion
in a variety of other ways. Obviously, it will not be possible to correct
all these different kinds of chromatic aberrations at the same time;
and, in fact, in order to have a distinct image (which is the primary
aim of an optical instrument), this will not be necessary, as some of
the chromatic aberrations are comparatively unimportant, depending
on the purpose which the apparatus is intended to fulfil.
An optical system which produces two coloured images of a given
object at the same place and of the same size is said to be compktdy
achromatic for these two colours. The images of other colours will,
however, generally be different as to both size and position, and the
effect on the resultant image usually appears in a coloured margin or
^'secondary spectrum" (§ 329).
But usually the best we can do is to contrive to obtain a partial
achromatism of some sort, and especially one of the two following
kinds: achromatism with respect to pla^e (so that the two coloured
images, although of unequal sizes, are both formed in the same image-
1 344.] Colour-Phenomena. 505
plane), or achromatism as to magnification (so that the two coloured
mages, although differently situated, are of equal size). In many
rases, indeed, it will be found quite sufficient to effect a partial achroma-
jsm of one or other of these two kinds. Thus, for example, it is
essential that the coloured images formed by the objectives of tele-
KX>pes and microscopes shall be situated as nearly as possible at the
iSLine place; whereas, since the images do not extend far from the
>ptical axis, the unequal colour-magnifications are comparatively neg-
ig^ble. On the other hand, in the case of the eye-pieces of these
nstniments, whose particular office is to produce extended images of
:he small images formed by the objectives, the main point is to ob-
tain achromatism as to magnification, whereas the slight differences
in the distances of the coloured images are of relatively small import-
ance. As a rule, it may be stated that for an optical system which
produces a real image, it is more desirable to have achromatism with
respect to the place of the image; whereas if the image is virtual,
achromatism with respect to the magnification is likely to be the more
important requirement.
In the following pages it is proposed to develop the formulae for the
numerical calculation of the more important of the chromatic aber-
rations, and to determine the conditions that are necessary in order to
abolish or diminish them. In this investigation it will be assumed
(except in the brief treatment, at the end of the chapter, of the chro-
matic variations of the spherical aberrations) that we are concerned
only with paraxial rays, so that between the object and its image in
any one definite colour there is complete coUinear correspondence.
As to notation, let us state here that the change of a magnitude x
in consequence of a finite variation of the wave-length of the light
from the value X to the value X will be indicated by the capital letter
D written immediately in front of the symbol of the magnitude, thus:
Dx = X — X.
ART. 109. THE CHROMATIC VARIATIOlfS OF THE POSITION AND SIZE
OF THB IMAGE, IN TERMS OF THE FOCAL LENGTHS AND
FOCAL DISTANCES OF THE OPTICAL SYSTEM.
344. Let A I and A^ designate the positions of the vertices of the
first and last surfaces, respectively, of a centered system of m spherical
refracting surfaces, and let F, £' and F, E' designate the positions
on the optical axis of the focal points of the system for the two colours
corresponding to light of wave-lengths X and X, respectively. At a
506 Geometrical Optics, Chapter XIII. [§515.
point M on the optical axis, erect the perpendicular MQ\ and let
M'Q' and M'!^' be the GAUSsian images in the two colours correspond-
ing to the object MQ. We shall employ here the following symbob:
z' = A^E\ z'^A^E\ Dz' ^z'-z'^E'E',
u = A,M, u'^A^M\ u'^A^M\ Du' ^u'-u'^M'M',
x^FM, txf ^ E'M\ X = FM, x! = E!M\
Denoting the focal lengths of the system for the two colours by
/, e' and /, e\ we have the following set of equations (§ 178):
xx' = fe\ xx' = Je\ 1
^ y x* y i 'J
where F, T" denote the lateral magnifications for the two colours.
Now
X ^ X - Dz, 3c' = x' + Du' - Dz'\
and, hence, eliminating x and x' and solving for Du'^ we obtain from
the upper pair of equations (405) :
D«- = Z>/ + ^-^' + ^^^. (406)
Similarly, eliminating x from the lower pair of equations (405), ^
obtain :
These Difference-Formulae, which are given by Koenig,* give the
variations of the position and magnification of the image of an object
corresponding to any arbitrary variations of the fundamental charact-
eristics of the optical system.
345. We may consider several special cases as follows:
(i) If the optical system is achromatic with respect to the position
of the image, then we shall have Du' = o, and from equation (406) we
obtain in this case the following quadratic equation with respect to **•
x^Dz' + {DUe') - Dz'Dz'}x+fe''Dz = o;
' See A. Koenig: ** Die Theorie der chromatischen Aberrationen ", Chapter VI ^
Die Theorie der opUschen Instrumente, Bd. I (Berlin. 1904), edited by M. voN ROOK*
See p. 345.
§ 345.1 Colour-Phenomena. 507
so that, in general, there will be two positions of the object for which
its images in the two given colours will be formed in the same trans-
versal plane; but if the roots of the quadratic are imaginary, there
will be no position of the object for which the system can have this
kind of achromatism. In the special case when
Dz = Dz' = DUe') = o,
the quadratic equation will be satisfied for all values of x\ and we
have then what is sometimes called stable achromatism with respect
to the place of the image. If, also, the first and last media are ident-
ical, we have Z>K = o, and the system will be completely achromatic,
in the sense in which this term was defined in § 343.
(2) If the system is achromatic with respect to the lateral magni-
fication, then Z>F = o. This condition is satisfied by x = i = 00,
F = F = o; and, also, according to equation (407), by the following
value of XI
If Df =^ Dz ^ o, then DY = o for all values of x.
If gt g denote the ordinates of the points where an incident ray
emanating from the axial object-point M crosses the primary focal
planes corresponding to the two colours, then
X :x == g :g;
and if the two magnifications F, F are equal, then
X :x = f :J;
and hence: _
g :| =/:/;
and, since (§ 178) _
g = ^'-tan^', g = g'-tan e\
where $', W denote the slopes of the pair of coloured image-rays cor-
responding to the given incident ray, we have :
e'-tan e' :e' tan's' = / : J.
If, therefore, the first and last media are identical, so that e' = — /,
e' = — 7» we obtain 6' = 6'; and hence when Y == Y, and n = «',
n = «', the pair of coloured emergent rays corresponding to a given inci-
dent ray will be parallel.
S 346.] Colour-Phenomena. 509
where the symbols with dashes above them relate to the same incident
ray for light of wave-length X. Accordingly, we find:
-(")--('^">
where, according to Abbe's system of notation, the operator A, written
before an expression, indicates, as always heretofore, the difference
of the values of the expression before and after refraction. Thus, for
the Jfeth surface of a centered system of spherical refracting surfaces,
we have:
fn'Du\ , fDn\ , ,
Since the distance measured along the optical axis between the Jfeth
and the {k + i)th spherical surfaces is
we have:
Moreover, if Aj^, 5^ denote the incidence-heights, for the rays of the
two colours, at the ifeth surface, then :
Taking note of these relations, and multiplying both sides of equation
(408) by Afc-Sfc, and then giving k in succession all integral values
from k ^ I to JE; = m, and adding together all the equations thus
formed, we obtain:
and if the object is without dispersion, that is, if Du^ = o, we derive
finally the following formula for the so-called chromatic longitudinal
aberration} of a centered system of m spherical surfaces for a given
position of the axial object-point:
^ ' «L'^l*^Aft K -r f J^^\ / X
Du^ = - -^'— Hr T^ ' Jk' ^\ — I • (409)
n„ *=i/»« hnt \ ^ Jk
* See A. Kobnig: Die Theorie der chromatischen Aberrationen: Chapter VI of Die
Theorie der opOschen Instrumente, Bd. I (Berlin. 1904), edited by M. von Rohr. See
page 341.
M
Qtlour-Phenomena.
511
differentiation, we obtain:
dY
Y
Z
dX
X
n.
«
m
"\
(41 S)
iiave evidently:
dti
and d4_i = Uj^^ — w^,
if we assume that the object is without dispersion {du^ » o), we
% therefore, write formulae (415) as follows:
du^
dY ^ dn^ _ dnl , ^^^ _ y
^*-i
«.
n.
««*
d^ dn, dn^ .
IT = — r* + 2
-I
►»-l»
*-i
dttU;
(416)
ch forms are more convenient than equations (415) in case we have
letermine the chromatic variations of the magnification for the
nal case when du^ = o.
: the first and last media are identical (n, = n^^ we have, ac-
ling to formulae (416):
I dX ^ du^ ^
dtt;_i,
the condition that dZ =» o is also in this case identical with the
ditions that (fKand dX shall vanish. Under these circumstances
§ 34S)» ^^^ pO'i^ of coloured emergent rays corresponding to a given
dent ray will be parallel. If, moreover, the distances {d) between
1 pair of successive surfaces are all vanishingly small (system
ifinitely thin lenses in contact), the condition
lentical with the condition du' = o.
§ 349.1 Colour-Phenomena. 513
of the second order of smallness, then
/ M'JirW . M'M'\
DY
Y
M'M' M'M'
~ M'M' + 19' W
U
"m
M'M'
'M' - (a'M' -M'M'\
'M''^^^ \ M'M'-a'M' )
M'M"
so that if the abscissae, with respect to the vertex of the last surface
of the centered system of spherical surfaces, of the points M', M' and
M' be denoted by «', u' and u', respectively, we have approximately:
DY Du' , ,
(417)
u'-u''
This formula is given by Koenig.'
AKT. 111. CHROMATIC VARUTIOITS IH SPECIAL CASBS.
349. Optical System consisting of a Single Lens, surrounded on
both sides by air.
If the optical system consists of a single lens (tn <= 2), surrounded
on both sides by air (», = »i = i, w| = «), the formulae for the chro-
matic longitudinal aberration, as derived from the difference-equation
(409)1 is as follows:
D«, = -«,.«,(^^.^-/.-/,j-; (4,8)
which vanishes when wj = o or «, = o (neither of which cases need
be considered), and also when:
A,^i/, = A,^,/,. (419)
Since (cf. § 126) A|/i = ai, A,/, = ^^2 = «i» ^his condition may also
be written as follows:
"a *i
> A. Koenig: Die Theorie der cfaromatischen Aberrationen: Chapter VI of Die
Theorie der opiischen Instrumente, Bd. I (Berlin. 1904), edited by M. von Rohr. See
page 34S-
34
'.] Colour Phenomena. 515
It of the lens (A') for the colour corresponding to the value n
.cides with the secondary focal point of the first surface of the
(JSJ) for the colour corresponding to the value n; and, similarly,
: the point A' (secondary principal point of the lens for the colour
esponding to the value n) coincides with the point E[ (secondary
1 point of the first surface of the lens for the colour corresponding
he value n).
1 the case of a lens surrounded by the same medium on both sides,
lateral magnifications for the two colours corresponding to the
les n and n are as follows:
mce we derive the following difference-formula:
Y u\ nr^ — (n — i){u\ — d)
- I. (422)
order, therefore, that the chromatic variation of the lateral magni-
ition shall vanish (Z>F = o), we have the following condition:
(n — n)u\"il\ — {(n — i)u\ — (n — i)u\]d , .
u = ^ — — ^^-^ ^ — ; (423)
that, provided the position of the axial object-point is assigned,
I the radius of the first surface of the lens is given, together with
values of n and n, this formula (423) gives fj and d as linear funct-
3 of each other.
Eliminating u[ and u[ by means of the relations given above just
;r formula (420), we derive from formula (423) the following equa-
i:
fjd - V
«i - nn{r, - r,) - {nn - i)d' ^^^^^
jreby for a given lens we can determine the position of the object-
le which for two given colours is portrayed by the lens in images
<\U3l dimensions,
f the thickness of the lens is
«w(r, - fj)
d =
nn — I
n, according to formula (424), we find ttj = 00; so that for such a
3 the object must be situated at infinity. We find also that the
516 Geometrical Optics, Chapter XIII. [§350.
focal lengths are equal; thus:
«n — I fyf,
/ J (n-i)(n-i) r,^r,'
and It may also be shown that the primary principal point of this lens
(A) for the first colour coincides with the secondary focal point of
the first surface of the lens {E[) for the second colour; and, similarly,
that A coincides with E[.
350. Infinitely Thin Lens.
If the lens is infinitely thin, we may put:
hi ^ hi = h^ ^^ h^
in formula (418) ; and if also we introduce here our special notation for
the case of an infinitely thin lens (§ 268) , so that jc = i /«, xf = i /«', and
i' = i/w' denote the reciprocals of the intercepts on the axis of the inci-
dent and emergent paraxial rays for the two colours; and if 9= i// de-
notes the "power" of the lens for the colour corresponding to the value
«; and, finally, if c= i/fp c'= i/r, denote the curvatures of the bound-
ing surfaces of the lens, then the lens-formulae may be written as
follows:
^ = (n — i)(c — c')» x' = X + ip.
Accordingly, we derive the following formula for the chromatic lonp-
tiidinal aberration of an infinitely thin lens :
where
n — I
'^ Dn
(4*6)
is the magnitude defined in § 329. The reciprocal value ijv is some-
times called the dispersor of the lens, and the quotient tpfv is called the
dispersive strength of the lens.
If M\ M' designate the positions of the image-points in the two
colours corresponding to the axial object-point Af, then
Du' = M'M' = - u'u' - ;
and hence (except in the merely theoretical case when «' = u' * 0)
it is not possible to abolish the chromatic longitudinal aberration 01
an infinitely thin lens.
•51.] Colour-Phenomena. 517
When the incident ray is parallel to the axis, we have u' = /, «' = /,
d in this case:
7
the lens is convergent Q>o)^ and if, as is assumed throughout,
> w, then Du' <o; so that the more refrangible (blue) rays are
ought to a focus £' nearer to the lens than the focus £' of the less
'rangible (red) rays; which is the case known as Chromatic Under-
rrection. The opposite effect, viz.. Chromatic Over- Correction^ is ex-
cited by an infinitely thin divergent lens.
The chromatic aberration of the lateral magnification of an infinitely
in lens is:
DY^—r^ -• (427)
U UP NT-#'
351. Chromatic Aberration of a System of Infinitely Thin Lenses.
The formula for the chromatic longitudinal aberration of a system
infinitely thin lenses with the centres of *their surfaces ranged along
e and the same straight line may be derived very easily from the
neral formula (409). However, as we use here a special notation
rresponding to that employed above in the case of a single infinitely
in lens, and as the subscripts attached to the symbols relate now to the
mber of the lens, and not to the number of the refracting surface, it is
3re convenient to deduce the formula independently. Consider,
erefore, the ifeth lens of the system, and let -4^ designate the position
the optical centre of this lens. Also, let Jli^i, M]^ designate
e points where a paraxial ray (emanating originally from the axial
ject-point Af,), of colour corresponding to the value w^, crosses the
tical axis before and after refraction, respectively, through this lens;
diet
d, similarly, for a paraxial ray of colour corresponding to the value
which emanates from the same axial object-point Mj, we shall
ite:
^ = «A = ^*-^-i» 5^ = «1 = ^iMl'
moting the strength or "power" of the feth lens for the two colours
tpf, and ^jt, we derive easily the following difference-relation :
Dxl = Dx^ + Dip^ = Dx^ + - ,
§ 352.] Colour-Phenomena. 519
and made of material whose y- value is:
^ = ^ (431)
will be equivalent to the system of m thin lenses in contact, in respect
both to the refraction and the dispersion of paraxial rays. Thus, for
a given value of ip and for a given value of y^, we may vary the strength
^4 of the jfeth lens, so that v has any arbitrary value whatever. This
fact is of immense importance to the optician; for although he has at
his disposal only a limited series of optical glasses with values of v
ranging from, say, y = 20 to y = 70, yet in case he needs for a certain
lens a certain i^-value that does not belong to any actual kind of glass,
he has merely to substitute for this lens a suitable combination of two
or more lenses.^
If the system of thin lenses in contact js achromatic, we have:
S?-o; (432)
in which case the l^-value of the combination is equal to infinity.
352. In particular, let us consider, first, a system consisting of Two
Infinitely Thin Lenses in Contact (m == 2). The condition of the
abolition of the chromatic aberration with respect to the place of the
image, as derived from formula (429), is as follows:
— I = 0.
If the differences of the curvatures of the two surfaces of the lenses
be denoted by Q and Cj, that is, if
C, = Ci - c\, C^^ C2- c\, (433)
then, since
the condition above may be expressed in the following form also:
C,Pni + CjPnj = o. (434)
* See A. Kobnig: Die Thcorie der chromatischen Aberrationen, Chapter VI of Vit
Theorit der optischen InstrumetUe, Bd. I (Berlin. 1904), edited by M. von Rohr. See
page 349*
520 Geometrical Optics, Chapter XIIL [ § 353.
Moreover, if
^ = ^l + ^2
denotes the strength of the combination of lenses, we find that, in
order to fulfil the condition of achromatism, the strengths of the two
lenses must be as follows:
also we obtain:
(p ip
According to these results, a system of two infinitely thin lenses
in contact can be achromatic only in case the I'-values of the two lenses
are different; that is, the two lenses must be made of different kinds
of glass. Moreover, the focal lengths of the lenses must have opposite
signs, and must be inversely proportional to the y-values. The focal
length of the combination has the same sign as that of the lens with
the greater y- value ; thus, the strength {<p) of the achromatic combina-
tion will be positive when the strength of the positive lens exceeds
that of the negative lens. For a prescribed value of ^, the strengths
of the individual lenses are smaller in proportion as the i^values are
smaller, and also in proportion as the difference of the y-values is
greater. By selecting two kinds of glass with as great difference of
y-values as possible, we can reduce the differences of the curvatures
of the two surfaces of the lenses.
353. System of Two Infinitely Thin Lenses Separated by a Finite
Interval {d).
According to formula (428), the chromatic longitudinal aberration
of a system of two infinitely thin lenses is as follows:
which, since
-4^*2 — L T ' - »
Hi hx Vi
(where d = A^A^ denotes the distance between the lenses), may be
written in the following form :
§ 353.] Colour-Phenomena. 521
It can be easily shown that, except in the case when the second
lens is so placed as to separate the two coloured images Mj, Mj formed
by the first lens, the strengths of the two lenses must have opposite
signs in order that Dx^ shall vanish. If we may assume that the
variations Dip^^tpjv^ and D<p2 = <P2/v2 are so small that their product
D<Pi'D<P2 may be neglected, the condition that the system of two thin
lenses shall have the same focal point for the two colours is found
by putting Xi = o in the equation Dxz = o; which condition is, there-
fore, as follows:
If the two lenses are made of the same kind of glass (V| = V2)* ^^^
condition that -Djc^ = o becomes:
Ai^i^i + h^2^2 ^ 05
which is analogous to the condition, expressed in formula (419), for
the abolition of the chromatic longitudinal aberration of a thick lens.
The angular magnification (Z) of a system of two infinitely thin
separated lenses is:
X| X2
whence, since
^i — *i "r ^i» ^2 ~ ^2 "t- ^2» ^2 "" I — v' ./£ »
we obtain:
X^Z = {Xy + ^i)(i - ^jd) + ^2-
Accordingly, the formula for the chromatic magnification-difference
(DZ) is as follows:
•"l "» \Vi ViVi V2J Vt
(436)
Assuming here also that the variations Dip^ and D(P2 are so small that
we may neglect their product, we can write the condition of the aboli-
tion of the chromatic magnification-difference as follows:
Mi^iyv}*-%^%-- (437)
which, in general, will not be independent of the position of the object.
If this condition is fulfilled, not only will the two coloured rays emerge
§ 354.] Colour-Phenomena. 523
AUT. 112. THE SECONDARY SPECTRUM.
354. In consequence of the so-called "irrationality of the dispers-
ion" (§ 329), it is evident that even when an optical system has been
designed so as to be achromatic with respect to one pair of colours,
it will, in general, not be achromatic for all colours. If, for example,
It is contrived so that the red and blue rays are again united in the
image, there will still be, perhaps, a slight dispersion of the yellow
and green rays; that is, an uncorrected residual colour-error, or, as
Blair termed it, a ^'secondary spectrum*'.^ The residual chromatic
longitudinal aberration of a system of thin lenses which is achromatic
with respect to two principal colours has been thoroughly investigated
by Koenig;^ whose methods will be used in the brief and elementary
treatment of this matter that is given here, wherein we consider only
the secondary spectrum of a system of thin lenses in contact.
If n, n denote the indices of refraction of an optical medium for
the two principal colours with respect to which the optical system is
assumed to be achromatic, the difference » — n = Dn is called the
fundamental dispersion of the medium; and if n denotes the index of
refraction of the same medium for a third colour, the difference
^n = n — n is called the partial dispersion; and the ratio
is called the relative partial dispersion of the medium (see § 329). In
general, the relative partial dispersion /3| of one medium will be dif-
ferent from the relative partial dispersion /S, of another medium.
The ratio
n - w""^
will be the y-value of the medium for the interval from the first to
the third colour.
If the chromatic longitudinal aberration with respect to the two
principal colours has been abolished, the paraxial image-rays corre-
sponding to these two colours which emanate originally from the axial
^ See S. CzAPSKi: Mittheilungen ueber das glastechnische Laboratorium in Jena und
die von ihm hergestellten neuen optischen Glaeser: Zft. f. InsttumetUenkunde, vi (1886),
393-299, 335-348* Also, see S. Czapski: Theorie der optischen Insirumenle nach Abbe
(Breslau, 1893), pages 128-132.
• A. Kobnig: Die Theorie der chromatischen Abeirationen: Chapter VI of Die Theorie
der opiischen Ins^rumetUe, Bd. I (Berlin, 1904), edited by M. von Rohr. See pages
357-366.
§ 355.] Colour Phenomena. 525
whence we see that the smaller the difference of the relative partial
dispersions of the two kinds of glass and the greater the difference of
their y-values, the less will be the magnitude of the secondary spectrum.
A number of pairs of gla,sses fulfilling these requirements will be found
listed in the catalogue of the ''glastechnische Laboratorium" in Jena.
3SS. The character and extent of the secondary spectrum of an
achromatic combination of lenses will evidently depend on the choice
of the two principal colours with respect to which the conditions of
achromatism are satisfied. A chief consideration in the determination
of the two colours that are to be united will be the mode of using the
instrument. Thus, if it is designed to be an optical instrument in the
literal sense of that term, we shall be concerned primarily with the
physiological actions of the rays on the retina of the eye; whereas
in the case, for example, of a photographic objective, in which the
rays are to be focussed on a sensitive plate, achromatism with respect
to the so-called actinic rays will be extremely desirable.
The rays that are most effective in their actions on the retina of
the eye are comprised between the Fraunhofer lines C and jF", with
a distinct maximum of brightness in the region between the lines D
and E. If, therefore, the instrument is intended to be used by the
eye, it is usual to design it so as to be achromatic with respect to the
colours corresponding to C and F. Assuming that the system is a
convergent combination of two thin lenses in contact, we shall find
then that the focal points corresponding to the colours between C and
jF" will lie nearer to the lens-system, and the focal points corresponding
to the other colours will lie farther from it, than the common focal
point of the two principal colours. Moreover, the secondary spectrum
will be approximately least for some colour very nearly corresponding
to the D'line, which is a very favourable circumstance, since this is the
brightest region of the spectrum for visual purposes.
For the purposes of astrophotography, it is found best to obtain
as great a concentration as possible of the actinic rays, especially as
here the object will usually be of relatively feeble light-intensity.
Moreover, since the celestial objects are infinitely distant, the focus-
sing of the instrument may be done once for all, so that the eye does
not have to judge of the perfection of this adjustment, and conse-
quently we may disregard the visual rays here entirely. Such an
instrument will be designed, therefore, to unite the rays corresponding
(say) to the Fraunhofer line F and the violet line in the spectrum of
mercury. The secondary spectrum with respect to the longer wave-
lengths will be very extensive, but in this case this will not matter.
Colour-Phenomena.
527
Fio. 155.
Optical System in which thb Chromatic I«onoi-
TUDiNAL Aberration of thb Central Red and Blub
Rays is abolished.
cal difference of the chromatic aberration", whereby this colour-
s regarded as due to the variation of the chromatic longitudinal
tion (§ 346) from zone to zone,
adjoining diagrams (Figs. 155 and 156), similar to those given
MMER^ in his treatment of this subject, will help to make the
r clear. In both figures the red rays and the blue rays repre-
g; the light of the longer wave-lengths and the shorter wave-
s, respectively, are shown on opposite sides of the optical axis;
ibove the axis the two rays selected are a red paraxial ray and
edge-ray; whereas below the axis the two corresponding rays
le. For some colour,
ellow (as being op-
the most intensive),
lediate between red
lue, the optical sys-
i both cases is sup-
to be spherically
ted, so that the edge-
orresponding to this
colour cross the axis
; same point as the
1 rays of this colour. In both cases also there is spherical
correction of the red rays and spherical over-correction of the
lys. In Fig. 155, however, the chromatic longitudinal aberra-
tion of the central red and
blue rays is abolished,
whereas in Fig. 156 the
chromatic longitudinal
aberration of the red and
blue edge-rays is abol-
ished. In both illustra-
tions there is a residual
chromatic aberration,
which in the case of sys-
tems of relatively large
re may be more injurious than the so-called "secondary spec-
(Art. 112) due to the disproportionality of the dispersion-ratios
: different parts of the spectrum.
.ummer: See Mueller-Pouillet's Lehrbuch der Physik und Meteorologie, Bd.
te Auflage (Braunschweig, 1909). Art. 169.
Fig. 156.
L System in which the Chromatic I«ONGiTn-
serration op the Red and Blue Edge-Rays
;bbd.
§ 357.] Colour-Phenomena. 529
to this colour cross the optical axis (u' = AM'); and where a' is the
characteristic aberration-co-efficient employed in the series-develop-
ment in formula (273). If here we use the symbol h to denote the
incidence-height of the cone of rays of this bundle of image-rays that
meet the axis at the point N\ we have also:
AN' =^u' + —^'h\
u'
and, hence, by equating these two expressions for the abscissa AN',
we find:
A =-1/3;
as given by Kerber; who concludes, according to this process of
reasoning, that the chromatic correction should be made for the zone
whose height above the axis is A = 0.866 • H.
A comprehensive view of the performance of a given optical system
with respect to the chromatic variations of the spherical aberrations
for rays of different colours and of different incidence-heights can be
obtained by means of the so-called isoplethic curves employed by M.
VON RoHR.* The wave-lengths of the light (expressed in iiy) are laid
off along the axis of abscissae, whereas the incidence-heights (in mm.)
are represented along the other of the two rectangular axes of the
diagram; so that to each point in the plane of the figure there corre-
sponds a certain ray of a definite colour and of a definite incidence-
height. In the object-space of the optical system the rays are all
assumed to be parallel to the optical axis. If M' designates the point
where a paraxial image-ray of mean refrangibility, corresponding, say,
to the Fraunhofer P-line, crosses the optical axis, and if I' desig-
nates the point where a ray of some other colour, say X, and of finite
incidence-height A, crosses the optical axis, we calculate (in thousandths
of a millimetre) the length M'JJ\ and if the point (X, A) in the diagram
is designated by P, we ascribe to this point P the number corresponding
to the numerical value of M'TJ, The curve drawn through all points
P which have the same numerical value will be one of the system of
isoplethic curves of the optical system. M. von Rohr gives diagrams
showing the system of isoplethic curves from the value M'L' = +0.050
mm, to the value M'JJ = — 0.050 mm. for a Petzval portrait-object-
ive and for the so-called "Planar" type of photographic objective of
P, Rudolph.
^ M. VoN Rohr: Theorie und CeschichU des phoiographiscken Objektivs (Berlin. 1899),
65-68.
35
!>.]
Colour-Phenomena.
531
^ee from aberrations, so that dS ^ o) we obtain:
tan^
dn dn'
n n'
59. If r, B denote the ray-co-ordinates of a ray of wave-length X
>re refraction at a given spherical surface of radius r , and if a, a'
Dte the angles of incidence and refraction, respectively, we have
following system of equations for determining the corresponding
co-ordinates (»', 6') of the refracted ray (§ 2ii):
sma = —
I
— sm Ot sin a = — ; sm a,
r w
fl' = « + «'- a,
r' — f = —
f-sma'
nee for an adjacent ray of wave-length X + ^ we derive immedi-
yr a series of differential formulae as follows:
da
de
tan a tan 0
+
dv
da' __ da f dn' dn\
tan a' "" tan a \ n' « /
dd' ^dB + da' -da.
dv'
da'
dS'
(44O
v' — r tan a' tan C
hat if we know the values of dv, dS before refraction at a given
^rical surface, we can find the corresponding variations dv', dS'
r refraction.
§ 361.J The Aperture and the Field of View. 533
instruments; but, on the other hand, it would lead us too far and tend
only to confuse the matter in hand if we attempted here to go into
all the intricate and special questions that are involved when the dif-
ferent aberrations are taken into account.
The bundles of rays that traverse an optical instrument are limited
either by the physical dimensions of the lenses themselves or by per-
forated diaphragms or ** stops'' interposed specially for this purpose.
In all cases that possess interest for us such stops are circular in form
and concentric with the optical axis. The direct and obvious effect
of a stop or lens-rim is two-fold, viz., first, to restrict the apertures of
the bundles of effective rays^ and, second, to limit the extent of the object
that is reproduced in the image. The mode and measure of these re-
strictions will depend on the sizes and positions of the stops and also
on the type of the optical apparatus itself.
361. The Aperture-Stop.
In the general and at the same time the most usual case, the dia-
phragm or stop is placed with its centre on the optical axis at some point
lying between two consecutive lenses of the optical system L; which
is thereby divided into two parts, a front component (Li) consisting
of the part of the lens-system in front of the interior stop, and a hinder
component (Lj) consisting of the remainder of the lens-system lying
on the other or far side of the stop. There may be also not merely
one but several such interior stops, either actual perforated diaphragms
or the rims of the lenses themselves; each of which, according to its
position, will divide the lens-system into two parts, as above-men-
tioned. Frequently a stop is placed in front of the entire system, in
which case it is called a front stop. And, similarly, a stop which is
placed behind, or towards the image-side of, the optical system (as is
also not uncommon) is called a rear stop. With respect to a front
stop, Lj = L, and with respect to a rear stop, L^ = L.
The apertures of the bundles of effective rays are conditioned by
these stops. In the simplest case of all when the optical system con-
sists of a single lens whose two surfaces intersect in the circular rim
of the lens, this circle is the common base of the cones of incident
and refracted rays that take part in the image-phenomena; and here
the bundles of effective rays are limited by the surface of the lens itself.
If now we interpose between the axial object-point M (Fig. 157)
and the lens a front stop with its centre on the axis at the pxjint desig-
nated in the figure by Af whose diameter subtends at Af an angle smaller
than that subtended at the same point by the diameter of the lens,
this stop will evidently limit the aperture of the bundle of object-
534
Geometrical Optics, Chapter XIV.
[§361.
rays emanating from the axial object-point M; and if the position
on the axis of the point which is conjugate to Af is designated byilf',
the GAUSsian image of the circular stop in the transversal plane 9
of the Object-Space with its centre at M will be a circle with its centre
at Af' lying in the transversal plane a' conjugate to v. Since all the
Pig. 157.
INPINITBLT Thin Coif vex I«bn8 with Front Stop. CD is the Aperture^top with its centre
on the optical axis at M. CD is here also the Hnttance-Pupil : CD^ the Bxit-PupiL At(/ is the
imaffe of the object AfQ.
MAf'^t JTAT-r, MD^P, Itlf^p\ IMMD^^. IM'M'Ef^^,
rays that before refraction go through the stop at M must after re-
fraction pass through the stop-image at Af' , we see that, whereas the
material stop placed in front of the lens at M limits the apertures of
the bundles of effective rays in the Object-Space, the stop-image at
M' performs the same office for the bundles of rays in the Image-Space;
or, in other words, the front stop lying in the transversal plane cr is
the common base of all the cones of object-rays, and, similarly, the
stop-image in the transversal plane c' is the common base of all the
cones of image-rays.
Proceeding now to the most general case, let us suppose that the
optical system L is composed of several lenses and provided with
one or more interior stops, either perforated diaphragms or lens-
rims. We begin by constructing the GAUSsian image of each stop 0
(Fig. 158) formed by that part Lj of the system that lies in front of
(or to the left of) 0. The stop that corresponds to that one of these
images that subtends the smallest angle at the selected axial object-
pxjint M is called the aperture-stop; because this is evidently the stop
that, with respect to M, conditions the apertures of the bundles d
§ 361.]
The Aperture and the Field of View.
535
effective rays. In the figure the aperture-stop is represented as the
one with its centre located at the point 0, whose image formed by
the front part Li of the optical system L in the transversal plane o"
that is crossed by the axis at the point M subtends a smaller angle at
M than the corresponding image of any of the other stops. Which
one of the perforated diaphragms or lens-rims plays the r61e of aperture-
stop will depend essentially on the position of the axial object-point Af .
PlO. 158.
CoKPOUKD Opticai. Ststbk consisting op Two Thin I«bnsb8 Zi. Z«. sbpa&atcd by Intbrior
Apbrturb-Stop with cbntrb at O. The axial point M conjugate to O with respect to L\ and the
axial point M* conjugate to O with respect to Zt (and therefore conjugate also to M with respect
to Zi + Zfl) are the centres of the Entrance-Pupil and Bxit-Pupil, respectively. M'(/ is the image
of the object MQ.
MAf'-'t M'Af"^, MD'^p, Jf//-/, IMMD^^, ^ IfAT// - e'.
In passing, it may be observed that a case may occur, such as that
shown in Fig. 159, in which the images of two (or more) of the material
stops formed by the parts of the optical system lying in front of them
subtend at the axial object-point M angles of equal magnitude;
so that (if this angle is also the smallest of all such angles) either of
these two stops may be regarded as the aperture-stop. The point
of intersection of the pair of straight lines joining the upper extremity
of one stop-image with the lower extremity of the other determines a
second point K on the optical axis at which the two stop-images also
subtend angles of equal magnitude. With respect to an axial object-
point situated anywhere between the two extreme positions M and
Ky the stop-image marked // in the diagram will subtend the smaller
angle of the two; whereas for an axial object-point lying anjrwhere
outside the segment MK the stop-image marked / will subtend the
536
Geometrical Optics, Chapter XIV.
[§361.
smaller angle.* It is apparent that the stop that acts as the aperture-
stop for an object in one position on the axis may not be the aperture-
stop for another position of the object. We must assume, therefore,
that the object has a fixed position or at any rate that it is movable
within certain prescribed limits if the stops are to retain their functions,
as is necessary, for example, in the case of such optical instruments as
the telescope and the microscope.
Returning to the consideration of Fig. 158, we see that, rfnoe the
aperture-stop at 0 must be the common base of all the cones of rays
after their emergence from the front part Ly of the optical system, the
stop-image in the transversal plane a must likewise be the common
base of all the cones of rays in the Object-Space. Moreover, if Af
designates the position of the point which, with respect to the hinder
part La of the optical system, is conjugate to the stop-centre 0, the
FlO. 159.
C48B OF Two BlTTRilNCB-PUPllA.
image of the stop formed by L^ will lie in the transversal plane ^
determined by the axial point Af' ; and, similarly, this stop-image will
evidently be the common base of all the cones of image-rays after
having traversed the entire compound system L = Li + L,. Evi-
dently, also, the transversal planes o", c' are a pair of conjugate planes,
so that the stop-images at M and M' are images of each other with
respect to the whole system L. Together they constitute a pair of
virtual stops (as distinguished from actual or material stops) that are
the measures of the apertures of the ray-bundles in the Object-Space
and Image-Space. A material stop of the same size and position as
* See M. VON Rohr: " Die Strahlenbegrenzung in optischen Systemen *•, Chapter IX
of Die Theorie der opUschen InsUrumente, Bd. I (Berlin. 1904)1 edited by M. voN RoBK.
See p. 469.
§ 362.1 The Aperture and the Field of View. 537
the stop-image at Af will act exactly in the same way with respect to
the limiting of the bundles of rays as a material stop identical in size
and position with the stop-image at Af' ; and either of them or both
together, so far as this effect is concerned, would be precisely equiva-
lent to the actual stop that we suppose to be situated at 0. Abbe/
who has done most to develop the theory of stops, calls the stop-images
at M and Af' , from an analogy with the optical system of the human
eye, the pupils of the system. The pupil of the eye is the contractile
aperture of the iris, the image of which produced by the cornea and
the aqueous humour lies in front of the eye (as can be seen by looking
directly into the eye) ; so that only such rays as are directed towards
this image can enter the eye through the iris-opening. From this
same analogy. Abbe calls also the aperture-stop at 0 the iris of the
optical system. The two pupils at Af and Af' are distinguished by
the names Entrance- Pupil and Exit- Pupil, respectively.*
362. An imagery is completely determined so soon as we know the
positions of the two pairs of conjugate transversal planes <r, a' and o", cr',
together with the values of the magnification-ratios Y and Y that
characterize these two pairs of planes. Thus, if the pupils of the sys-
tem are given in both size and position, and if also the image M'Q^
corresponding to a given object-line MQ at right angles to the optical
axis has been constructed, the procedure of every ray that traverses
the system can be ascertained immediately without taking farther ac-
count of the special construction of the apparatus. For example, to
an object-ray QD which, originating at the object-point Q crosses the
(T-plane at a point D in the circumference of the entrance-pupil there
must correspond an image-ray directed toward the image-point Q' and
going through the point D' of the circumference of the exit-pupil that
is conjugate to the point D. It is evident also that the totality of the
effective rays in the Object-Space may be regarded in either of two
wayg, viz.: (i) As cones of rays emanating from points in the object
MQ and having the entrance-pupil as a common cross-section; or
(2) As cones of rays with their vertices at points of the entrance-pupil
and a common base in the object MQ; so that the r61es of object and
entrance-pupil are interchangeable. This same reciprocity exists like-
wise between the image and the exit-pupil.
' See E. Abbe: Beitraege zur Theorie des Mikroskops und der mikroskopischen Wahr-
nehmung: Archiv f. mikr. Anal,, ix (1873), 413-468. Also, Cesammelle Abhandlungen,
Bd. I (Jena. 1904)* 45~ioo.
* See also E. Abbe: Ueber die Bestimmung der Lichtstaerke optischer Instrumente:
Jen. ZfLf. Med, u, Naturw., vl (1871). 263-291. Also. Gesammelte Abhandlungen, Bd. I
Gena. 1904). i4-44-
538 Geometrical Optics. Chapter XIV, [ § 36i
363. The Aperture-Angle.
The angle MMD = G, defined more precisely by the relation
MD
tane = ^^,
where D designates the position of a point in the meridian plane lying
in the circumference of the entrance-pupil, is called the aperture-angk
of the optical system. If /> = MD denotes the radius of the entrance-
pupil (reckoned positive or negative according as D lies above or below
the optical axis), and if MM = f denotes the abscissa of the axial
object-point M with respect to the centre M of the entrance-pupil as
origin, we may write:
P
tane= — 7. (442)
Similarly, if 6' = LM'WD\ p' = M'D\ {' = M'M\ where the points
designated by M\ M\ D' are conjugate to the points in the Object-
Space designated by the same letters without the primes, we have also:
tan0'= -^. (443)
364. The Numerical Aperture.
Although the size of the aperture-angle 6 is in a certain more or less
geometrical sense a measure of the number of effective rays emanating
from the axial object-point M, this angle by itself, from an optical
standpoint, is not a true criterion of the aperture of the optical s>'s-
tem. All the rays of a bundle are not of equal optical value, and on
this account the quantity of light-energy that is transmitted through
the optical system from an object-point to its conjugate image-point
depends on something more than just the size of the aperture angle.
A luminous surface-element emits more energy along some directions
than along others, the intensity of radiation (§ 388), according to Lam-
bert's law, being proportional to the cosine of the angle of emission; so
that the most energetic ray is the one that is directed along the normal
to the surface-element at the origin-point of the ray. Consequently
different rays emanating from the same object-point will be the routes
through the entrance-pupil of the optical system of different cargoes
of light-energy.
According to Abbe,* the proper and rational measure of the aperture
* £. Abbe: Die optischen Huelfsmittel der Mikroskopie: GesamnuUe Ahkandbtntt^
Bd. I (Jena, 1904), 1 19-164; especially, p. 142. (This paper was published originaliy
i 364.1 The Aperture and the Field of View. 539
of an optical system — the only one indeed that affords a just idea of
its efficiency — is given by the product of the refractive index of the
first medium (n) and the sine of the aperture-angle; this product, to
which Abbe gives the name numerical aperture, and which is denoted
here by the symbol A, has therefore the following expression:
A = »-sin6. (444)
It would derange too much the plan of this treatise if we paused here
to explain fully the basis of this definition, especially also as such an
exposition belongs rather to the special theory of optical instruments
and to the theory of the microscope in particular where the numerical
aperture has an exceedingly important r61e. In the case of the instru-
ment just mentioned, the conjugate axial points M, M' are the apla-
natic pair of points of the optical system (§ 279), and under these cir-
cumstances it would be easy to show that the quantity of radiant energy
transmitted from M to M ' is proportional to the numerical aperture.
It may be remarked that the magnitude denoted by A is propor-
tional, not to the aperture-angle G, but to the sine of this angle; so
that, for example, if G were increased from, say, 30** to 90**, the numeri-
cal aperture would be only doubled, since sin 90** : sin 30** = 2 : i . The
numerical aperture is also proportional to the refractive index, so that
its value can be altered merely by immersing the object in a different
medium for which n has a different value; and, hence, as Abbe has
observed, this measure A enables us to compare the apertures of the
so-called "dry" and "immersion" optical systems.
The relation between the numerical aperture and the radius (p)
of the entrance-pupil and the abscissa { = MM is exhibited by the
formula:
whence also we can see the effect on the aperture of a displacement
^{ of the object-point M, Whether the aperture will be increased or
diminished by such a variation of the pxjsition of the axial object-
point M, will depend on the signs of both ( and 6^,
If Z denotes the angular magnification of the system with respect
in Braunschweig in 1878.) Also:
E. Abbe: Ueber die Bedingungen des Aplanatismus der Linsensysteme: Sitzungsber,
d. Jen. Gesellsckafi /. Med, u. Naturw., 1879. 129-142. See Gesammelle Ahhandlungen,
Bd. I (Jena, 1904). 213-226; especially, pages 225 and 226. Also:
E. Abbe: On the Estimation of Aperture in the Microscope: Journ. Roy. Micr. Soc.^
(3), i (1881), 388-433: especially, pages 395 & 396. (A German translation of this
paper is in Gesammelle Abhandlungen, Bd. I. 325-374-)
540 Geometrical Optics, Chapter XIV. [ § 365
to the pupil-centres Af , M\ and if Y denotes the lateral magnification
with respect to the pair of conjugate points M, M\ then, according
to the last of the image-equations (127), we shall have:
{' Y
and, hence, in the special case when the points JIf , M' are the aplanatic
pair of points, so that
nsinO _ ^ _ y.
n'.sine'""i4'" ^'
we obtain the relation :
j, = fz: (447)
which will be found to be a very useful formula in the special theory
of optical instruments.
ART. 115. THE CHIEF RATS AND THE RAT-PROCEDURE.
365. Chief Ray as Representative of Bundle of Rajrs. The rays
which, emanating from all the points of the object, are directed
towards the centre M of the entrance-pupil constitute the bundle of
so-called chiej rays in the Object-Space; to which in the Image-
Space there corresponds also a conjugate bundle of chief rays which
all meet at the centre M' of the exit-pupil. Accordingly, the pupil-
centres Af , M' are to be considered as the centres of perspective of the
Object-Space and Image-Space, since to any object-point P lying on
the chief object-ray PM there corresponds an image-point P' lying
on the conjugate chief image-ray P'M', The chief ray is the axis
of symmetry of the cone of rays, and, therefore, espedally when the
circular aperture-stop is very small, it may be regarded as the repre-
sentative ray of the bundle (cf. § 286) ; and, hence, a knowledge of the'
procedures of the chief rays will often afford an accurate idea of the
entire image-process.
Since the pupil-centres are the centres of perspective of the Object-
Space and Image-Space, object-points which He along a chief ray in
the Object-Space will be reproduced by image-points which lie along
the conjugate chief ray in the Image-Space, and which, therefore, if
viewed by an eye placed at the exit-pupil (which is the usual place for
the eye in order that the entire image may be all commanded at the
same time), will appear to lie all at the same place. If the image is
received on a plane screen, placed at right angles to the optical axis,
§366.]
The Aperture and the Field of View.
541
and if this screen does not coincide exactly with the transversal image-
plane c' which is conjugate to the transversal plane <r in the Object-
Space that contains the object-point Q (Fig. i6o), the image of Q on
Fiu. 160.
BLUR-CntCLBS IN THE SCKBBN*Pl.ANB DUB TO IMPERFECT FOCUSSZNO. X3^ is the optical azis
of the system Z. CD, C*£f diameters of Entrance-Pupil and Exit-Pupil. Af*(/ is the imaire of
MQ, and At' and (/' are the centres of the blur-drdes in the Screen-Plane corresponding to the
object-points M and Q, respectively.
the screen will not be a point but an aberration-figure coinciding with
the section of the bundle of image-rays made by the screen-plane. If
the aperture-stop is circular in form, this aberration-figure will be a
circle (so-called **blur-circle**)f and the centre of the circle where the
chief ray crosses the screen-plane will be regarded as the place on the
screen of the image corresponding to the object-point Q. The smaller
the diameter of the exit-pupil, the smaller will be the diameter of the
blur-drde; and if the diameter of the aperture-stop is infinitely small,
the blur-drcles will all contract into points at their centres.
366. Optical Measuring Instruments.
The importance of taking into consideration the procedures of the
chief rays may be illustrated by investigating the class of optical in-
struments that are espedally contrived for determining the size of an
object by measuring the size of the image. The image may be cast
on a screen which is provided with a scale or the image may be formed
in the air in a plane containing a material scale or a scale-image. But
here, owing partly perhaps to the unavoidable dioptric imperfections
of the image itself but above all to the difficulty of focussing the instru-
ment exactly so that the true image-plane coinddes with the scale-
plane, there is a source of error in the method, since, instead of measur-
§ 367.1 The Aperture and the Field of View. 543
entranoe-pupil will be the infinitely distant point of the optical axis,
and the chief rays in the Object-Space will, therefore, be parallel to
the axis.
An optical system in which the centre 0 of the aperture-stop coin-
cides with one or other of the two focal points that are here designated
by E[ and F^ is called by Abbe' a telecentric system. According as
it is the entrance-pupil or the exit-pupil which is the infinitely distant
one of the two pupils, the system is said to be "telecentric on the side
of the object" or "telecentric on the side of the image", respectively.
In the special case when the focal points E\ and F^ coincide with each
other the system will be telescopic (§ i86, Case i); and if, moreover,
the centre 0 of the aperture-stop coincides with both of these focal
points, the system will be "telecentric on both sides".
367. If the positions of the two focal points of the optical system
are designated by F and £', and if the magnitudes of the focal lengths
are denoted by / and e\ and if, finally, x = FM, x' = E'M' denote
the abscissae, with respect to the focal points, of the pair of conjugate
axial points Jlf, M'\ then, on the assumption of perfect coUinear cor-
respondence, we have, according to the second of formulae (i 15), for the
lateral magnification of the system with respect to the points Jlf, M'l
X e
In the special case, therefore, when the centre M of the entrance-pupil
coincides with the position F of the primary focal point, so that
X = FM = MM = {, we obtain:
r
and, hence, when the system is telecentric on the side of the image,
the magnification Y will not depend on the position of the scale-plane,
but only on the position of the object-plane <r. Similarly, when the
centre M' of the exit-pupil coincides with the position E' of the sec-
ondary focal point (x' = E'M' = M'M' = {')» we find:
which shows that when the system is telecentric on the side of the
' E. Abbe: Ueber mikrometrische Messung mittelst optischer Bilder: Silzungiber,
d. Jen. Cesellschafl f, Med, u. Naiurw., 1878, 11-17. See also: Cesammelte Abhandlungen,
Bd. I (Jena. 1904), 165-172.
§ 369.1 The Aperture and the Field of View. 545
Since
y
tane =^,
where { = MM denotes the abscissa of the axial object-point M with
respect to the centre M of the entrance-pupil, and since, moreover,
y X X + f *
where x = FM and x = FM denote the abscissae, with respect to the
primary focal point F, of the points M and Af , respectively, and where
/ denotes the primary focal length of the optical system; we obtain
finally:
£-, - 4-/. (448)
If the object is at a great distance, x will be very small compared with
{, so that the fraction {/(x -|- {) will be very nearly equal to unity;
and hence we may write:
y
^^^ « /, approximately;
which will be not only approximately but strictly true in case either
the object is infinitely distant ({ = oo) or the plane <r of the entrance-
pupil coincides with the primary focal plane (x = o).
In making geodetic measurements it often happens that one wishes
to determine the distance of the object (a surveyor's rod, for example)
by measuring the size of its image. If the entrance-pupil of the optical
instrument is situated in the primary focal plane, the angle 0 can be
determined by the relation found above:
y
tan e = J ,
and hence the distance of the object may be computed by the formula:
^ tane ^' y"
provided we know the values of the magnitudes denoted by y, y' and /.
369. The Subjective Magnifying Power.
If, however, the optical instrument is designed to be used subject-
ively in conjunction with the eye for the purpose of reinforcing vision
36
§ 369 J The Aperture and the Field of View. 547
magnifying power of an optical instrument belonging to the same
general class as the microscope is the ratio of the visual angles (or
trigonometric tangents of the angles) subtended at the eye, on the one
hand, by the image as viewed in the instrument, and, on the other hand,
by the object as seen by the naked eye at the distance of distinct vision.
Denoting this ratio by the symbol W, we have therefore:
^ tanO' a y
pr = T — = 77-^- (449
tan n < y ^1-1-7/
Although this definition of the subjective magnifying power com-
bines the two merits of simplicity and clearness, it is open to objection
on account of the fact that it involves essentially the magnitude de-
noted here by a, the so-called "distance of distinct vision", which has
no connection with the instrument itself and which is different for
different individuals. It is a well-known fact of experience that by
virtue of its power of accommodation the normal eye is capable of
seeing distinctly at almost any distance; but what is here meant by
the distance of distinct vision is the distance from the eye at which
an observer would naturally place an object in order to view it intently ;
which in the case of a normal eye is usually reckoned as about 25 cm.
or 10 in. Accordingly, whereas the magnification as defined by the
ratio W will be different for a near-sighted observer for whom a = 10
cm. and for a far-sighted observer for whom a = 50 cm., yet, as Abbe*
has pointed out, both observers looking through the instrument will,
as a matter of fact, view the image of the same object under the same
visual angle; so that whatever difference there may be in the magnifi-
cation is to be found, not in the instrument itself, but in the different
organs of sight that are employed in conjunction with the apparatus.
Eliminating the angle 17 which has nothing to do with the optical
instrument, we may write the formula for W in the following form:
.^ tan 6'
PT = a • — — = a • F, (450)
whereby the magnifying power W is expressed now as the product of
two factors, viz., the factor a, which depends entirely on the eye of the
' E. Abbe: Note on the Proper Definition of the Amplifying Power of a Lens or a
Leii»-83rstem: Joum. Roy. Micr, Soc^ (a), iv (1884), 348-351. See German translation
In CtsammelU Abhandlungen, Bd. I (Jena. 1904). 445-449*
See also S. Czapski: Theorie der oplischen Instrumente nach Abbe (Breslau. 1893),
160-164.
548 Geometrical Optics, Chapter XIV. [ { 369.
observer, and the factor
V^——. (4S0
which, notwithstanding the fact that the distance of the image from
the eye is involved in the definition of the apparent size tan 6' of the
image, depends essentially, as we shall show, on the structure of the
optical system alone.
Since
tane =^ and ^- = -,-— ^,
where x' = E'M', x' = E'M' denote the abscissae, with respect to the
secondary focal point E\ of the points M' and Af' respectively, and
where e' denotes the secondary focal length of the optical system, we
obtain :
Now almost without exception in the case of all optical instruments
that are employed subjectively in conjunction with the eye, no matter
how the image may be focussed by the eye, the distance x' is so small
in comparison with the distance { that the fraction x'/f' is practically
negligible. Under these circumstances we may write therefore:
V = — — - = -„ approximately; (453)
y c
and in the special case when the plane a' of the exit-pupil coincides
with the secondary focal plane (x' = o) and the eye is situated at
the secondary focal point £', the formula*F = i/e' will be strictly true.
Accordingly, as above stated, the magnitude denoted by V depends
solely on the structure of the optical instrument provided it is to be
used subjectively.
According to Abbe, this magnitude V defined as the ratio of the
visual angle subtended at the eye by the image viewed through the instru-
ment to the corresponding linear dimension of the object is therefore a
proper measure of the characteristic or intrinsic magnifying power of
an optical system on the order of the microscope. For every such
system it has a perfectly definite value, viz., i/e', and thus is entirdy
independent of all the more or less accidental circumstances that may
affect the magnification, such as the distance from the image of the
observer's eye, the distance from the focal plane of the exit-pupil, etc
§ 370.1 The Aperture and the Field of View. 549
Abbe's definition V of the Subjective Magnifying Power is obtained
from the ordinary definition W by merely dividing W by the distance
a of distinct vision of the observer; thus,
W
V--. (454)
Since W is proportional to a, the popular use of the term "magnifying
power", which corresponds to the magnitude TF, expresses the fact
that the advantage gained by the use of an optical instrument is
proportional to the observer's distance of distinct vision and is there-
fore greater for a far-sighted than for a near-sighted observer. From
the scientific point of view, Abbe's definition V is far superior, inas-
much as F is a constant of the instrument itself. The subjective
magnifying power V in the case of an instrument on the order of the
microscope is seen to be completely analogous to the objective magni-
fying power y/tan 6 in the case of the image of an infinitely distant ob-
ject formed by an optical instrument on the order of the photographic
objective or the objective of the telescope.
ART. 117. THE FIELD OF VIEW.
370. Entrance-Port and Exit-Port.
The limiting of the bundles of rays that are permitted to traverse
the optical system is not the only duty performed by the stops and
lens-fastenings; but these serve also to define the extent of the object
that is reproduced in the image. For the sake of simplicity, let us
assume for the present that the aperture-stop at 0 is infinitely small, so
that the pupil-openings at M and M' (Fig. 162) are reduced to mere
points (9 = 0' = o, /> = />' = o). In this case the chief ray of a
bundle will be the only effective ray, and the bundle of chief rays will
constitute therefore the entire system of effective rays.
In order now to ascertain which one of the stops present is the one
that determines the expanse of object that will be depicted, we con-
struct, as before (§ 361), the image of each stop formed by that part
of the optical system which is in front of it. That one whose image
thus constructed subtends at the centre M of the entrance-pupil the
smallest angle is the stop that limits the field of view of the object. In
the diagram this stop-image is represented as situated with its centre
on the optical axis at the point designated by S. The cone of chief
object-rays whose transversal cross-section at 5 coincides with this
stop-image divides the transversal object-plane <r into two regions, an
§ 372.1 The Aperture and the Field of View. 551
also, the angle S'M'T' — 0', where S\ T' designate the positions of
the points conjugate, with respect to the entire system, to the points
designated above by 5, 7*, respectively, is the angular measure (or the
semi-angular diameter) oj the field of view of the image.
It is possible, of course, that an optical system may have two or
more entrance-ports. An obvious illustration is suggested by the
familiar type of photographic double-objective in which the two parts
of the system are symmetrical with respect to the aperture-stop in
the middle (as in the case of the "Aplanats"), so that the rims of the
two lens-systems subtend equal angles at the centre 0 of the aperture-
stop; and hence, since the rim of the front component and the image
of the rim of the hinder component produced by the front component
subtend equal angles at the centre M of the entrance-pupil, either of
these two may be regarded as the entrance-port. This fact will be
found to possess a certain importance in the case of an optical system
of finite aperture, as we shall have occasion to see (§383).
371. In the special case when the extent of the object MQ is so
small that the angle subtended at the centre M of the entrance-pupil
is smaller than the angle subtended at the same point by the entrance-
port (that is, ZMMQ < Z SMT), the field of view is limited by the
object itself. In any case if we designate by Q the object-point in
the transversal plane a that is farthest from the axis, the angular
measure of the field of view of the object is ZMMQ = 0, where 0
denotes always the slope-angle of the outermost ray of the bundle of
chief rays in the Object-Space. If we put MQ = y, MM = {, we
can write:
tan 0 = ^. (456)
ART. 118. PROJBCTION-STSTEMS WITH INFINITELT NARROW APERTURE
(0 = 0).
372. Focus-Plane and Screen-Plane. According to the geometri-
cal theory of cdllinear correspondence, the image of a 3 -dimensional
object is itself 3 -dimensional; but by the image produced by an
optical instrument is usually meant not this geometrical image-relief
in space, but almost without exception the projection thereof on some
specified surface, such as the retina of the eye itself in the case of the
class of optical instruments that are used subjectively in conjunction
with the eye, or such as a screen or sensitive photographic plate in
on Die SirakUnbegrentung in opiischen Systemen. (See Die Theorie der optischen /ustm-
menie, Bd. I (Berlin, 1904), edited by M. von Rohr: Chapter IX, 466-507.)
§ 373.] The Aperture and the Field of View. 553
the Screen- Plane, respectively. In case the aperture is infinitely nar-
row, as is here assumed, the chief ray is the only ray of the bundle
that is effective; and the figure in the focus-plane corresponding to
that which is actually visible on the screen-plane may be constructed
point by point by tracing backwards the path of each chief ray from
the point P' where it crosses the screen-plane to the point P where
the corresponding ray in the Object-Space crosses the focus-plane.
Practically, this process amounts simply to projecting all the points
of the object from the centre M of the entrance-pupil on to the chosen
focus-plane; and this projection-figure, which may be called the **/>ro-
jecied objeci*\ is the object that is in reality reproduced in the '^pro-
jected image" in the screen-plane; which latter may also be constructed
in the same way by projecting all the points of the relief-image from
the centre M' of the exit-pupil on to the screen-plane, as shown in
Fig. 163.
373. Perspective-Elongation.
If MQ (Fig. 164) is the projection from Af on to the focus-plane <r
of an object-line NR perpendicular to the optical axis at N, we have:
Fio. 164.
Pbrbpbctivb Elongation op the Object. O is the projection from centre Jf of the Entrance-
Pupil of the object-point H on to the Pocus-Plane <r.
^ MM 0^9,
..^ .Tr> ^^M ,,„ MM
or, sinoe MN is usually small in comparison with MM,
MQ MN . .
- ^ = I + ]|^» approximately.
The difference MQ — NR is the measure of the perspective elonga-
tion of the object NR, and the ratio
MQ - NR MN
NR - NM
is called the relative perspective elongation of the object NR.
§ 375.1 The Aperture and the Field of View. 555
ART. 119. OPTICAL SYSTEMS WITH FINITE APERTURE.
375. Projected Object and Projected Image in the case of Pro-
jection-Systems of Finite Aperture.
So long as the aperture of the system was infinitely narrow, we had
to consider merely the procedures of the chief rays; but advancing
now to the study of optical projection-systems of finite aperture, we
must take account of other rays besides just those that in the Object-
Space are directed towards the centre of the entrance-pupil. Every
point of the object is the vertex of a cone of rays whose paths lie along
straight lines which, produced if necessary, must first of all go through
points in the transversal plane o* contained within the circular opening
of the entrance-pupil. Some of the rays of such a bundle, possibly
all of them, may be intercepted at the entrance-port, and in this event
only a portion of the bundle at most will be effective. To each cone
of rays in the Object-Space corresponds also a cone of rays in the
Image-Space, whose paths likewise lie along straight lines which, pro-
duced if necessary, must pass through points in the transversal plane
& comprised within the circular opening of the exit-pupil ; and to an
incomplete cone of object-rays corresponds, of course, an incomplete
cone of image-rays. The relief-image of a 3 -dimensional object is the
configuration of image-points which are at the vertices of all these
cones or partial cones of image-rays. Some of these vertices may fall
in the transversal screen-plane <r'; and these will be the image-points
corresponding to such of the points of the object as lie in the trans-
versal focus-plane <r. But all the other points of the object, which
lie to one side or other of the focus-plane, will be represented in the
projected image on the screen-plane, not by points at all, but by the
circular discs or patches — so-called "diffusion-circles" or "blur-circles"
(see § 365) — which are the sections of the cones of image-rays made
by the screen-plane. In the case of an incomplete cone of image-rays,
the image of the corresponding object-point will be represented on the
screen-plane by only a piece of a blur-circle. These ideas will be made
dear by the consideration of the diagram (Fig. 166) which represents
a meridian section of an optical system consisting of an infinitely thin
convex lens VT with a front stop CD with its centre on the optical
axis at M. In this illustration the rim of the lens is the circumference
of both the entrance-port and the exit-port.
The real object corresponding to the above-described projected
image in the screen-plane <r' is the figure in the focus-plane a obtained
by projecting the entrance-pupil on to this plane from each point of
the actual object. In the case of those object-points so situated that.
556
Geometrical Optics, Chapter XIV.
[§376.
on account of the limited opening of the entrance-port, they can
utilize only a part of the area of the entrance-pupil, we must project
on to the focus-plane only the part of the entrance-pupil that is util-
ized. The centres of these circular discs and disc-portions which are
the sections of the bundles of effective rays made by the focus-plane
<r and the screen-plane a' in the Object-Space and Image-Space, re-
spectively, are at the points where the chief rays cross these planes.
This last statement suggests also, that in regard to this vicarious
object-figure in the focus-plane <r, there is an important difference to
be remarked between the case of a point inside of one of these object-
Projkctbd Object and Imaob in Projection-System of Piwitb Apb&turb. The Ent^l1lC^
Pupil CD is projected from the object-point ^ on to the Focus-Plane v In the blur-circle with centre
at (?; and. similarly, the Exit-Pupil C'l^ is projected from the image-point ^ on to the Scfeefi-
Plane cr' in the blur-circle with centre at the point (/ conjugate to Q,
side blur-circles and the case of an ordinary object-point lying in the
focus-plane; for whereas the latter emits rays in all directiqns, the
former is to be regarded as sending out only one single ray coinciding
with the actual object-ray which crosses the focus-plane at this point.
If the aperture of the optical system is not only finite but rela-
tively large, the transversal planes <r, a' must be a pair of aplanatic
planes in order that there may be a point-to-point correspondence
between the focus-plane and the screen-plane; and when this is the
case, the image of an object-point which lies outside the focus-plane
will not be a point, since the so-called HERSCHEL-Condition (cf. § 3^4)
is incompatible with the Sine-Condition. Under such circumstances,
where, in general, the bundles of image-rays are no longer homocentric,
it is particularly advantageous to represent the image of a 3-dimen-
sional object by means of its projected image on the screen-plane.
376. The centres of the blur-circles on the screen-plane are to be
regarded as the positions of the image-points; and since, even in the
extreme case just mentioned of a system of very large aperture, these
§ 378.) The Aperture and the Field of View. 557
are the places where the chief image-rays cross this plane, the perspect-
ive IS exactly the same here as for the case of a system of infinitely
narrow aperture (§373), so that nothing needs to be added to what
has been said already in the treatment of the perspective in the pre-
ceding case.
377. Focus-Depth of Projection-System of Fmite Aperture.
With regard to the distinctness of the image on the screen-plane,
that is a matter that will depend very largely on the acuteness of
vision of the observer. If the resolving power of the eye were ab-
solutely perfect, this screen-image composed partly of image-points
and partly of blur-circles and pieces of such circles would appear
faulty on the mere ground that it was not a faithful reproduction of
the original. But the resolving power of the eye is limited {cf, § 252),
depending on a variety of conditions, both physical and physiological.
Under average conditions the human eye is able to distinguish as sepa-
rate and distinct two points whose angular distance apart varies for
different individuals between the limits of one and five minutes of arc;*
and hence the blur-circles in the projection-image will not be dis-
tinguishable from points provided their angular diameters do not ex-
ceed this limiting angular measure («) of the resolving power of the eye.
Similarly, also, in regard to the projection-figure of the object on the
focus-plane, in order that this may appear sharp and distinct as viewed
by an eye at the centre M of the entrance-pupil, the diameters of the
blur-cirdes must subtend at M angles that are smaller than the limiting
angle €. Since the diameters of these blur-circles will depend on the
distances of the actual object-points from the focus-plane, the question
arises how far from this plane can such an object-point be in order that
its image in the screen-plane shall still appear to be a point and not
a fleck of light. This distance, as we shall see, will be different ac-
cording as the object-point lies on one side or the other of the focus-
plane, so that all object-points which are comprised within the space
between two determinate transversal planes at unequal distances from
the focus-plane and on opposite sides of it will be reproduced distinctly
in the projection-image in the screen-plane. The distance between
this pair of transversal planes, called the Focus-Depth, we propose now
to investigate.
378. Let Qx (Fig. 167) designate the position of the point where
the chief ray R^M of the object-point iZ, crosses the focus-plane <r,
so that Qi is therefore the centre of the blur-circle that represents R^
^ Sec. for example, E. Abbe: Beschreibung eines neuen stereosko0i8chen Oculars:
Carls Rep.f, Exp.-Phys., xvii (1881). 197-224. See p. 219. This paper will be found
ibo in GesamnuUe AbhandluHgen, Bd. I (Jena. 1904). 244-272.
378.1 The Aperture and the Field of View. 559
he blur-circle does not depend on the distance of the object-point
rom the optical axis, so that all object-points in the same transversal
Jane will be represented in the projection-figure on the focus-plane
)y blur-circles of equal diameters. Thus, for example, the blur-drcle
>f the axial object-point Ni is equal to that of R^; in the figure
MV = <2i<?a = dy.
The straight lines QiD and Q2M determine by their intersection a
)oint R2 on the opposite side of the focus-plane from i?i, which, re-
garded as an object-point, will be represented in the projection-figure
>n the focus-plane by a blur-circle whose centre is at the point Q2
ind whose radius QiQi = — dy has the same absolute magnitude as
hat of the object-point R^. Thus, on either side of the focus-plane
here is a certain transversal plane characterized by the fact that all
>bject-points in this plane will be projected on to the focus-plane in
>lur-circles all of the same prescribed size. If we put MN^ = Sf,,
irhere N^ is used to designate the point where the optical axis crosses
he transversal plane of the object-point R^, we obtain from the figure,
xactly as in the case of the similar formula above:
dy 5fe
P { + «{,•
lence, also, we find for the distances from the focus-plane of this pair
f transversal planes:
nd, accordingly, we see also that the two transversal planes deter-
lined by these formulae are at unequal distances from the focus-
lane, and, in fact, that the front one of the two planes (in the figure
he one containing the object-point R^ is always nearer to the focus-
lane than the other plane.
Now if the magnitude dy is such that
dy €
-| = tan-.
irhere € denotes the angular measure of the resolving power of the
ye (in the figure e/2 = /.VMM), the blur-drcle on the focus-plane
orresponding to an object-point lying anywhere in the space com-
>rised between the pair of transversal planes belonging to Ri and i?.
§ 581.] The Aperture and the Field of View. 561
reinforcement of vision, and if the pupil of the passive eye is supposed
to be placed at the exit-pupil of the instrument, the image is presented
to the eye at the distance M'M' = {'.
The absolute linear diameter of the blur-circle in the image-plane
corresponding to a non-focussed point is:
2dy^ = 2Y'dy = — 27* irril.' tanG = — 2F-5{'tan0, approx.,
where Y denotes the lateral magnification of the aplanatic pair of
axial points JIf, M\ and where, in obtaining the final approximate
expression, the distance 5( is supposed to be small as compared with (,
as is the fact with an optical instrument of high magnifying power.
If
€ = -^ = - 2F. -^ -tan e
denotes the visual angle subtended at the eye by the blur-cirde, and
if we recall from § 369 that
tan e; _ y 1 _ K
we find
€ = - 2y-5{-tane,
or
Thus, if yly = 100, and if the absolute value of {' is equal to the con-
ventional distance of distinct vision, viz., 250 mm., so that
V I
F = ^-- = o.4,
and if we take 9 = — 30®, « = 3' = 0.00087 radian, we obtain for the
Focus- Depth: 25{ = 0.0037 mm.
381. Accommodation-Depth.
By virtue of its power of accommodation, the eye can be focussed
at will on different points of the image-relief, and provided these
image-points are within the range of distinct vision, and also provided
the imagery is ideal, the different parts of the image can be viewed with
perfect exactness; so that, owing to this property inherent in the eye
to a greater or less degree in different individuals, a certain depth of
the object called the accommodation-depth will be seen distinctly in
37
562 Geometrical Optics, Chapter XIV. [ § 381.
its image, which measured along the axis may be denoted by MiMj.
The depth of vision is extended beyond these points by the focus-
depth 5f , in one direction from M^ and the focus-depth Sf, in the other
direction from M^, since within these extended parts the blur-circles are
too small to be resolved by the eye; and hence the Entire Depth of
Vision is equal to the sum of the Accommodation-Depth and Focus-
Depth, viz. = Jlf lilf, + 5{i + 5^2.
If the eye is placed at the exit-pupil of the instrument, whose centre
is at the point designated by Af ', and if the positions on the optical
axis in the Image-Space of the "near-point" and "far-point" of the
eye of the observer are designated by Afj and Af^, respectively, the
range of distinct vision is equal to the piece AfjAfJ of the optical axis.
The points designated above by M^ and Jf, are the axial object-points
conjugate to Af^ and Af^, respectively. If the focal points of the
optical system are designated by F and E\ and if we put
and, finally, if the focal lengths are denoted by / and ^, then, on the
assumption of coUinear correspondence, we have:
and hence:
/ /
where dx' = Afjilf^. If (§ 179)
denote the magnification-ratios of the two pairs of conjugate axial
points, and if we introduce also the relation (§ 193) :
n'f + ne' =^ Of
where n, n' denote the indices of refraction of the first and last media
of the optical system, we obtain:
n 5x'
Sx =
n''Y,'Y,'
If Af 'Jlfj = f I and Af 'Jlf^ = fj denote the least and greatest distances
of distinct vision of the eye, then, according to Donders, the magni-
tude
§ 382.1 The Aperture and the Field of View. 563
is the rational measure of the power of accommodation of the eye-/
and hence we obtain the following expression for the accommodation-
depth:
ix^—rA-y-^] (464)
and if Yi and Fj ^^ ^^^ much different from each other, we can
replace each of them by a certain mean value F, which, to be perfectly
accurate, should be the geometric mean between Y^ and Y^; and,
similarly, we can introduce in place of i[ and {j a mean value {'; so
that the final form of the expression becomes :
*=j-^(l^y. (46s)
where usually {' is put = 250 mm., the conventional distance of dis-
tinct vision. Thus, for example, in the case of a myopic eye, for
which i[ = 150 mm., {^ = 300 mm., so that A = 1/300, we obtain for
a magnification of F = 100 (assuming n = n' = i) :
« = 1/48 = 0.021 mm;
Abbe,* who has investigated this subject very exhaustively, espe-
cially in connection with the microscope, gives several tables (which
are given also by Czapski*) exhibiting the relations between the
Focus- Depth- and the Accommodation- Depth for different values of
the magnification-ratio Y; whereby it appears that, although for low
magnifications the accommodation-depth is far more important than
the focus-depth, the reverse is true in the case of high magnifications.
ART. 120. THB FIELD OF VIEW IN THE CASE OF PROJECTION-STSTEMS
OF FINITE APERTURE.
382. Case of a Single Entrance-Port.
The characteristic effect of a finite aperture in dividing the field of
view into separate regions distinguished by the different magnitudes
of the apertures of the bundles of rays that have their vertices at
' The measure of the power of accommodation of the eye is the strength of an infinitely
thin lena placed where the eye is, for which the far-point and near-point are conjugate
points.
* E. Abbb: Beschreibung etnes neuen stereoskopischen Oculars: Carls Rep. /. Exp,-
Phys., xvii (1881), 197-224: also GesammeUe Abhatidlungen, Bd. I (Jena, 1904), 244-
272. Section III of this paper treats of the special matters here referred to. See also:
E. Abbb: Joum, Roy. Micr. Soc. (2), I (i88i). 687-689.
*S. CzAPSKi: Thtorie der optischen InstrumenU nach Abbb (Breslau, 1893). p. 173.
564
Geometrical Optics, Chapter XIV.
[§381
points comprised within these regions was remarked by J. Petvzal*
in the case of a photographic double-objective in which there was no
material diaphragm other than the lens-fastenings themselves. The
investigation of this effect in the general case of an optical projection-
system of finite aperture will be different according as the field of
view is limited by one or by two ports; and hence we shall treat, first,
the simpler case of an optical system with a single entrance-port.
In the diagram (Fig. i68) the plane of the paper represents a meri-
dian section in the Object-Space; so that, in order to have a complete
representation, the entire figure should be imagined as revolved
Fio. 168.
PxBLD OP View of Omscr in casb op Projbction-Ststbk of Finitb Apbrturb wttb a
Single Entrance- Port.
MM^t MS^c, MD'-P, Sr^c, I SMT» l MMY^B, I SHT" I AfHX*^,
I SGT" I MGZ" A, Z MMX^ x. ^ MMZ'^ ^, I MMD - e.
around the optical axis xx. The positions on the axis of the centres
of the entrance-pupil and entrance-port are designated by M and 5,
respectively. The end-points, on the same side of the axis, of the
diameters, in the meridian plane of the figure, of the entrance-pupB
and entrance-port are designated by D and 7", respectively. Finallyi
the point M designates the point where the optical axis crosses the
focus-plane <r.
The straight line DT joining the end-points, on the same side of
^ J. Pbtzval: Bericht ueber dioptrischc Untersuchungen: SihungshaidiU der meA.'
naiurw, CI. der kaiserl. Akad. der Wissenschaflen (Wien), xxvi (1857), 33-90. See p. SI'
% 382.] The Aperture and the Field of View. 565
the axis, of the diameters of the entrance-pupil and entrance-port meets
the optical axis at the point designated by H and crosses the focus-
plane at the point designated by X, The region of the field of view of
the object defined by the circle described in the focus-plane around M
as centre with radius equal to MX is distinguished by the fact that with-
in this circular space are contained all the points of the focus-plane a
that are the vertices of cones that have the entire opening of the en-
trance-pupil as common base; so that no object-ray emanating from
a point of this central region of the focus-plane and directed towards
a point of the circular opening of the entrance-pupil will be intercepted.
The straight line MT crosses the focus-plane at a point designated
by y, which, since the entrance-pupil, in consequence of its definition
(§ 361), must subtend at Jkf a smaller angle than is subtended there by
the entrance-port, will lie always on the same side of the optical axis
as the point X and at a distance MY greater than MX. The annular
region of the field of view comprised between the circumferences of
the two concentric circles described around M as centre with radii
equal to MX and MY contains all points which, regarded as object-
points, are in a position to utilize one half or more of the total aperture
of the entrance-pupil. Not more than half of the rays of a bundle of
rays emitted from an object-point in this annular region of the focus-
plane and directed towards all the points of the entrance-pupil will
be intercepted, and in general less than half.
Finally, the straight line joining the extremity T of the diameter
of the entrance-port with the opposite extremity of the diameter of
the entrance-pupil will determine by its intersection with the focus-
plane a a third point Z, also on the same side of the axis as the points
X and F, but the most distant one of the three, which marks the
extreme limit on that side of the axis of the field of view. More than
half of the rays emitted by an object-point lying within this outside
annular space of the focus-plane that are directed towards all the
points of the entrance-pupil will be cut off; and a point lying in the
focus-plane at a distance from the axis greater than MZ can send
through the system no ray at all.
The /.MHXxs called by von Rohr* the vignette-angle. Employing
symbols as follows:
ZMHX = ZSHT = M. MM = f. Af 5 = c, MD = />, ST = q,
we obtain from the figure :
p-q p- MX
tanM=-V = "— r"'
1 M. VON Robr: Die Theorie der optischen Instrumenie, Bd. I (Berlin. 1904). p .485,
566 Geometrical Optics, Chapter XIV. [ § 382.
whence also we find the following expression for the radius of the
central region of the field of view:
MX^p-^-l (+66)
The abscissa of the point H with respect to the centre Af of the en-
trance-pupil is:
MH = — = -^ . (467)
tanM p-q ^^'^
From the figure also we obtain the following relations:
MX MD MH MD MD
tanZAfilfX' =
MM " MM ' MH MM MH
= - tan LMMD + tan Z.MHX.
The LMMD = 9 is the aperture-angle, and if we put Z.MMX = x»
the result just obtained may be written as follows:
tan X = tan m — tan 9; (468)
and hence the tangent of the angle x subtended at the centre M o(
the entrance-pupil by the radius MX of the central region of the field
of view is equal to the algebraic difference of the tangents of the
vignette-angle /li and the aperture-angle 9. In terms of the given
linear magnitudes, we can write also:
tanx=-^^ + ^. (469)
If G designates the position of the point where the straight line TZ
crosses the optical axis, and if we put ZSGT = AMGZ = X, ve
obtain from the figure exactly as above:
tanX = -^ ^ .
and hence for the radius of the entire field of view we find:
MZ^^^i'-p. (470)
The abscissa of the point G with respect to the centre M of the entrance-
pupil is:
ilfG = -^ = -^. (471)
tan X p + q ^^'
§ 382.1 The Aperture and the Field of View. 567
Moreover, from the figure :
MZ MP MG MP _ MP
tan AMMZ - ^^ - j^^ ' MG' MG MM
= tan Z SGT + tan /.MMP.
If we put AMMZ — ^, this result may be written as follows:
tan ^ = tan X + tan 9. (472)
Hence, the tangent of the angle ^ subtended at the centre M by the
radius MZ of the entire extent of the field of view is equal to the
algebraic sum of the tangent of the angle X subtended at G by the same
radius and the tangent of the aperture-angle 9. In terms of the
given linear magnitudes, we can write also:
tan^=-^ + ^. (473)
In the case of an object-point in the focus-plane a whose chief ray
has a slope-angle 8 greater than the angle denoted by Xi a part of
the bundle of object-rays will be intercepted at the entrance-port.
The chief rays will be absent from the bundles of effective rays that
come from the object-points in the focus-plane that are farther from
the axis than the point designated by F.
Of all possible object-rays that pass unimpeded through the centre
M of the entrance-pupil, those (such as YT) that graze the "rim" of
the entrance-port have the greatest slopes, viz.:
Z3fAfF= z5Afr = e,
where 6, in the case of an infinitely narrow aperture, is the angular
measure of the field of view (§ 370). The three angles at M denoted
by x> 6 and ^ define the limits of the thiiee parts of the field of view of
the object.
In connection with the case of an optical system of finite aperture
with a single entrance-port, one point remains to be particularly men-
tioned, viz., with respect to the projection-figures on the focus-plane
of object-points that lie outside this plane. If the slope-angle 8 of
the chief ray from an object-point R not in the focus-plane is greater
than the angle Xi the projection-figure on the focus-plane will not be
a circle but a lune, and the chief ray will not be the representative
ray of the bundle, and indeed, if Z.MMR > 6, the so-called chief ray
will be absent from the bundle of effective rays emitted by the object-
568
Geometrical Optics, Chapter XIV.
[§383.
point R. Hence, also, in the case of object-points thus situated, it is
obviously not correct to consider the points where their chief rays
cross the focus-plane as the representative points of their projection-
figures, especially since the former may not even lie within the bound-
aries of the corresponding projection-figures at all.
383. Case of Two Entrance-Ports.
Proceeding now to consider the case of an optical projection-system
of finite aperture with two entrance-ports, we may regard as typical
thereof the case shown in the diagram (Fig. 169)., which represents a
Pxo. 169.
Field OP ViBW OP Oajacr in casb of Projbctxon-Ststbm of FxxfiTBAPB&TU&B with Two
EXtTRANCB-PORTS.
JfJI/-^ Jf^i-^ Jf^-n. MD^p, S\T\^9\, StTt^Qt,
meridian section of the Object-Space. Let 5i, 5, designate the centres
and 7*1, 7*2 the extremities above the optical axis xx of the diameters,
in the meridian plane of the figure, of the two entrance-ports; and let
I/i, I/j designate the other two ends of these diameters. According
to our previous definitions, the centre of the entrance-pupil must lie
at the point M where the straight line 7*11/2 which joins a pair of
opposite ends of the diameters UiT^, U2T2 crosses the optical axis.
The point M where the optical axis crosses the focus-plane <t is deter-
mined by the intersection of the straight line T^T^ with the straight
line XX. It is obvious from the figure that the points M, M axe har-
monically separated by the points 5i, 5,, so that we have the relation:
(JlfAf5i5,) = - I.
(474)
§ 384J The Aperture and the Field of View. 569
If, therefore, the positions of the two entrance-ports with reference
to the entrance-pupil are given, and if we put:
MSi = Cp MS2 = C2,
the position of the focus-plane is determined by the relation:
2C,C^
^ c, + c,'
where { = MM denotes the abscissa of the point M with respect to
the centre Af of the entrance-pupil.
Let C and D designate the lower and upper ends, respectively, of
the diameter of the entrance-pupil which lies in the meridian plane of
the figure, and through the upper ends D and Ti of the diameters CD
and UiTi draw the straight line DT^ crossing the optical axis at the
point designated by AT,; and, similarly, through the lower ends of
the diameters CD and t/2^2 draw a straight line CU^ crossing the
optical axis at the point designated by flT,- Le^ ^ designate the point
of intersection of the straight lines DT^ and CZ/,; we wish to show
that this point -X" will fall in the focus-plane a. Suppose it does not,
and that we draw through -Y a straight line parallel to CD meeting
the optical axis in a point not marked in the diagram which we shall
call iV, and meeting the straight line UJt/ITi in a point Y, According
to this construction, it is plain that the pair of points N, M will be
harmonically separated by the pair of points H2, H^, and that from
the point -X" the harmonic point-range N, Af, H,, H^ will be projected
on to the straight line I/,^! ^^ ^^e harmonic point-range F, Af, I/,* ^n
which latter projected on to the optical axis from the infinitely distant
point of the straight line CD will give:
{NMS^,) = - I.
But since, as a matter of fact, we know that
{MMS2S,) = - I,
it follows that the point designated by N must be coincident with the
point M, and hence the point JY must lie in the focus-plane a, as shown
in the figure. Moreover,
(MMH2H,) = - I. (475)
384. Any object-point lying in the focus-plane within the central
region defined by the circle described around M as centre with radius
570 Geometrical Optics. Chapter XIV. I $ 384.
equal to MX will be in a position to send out rays that will go through
every point of the opening of the entrance-pupil; whereas any point
in the focus-plane at a greater distance from M than X will be in a
position to send out rays that will go through some, but not all, of the
points of the entrance-pupil, provided its distance from M does not
exceed the distance MY; in which latter case it cannot send any rays
through the optical system.
If we put
Z S,H,T, = Z MH,X = Ml, ^ S^^T^ = Z MH^ = ^2.
we obtain from the figure :
Qi-P P MX - p .
tan Hi =
tan iM, = —
Q2-P P MX + p
^2 MH^ {
whence also we find for the radius of the central region of the field
of view:
MX^P + ^-^ I - - ^ - ^^^€; (476)
and for the abscissae, with respect to M, of the points H|, H,:
Likewise from the figure we obtain also the following relations:
.,,.Mv ^X MD . MD MD MD
^^^^^^im^-jm+im'^MwrMM'
and hence if ZMMD = 9, ZMMX = Xi we have here:
tan X == tan mi — tan 9 = tan Ms + tan 9. (478)
If we put ZMMY = 0, we obtain evidently also:
c, c,'
and for the radius of the entire field of view:
Jlf F= ?? { = ^* f (479)
Ci C2
Thus, we see that, whereas the position of the point Y is entirely
independent of the diameter of the entrance-pupil, this is not tnie
$ 387.] Intensity of Illumination. 571
with regard to the position of the point X; for the greater this diameter
IS, the nearer -X" will be to the axial point M\ and in the limiting case
when the end-point D of the diameter of the entrance-pupil lies in the
straight line MT^T^^ the point X will coincide with M.
385. By placing in the focus-plane a circular diaphragm with its
centre at M and with an opening of radius equal to MX^ all of the
field of view outside the central part will be screened off; and then, pro-
vided the object lies wholly in the focus-plane, all the points of the
object will send through the system cones of rays that fill completely
the opening of the entrance-pupil. The same result will be obtained
by placing in the screen-plane or image-plane <t' a diaphragm with its
centre at the point M' conjugate to M and with an opening of radius
M'X' = Y • MX^ where Y denotes the magnification-ratio of the
pair of conjugate transversal planes a and a'. Thus, for example,
in the case of the astronomical telescope, a diaphragm of this kind is
placed in the focal plane of the objective. This simple method is
applicable to all cases in which the depth of the object is negligible,
especially when the object-distance is prescribed and the points Jkf, if '
are the pair of aplanatic points of the optical system (§ 279). But if
the points of the object are situated at finite distances from the focus-
plane, a stop such as above described will not avail for this purpose.
386. Consider an object-point in the plane of the figure above the
optical axis; if it lies to the right of the focus-plane within the angle
MH^ = /i2 or on the other side of this plane within the angle MH^X = /n^
it will be in a position to send through the optical system a cone of rays
completely filling the opening of the entrance-pupil. If the object-
point lies within the angle subtended at X by the diameter CD of the
entrance-pupil, some of the rays of the cone which has the opening of
the entrance-pupil for its base will be intercepted at the entrance-
port 52 if the vertex of the cone lies to the right of the focus-plane, and
at the entrance-port Si if the vertex of the cone lies on the other side
of the focus-plane. And, finally, if the object-point lies to the right
of the focus-plane and outside the angle /nj, or on the other side of the
focus-plane outside the angle ^2* 21II the rays will be intercepted.
INTENSITY OF ILLUMINATION AND BRIGHTNESS.
ART. 121. FUNDAMENTAL LAWS OF RADIATION.
387. Radiation of Point-Source.
Regarding the light-rays as the routes of propagation of light-energy,
we may call a bundle of rays a "tube of light";* and it is assumed
' See P. Drudb: Lehrbuch der Oplik (Leipzig, 1900), p. 72. See also P. G. Tait: Lighl
{Edinburgh, 1889). Chapter V.
572 Geometrical Optics, Chapter XIV. [ § 387.
in the theory of radiation that with a steady source of light equal
quantities of light-energy traverse every cross-section of such a tube
in unit-time. If the source is a radiant point P or a luminous body
of such relatively minute dimensions that it may be considered as
physiologically a mere point (or centre) of light, the light-tubes will
be cones with their vertices at the point-source. The quantity of
light radiated in a given time from a steady source may be expressed
generally as the product of two factors, one of which has to do with
the purely geometrical relations, whereas the other depends on the
physical nature and condition of the radiating body. Thus, in the
simplest case, when we have a point-source at the point P, the quantity
of light which in unit-time "flows" through any cross-section of an
elementary tube of light may be represented as follows:
dL « C • do), (480)
where do) denotes the magnitude of the solid angle of the narrow cone
of rays emanating from P, and where C denotes a certain magnitude
called the *^ candle-power^* of the point-source in the direction of the
axis of the cone. If around P as centre a sphere of unit-radius is
described, the quantity of light that falls on a unit area of this sphere
will be numerically equal to the factor here denoted by C In general,
the value of C will vary with the directions of the light-rays; but if
we may assume that the point-source radiates light-energy at approxi-
mately the same rate in all directions, the total quantity of light-
energy per second that traverses any closed surface surrounding the
point P will be equal to 4irC.
If P' designates the position of a point within the elementary conical
light-tube of solid angle do) which lies on a surface a' at a distance from
the radiant point P denoted by r = PP\ and if da' denotes the area
of the surface-element that is cut out of the surface a' by the conCi
and, finally, if ^' denotes the acute angle between the normal to the
surface a' at the point P' and the straight line PP\ then
r^do) = da'-cos ^';
and accordingly we can write:
dL^Cd^^C-^^'-^^, (48.)
where dL denotes the quantity of light emanating from P that falls
every second on the surface-element da\ The quantity of light-
energy which is received by unit-area of the illuminated surface in
§388.] Intensity of Illumination. 573
unit-time is called the intensity of illumination of the surface <r' at
the point P'; and, since this magnitude is defined by the equation
^L ^ cos tp' / o \
^7, = C--^, (482)
we see that the intensity of illumination is inversely proportional to
the square of the distance from the point-source and directly pro-
portional to the cosine of the angle of incidence and to the candle-
power of the source in the given direction.
388. Radiation of a Ltuninous Surface-Element.
If the light-source at P must be regarded as a luminous element of
surface (da) rather than as a mathematical point, the quantity of
light-energy dL that is emitted in a given direction in the unit of time
will depend not only on the magnitude of da but also on the angle of
emission (tp) between the normal to da at P and the given direction
PP\ Thus, according to Lambert's Law, the specific energy of the
radiation of the luminous surface-element da in the direction PP'
will be expressed by the formula:
C = i-da-cos ^, (483)
where the co-efficient i denotes a magnitude depending on the physical
nature of the light-source (for example, its temperature, radiating
power, etc.) which is called the specific intensity or the intensity of
radiation of the luminous surface a at the point P. The apparent
umformity of the brightness of the sun's disc is in agreement with
this "cosine-law". Thus, near the margin of the sun's disc, areas
which appear to be of the same size as areas nearer the centre, but
which in reality are larger than their oblique projections, do not radiate
any more energy than the smaller but more central areas of the same
apparent size.
Hence, according to the so-called "cosine-law of emission", the
quantity of light-energy radiated per unit of time from the luminous
surface-element da to the illuminated element da' in the direction
PP' is:
. da ' da' ' cos <p ' cos ip' , ^ .
dL^t ;^ . (484)
By means of this fundamental formula of photometry, due originally
to Lambert,^ the factor denoted by i may also be defined as the quan-
> J. H. Lambbrt: Phototnetria sive de mensura et gradibus luminis colorum et umbrae
(Augsbitrg, 1 760). See also German tramdation by E. Anding in Nos. 31-33 of Ostwald's
'* Klaasiker der exakten Wissenschaften " (Leipzig, 1892). Also, see A. Beer: Grund*
riss des photomeirischen CalcueUs (Braunschweig, 1854).
574 Geometrical Optics, Chapter XIV. [ § 389.
tity of light which in the unit of time is radiated from a unit-area of
the radiating surface to another unit-area at unit-distance from it,
when the line PP' is a common normal to the radiating surface at
P and to the illuminated surface at P'. As a matter of fact, it is
found by experiment that the specific intensity i varies with the angle
of emission ^ and according to a peculiar law for each different sub-
stance; but in the following discussion it will be simpler to disregard
this variation and to assume therefore that the value of i is independ-
ent of the angle tp.
The symmetry of the expression on the right-hand side of the above
equation cannot fail to be remarked. Thus, for example, the quantity
of light conveyed from da to da' in a given time is the same as would
be transmitted in this same time from da' to da in case the rdles of
the two surfaces were interchanged, so that da' was the radiating ele-
ment of specific intensity equal to i and da was the illuminated element.
Since
_ da'- cos ip'
d«« p ,
and since, also, if dw' denotes the solid angle subtended at P' by the
radiating surface-element da^
. , da- cos 4p
d<a =• p ,
formula (484) may be written likewise in either of the two following
forms:
dL = i'dacos^'dia = i-da' -cos ^'•dw'. (485)
389. Equivalent Light-Source. The intensity of iUumination at P'
due to the radiating element da at P, viz.,
-3-7 = i ' cos ip' • id)', (486)
aa
is proportional to the specific intensity (i) of the source and also to
the solid angle {dca') subtended at P' by the radiating surface-element
(da) and the cosine of the angle of incidence (tp') of the rays. With
respect to the illumination at P', the most important deduction to be
made here is that, so far as the resultant effect at P' is concerned, the
surface-element da may be supposed to be replaced by its central pro-
jection from P' on to any other surface in the same optical mediumi
provided we ascribe the same specific intensity i to the corresponding
§ 390.] Intensity of Illumination. 575
points of the projection-surface.^ Accordingly, a fictitious source of
light, or rather an imaginary distribution of the specific intensity, can
be thus substituted in place of the actual distribution so as to have
precisely the same effect at a prescribed point P'. However, this
so-called equivalent surface-distribution of the specific intensity— or
**eguivalent Ught-source'' —vnllt in general, produce a different effect
from that produced by the actual light-source at any point other than
the given point P'.
ART. 122. INTENSITY OF RADIATION OF OPTICAL IMAGES.
390. Optical System of Infinitely Narrow Aperture (Paraxial Rays).
Let Af, M' designate the positions of a pair of conjugate axial points
of a centered system of spherical surfaces, and let us suppose, at first,
that the aperture of the system is infinitely narrow, so that only the
so-called paraxial rays emanating from the luminous point-source M
can traverse the system. To the bundle of paraxial rays in the Object-
Space of solid angle do) corresponds also a bundle of paraxial rays in
the Image-Space of solid angle dw'; and if C denotes the candle-power
of the point-source, the quantity of light radiated from M in unit-time
will be dL = C*d<a; and, similarly, the quantity of light radiated in
the same time from the conjugate image-point M' will hedL' = C''d<a\
where C denotes the candle-power of the image-point M' regarded
as a source of light in the Image-Space. Moreover, for the sake of
simplicity, let us assume here that no light-energy is ''lost** either by
absorption in traversing the various media or by undesirable reflexions
at the spherical surfaces; and although this assumption is notoriously
contrary to the fact, it will not materially affect the conclusions which
we have here in view. Accordingly, putting dU = dL, we obtain
therefore:
C'do) = C'do)'.
The following relation may easily be deduced :
do)'
'-.-tm-
do)
where f*^, ul denote the abscissae of the points where the rays cross
the optical axis before and after refraction, respectively, at the jfeth
surface of the centered system of m spherical surfaces. If, therefore,
* See E. Abbb: Ueber die Bestimmung der Lichtstaerke optischer Instrumente: Jen.
Zfi.f. Med. M. Natw., vi (1871), 263-291. Also, GesammeUe Abhandlungen, Bd. I (Jena*
1904). 14-44.
576 Geometrical Optics, Chapter XIV. I § 391.
Y denotes the lateral magnification of the system with respect to the
pair of conjugate axial points Jlf , M\ and if also n, n' denote the indices
of refraction of the media of the Object-Space and Image-Space, re-
spectively, we obtain by the employment of formula (93) :
dw' I «*
d<j>~Y*' „'"
and, hence:
C ' n*'
(487)
whereby, knowing the candle-power {C) oi the point-source on the
axis of the optical system, and knowing also the constants of the sys-
tem, we are enabled to determine the corresponding candle-power (CO
of the image-point Jlf '.
If, instead of a point-source at the axial point Jkf, we have a luminous
surface-element da at right angles to the optical axis at if , the image
thereof will be a surface-element da' at right angles to the optical axis
at the point M\ of such dimensions that
da' = F*.d<7.
Hence, since here we have
dL = i*da*d(a = dU = V *da' *dw\
where i, V denote the specific intensities in the direction of the axis
of the radiating elements d<r, da', respectively, we obtain in this case
the following striking relation :
391. Optical System of Finite Aperture.
Finally, let us now proceed to the more general case and assume
that the aperture of the optical system is finite ; and let us denote by
i the specific intensity of radiation, in a direction defined by the slope-
angle By of a luminous surface-element da placed at right angles to
the optical axis at the point M, The quantity of light radiated in
unit-time from the element da to an elementary annular ring of the
entrance-pupil whose inner and outer radii subtend at the axial object-
point M angles denoted by ^ and ^ •\- d$, respectively, may be easily
calculated from the fundamental formula (484) and will be found to be;
dL = ivi'da'sin 6'd (sin 6).
§ 392.] Intensity of Illumination. 577
Employing the same symbols with primes to denote the corresponding
magnitudes in the Image-Space, we shall find also a precisely analogous
expression for the quantity of light that is radiated per unit of time
from the image-element da' to the corresponding elementary annular
ring of the exit-pupil, viz.:
dV = 2Ti'-da'-sin B'-d (sin B').
Now if da' is to be a correct image of the object-element da, it is neces-
sary to suppose that Jlf, M' are an aplanatic pair of points, so that
the Sine-Condition is satisfied, whereby we must have (§ 277) :
n«sin d = n'-F'sin 6'.
Introducing this condition, and employing here also the relation
da' = r-da,
and, finally, assuming, as before, that dL' = dL, we derive again the
same relation as above, viz. :
7 = ^- (488)
Accordingly, no matter how the specific intensities of radiation of
object and image may vary for different angles of emission, their ratio is
ike same for every pair of values of 6 and $\ This constant ratio de-
pends only on the indices of refraction of the media in which the
object and image are situated; and the specific intensity of radiation
(f ') of any element of the image in a given direction {$') is equal always to
(n'/n)* times that of the corresponding object-element in the conjugate
direction {d).^
392. In deriving the above results, it was assumed that there were
no losses of light by absorption, reflexion, etc., so that we could put
dL' = dL. It would have been more correct to have written:
dL' = (i - i?)dL,
where ri denotes the fraction of the original quantity of light that is
1 This result is identical with Kirchhoff's well-known law of radiation. See G.
Kxrchbotf: Ueber das Verhaeltniss zwischen dem Emissionsvermoegen und dem Ab-
aorptionsvermoegen der Koerper tttr Waerme und Licht. Pogg. Ann., cix (i860), 275-
30X. Also. R. Clausius: Ueber die Concentration von Waerme- und Lichtstrahlen und
die Grenzen ihrer Wirkung. Pogg. Ann., cxxi (1864); also, Browne's English trans-
lation of Clausrjs's Mechanical Theory of Heat (I^ndon, 1879), Chapter XII.
Starting from Kxrchhopf's law of radiation. Helbiholtz deduced the Sine-Law; see
H. Helmholtz: Die theoretische Grenze fUr die Leistungsfaehigkeit der Mikroskope:
Pogg. Ann, Jubelband, 1874. 557-584-
38
578 Geometrical Optics, Chapter XIV. [ § 393.
dissipated in its passage through the system, and is a function of the
angle of emission (^), which may be determined in any given special
case. Under these circumstances, formula (488) would be modified
as follows:
7=(i -i|)— 1; (489)
which shows also that the ratio V 1% is in reality a function of the
angle Q}
In nearly all actual optical instruments the first and last media are
both air (n = n' = i); even in the so-called * 'immersion-systems"
the source is not the object immersed in the fluid, inasmuch as the
object is illuminated from without. The case when n' > n is hardly
realizable. Thus, under the most favourable conditions, the specific
intensity of radiation from a definite part of the image in a given direct-
ion will always be less than the specific intensity of radiation from the
corresponding part of the object in the conjugate direction. For ex-
ample, the intensity of radiation of the sun's image at the focus of a
convex lens can never be greater than that of the sun itself, although
the intensity of illumination of a screen placed at the focus of the
glass may be much greater with the lens than without it.
393. The Illumination in the Image-Space.
The image M'Q' of a luminous object MQ may be regarded as the
source of all the illumination in the Image-Space; and in case we wish
to ascertain the intensity of the illumina-
tion produced at any point B! of the
Image-Space by an element of the image
at P', we have merely to trace backwards
^ through the optical system the path of the
image-ray P'R' and thereby determine the
point P of the object that is conjugate to
the image-point P'. The specific intensity
of the radiation from P' in the direction
Fio. 170. pi^f is (^Y^)i tjjn^s ^hat from the object-
ExiT-PupiL iia KQuivALBNT p^i^t p in the conjugate direction in the
I^UMiNous Surface. >vi . o» • t «
Object-Space; provided we assume that
none of the light is dissipated in its passage through the system.
The part of the image M*Q' (Fig. 170) that is effective in producing
illumination at a point 2?' of the Image-Space is easily found by pro-
jecting the exit-pupil CD' on to the image-plane a'; thus, in the
' See S. CzAPSKi: Theorie der optiscken InstrumenU nach Abbr (Breslaa, 1893), p. I79*
§ 394.] Brightness. 579
diagram, U'V represents the effective part of the image with respect to
the illumination at the point R'. In place of the portion of the image
U'V\ we may substitute an equivalent distribution of light (§ 389) by
considering the specific intensity of the parts of the image comprised
between U' and V as localized at the corresponding parts of the exit-
pupil ; and this distribution of light supposed to be spread over the
exit-pupil would produce exactly the same effect at R' as is produced
there by the image of the luminous object. This ingenious method,
due to Abbe,* enables us to determine the intensity of illumination
at any point of the image itself. For example, the nearer the point
R' is to the point P' of the image, the smaller will be the circular space
around P' that is obtained by projecting the exit-pupil on to the plane
of the image; and, finally, when the point R' coincides with P', so
that the exit-pupil is projected on to the image-plane in the point P'
itself, the intensity of the illumination at P' can be found by regarding
the illumination there as due to a distribution of light over the exit-
pupil of the same specific intensity of radiation as that of the point
P', viz., (n'/n)H, where 1 denotes the specific intensity of radiation in
any given direction (fi) of the object-point P conjugate to P\
ART. 123. BRIGHTNESS OF OPTICAL IMAGES.
394. Brightness of a Luminous Object.
In connection with the definition of the objective intensity of illumi-
nation (§ 389) at a given place of an illuminated surface, we can derive
also an idea of what is meant by the Brightness of the source as seen
by an eye situated at the place in question. The brightness of an
element d<r of a radiating surface is defined as the quantity of light-
energy which in the unit of time falls on unit-area of the image d<r'
that is formed on the retina of the eye; in other words, it is the inten-
sity of illumination of the element of the retina-surface that is affected
by the given element of the luminous body. Thus, if dL denotes the
quantity of light which is radiated per unit of time from the element
d<r into the eye, the brightness of this element is defined by the equa-
tion:
dL
5 = ^,. (490)
If we assume that there is no loss of light in traversing the optical
* E. Abbb: Ueber die Bestimmung der Lichtstaerke optischer Instrumente. Jen. Zft,
/. Aled. u. N<Uw., vi (1871), 263-291. Also Gesammelte Abhandlunggn, Bd. I (Jena,
igo4). Z4->44«
580 Geometrical Optics, Chapter XIV. [ § 395.
media of the eye, then
2iri''d<T''sine''d{sine')
will be the quantity of light that is radiated per unit of time across an
elementary annular ring of the exit-pupil of the eye, where 6' denotes
the angle subtended at the retina by the inner radius of this ring and
i' denotes the specific intensity of the image da' on the retina; and
hence the total quantity of light that enters the pupil will be
dL = Tt' -da' -sin* 9^,
where 6^ denotes the angle subtended at the image on the retina by
the radius of the exit-pupil of the eye, which usually does not exceed
about 5®. If, therefore, the object is viewed by the unaided eye, we
find for the so-called natural brightness (B^) of the luminous surface-
element d<r (supposed to be situated in air, so that n = i) :
Bo = ^, = Tn'*.i.sin*e;, (491)
where i denotes the specific intensity of radiation of the source, and
n' denotes the refractive index of the vitreous humour of the eye. It
follows immediately from this expression that the natural brightness
of a uniformly radiating surface depends only on the intensity of
radiation of the light-source and is entirely independent of the dis-
tance of the luminous object from the eye; as is found to be practically
the case.
ii 395. In the next place, let us suppose that this same object is
viewed through an optical instrument by an eye placed at the exit-
pupil of the instrument. Everything is the same as before, except
that now, instead of the mere optical system of the eye, we have a
compound optical system formed by the combination of the eye with
the optical instrument. If we disregard all losses of light by reflexion
and absorption, and assume, as before, that the luminous object is in
air, the brightness B of the optical image as seen through the instni-
ment will be equal to the natural brightness Bq of the object as viewed
by the naked eye, provided the exit-pupil of the eye is smaller than
that of the instrument. But if, on the other hand, the diameter of
the exit-pupil of the instrument is smaller than that of the eye, the
aperture-angle will be an angle 6' < 9^, so that in this case we shall
have:
B:Bo = sin'e':sin'e;.
Since the angles 9^, 9' are so small that we may substitute the tangents
§ 396.] Brightness. 581
of these angles in place of their sines, and since, moreover, the exit-
pupil of the eye coincides very nearly with the eye-pupil (or iris),
we obtain:
B:Bo^p'':pl (492)
where p^, p' denote the radii of the iris-opening and exit-pupil
of the instrument, respectively; so that the brightness of the image
compared with the natural brightness of the object is diminished in
the ratio of the size of the exit-pupil of the instrument to the size of
the eye-pupil. It is, therefore, impossible by means of any optical
instrument to increase the natural brightness of an object as seen by
the unaided eye. Thus, the only function of an optical instrument is
by means of a light-source either of small dimensions or very far away
to produce an effect equal to that which could be produced without
the instrument only by a larger or nearer source of light radiating with
equal specific intensity.^
396. Brightness of a Point-Source.
If the luminous object is so small or so far away that it has no
sensible apparent size, the definition of brightness given above (§ 394)
ceases to have any meaning; for the image on the retina of the eye
will in this case be itself a mere point without appreciable area. If,
therefore, the source of illumination is a point, for example, a fixed
star, the brightness is defined as equal or proportional to the quantity
of light which comes to us from it. Thus, when we speak of a star
of the "first magnitude", this expression refers merely to the amount
of light we receive from it and has nothing to do with the size of
the star.
If in formula (481) we put da' = vpl (where p^ denotes the radius
of the eye-pupil) and cos ^' = i (since the rays are supposed to fall
normally on the retina when the eye is directed towards the point-
source), we obtain:
B = Cr-p. (493)
Hence, the brightness of an object which appears like a point is inversely
proportioned to the square of its distance from the eye, and directly pro-
portional to the size of the eye-pupil.
Thus, stars which are invisible to the naked eye may be brought to
' Lord Raylbigh, in his brilliant article on Optics in the ninth edition of the Encyclo-
paedia Britannica, has pointed out that *' the general law that the apparent brightness
depends only on the area of the pupil filled with light " was stated and demonstrated by
ROBBRT Smith. See Smith's Optics (Cambridge, 1738), Vol. I, Sections 255 and 261.
582 Geometrical Optics, Chapter XIV. [ § 396.
view by the aid of a telescope, whereby the eye receives a greater
quantity of light from the star than before, so that the brightness
(in this latter sense of the term) is increased; whereas, on the other
hand, the brightness of the background of the sky (using the word
"brightness" in its original sense, as defined in § 394) will be di-
minished. This is the reason why a powerful telescope, of large
aperture and great magnifying power, may enable an observer to
view the stars even in the noon-day glare.
APPENDIX.
EXPLANATIONS OF LETTERS, SYMBOLS. ETC.
The meanings of the principal letters and symbols both in the text
and in the diagrams are here set forth as briefly as possible; but such
uses as are occasional or merely incidental are generally not noted at
all. In consulting these tables, it is important to bear in mind this
last statement.
L DESIGNATIONS OF POINTS IN THB DIAGRAMS.
As a rule (but not without exception), the positions of the points
in the diagrams are designated by Latin capital letters. The most
important uses of these letters are explained below.
1. AfA' are used to designate the primary and secondary principal
points, respectively, of two coUinear space-systems; see Fig. 92. Simi-
larly, as in Fig. 99, Aj^, A\ designate the principal points of the *th
component of a compound optical system.
In Chap. XIII, A, A' and A, A' designate the positions on the
optical axis of tl^e two pairs of principal points of the system for rays
of light of wave-lengths X and X, respectively.
In the case of two centrally collinear plane-fields, A designates the
position of the point of intersection with the axis of collineation (y)
of the self-corresponding ray (jc, x') that meets this axis at right angles.
2. Especially, the letter A is used to designate the vertex of a spher-
ical refracting (or reflecting) surface. Similarly, A may be used to
designate the position of the foot of the perpendicular let fall on to a
plane refracting (or reflecting) surface from a point on the incident
ray regarded as object-point, as in Fig. 8.
The letter A designates the optical centre of an Infinitely Thin Lens.
The vertex of the Jfeth surface of a centered system of spherical
surfaces is designated by Aj^; also, the optical centre of the Jfeth lens
of a centered system of Infinitely Thin Lenses.
3. In Chap. I X, in the determination of the path of a ray refracted
obliquely at a spherical surface, A^, A^ are used to designate the
points of intersection with the surface of the radii drawn through the
points designated by G (14) and / (18), respectively (Fig. 122). In
583
584 Geometrical Optics, Appendix.
Chap. X, in the case of a ray refracted obliquely through a centered
system of spherical surfaces, -4,,*, i4<,» have the same meanings as
above with respect to the kth surface.
4. 5, 5' are used to designate the points of intersection of a pair
of conjugate rays with the principal planes of two coUinear space-
systems. In Fig. 99, for example, Bj^, B^ designate the points where
a meridian ray crosses the principal planes of the Jfeth component of
a compound optical system.
In particular, B designates the point of intersection of a ray with
the axis of collineation (y) of two centrally collinear plane-fields.
5. Especially, B designates the position on the refracting (or re-
flecting) surface of the incidence-point of a ray. In the case of a
centered system of spherical surfaces or a prism-system, B^ designates
the incidence-point of the ray at the Jfeth surface.
In Chap. XIII, Sj^, Bf^ designate the incidence-points at the Jfcth
spherical surface of rays of light of wave-lengths X, X, respectively,
whose paths in the Object-Space are identical.
If we are concerned with a pair of rays from two different sources,
whose paths lie in the plane of a principal section, their incidence-
points may be designated by B and B (or by B^ and JB^), as, for
example, in Chap. VIII.
Usually, however, B or B^ designates the position of the incidence-
point of the chief ray of a bundle.
C
6. C is used primarily to designate the centre of a spherical refracting
or reflecting surface. The centre of the Jfeth surface of a centered
system of spherical surfaces is designated by Cj^.
This letter is used also to designate the centre of collineation of two
centrally collinear plane-fields, as in Fig. 66.
7. In Chap. XIV, C, C are used to designate corresponding ex-
tremities of conjugate diameters of the entrance-pupil and exit-pupil,
respectively, of an optical system. C designates the lower extremity
of the diameter, in the meridian plane, of the entrance-pupil.
8. D designates the foot of the perpendicular BD let fall from the
incidence-point B on to the optical axis; in a centered system of
spherical surfaces, Dj^ designates the foot of the perpendicular let fall
on to the optical axis from the point B^^ where the ray meets the feth
surface.
Designations of Points. 585
The foot of the perpendicular let fall on to the optical axis from the
point B (see 5) is designated by D, as in Fig. 140.
The feet of the perpendiculars let fall on to the optical axis from the
points Bf^t B^ (see 5) are designated by D^, 5^, respectively.
9. Dj D' are used (in Chap. XIV) to designate corresponding ex-
tremities of conjugate diameters of the entrance-pupil and exit-pupil ^
respectively, of an optical system. Generally, D designates the upper
extremity of the diameter, in the meridian plane, of the entrance-
pupil (see 7).
10. In certain of the prism-diagrams of Chap. IV, D designates
the point of intersection of the incident and emergent rays.
E
11. Et E' are used to designate the infinitely distant point of the
optical axis (x) in the Object-Space and its conjugate point in the
Image-Space, respectively. E' designates, therefore, the secondary focal
point of the optical system.
£', E' designate the secondary focal points of the optical system
for rays of light of wave-lengths X, X, respectively.
£4 is used to designate the secondary focal point of the Jfeth com-
ponent of a compound optical system, as in Fig. 99.
F
12. F, F' designate the primary focal point in the Object-Space
and the infinitely distant point of the optical axis (jc') in the Image-
Space, respectively.
F, F designate the primary focal points of the optical system for
rays of light of wave-lengths X, X, respectively.
Fj^ designates the primary focal point of the Jfeth component of a
compound optical system, as in Fig. 99.
G
13. G designates the incidence-point of a ray in the meridian section
of an infinitely narrow bundle of rays (Fig. 127).
14. In Chap. IX, G, G' are used to designate the points where
an oblique ray crosses the plane of the principal section (:ic>'-plane) of
the spherical refracting surface, before and after refraction, respect-
ively (Figs. 122 and 123).
H
15. Hj H' designate the points where an oblique ray crosses the
central transversal plane (yz-plane), before and after refraction, respect-
586 Geometrical Optics, Appendix.
ively, at a spherical surface (Fig. 123). Similarly, Hf^, H^ are used
as above stated, with respect to the jfcth surface of a centered system
of spherical surfaces.
In particular, H, H' designate the points where a ray, lying in the
principal section of a spherical refracting surface, crosses the central
perpendicular, before and after refraction, respectively (Fig. I2q).
16. In certain diagrams in Chapters V, VI and VII, /, /' are used
to designate the infinitely distant point of an object-ray s and the
**Fluchi'' Point of the conjugate image-ray s\ respectively, of two
coUinear plane-fields. See, for example, Fig. 67.
17. Especially, in Chap. XI, in connection with the theory of the
refraction of an infinitely narrow bundle of rays at a spherical surface,
/, /' designate the infinitely distant point of the range of primary
object-points lying on the chief incident ray and the "Flucht" Point
of the conjugate range of primary image-points lying on the chief
refracted ray, respectively. Thus, /' designates the secondary focd
point of the collinear plane-fields of the meridian sections of the bundles
of incident and refracted rays. See Fig. 128.
Similarly, 7, T designate the infinitely distant point of the range
of secondary object-points lying on the chief incident ray and the
"Flucht" Point of the conjugate range of secondary image-points lying
on the chief refracted ray, respectively. Thus, also I' designates the
secondary focal point of the collinear plane-fields of the sagittal sections
of the bundle of incident and refracted rays. See Fig. 128.
According, therefore, as the chief incident ray is regarded as the
base of a range of primary or secondary object-points, the infinitely
distant point of this ray is designated by / or /.
Moreover, i^, 7^ designate the focal points of the systems of meridian
and sagittal rays, respectively, with respect to a given chief ray re-
fracted at the Jfeth surface of a centered system of spherical surfaces.
18. In Chap. I X, 7, 7' designate the points where an oblique ray
crosses the (horizontal) meridian xz-plane, before and after refraction,
respectively, at a spherical surface (Fig. 122).
Similarly, in Chap. X, 7;^, 7]^ (or 7;^+,) are used in the same way
as above, with respect to the Jfeth surface of a centered system of
spherical refracting surfaces.
/
19. In certain diagrams in Chapters V, VI and VII, /, /' designate
the **Flucht'' Point of an object-ray s and the infinitely distant point
Designations of Points. 587
of the conjugate image-ray s\ respectively, of two colHnear plane-
fields. See, for example, Fig. 67.
20. In Chap. XI, in connection with the theory of the refraction
of an infinitely narrow bundle of rays at a spherical surface, /, /'
designate the ^'Flucht'* Point of the range of primary object-points
lying on the chief incident ray and the infinitely distant point of the
corresponding range of primary image-points lying on the chief re-
frjtcted ray, respectively. Thus, / designates the primary focal point
of the collinear plane-fields of the meridian sections of the bundles of
incident and refracted rays (Fig. 128O.
Similarly, J, J' designate the *'Flucht** Point of the range of sec-
ondary object-points lying on the chief incident ray and the infinitely
distant point of the corresponding range of secondary image-points
lying on the chief refracted ray, respectively. Thus, also, J designates
the primary focal point of the collinear plane-fields of the sagittal
sections of the bundles of incident and refracted rays (Fig. 128).
According, therefore, as the chief refracted ray is regarded as the
base of a range of primary or secondary image-points, the infinitely
distant point of this ray is designated by J' or J\
Moreover, Jj^, Jj^ designate the primary focal points of the systems
of meridian and sagittal rays, respectively, with respect to a given
chief ray refracted at the Jfeth surface of a centered system of spherical
surfaces.
K
21. In Fig. 128, X designates the centre of perspective of the range
of object-points lying on the chief incident ray and the range of pri-
mary image-points lying on the corresponding refracted ray.
22. In Chap. XII, K, K and K'^ K' designate the centres of
curvature of the two astigmatic image-surfaces, before and after re-
fraction, respectively, at one of a centered system of spherical surfaces
(Figs. 141, 142).
23. L, L' designate the points where a ray, lying in the principal
section of a spherical refracting surface, crosses the axis, before and
after refraction, respectively (Fig. 120).
L, V designate the points where the chief ray of a bundle crosses
the axis, before and after refraction, respectively, at a spherical surface.
In certain cases, also, L, V (or L, L') are used to designate the
points where an object-ray and the corresponding image-ray, respect-
ively, cross the optical axis of a centered system of spherical refracting
surfaces.
588 Geometrical Optics, Appendix.
Sometimes V is used to designate the point where a second ray
emanating from the axial object-point L crosses the axis after emerging
from the optical system (Fig. 117).
l!j^ (or Lfc+i) designates the point where a ray lying in the prin-
cipal section crosses the optical axis after refraction at the Jfeth surface
of a centered system of spherical surfaces. If the ray is the chiej ray
of the bundle, the point in question is designated by L]^ (or L^,).
The point where the ray crosses the axis in the Object-Space is desig-
nated by L4 (or Li).
Af, Af, m
24. JIf , JIf ' designate a pair of conjugate axial points of an optical
system; especially, a pair of points where a paraxial ray crosses the
optical axis in the Object-Space and Image-Space, respectively.
In Chap. XIII, Af' , M' and 3W' designate the points where paraxial
rays of light of wave-lengths X, X and I, respectively, all emanating
originally from the same axial object-point JIf , cross the optical axis
in the Image-Space.
AfJ^ (or Jf^+i) designates the point where a paraxial ray, emanat-
ing from the axial object-point JIf p crosses the optical axis of a centered
system of spherical surfaces after refraction at the fcth surface (see
Fig. 71). Here also j^ has a meaning corresponding to that of J?'
above.
25. Especially, M, M' designate the points where the optical axis
crosses the transversal object-plane (a) and the conjugate (GAUSsian)
image-plane (<r') ; or the points where the optical axis crosses the focus-
plane and the screen-plane^ respectively (Chap. XIV). See 69. ^
may be defined as the foot of the perpendicular let fall on to the
optical axis from the extra-axial object-point Q\ and if Q' designates
the GAUSSian image-point corresponding to Q, M' will designate also
the foot of the perpendicular let fall on to the optical axis from Q'-
26. Af, M' designate a second pair of conjugate axial points, with
respect either to a single spherical surface or a centered system of
spherical surfaces. The meanings of Af' , Af'; Af|^ (or Af^^,); and
Afl correspond exactly with the meanings given above (24) of Af' , S'\
M'f^ (or M;fc+,); and Af]^, respectively.
27. Especially, Af, Af' designate the positions on the axis of the
centres of entrance-pupil and exit-pupil, respectively, of the optical
system. In an optical system of m centered spherical surfaces, the
centres of the pupils may be designated by Af i and Af|„.
If Af , Af ' designate the pupil-centres of the optical system for rays
of wave-length X, Af , Af ' may be used to designate the pupil-centres
for rays of wave-length X.
Designations of Points. 589
28. In an Infinitely Thin Lens, Af, Af' are used (as in Fig. 75) to
designate the points where a paraxial ray crosses the optical axis,
before and after passing through the lens. And, especially, in the
case of a centered System of Infinitely Thin Lenses, M^, M[ are used
in this way with respect to the Jfeth lens. Exactly, the same statements
can be made here with reference to the use of Af , iW' and M^, M[.
29. M" is used in various ways; for example, to designate the
point where the focussing screen is crossed by the optical axis, or, as
in Fig. 161 (and elsewhere), to designate the centre of the ^^blur-circle**
corresponding to an axial object-point M.
N
30. N, N' are used to designate points on the normal to a refracting
surface at the incidence-point B in the first and second medium,
respectively (as in Fig. 5).
31. Especially, iV, N' designate the pair of nodcU points of an
optical system (Fig. 92).
O
32. 0 designates the position on the optical axis of the centre of
the aperture-stop,
33. In the prism-diagrams of Chap. IV, 0 designates the point of
intersection of the normals to the two faces of the prism at the points
of entry and exit. In the case of a train of prisms, a numerical sub-
script indicates the prism to which the letter refers (as in Fig. 45).
34. O is used also to designate the position of the optical centre of
a thick lens (Fig. 74).
35. In Chapters V, VI and VII, 0, 0' occur frequently to designate
a pair of conjugate points,
p,p
36. P, P' designate the point where the object-ray crosses the
transversal object-plane a and the point where the corresponding re-
fracted ray crosses the transversal image-plane a', respectively (69).
P4 (or Pj^+i) designates the point where a ray crosses the trans-
versal plane a^ after refraction at the fcth surface of a centered system
of spherical surfaces.
Pi designates the point where the rectilinear path of the ray in the
Object-Space crosses the transversal object-plane <ri; especially, it des-
ignates the position of the object-point in this plane, and, in general,
the same extra-axial object-point as is designated by Qi (see 39).
In a certain sense (see Chap. XII) the point P^ may be regarded as
the image of the object-point Pj.
590 Geometrical Optics, Appendix.
37. P, P' are used in Chap. XII to designate the point where the
object-ray crosses the transversal plane cr in the Object-Space and the
point where the corresponding image-ray crosses the conjugate plane
cr' in the Image-Space, respectively (71). The object-ray here men-
tioned is a ray that goes through the point P (36).
Similarly, also, P]^ (or Pi,^^ designates the point where a ray
which in the Object-Space goes through Pj crosses the transversal
plane cr]^ after refraction at the Jfeth surface of a centered system of
spherical surfaces. The point where the object-ray crosses the first
one (o-j) of this series of transversal planes is designated by P, (or Q,).
38. P, P'; 0, 0'; P, P'; 5, 5' and P, P'; 5, p'; P, P'; 5, ^
are used frequently to designate pairs of corresponding points of pro-
jective point-ranges. Thus, for example, in Chap. XI, P, P' designate
a pair of corresponding points of the ranges of primary object-points
and image-points lying along the chief incident ray of a narrow bundle
of rays and the corresponding refracted ray, respectively; and, sim-
ilarly, P, P' designate corresponding points of the ranges of secondary
object-points and image-points lying along the same chief incident
and refracted rays, respectively.
39. Q, Q' designate a pair of conjugate points, especially a pair
of extra-axial conjugate points, of two collinear systems.
In general, Q, Q' designate a pair of points, lying outside the axis
of the optical system, which, by Gauss's Theory, are conjugate to
each other, with respect to either a single spherical surface or a centered
system of spherical surfaces. Especially, Q, Q' designate a pair of
conjugate points lying in the transversal planes a, a', respectively.
Q, Q' are used also (Chap. XIV) to designate the centres of the
**blur'Circles** in the focus-plane (cr) and the screen-plane (a') i respect-
ively.
In case we have to do with rays of light of two different colours
(as in Chap. XIII), Q\ 'Q' designate the points conjugate to Q for
rays of light of wave-lengths X, X, respectively.
Q'k (or Qk+i) designates the point where, according to Gauss's
Theory, a ray, emanating originally from the object-point Qi, crosses
the transversal plane cr^ after refraction at the Jfeth surface of a centered
system of spherical surfaces.
40. The point where an object-ray, which goes through the object-
point Q (or P), crosses the plane (<r) of the entrance-pupil is desig-
nated by Q or P; and the point in the plane (<r') of the exit-pupilt
which, by Gauss's theory, is conjugate to Q (or P), is designated by (?'.
Designations of Points. 591
In a centered system of spherical surfaces, Q^ (or Pj) designates the
point where an object-ray, containing the object-point Q^ (or Pj),
crosses the plane <ri of the entrance-pupil; and Ql (or Qjh-i) desig-
nates the position, in the transversal plane <ri^ (71), of the point which,
by Gauss's Theory, is conjugate to the point Qi after refraction at the
ifcth surface.
41. C, ^" are used to designate the points where a pair of paraxial
rays, of colours X, X, respectively, both emanating from the same
extra-axial object-point Qj cross the focussing plane in the Image-
Space of the optical system (Chap. XIII).
In 'Fig. 161, Q" designates the centre of the ^'blur-circle'^ in the
scale-plane a" of an optical measuring instrument, which corresponds
to the extra-axial image-point Q', '
42. See use of this letter in conjunction with P, R and 5 (38).
R
43. The letter R is used, especially in Chap. XIV, to designate a
point of a 3-dimensional object, and R' to designate the conjugate
point of the relief-image.
44. See 38 for use of this letter in conjunction with P, Q, 5.
S
45. 5, S and 5', S' are used, especially in the theory of the refract-
ion of a narrow bundle of rays, to designate the primary and secondary
object-points and image-points, respectively. Thus, 5, 5' designate a
pair of conjugate points on the chief ray of a pencil of meridian rays,
before and after refraction, respectively; and, similarly, 5, 5' desig-
nate a pair of conjugate points on the chief ray of a pencil of sagittal
rays, before and after refraction, respectively. If the bundle of inci-
dent rays is homocentricj the points 5, 5 coincide at the vertex of the
bundle.
5i (or Sj^^i) and ^l (or 5^^.,) designate the positions on the chief
ray of the primary and secondary image-points, respectively, after
refraction of the narrow bundle of rays at the Jfeth surface of a system
of refracting surfaces.
46. 5, 5' and 5, 5' are used also as explained in 38 above. 5, S'
occur frequently to designate a pair of conjugate points of two collinear
systems.
47. Especially, the letters 5, 5' designate the positions on the axis
of the centres of the entrance-port and exit-port, respectively, of an
optical system. If the system has two entrance-ports, the centres are
designated by 5, and Sj.
592 Geometrical Optics, Appendix.
48. r, V are used in Chap. I X to designate the points of inter-
section of a pair of incident rays lying in the plane of a principal section
of the spherical refracting surface and the pair of corresponding re-
fracted rays, respectively. See Fig. 121.
49. T is used also to designate the upper end of the diameter, in
the plane of the principal section, of the entrance-port of the optical
system; and T' designates the point in the circumference of the exit-
port which is conjugate to T, If the system has two entrance-ports,
the upper ends of the diameters, in the plane of the principal section,
are designated by T^ and T^.
50. In certain diagrams ofc Chap. V, the letter T is used to desig-
nate the infinitely distant point of the ^^-axis.
U
51. This letter occurs in various uses. We mention here only one
of these, viz. : C/ designates the lower end of the diameter, in the plane of
the principal section, of the entrance-port of an optical system; and U'
designates the point in the circumference of the exit-port which is con-
jugate to U. If the system has two entrance-ports, the lower ends
of the diameters are designated by J7|, U^.
V
52. In the prism-diagrams, V designates the vertex of the prism.
In a system of prisms whose refracting edges are all parallel, F^ desig-
nates the point where the refracting edge of the Jfeth prism meets the
plane of the principal section.
There are also various other uses of this letter which it is not neces-
sary to enumerate.
W
53. This letter occurs frequently in various ways.
X
54. This letter occurs in various ways.
55. F, y are used to designate the feet of the perpendiculars let
fall from the centre of the spherical refracting surface on the incident
and refracted rays, respectively (Fig. 120).
The letter Y occurs also in various other connections.
£>esignations of Lines. 593
Z
56. In Chap. IX, Z, Z' designate the points where the incident
and refracted rays cross the auxiliary concentric spherical surfaces
T, t' (72), respectively, which are used in Young's construction of
the path of a ray refracted at a spherical surface (Figs. 114, 115).
In particular, Z, Z' designate the positions on the optical axis of
the pair of aplanatic points of a spherical refracting surface (Fig. 116).
The letter Z is used also in various other ways.
n. DESIGNATIONS OF LINES.
Lines in the diagrams are designated generally by italic small letters.
Without undertaking to enumerate all the uses of these letters, we
may mention here the following as among the most important.
57. In Chapters V, VI and VII, the letters/ and e' are frequently
used to designate the Focal Lines (or ^^Fluchi** Lines) of two collinear
plane-fields; or the lines in which the Focal Planes 0, €' (73, 65) are
intersected by conjugate meridian planes containing the principal axes
X, x', respectively, of two collinear space-systems. See Figs. 64, a
and bi and 65.
58. In Chap. VII, the letters f, i' (and similarly also the letters
;, /) are used to designate the infinitely distant straight line and the
**Fluchf* Line, respectively, of two collinear plane-fields.
Sf s'
59. In Chapters V, VI and VII, 5, s' are used to designate a pair
of conjugate rays of two collinear systems; as in Fig. 66.
60. Throughout Chap. XI, «, u' are used to designate the chief
incident ray and the corresponding refracted ray, respectively, of an
infinitely narrow bundle of rays refracted at a spherical surface.
X, x'\ y, y'\ 2, z'
61. X, x' designate the Principal Axes of two collinear systems.
In an optical system of centered spherical surfaces, the optical axis
is designated by x or x' according as it is regarded as belonging to
the Object-Space or Image-Space, respectively.
In Chap. VII, Xj^, xi designate the Principal Axes of the jfeth com-
ponent of a compound optical system.
39
594 Geometrical Optics, Appendix.
62. X, x'\ y, y'\ 2, z' are used also to designate corresponding (but
not necessarily conjugate) pairs of rectangular aoces of coordinates in
the Object-Space and Image-Space.
63. y designates the axis of collineation of two centrally collinear
plane-fields. It designates especially the tangent-line in the meridian
plane at the vertex of a spherical refracting surface.
64. It may also be mentioned that z is used (with suitable primes,
subscripts, etc.) to designate the chief ray of a narrow bundle of rays
refracted at the edge of a prism. See Fig. 42.
m. DESIGNATIONS OF SURFACES.
Surfaces, plane or curved, are designated by small letters of the
Greek alphabet. Of these the following are the more important.
65. €, e' are used to designate the infinitely distant plane of the
Object-Space and the Focal Plane of the Image-Space, respectively,
of two collinear space-systems. Similarly, in the case of a compound
optical system, e^ designates the secondary focal plane of the kth com-
ponent.
66. 71, ri' are used sometimes (see Chap. VII) to designate a pair
of conjugate plane-fields of two collinear space-systems.
67. 11 is used to designate the refracting or reflecting surface. If
there are a series of such surfaces, fif^ designates the kth surface of
the series reckoned in the order in which they are encountered by the
rays of light.
IT
68. T, t' are used frequently, especially in Chap. VII, to designate
two collinear plane-fields.
These symbols are also employed, especially in Chap. XI, to desig-
nate the coincident planes of incidence and refraction of the chief
incident ray and the corresponding refracted ray, respectively, of an
infinitely narrow bundle of rays refracted at a spherical (or plane)
surface. In particular ir, v' designate the collinear plane-fields of the
meridian sections of a narrow bundle of incident rays and the bundle
of corresponding refracted rays.
In the same way, also, v, v' are used to designate the pair of planes,
both at right angles to the plane of incidence of the chief ray of a
Designations of Surfaces. 595
narrow bundle of rays refracted at a spherical (or plane) surface,
which contain the chief incident ray and the corresponding refracted
ray, respectively. And, especially, J, i' designate the two coUinear
plane-systems of the sagittal sections of the bundles of incident and
refracted rays, respectively.
Moreover, the symbols ir]^, Vj^ designate the plane-systems of the
meridian and sagittal sections, respectively, after refraction of a narrow
bundle of rays at the jfeth surface of a series of refracting surfaces.
a, o*
69* <r, a' are used in Chap. VII to designate a pair of conjugate
planes parallel to the Focal Planes.
Especially, the symbols <r, <r' are used to designate a pair of trans-
versal planes which are conjugate, in the sense of Gauss's Theory,
with respect to either a single spherical refracting (or reflecting) surface
or a centered system of spherical surfaces. In this case, a designates
the so-called Object-Plane (Chap. XII) which is defined as the trans-
versal plane (perpendicular to the optical axis) which contains the
object-point P (or Q); see 36, 39. The axial point M' conjugate,
by Gauss's Theory, to the point M (24) where the optical axis crosses
the Object-Plane a determines the position of the transversal Image-
Plane a'. In Chap. XIV, the planes a, a' are usually called the
Focus- Plane and the Screen- Plane, respectively.
In the case of a centered system of spherical surfaces, a]^ is used to
designate the transversal plane which, by Gauss's Theory, is conjugate
to the Object-Plane Cj with respect to the optical system cdtnposed
of the first k surfaces.
70. a' is employed to designate a transversal plane of the Image-
Space of an optical system which is usually not far from the Image-
Plane <r'. For example, in Fig. 161, a* designates the so-called Scale-
Plane of an optical measuring instrument.
71. The symbols o*, o*' are used to designate a second pair of trans-
versal planes conjugate to each other in the same way as <r, a' above.
Generally, o*, o*' designate (as always in Chapters XII and XIV) the
planes of the Entrance- Pupil and Exit- Pupil, respectively, of the
optical system.
In the case of a centered system of spherical surfaces, o*]^ designates
the transversal plane which, by Gauss's Theory, is conjugate to the
initial plane 0*1 in the Object-Space, with respect to the optical system
composed of the first k surfaces.
596 Geometrical Optics, Appendix.
72. r, t' are used to designate the auxiliary spherical surfaces,
concentric with the spherical refracting surface, used in Young's Con-
struction of the path of the refracted ray (Figs. 114 and 115).
73. ^, fp' are used to designate the Focal Plane (or **Fluchl** Plane)
of the Object-Space and the infinitely distant plane of the Image-
Space, respectively, of two collinear space-systems. Similarly, in the
case of a compound optical system, tpj^ designates the primary focal
plane of the jfeth component.
IV. SYMBOLS OF UHSAR ICAGHITUDBS.
Introduction. A straight line is divided into two segments by a
pair of actual (or "finite") points A, B on the line, viz., a segment of
finite magnitude which is the shortest distance between the two points
and another segment of unlimited length which is the *iong way"
between the two points via the infinitely distant point / of the straight
line. Three actual points i4 , 5, C lying along a straight line determine
a certain "sense" ABC along the line or direction in which the line
has to be traversed in order to go from i4 to -B without passing through
C. As we shall exclude infinitely great line-segments, the segment
i4 -B is to be understood therefore as meaning always the finite one of
the two above-mentioned ; and as indicating also not merely the dis-
tance from AtoB but the segment A Bin the sense ABI. Evidently,
therefore, we have the following relation :
AB + BA = o.
Also, ii At B, C are three points ranged along a straight line in any
order whatever, we may write according to the above:
AB + BC + CA = o;
and, generally, in the case of any number of points lying on one
straight line, a similar relation will exist.
If ^, B, C, D, • • • designate a series of points ranged along a
straight line, the segments AB, AC, AD, • • • are called here (for
lack of a better term) the ^^abscissse" of the points 5, C, D, •••!
respectively, with respect to the point A as origin.
As a rule, to which, however, there are some notable exceptions
(as will be seen in the following), linear magnitudes are denoted by
italic small letters. Italic capital letters and Greek letters occur some-
Symbols of Linear Magnitudes. 597
times as symbols of linear magnitudes. The more important of these
magnitudes will be found in the following list.
a
74. In Chap. X, the symbol aj, is used to denote the abscissa of
the centre C^+j of the (jfe + i)th surface with respect to the centre
Cj^ of the jfeth surface of a centered system of spherical surfaces; thus
&, B
75. In Chap. I X, 6, b' denote the intercepts of a ray, lying in the
principal section of a spherical refracting surface, on the central per-
pendicular, before and after refraction, respectively; thus b = CH,
V = CH' (6 and 15). Similarly, in Chap. X, bj, = C^H^, &; = C.If^.
76. In Chap. XIII, &, V are used to denote the widths of a pencil
of parallel meridian rays before and after refraction, respectively, at
a plane surface. Similarly, b'j^ denotes the width of a pencil of parallel
meridian rays after refraction at the ith surface of a system of prisms
with their refracting edges all parallel.
c
77. In Chap. IX, c, c' denote the abscissae, with respect to the
centre C of the spherical refracting surface, of the points designated
by L, V (23); thus, c = Ci, c' = CL'.
78. In Chap. XIV, c, c' denote the abscissae, with respect to the
centres of the pupils, of the centres of the ports (27 and 47); thus,
c = MS, d = M'S'. Also, c^ = MS^, c, = MS^.
79. d denotes the axial thickness of an optical medium comprised
between two consecutive surfaces of a centered system of spherical
surfaces. Particularly, d^ = Aj^Ak+i (see 2).
In an optical system composed of a single lens, the thickness of the
lens is denoted by d; thus, d = A1A2.
80. In a centered System of Infinitely Thin Lenses, dj, denotes the
distance of the (jfe + i)th lens from the jfeth lens; thus, d^ = Aj^A^^^
(see 2).
81. In Chap. VIII, in an optical system consisting of a combina-
tion of two lenses, d is used to denote the abscissa, with respect to
the secondary principal point of the first lens, of the primary prin-
cipal point of the second lens; thus, d = A'^A^ (see i).
82. The symbol hj^ is employed to denote the length of the ray-path
comprised between the jfeth and the {k -f i)th refracting surfaces; thus,
«* = ^*5jH.i (see s).
598 Geometrical Optics, Appendix.
83, Here also we note the use of the symbol A^ to denote the so-
called '* optical interval between the jfeth and the (Jk + i)th components
of a compound optical system; thus, A^ = -EI-Fjh-i (see ii and 12).
If the compound system has only two parts, we write: A = E\Fy
84. The secondary focal length of an optical system is denoted by
e*\ that is, e' = E'A' (i and 11). Also, in a compound optical system,
e^ = E;^;. Also, in Chap. XIII, e' = E'A'.
85. In the theory of the refraction of a narrow bundle of rays at
a spherical surface (or through a centered system of spherical surfaces),
the symbols e^ and e^ are used in Chap. XI to denote the secondary
focal lengths of the two collinear plane-systems ir, ir' and J, v', re-
spectively (68). The subscript u refers to the chief ray of the bundle
of incident rays (60).
Similarly, the secondary focal lengths of the systems of meridian
and sagittal rays of an infinitely narrow bundle of rays which are
refracted at the kth surface of a centered system of spherical surfaces
are denoted by el,*, ^m,*, where u designates the chief ray of the bundle
of object-rays.
/
86. The primary focal length of an optical system is denoted by
/; thus,/ = FA (see i and 12). Also, in a compound optical system,
A = Fj^Aj,. Also,/ = FA (see Chap. XIII).
87. The symbols /^ and /^ are used in the same connection as e^
and e^ (85) to denote the primary focal lengths of the systems t, t'
and TT, tt', respectively. Similarly also the symbols /., ^k, 7«.*> corre-
sponding to el, J, el. t, respectively.
88. The symbols g, i are used in Chap. XIII to denote the ordi-
nates of the points where an incident paraxial ray emanating from the
axial object-point M crosses the primary focal planes of an optical
system which correspond to light of wave-lengths X, X, respectively.
Also, in Chapters V, VI and VII, the symbol g is employed in a sense
similar to the above. See Figs. 65 and 90, where g — FR.
89. The symbol h is used to denote the incidence-height (or ordinate
of the incidence-point B) of a ray refracted (or reflected) at a spherical
surface; thus, h = DB {$ and 8). With respect to a centered system
of spherical surfaces, hf^ = Df^Bj^ denotes the incidence-height at the
Symbols of Linear Magnitudes. 599
ifeth surface of a ray lying in the principal section. In Chap. XIII,
we have also h^ = -^ A-
90. Similarly, A^ = Djfif, (5 and 8) denotes the incidence-height
of a second ray, usually the chief ray, at the jfeth surface of a centered
system of spherical surfaces.
91. The symbol h is used to denote the incidence-height of a ray
refracted through an Infinitely Thin Lens. In a centered system of
Infinitely Thin Lenses, A^ denotes the incidence-height at the jfeth lens.
92. In the case of two coUinear space-systems (Chap. VII), the
symbols A, h' are used to denote the ordinates of the points where a
pair of conjugate rays cross the primary and secondary principal planes,
respectively; A' = A. In a compound optical system, A^, A]^ (= hj)
are used in this same sense with respect to the Ath component; see
Fig. 99-
k
93. The symbol A' is used, always in connection with the symbol
g (88), to denote the ordinate of the point where a paraxial image-ray
lying in the plane of the principal section crosses the secondary focal
plane; see Fig. 65.
I
94. The symbols /, /' are used to denote the so-called **ray-lengths**
of a ray lying in the principal section of a spherical refracting surface,
before and after refraction, respectively; reckoned in each case from
the incidence-point B to the point where the ray crosses the optical
axis; thus, / == 5L, T = 5i' (23). In the case of a ray lying in
the princiapl section of a centered system of spherical refracting surf-
aces, l,^ = Bf^L'f^^^, I'l^ = BJJj, denote the ray-lengths, before and after
refraction, respectively, at the Ath surface.
P
95. In Chap. I X, />, p' denote the radii vectores of the points H, IV
(is); thus p = CU, p' = CH' (Fig. 123). In Chap. X, p^ = C^Hj,
96. In Chap. XIV, />, p' denote a pair of conjugate radii of the
Entrance-Pupil and Exit-Pupil, respectively, of the optical system;
thus, p = MD, p' = M'D' (9 and 27).
The symbol p^ occurs to denote the radius of the iris-opening of
the eye.
a
97. In Chap. XIV, g, g^ denote a pair of conjugate radii of the
600 Geometrical Optics, Appendix.
Entrance- Port and Exit- Port, respectively, of the optical system; thus,
q = ST, g! = S'T (47 and 49). Also, g^ = S^T^, & = SJ"^.
98. The symbol r is used to denote the radius of the spherical
refracting surface; or, more exactly, to denote the abscissa of the centre
C with respect to the vertex A; r = AC. Similarly, r^ = il^^C^ (2
and 6) denotes the radius of the ifcth surface.
99. The symbols i?, R and R\ R' are used to denote the radii of
curvature at the axial points M and Jkf' of the I. and II. image-surfaces,
before and after refraction, respectively, at a spherical surface (Chap.
XII); R = AfX, R' = M'K\ R = MK, R' = M'K' (22 and 24).
Also, R'f^, Rl are used in the same way, with respect to the astigmatic
image-surfaces after refraction at the Jfeth spherical surface.
100. The symbols 5, s' are used to denote the distances, reckoned
in each case from the incidence-point B of the chief ray, of the vertex
S of an infinitely narrow pencil of meridian rays and the vertex S'
of the pencil of corresponding refracted rays, respectively; thus, 5= jB5,
s' = BS' (5 and 45). Similarly 5, s' denote the distances, from the
incidence-point B of the chief ray, of the vertex S of an infinitely nar-
row pencil of sagittal rays and the vertex 5' of the pencil of corre-
sponding refracted rays, respectively; J = 55, T = 55' (5 and 45).
If the rays traverse a system of prisms or a centered system of
spherical surfaces, we have with respect to the feth surface:
t-i»
^k — ^k^k = ^k^k-v h = ^k^k =* ^k^k
^k ^ ^k^k — ^k^h^V ^k ~ ^k^k ^ -S^Sji+i.
101. In Chap. IX, /, t' denote the distances from the inddence-
point A of a chief ray lying in the plane of the principal section of a
spherical surface of the points T, T' of intersection with this ray of
another meridian ray, before and after refraction, respectively; thus,
/ = BT, t' == BV, as in Fig. 121. See 5 and 48.
tt, u, U
102. The symbols w, u' are used to denote the abscissae, with respect
to the principal points A, A* oi two collinear systems, of a pair of
conjugate axial points M, M' respectively; thus, u ^ AM^v! ^ A'M!
(2 and 24).
Symbols of Linear Magnitudes. 601
103. Especially, w, u' denote the abscissae, with respect to the
vertex A of the spherical surface, of the points Af, M' where a paraxial
ray crosses the optical axis, before and after refraction (or reflexion),
respectively; « = i4M, «' = i4M' (2 and 24, 25 ).
If (as in Chap. VIII, § 195) we have a pair of paraxial rays of dif-
ferent origins, the abscissae of the points Af , M' where the second ray
crosses the axis before and after refraction are denoted by u, u',
respectively; u = AM,u' ^^ AM' (2 and 26).
In case we are concerned with paraxial rays of two different colours
emanating from a common source, the symbols u, u' and m, v! are used
as above described with reference to rays of light of wave-lengths X, X,
respectively; and if also there is a ray of a third colour I, the abscissae
of the points where this ray crosses the axis are denoted by tt, 11'
(see Chap. XIII).
In an optical system ccmsisting of a centered system of spherical
surfaces, the symbols m^, tt]^; u^^, a^; Uk, u\\ etc., are used precisely in
the same way as described above, with respect to the jfeth surface of
the system; so that
«ik = AJd^ = i4jAf;.„ «; = i4*M ; = il^il/j+i, etc.
In particular, the symbol u^ = A^M^ denotes the abscissa, with
respect to the vertex Ap of the axial object-point Mp Frequently,
however, the symbol w, without any addition, is used to denote the
abscissa of the point where a paraxial object-ray crosses the optical
axis of a centered system of spherical surfaces; in which case u' de-
notes the abscissa, with respect to the vertex of the last surface, of
the point where the conjugate image-ray crosses the axis.
104. In the case of an Infinitely Thin Lens, the symbols «, u'
denote the abscissae, with respect to the optical centre of the lens, of
the points where a paraxial ray crosses the axis before entering the
lens and after emerging from it, respectively. The symbols u, a' and
«, «' are used also in this way.
Similarly, in the case of a centered system of Infinitely Thin Lenses,
the symbols U/^, u^ are used as just stated, with respect to the Jfeth lens.
So also ttjt, Uf^ and u,^, ii[.
105. In general, the symbol Uj = -4,M, denotes the abscissa, with
respect to the vertex ^, of the centre M^ of the Entrance- Pupil of
the system. If the centres of the Entrance-Pupil and Exit-Pupil are
designated by Af, M' (27), then u = A^M, u' = A^M' are used to
denote the abscisses of the pupil-centres.
602 Geometrical Optics, Appendix.
106. The symbols v, v' denote the abscissae, with respect to the
vertex A of the spherical refracting surface, of the points L, L' (23)
where a ray lying in the plane of the principal section crosses the optical
axis, before and after refraction, respectively; v = AL, rf ^ AV.
Also: v^ = AJ.^ = ^*il-.i, v\ ^ AJ.I = ^4*^*+,.
The symbols r, v' have meanings with respect to the chief ray pre-
cisely the same as above; thus, v = AL, t/ = AV; also, v^ = A^L'i^if
v\ = AjLk (see 23).
In Chap. IX, in Kerber's formulae for the path of an oblique ray
refracted at a spherical surface, we have:
Vg = Afi, Vi = AJ, Vg^ Afii\ v\ = Af\
where the points designated by A^y A^, G, G\ I and /' are points
described in 3, 14 and 18. See Fig. 122. Also, in Chap. X, in the
same connection we have:
107. X, x' denote especially the abscissae, with respect to the focal
points F, E\ of a pair of conjugate axial points Jkf, M\ respectively,
of two coUinear systems; thus, x = FJIf, x' = E'M'. Similarly, with
reference to the jfeth component of a compound optical system, we
have: x^ = F^Jlf]^, x^ = £1-8^ (iif 12 and 24).
In Chap. XIII, x, x' occur in connection with the Focal Points
?, £' (11 and 12).
In Chap. VII, the letters, x, x' occur also with special subscripts.
108. X, x' denote the abscissae, with respect to the focal points
F, £', of the pupil-centres; thus, x = FM, x' = £'Af' (11, 12 and 27).
109. The letters x, y, z and x', y', 2' are used to denote the rectangu-
lar co-ordinates of a pair of conjugate points of two colKnear space-
systems.
110. In Chap. I X, Xg, Xg and x^, x'^ denote the x-co-ordinates, with
respect to the centre C of the spherical refracting surface, of the points
designated by G, G' and 7, I\ respectively; and, similarly, in Chap.
X, x^, ft, Xg, kf and X|, *, x<, » denote the x-co-ordinates, with respect toQ,
of the points Of,, Gl (or G,,+i) and 7^, 7^ (or Ii^+i)t respectively (6, 14
and 18).
Symbols of Linear Magnitudes. 603
y, y
111. y, y denote the y-co-ordinates of a pair of conjugate points
of two collinear space-systems; especially, the ordinates of the extra-
axial conjugate points Q, Q* lying in the meridian xy-plane; y = MQ^
y = M'Q' (24 and 39).
y' denotes the ordinate of the vertex, after refraction at the Jkth
surface of a centered system of spherical surfaces, of a bundle of
paraxial rays which emanate originally from the extra-axial object-
point Ci lying in the plane of the principal section; or the ordinate of
the point C* (or 0*+i)f where, according to Gauss's Theory, a ray
emanating from the object-point Q^ (or P,) would cross the trans-
versal plane a^ after refraction at the Jfeth surface of a centered system
of spherical surfaces. See 36, 39 and 69. Thus, y'j^ = ^kQlw
In Chap. XIII, where we have to do with rays of light of two or
more different colours, the symbols 5» y' denote the ordinates of the
pair of extra-axial conjugate points 5, Q for rays of wave-length \;
y = MQ, y* = M*7^ (24 and 39); usually, y = y- ^^ the same way,
112. y, y' denote the y-co-ordinates of the points Q, Q', respect-
ively (40). The symbol y\ denotes the y-co-ordinate of the point
Q]^ (see Chap. XII).
113. In Chap. IX, y^, y\ and y^^, y\ denote the y-co-ordinates of
the points designated by G, G' and fl", fl"', respectively (Figs. 122 and
123); see also Chap. XII. In Chap. X, y^,», y^.* and y»,», yl,» denote
the y-co-ordinates of the points designated by G^, G'^ (or G^^.,) and -ff^,
H]^, respectively (14 and 15).
2, z
114. The symbols 2, 2' denote the 2-co-ordinates of a pair of con-
jugate points of two collinear space-systems.
115. Especially in Chap. XII, 2, 2' denote the 2-co-ordinates of
the points Q^ Q\ If the object-point Q lies in the meridian xy-plane,
2 = 2' = o. z^ denotes the 2-co-ordinate of the point C* (39)«
116. z, z\ Zj, denote the 2-co-ordinates of the points designated
by Q, Q', Qi, respectively; see 40.
117. In Chap. IX, z^, z^ and 2,., z\ denote the 2-co-ordinates of
the points designated by H, H' and 7, /', respectively (Figs. 122 and
123). Also, in Chap. X, 2*. », 2^, ^ and Zi^ *, z\^ * denote the 2-co-ordinates
of the points Hj^, Wf, and /j^, 7]^, respectively (15 and 18).
118. Finally, in Chap. XIII, the symbols 2, 2' are used in a special
sense to denote the abscissae, with respect to the vertices of the first
604 Geometrical Optics, Appendix.
and last surfaces, of the primary and secondary focal points, respect-
ively, for rays of light of wave-length X; so that 2 = A^F.z' = A^E\
Similarly, for rays of light of wave-length X, we have: z ^ A^F,
z' = A^E' (11 and 12).
£, i?» r; n. 5
119. In Chap. XIV, the Greek letters {, f' are used to denote the
abscissae, with respect to the centres of the entrance-pupil and exit-
pupil, of the pair of conjugate axial points Jlf, M\ respectively; thus,
f = MM, f = M'M' (24 and 27).
120. In Chap. XII, the rectangular co-ordinates of the points
designated by P and P' are denoted by {, 17, f and {', 17', f ', respect-
ively. Also, the co-ordinates of Pj^ are f]^, ly^, f]^ (36).
121. In Chap. XII, n, r( and J, J' are used to denote the y- and
«-co-ordinates of the points designated by P, P\ respectively; simi-
larly, ij]^, X^ are used with reference to the point P'j^ (37).
V. SYMBOLS OF ANGXTLAR ICAGNITUDSS.
If i4 , 5, C designate the positions of three points not in a straight
line, the LABC is the angle through which the straight line AB
must be turned in order that the point A may be brought to lie in the
same direction from the turning-point B as the point C is; thus,
LABC ^^ ZCBA = o.
Throughout this volume, counter-clockwise rotation is reckoned al-
ways as positive rotation.
With rare exceptions, angular magnitudes are denoted by the letters
of the Greek alphabet. The more important of these angles are
enumerated in the following list.
a, a, A
122. The angles of incidence and refraction, as defined in Chap. II
(see Fig. 5), are denoted by a, a', respectively. When a ray of light
traverses a series of optically isotropic media, the symbols a^, a]^ denote
the angles of incidence and refraction, respectively, at the kxh refracting
surface.
123. The capital Greek letter A denotes the critical angle of inci-
dence of a ray refracted into a less dense medium, and A' denotes the
critical angle of refraction of a ray refracted into a more dense medium*
124. The angles of incidence and refraction at a spherical surface
of the so-called chief ray are denoted by a, a', respectively. Similarly,
with respect to the kth refracting surface of a series of such surfaces,
the symbols a^, a^ are employed.
Symbols of Angular Magnitudes. 605
125. The refracting angle of a prism is denoted by j8. In a train
of prisms, ft = Z V,,^iVf,V,,+i denotes the refracting angle of the feth
prism (52).
B
126. In Kerber's Refraction-Formulae (Chap. IX), 6, 6' are used
to denote a certain pair of auxiliary angular magnitudes relating to
the ray before and after refraction, respectively (Fig. 122). In Chap.
X, 6[. is employed in the same way with reference to the ray after
refraction at the jfeth surface of a centered system of spherical surfaces.
127. The acute angle through which the refracted ray has to be
turned in order to bring it into coincidence with the corresponding
incident ray, the so-called angle of deviation^ is denoted by €; in Fig. 9
Z, P'BP = €. Thus, also, c^ denotes the angle of deviation at the
ith refracting surface. The total deviation of a ray after traversing a
train of prisms with their edges all parallel is denoted by c = £^=7 «*f
where m denotes the total number of refracting planes.
The angle of minimum deviation of a prism or prism-system is de-
noted by €q.
128. In Kerber's Refraction-Formulae (Chap. IX), €, e' are used
to denote a certain pair of auxiliary angular magnitudes relating to
the ray before and after refraction, respectively (Fig. 122). Also,
in Chap. X, in the same connection, e^ has reference to the ray after
refraction at the jfeth surface of a centered system of spherical surfaces.
e, e and 6, 9
129. e = j^AMB or ZALB, 6' = ZAM'B or ZAL'B, where
the points designated by i4, B, Af, M\ L, L' have the meanings
explained in 2, 5, 23 and 24. Also, ^i = ^»+i= Zil^Afj^-B^ or ZAf^L^Bi^.
The angles 6, B' are the so-called slope-angles of the ray before and
after refraction, respectively, at a spherical surface.
If we have a pair of rays of two different colours emanating origi-
nally from the same point on the optical axis of a centered system,
^i, dl denote the slope-angles, after refraction at the jfeth surface, of
therays of wave-lengths X, X, respectively.
The symbol 0 is used to denote the slope-angle of an object-ray
proceeding from the axial object-point M (24 and 25) and the symbol
$' to denote the slope-angle of the conjugate image-ray, especially
606 Geometrical Optics, Appendix.
on the assumption of collinear correspondence between Object-Space
and Image-Space.
130. • The symbols 6, 6' are used to denote the slope-angles of the
chief ray, before and after refraction, respectively, at a spherical surf-
ace; thus, e = LALB, V = A ALB. Similarly,
See 2, s and 23.
6, 6' denote the slope-angles of a chief object-ray and its conjugate
image-ray, respectively; especially, on the assumption of collinear
correspondence between Object-Space and Image-Space.
131. In Chap. XIV, 9, G' are used to denote the semi-angular
diameters of the aperture of the optical system in the Object-Space
and Image-Space, respectively; 9 = AMMD, 8' = ZM'M'D' (9, 24,
25, 26).
In this same chapter, 8', 80 are used to denote the angles subtended
at the centre of the image on the retina of the eye by the radius of
the exit-pupil of the instnunent and the radius of the eye-pupil^ re-
spectively.
132. In Chap. XIV, 9, 9' denote the semi-angular diameters of
Xh^ field oj view of the object and image, respectively; thus,
9 = ASMT, 9' = LS'M'V (26, 47, 49).
133. Finally, in connection with Kerber's Refraction-Formulae
(Chap. IX), we have:
^^ = Z AfiB, ^; = Z Afi'B, Si = Z.AJB, e\ = Z AJ'B,
where the points designated by Ag, A^, B, G, G' and 7, 7' are the
points explained in 3, 5, 14 and 18. See Fig. 122.
Similarly, in Chap. X,
Og^h = ^Ag^ kGkB^, 6i^ k = AAi^ iJkBk*
X
134. In Chap. I X, X, X' are used to denote the angles between a
pair of meridian incident rays and the pair of corresponding refracted
rays, respectively; see Fig. 121.
Especially, in Chaps. XI and XII, the symbols rfX, d\' are used
to denote the angular apertures of an infinitely narrow pencil of merid-
ian incident rays and the pencil of corresponding refracted rays, re-
spectively; thus, d\ = ZBSGf dX' = ZBS'G (see 5, 13 and 45), for
example, in Fig. 127.
Symbols of Angular Magnitudes. 607
Similarly, d\, d\' denote the angular apertures of a narrow pencil
of sagittal incident rays and the pencil of corresponding refracted
rays, respectively.
The symbols dk^, dX^ are employed in the same way as above, with
respect to the Jfeth surface of a centered optical system.
135. In Seidel's Refraction-Formulae, the symbols /yi, /*' are em-
ployed to denote a pair of auxiliary angles, viz., the angles at H, H'
of the triangles BHC, BH'C, respectively (5, 6 and 15). For the'
exact definitions of these angles, see Chap. IX. In Chap. X, the
symbols m^, m]^ are used in the same sense.
136. In Chap. I X, in Seidel's Refraction-Formulae, x, x' denote
the polar angles of the points 21, H'^ respectively; thus, v — L HCy^
t' = Z H'Cy, where y designates a point on the positive half of the
y-axis of co-ordinates and C, -ff, H' have the meanings given in 6
and 15. See Fig. 123. Similarly, also, in Chap. X:
137. In Chap. I X, in Seidel's Refraction-Formulae, r, r' are em-
ployed to denote the positive acute angles between the direction of
the optical axis (x-axis) and the path of an oblique ray, before and
after refraction, respectively, at a spherical surface (Fig. 123). Simi-
larly, in Chap. X, the symbols r^, t^ are used.
138. 0 is used to denote the central angle subtended at the centre
C of the spherical refracting (or reflecting) surface by the arc B C;
thus, 0 = ZBCA. Also, ^^^ I. B^^C^Aj^ (2, 5 and 6).
139. Similarly, 4> = LBCA or ^^^ = /.B^^Cj^A^ denotes the central
angle with respect to the so-called chiej ray (Fig. 121). See 2, 5, 6.
140. In Chap. IX, in Kerber's Refraction-Formulae, we have:
0^ = /.ACAg, 0,. = AACAi, where the letters A, A^, A^ and C have
the meanings given in 2, 3 and 6. See Fig. 122. Similarly, in Chap.
X: 4»9,h = Zi4*C»-4^,t, ^i,* = Zi4kC*i4|,».
141. In Chap. XIV, <^, 4>' are used in the radiation-formulae to
denote the angles of emission and radiation, respectively.
608 Geometrical Optics, Appendix.
X
142. In Chap. I X, the symbol x is used to denote the angle B CB,
where B, B designate the incidence-points on a spherical refracting
surface of a pair of meridian rays. See Fig. 121.
143. In Chap. I X, in Seidel's Refraction-Formulae, ^, ^' denote
a certain pair of angular magnitudes (see Fig. 123); also, in Chap. X,
^A> ^* ^^^ ^sed in same sense.
VL SYMBOLS OF NON-GEOMETRICAL MAGNITUDES (CONSTANTS,
CO-EFFICIENTS, FUNCTIONS, ETC.).
Among the more important magnitudes under this head may be
mentioned the following:
A
144. The numerical aperture of the optical system, in the Object-
Space and in the Image-Space, is denoted by A, A\ respectively;
thus, A = n-sinO, A' = n'-sinO' (131, 155).
B
145. B = na = hJ denotes the optical invariant in the case of
paraxial rays (89, 122, 150, 155).
c, C
146. The symbols c, c' (sometimes also c,, c^ are used to denote
the curvatures of the surfaces of an Infinitely Thin Lens; thus,
c = i/fi, c' = i/r, (see 98).
In a centered system of infinitely thin lenses, the symbols c^, c[ denote
the curvatures of the feth lens. Moreover, in Chap. XIII, Cj^ = Cj^ — c^.
147. In Chap. XIV, C, C denote the candle-powers of a point-
source of light in a given direction and the corresponding point of
the image in the conjugate direction, respectively.
I.i
148. I denotes the invariant of refraction in the case of the refract-
ion at a spherical surface of a ray of finite slope lying in the principal
section: / = n(v — r)/rl = n'{v' — r)lrV ; see 94, 98, 106 and 155.
149. In Chap. XIV, i, V denote the specific intensities of radiation
of a luminous surface-element in a given direction and the corre-
sponding element of the image in the conjugate direction, respectively.
Symbols of Non-Geometrical Magnitudes. 609
J, J
150. The symbols 7, J denote the so-called ^'zero-invariants'^ in
the case of the refraction of paraxial rays at a spherical surface, with
respect to the two pairs of conjugate axial points Jlf, M' and Af, Af' ,
respectively (24, 25, 26 and 27); thus,
J = n(i/r — i/tt) = »'(i/r— i/tt'); ^=»(i/r— i/u) = n'(i/r— i/u');
(see 98, 103, iss).
In the case of a centered system of spherical surfaces, Jj^, J^ denote
the zero-invariants for the kth surface, with respect to the pairs of
conjugate axial points Af^_p M'^ and Af]^_i, Af]^, respectively.
In Chap. XIII, J^, Jj^ denote the zero-invariants, with respect to
the feth surface, for paraxial rays of light of colours X, X, respectively,
emanating originally from the same axial object-point.
151. X = n-sin a = n'-sin a' denotes the magnitude of the optical
invariant in the refraction of a ray of light (122 and 155).
152. The symbol k, which occurs usually as a subscript, denotes
the series-number of any one of a system of refracting (or reflecting)
surfaces; or of any integral part or component of a compound optical
system. In certain prism-formulae, the subscripts i and r occur also
in this same sense.
L
153. In Chap. XIV, L, V are used to denote the quantities of
light-energy emitted in unit-time by a certain portion of a luminous
object and the corresponding portion of the image, respectively.
m
154. The total number of refracting surfaces of a system is denoted
by m; also, the total number of components (prisms, lenses or lens-
combinations) of a compound optical system.
n, n
155. The absolute indices of refraction of the first and second medium
are denoted by n, n', respectively.
When a ray traverses a series of media, the symbol n'f^ = n^^.|
is used to denote the absolute index of refraction of the {k -t- i)th
medium. Note that n^ = n, denotes the absolute index of refraction^
of the first medium.
40
610 Geometrical Optics, Appendix.
Often, also, the symbols n, n' are used to denote the absolute indices
of refraction of the first and last medium, respectively.
The symbols n, n and n are used to denote the absolute indices of
refraction of a medium for rays of light of wave-lengths X, X and f,
respectively. The symbols n, n, n and n', n', u' refer to the first and
second (or to the first and last) medium, respectively.
So, also, Ha, nst nc etc. are used to denote the absolute indices of
refraction of a medium for rays of light corresponding to the Fraun-
HOFER lines -4,5, C, etc., respectively.
156. In an optical system wherein there are only two different
media, as, for example, in a glass lens (or prism) surrounded by air,
the relative index of refraction from the first medium to the second is
usually denoted by n; thus, n = n\ln^ = n^jn^. In this sense, the
symbol n^^ is regularly employed to denote the index of refraction of
the material of the kxh. lens of a System of Infinitely Thin Lenses,
each of which is surrounded by air.
157. In Chap. XIII, P, P^ denote the ''purity'' and the ''ideal
purity'^ respectively, of the spectrum. In this chapter, also, the
resolving power of a prism or prism-system is denoted by p.
Q
158. The invariant-functions of the chief ray of an infinitely narrow
bundle of rays refracted at a spherical surface are denoted by Qy 5
(or Q^, ^a); see Chap. XII, §299.
T
159. The function T = A/AjkCA — ^i) denotes a certain constant
which has the same value for each surface of a centered system of
spherical surfaces; see Chap. XII, §323.
V
160. In Chap. XIV, the symbol V is used to denote the characUr-
istic magnifying power of an optical instrument which is intended to
be used subjectively in conjunction with the eye.
161. In Chap. XI, § 247, Vf, denotes the so-called "constant of astig-
matism'' for the jfeth surface of a centered system of spherical surfaces.
W
162. In Chap. XIV, W denotes the ratio of the visual angles sub-
tended at the eye, on the one hand, by the image as viewed through
Symbols of Non-Geometrical Magnitudes. 611
the instrument, and, on the other hand, by the object as seen by the
naked eye at the distance of distinct vision.
X, X, X
163. In an Infinitely Thin Lens, the symbols x, x' are used to
denote the reciprocals of the abscissae, with respect to the optical
centre A (2), of the points Jlf, M' (28) where a paraxial ray crosses
the axis before entering the lens and after leaving it, respectively;
thus, X ^ i/Uf x' = i/w' (104). Similarly, in a centered system of
Infinitely Thin Lenses, x^^, x\ denote the same reciprocals with respect
to the jfeth lens.
Similarly, also, the symbols S, x are employed as follows (see 104) :
X = i/w, x! = i/tt'; x^ = 1/W4, Xj, = i/w^;
X = i/u, x' = i/u'; «; = i/u^, «; = i/u;.
164. The so-called **axial magnification^* (or ''depth magnification**)
with respect to a pair of conjugate axial points Af, M' of two col-
linear systems is denoted by X\ thus, if FM = x, E'M' = x' (11, 12,
24 and 107), we have: X = dx'/dx.
The symbols X, X denote the axial magnifications of an optical
system with respect to a given axial object-point M for rays of light
of wave-lengths X, X, respectively.
In Chap. VII, Xq is used to denote the axial magnification at the
point 0 (see 35).
Y.Y
165. Y denotes the so-called ''lateral magnification** at a pair of
conjugate axial points Jlf, Jlf ' (24) of two collinear systems: Y = y'/y»
In an optical system composed of a centered system of m spherical
surfaces, Y = y^/yi. See iii.
F, Y denote the lateral magnifications of an optical system with
respect to a given axial object-point M for rays of light of wave-lengths
X, X, respectively.
Y denotes the lateral magnification at the pupil-centres Af , Af' .
In Chap. VII, Yq denotes the lateral magnification at the axial
object-point 0 (see 35).
166. In the theory of the refraction of an infinitely narrow bundle
of rays, F, Y^ denote the lateral magnifications of the collinear plane-
systems X, x' and x, x', respectively (68).
612 Geometrical Optics, Appendix.
2, Z
167. The so-called ''angular magnification'^ (or ''convergence-ratio")
at a pair of conjugate axial points M, M' (24) of two collinear systems
is denoted by Z; Z = tan^/tand (129). In an optical system com-
posed of a centered system of m spherical surfaces, Z = tan 9'^ /tan 6^.
Z, 2 denote the angular magnifications of an optical system with
respect to a given axial object-point M for rays of light of wave-
lengths X, X, respectively.
Z denotes the angular magnification at the pupU-centres M , M'.
In Chap. VII, Zq denotes the angular magnification at the axial
object-point 0 (see 35).
168. In the theory of the refraction of an infinitely narrow bundle
of rays, Z„, Z„ denote the convergence-ratios of the meridian and sagittal
rays, respectively, where u designates the chief ray of the bundle;
Z^ = dX'/dX, Z„ = dV/d\ (134).
169. In Chap. XIII, fi is used to denote the so-called "relative
partial dispersion'' of an optical medium. For the jfeth medium, this
magnitude is denoted by /S^.
V
170. The symbol v is employed to denote the so-called "relative
dispersion" of an optical medium. See formulae (366), (426).
171. The symbol tp is used to denote the reciprocal of the primary
focal length of an Infinitely Thin Lens — the so-called "power" or
"strength" of the lens.
In a system of infinitely thin lenses, tp^ denotes the power of the kth
lens, and tp is used to denote the power of the Lens-System.
INDEX.
The numbers refer to the pages.
Abbe, E., 86, 104, 114, 159. 179, 201. 318,
223, 233, 246, 262. 299, 342, 346. 349,
353. 354. 374. 380, 382. 383, 385, 395. 397.
398, 401. 405, 406. 407. 422, 434, 437. 448,
450, 451. 467. 469. 478. 479. 480. 482. 48s,
492, 493. 509. 510. 523. 526, 528, 530.
537. 538. 539. 543. 547. 548. 549. 557. 558.
560. 563, 575. 578. 579; theory of optical
imagery. 198-201; definitions of the focal
lengths, 233; measure of the indistinct-
ness or lack of detail of the image. 385.
395; explanation and proof of the sine-
condition. 400-405: use of term *'aplan-
atic". 407; test of aplanatism. 407. 422;
method of invariants. 434. 448; use of
term "numerical aperture", 538; aboli-
tion of chromatic difference of spherical
aberration. 527; use of the terms "pupils"
and "iris", 537; optical measuring instru-
ments ("telecentric" systems). 541-544;
definition of magnifying power. 548; in-
vestigation of focus-depth and accommo-
dation-depth. 557-563; and of illumina-
tion in the Image-Space. 578.
Aberration. Chromatic: see Chromatic A&-
errations. Achromatism, etc.; see also
Table of Contents. Chap. XIII.
Aberration. Lateral: see Lateral Aberration,
Spherical Aberration.
Aberration, Least Circle of, 378.
Aberration, Longitudinal: see Spherical
Aberration, Chromatic Aberration.
Aberration-Curve, 398.
Aberration-Lines: comatic, 448-455; of as-
tigmatic bundle of image-rays. 430-434,
and approximate formulae therefor. 432,
440.
Aberrations or Image- Defects, 368; of the
third order, 373; sagittal and tangential
(or »- and y-aberrations), 374; series-
developments of, 371-376, 397-400, 456-
468.
Aberrations, Spherical: see Spherical Aber^
rations; also. Table of Contents, Chap.
XIL
Aberrations, Theory of: see Spherical Aber-
rations. Chromatic Aberrations; also Table
of Contenu, Chaps. XII and XIII.
Abscissa, Special use of this term, 52, 213,
596.
Absorption, 10, 12.
Acanonical system of co-ordinate axes, 223.
Accommodation-Depth, 561.
Accommodation of the eye. 563.
Achromatic Combinations. Early attempts
at contriving. 476.
Achromatic Combination of prisms. 489;
of two thin prisms, 4S3.
Achromatic Optical System, 504.
Achromatism. 476; difTerent kinds of, 503;
complete and partial. 504; stable, 507.
See Chromatic Aberrations.
Achromatism with respect to the visual,
and with respect to the actinic rays, 525.
Affinity-relation between Object-Space and
Image-Space, 209, 243; of two plane-
fields, 206; of conjugate planes parallel
to the focal planes. 211.
Affinity-relation in case of refraction at a
plane or through a prism. 59. 71, 91, 100,
123.
Airy, Sir G. B., 348, 420. 438; his tangent-
condition, 420.
Alhazbn. 15.
Amici, J. B.. 491. 492. 502, 503; direct-
vision prism-system, 492.
Anderson. A., 127.
Anding. E., 573.
Angles of incidence, reflexion and refract-
ion. 13; angle of deviation. 27; slope-
angle. 135. 296. 316; critical angle of
refraction. 24.
Angle-true delineation. 418.
Angular Magnification (Z): see Convert
gence-Ratio.
Angular Magnitudes. Symbols of, 604-608.
Angular (or Inclined) Mirrors, 54.
Anomalous Dispersion, 475.
Aperture. Numerical, 538.
Aperture of Objective. Choice of suitable,
400.
Aperture-angle, 538.
Aperture-stop, 533.
Aplanatic. Meaning of the term. 407.
Aplanatic Points of an optical system. 407;
only one pair of such points. 409.
Aplanatic Points (Z. Z') of refracting
sphere, 290, 300, 346, 348, 387, 400, 405;
613
614
Index.
sine-condition fulfilled with respect to.
291, 301, 401 ; not conjugate points in the
sense of coliinear imagery. 401; comatic
aberrations vanish for this pair of points,
455.
Aplanatism, 407; Abbe's method of testing
for, 407, 422.
Apochromatic Optical System, 530.
Apparent Distance, in sense used by Cotes.
192; "apparent distance" of object
viewed through a system of thin lenses,
191-197.
Apparent size of object, 544, 546; of image,
546; of slit-image as seen through a
prism or prism-system, 105.
Arago. D. F. J., 19.
Astigmatic Bundle of Rays, 44-50; merid-
ian and sagittal sections of. 46; primary
and secondary image-points and image-
lines of. 46. See Meridian Rays Sagittal
RaySt Image-Points, Image-Lines, Infi-
nitely Narrow Bundle of Rays, Astigma-
tism, etc.
Astigmatic Bundles of Rays, Imagery by
means of, 349-356, 402-405.
Astigmatic Constant, 357.
Astigmatic Difference, in case of narrow
bundle of rays refracted (x) at a plane, 60;
(2) through a prism, 94-97; (3) across a
slab, 108; (4) through a system of prisms,
121; (5) at a spherical surface. 345; and
(6) through a centered system of spheri-
cal surfaces, 358.
Astigmatic Image-Surfaces, 416. 429. 430.
Astigmatic Refraction: (i) at a plane, 64-
73. 360. 361; (2) through a prism. 90-
106; (3) across a slab. 106-111; (4)
through a prism-system, 11 5-1 23; (5)
at a spherical surface or through a cen-
tered system of spherical surfaces, see
Table of Contents. Chaps. XI and XII;
and (6) through an infinitely thin lens.
363-366.
Astigmatism, Sturm's Theory of, 44-50;
measure of the, 346; historical note con-
cerning, 347; condition of the abolition
of astigmatism in the case of a centered
optical system, 439. See Curvature of
Image; see also Table of Contents,
Cliaps. Ill, IV, XI and XII.
Axes of co-ordinates of Object-Space and
Image-Space, 212; positive directions of,
220, 221, 227; canonical and acanonical
systems, 223.
Axes, Principal: see Principal Axes.
Axial or Depth-Magnification (X), 234; in
case of telescopic imagery, 244; of a
centered optical system, 510; chromatic
variation of, 511.
Axis of collineation, 163.
Axis of reflecting or refracting sphere, 134;
see also Optical Axis.
B.
Barrel-shaped distortion. 421.
Barrow, I.. 347.
Beck, A.. 199.
Beer. A., 573.
Bending of lens, 390.
Bessel, F. W., 262, 263.
Blair. R., 523.
Blur-circle. 541. 555-56o.
Bow. R. H., 421; Bow-SUTTON condition.
421.
Brandbs. 400.
Bravais, 31. 127, 128.
Breton de Champ. P.. 438.
Brewster. Sir D.. 55.
Brightness, Definition of, 579; of a point-
source. 581; of a luminous object. 579;
of optical image, 580; natural brightness.
580.
Browne, W. R.. 406, 577.
Bruns. H., 38, 472.
Bundle of rasrs, 41; bundles of mys and
planes. 202; homocentric (or monocen-
tric) bundle of rays, 44; astigmatic bun-
dle of rays. 44-50; general characteristic
of infinitely narrow bundle of rajrs. 42-50.
See also Astigmatic Bundle of Rays, In-
finitely Narrow BundU of Rays,
Bundle of Rays. Character of, in case of
direct refraction at a spherical suriace.
376-380.
Bundle of Rajrs. Wide-angle, necessary for
formation of image, 42, 287. 367.
Bunsen. R.. 477.
Burmester. L.. 94. 97. 98. 99, 104. III. 123,
128. 336. 358; homocentric refraction
through prism or prism-system, see Table
of Contents. Chap. IV; homocentric re-
fraction through a lens, 358.
C.
Calculation of the path of a ray refracted
at a spherical suriace, (i) in a principal
section. 298, 299. 302; (2) not in a prin-
cipal section, 304-315. See A. Kerbbr,
L. Seidel; see also Table of Contents.
Chap. IX.
Calculation of the path of a ray refracted
through a centered system of spherical
suriaces. (i) in the principal section. 316-
321; numerical illustration. 318-321; (2)
not in the principal section. 322-330.
See A. Kerbbr, L. Sbidbl; see also
Table of Contents. Chap. X.
Camera. Pin-hole, 288.
Candle-power of point-source. 572.
Canonical ssrstem of axes of co-ordinates,
223.
Cardinal points of optical system, 179. 236.
Cauchy. a. L., 474. 501.
Caustic Suriaces. in general. 42-44; caostic
Index.
615
curves, 43; caustic by refraction at a
plane, 59-^4; caustic surfaces in case of a
direct bundle of rays refracted at a
sphere, 376.
Centered System of Spherical Surfaces,
Astigmatic Refraction of narrow bundle
of rays through a: see Table of Contents,
Chaps. XI and XII.
Centered System of Spherical Surfaces,
Calculation of the path of a ray through
a: see Table of Contents, Chap. X.
Centered System of Spherical Surfaces, Re-
fraction of paraxial rays through a, 174-
179; law of R. Smith, 267; formulx of
L. Sbidel, 269-273; focal lengths, 264-
267, 271; focal points and principal
points, 177, 178, 271; angular magnifi-
cation (Z) and axial magnification (X),
510; lateral magnification (K). 178, 510.
See Table of Contents, Chaps. VI and
VIII.
Centered Sjrstem of Spherical Surfaces,
Spherical and Chromatic Aberrations:
see Table of Contents, Chaps. XII and
XIII.
Central Collineation of two plane-fields,
162-173; characteristics of , 163; project-
ive relations. 163; geometrical construct*
ions, 165; invariant (c), 168; character-
istic equation, 170; cases that occur in
Optics, 1 71-173.
Central perpendicular, 295.
Centre of Collineation, 163.
Centres of Perspective (K and C) of ranges
of I. and II. object-points and image-
points on chief rajrs of narrow bundles
of incident and refracted rays in case of
refraction at a sphere, 339, 340, 343, 348;
also in case of refraction at a plane, 360,
361 ; and of reflexion at a spherical mirror,
362. 363.
Centres of perspective of Object-Space and
Image-Space, 540.
Characteristic Function of Hamilton, 36-
39.
Charlier, C. V. L., 473.
Chaulnbs, Due de, no.
Chief Ray, as representative of bundle of
rays, 41, 540; defined as ray that goes
through the centre of the aperture-stop,
335, 375* 540; regarded as determining
the place of the image-point in the image-
plane, 416, 540-544.
Christib, W. H. M., 128.
Chromatic Aberration of a system of thin
lenses, 517-522; of two thin lenses in con-
tact, 519; of two separated thin lenses,
520. See also Chromatic Variations.
Chromatic Aberrations, Image affected
with, 503.
Chromatic Aberrations, Theory of: see
Table of Contents, Chap. XIII.
Chromatic Axial or Longitudinal Aberra-
tion of a centered optical system, 508.
Chromatic Dispersion, 475 ; see Dispersion,
Chromatic Under- and Over-Corrections,
517.
Chromatic Variations of the position and
size of the image, difference-form ulse of
the, 505-510, 512; differential formulae
of the, 510. 511.
Chromatic Variations of the focal lengths,
505, foil.
Chromatic Variations in special cases: (i)
single lens in air, 513-516; (2) infinitely
thin lens, 516, 517. See also Chromatic
Aberrations.
Chromatic Variations of the Spherical Aber-
rations, 504, 526-531; of the longitudinal
aberration along the axis, 526-530; of
the sine-ratio, 526, 530.
Circle (or Place) of Least Confusion, 48,
349, 429, 433.
Clairaut, a. C, 477.
Classen, J., 262, 472.
Clausius, R.. 406, 577; sine-condition,
406.
CODDINGTON, H., 348, 407, 438.
CoUinear Imagery, essentially different
from "sine-condition" imagery, 401, 408,
411.
Collinear Optical Systems, 218-262.
Collinear Plane-Fields, 162-173, 201-206.
Collinear relations in the case of the refract-
ion of a narrow bundle of rays at a
spherical surface, 351-356, 402-405; and
through a centered optical system, 358-
360.
Collinear Space-Systems, 162, 206-210;
conjugate planes of, 210; metric rela-
tions, 213-217; lateral magnifications,
214.
Collineation, Central: see Central CoUineo'
tion.
Collineation, Centre and Axis of, 163.
Collineation, Definition of. 162, 201, foil.
Collineation, Theory of, as applied to
Geometrical Optics, 201-217; see Table
of Contents, Chap. VII; see also Central
Collineation.
Colour of a body due to selective absorp-
tion, 10.
Colour-phenomena: see Table of Contents,
Chap. XIII.
Coma. Origin and meaning of the term, 445.
Coma- Aberrations, in general, 444; for-
mulae for the comatic aberration-lines of
the meridian rays, 448-455; condition of
the abolition of coma. 455; comatic aber-
ration in case of a refracting sphere. 455,
and of an infinitely thin lens, 455.
Combination of two or more optical sys-
tems, 245-262; special cases of combina-
tions of two optical systems, 251-255;
616
Index.
focal points and focal lengths of com-
pound systems, 245-250, 255-262.
Compound Optical Systems: see Combina-
tion of optical systems.
Confusion, Circle of least, 48, 349, 429, 433.
Congruence and sjrmmetry, in special sense,
223.
Conjugate Abscissae of projective point-
ranges, 213.
Conjugate Planes of Object-Space and
Image-Space, 210.
Conjugate Points (or Foci), 41; construct-
ions of conjugate points of optical sys-
tem, 241.
Conjugate Rays of Object-Si)ace and
Image-Space, Analytical investigation of,
229.
Convergence of the meridian and sagittal
rays of a narrow bundle of rays re-
fracted at a sphere. Different degrees of.
333.
Convergence-Ratio or Angular Magnifica-
tion (Z), 234; in the case of telescopic
imagery, 245; in the case of a single
spherical refracting surface, 264; and of
a centered system of spherical surfaces,
510; chromatic variation of^ 1 1.
Convergence-Ratios (Zh and Zn) of narrow
pencils of meridian and sagittal rays re-
' fracted (i) at a plane, 68, 69; (2) through
a prism, 93, 94; (3) across a slab, 108;
(4) through a prism-system, 120; and (5)
at a spherical surface, 342, 345, 403-405.
Convergent and Divergent Optical Systems,
228.
Co-ordinates, Axes of: see Axes of co-ordi-
nates.
CoRNU, A., 31. 127. 339.
Correction-terms of 3rd order, in Theory of
Spherical Aberrations, 374-376, 458, foil.
Cotes, R., 192, 193, 195, 198, 268; formula
for the "apparent distance", 191-197.
Critical Angle of Refraction (A) with re-
spect to two media, 24.
Crova, a., 128.
CULMANN, P., 192, 268, 348, 350. 353, 366.
Curvature, Lines of, of a surface, 43.
Curvature of Image, 429-444; development
of formulae for the curvatures of the
astigmatic image-surfaces, 434, 441; cur-
vature of the stigmatic image, 439; cur-
vature of image in case of refracting
sphere. 442 ; and of an infinitely thin lens,
443. See also Astigmatism.
Cushion-shaped distortion, 421.
CzAPSKi, S., 48, 50. 114, 201, 217, 218, 233.
246, 262, 335, 336, 349. 350. 353. 379, 400,
448, 479, 480, 485, 492, 493. 523. 528,
530, 547. 558, 560, 563, 578; his great
work on the theory of optical instru-
ments, 201; arguments in favour of the
image-lines of Sturm, 50, 335; imagery
by means of astigmatic bundles of rays,
349.
D.
D'Alembert, J., 477.
Depth-Magnification: see Axial Magnifica-
tion.
Depth of Accommodation, 561.
Depth of Focus: see Focus-Depth,
Depth of Vision, 562.
Descartes, R., 15.
Deviation of refracted ray, 27; deviation
of ray refracted through a prism (i) in
principal section, 78-81; (2) obliquely,
125; through a prism-system. 113, 114.
See also Minimum Deviation.
Deviation without dispersion in a prism-
system, 489; in a combination of two thin
prisms, 483.
Diagrams, Designations of points, lines and
surfaces in the, 583-596.
Diagrams for showing procedures of paraxial
rays, 142.
Diffraction-effects, 4; diffraction-pattern as
image, 42.
Direction of ray or straight line: see Posi-
tive Direction.
Direct- vision prism-S3rstem, 491; of Ahici,
492, 502 ; combination of two thin prisms,
484.
Dispersion, Analytical formula for. 485;
anomalous, 475; characteristic dispersion
of a medium, 485; chromatic dispersion,
475; irrationality of dispersion, 482;
mean dispersion, 481; partial dispersion.
481, 523; relative dispersion (^i/"), 481;
relative partial dispersion (fi) . 523; resi-
dual dispersion. 523.
Dispersion in case of (i) a single prism in
air, 487; (2) a S3rstem of prisms, 484-492;
especially, a train of prisms composed
alternately of glass and air. 488.
Dispersion, Minimum, of a prism. 487.
Dispersion without deviation: see Direct^
vision prism-system.
Dispersive strength and "dispersor** of lens.
516.
Distinct Vision. Distance of. 547.
Distortion of image, and conditions of its
abolition. 415-429, 467; measure of the
distortion, 417; typical kinds of distor-
tion, 421: distortion in case the pupU-
centres arc the pair of aplanatic points,
421. See also Orthoscopy.
Distortion-aberration, Development of the
formula for, 422-427; in the case of a
refracting sphere, 427, and of an infinitely
thin lens, 428.
Ditscheinbr, L., 128.
Divergent and Convergent Optical Sys-
tems, 228.
DoLLOND, J., 476, 477.
Index.
617
DONDERS, 562.
Drude, p., 21. 407. S7I.
E.
EflFective Rays. 41, 537.
Electromagnetic Theory of Light. 2.
Emission Theory of Light, i, 19.
Entrance-port. 550; optical system with
two entrance-ports, 551; optical project-
ion-system with one entrance-port, 563-
$68; and with two entrance-ports, 568-
571.
Entrance-pupil, 323. 374. 537; reciprocity
between object and entrance-pupil, 537.
Equivalent Light-Source, 574.
EuLER, L., 36, 42. 370, 476. 477.
Euclid, 15.
Everett. J. D., 407.
Exit-port, 550; see Entrance-port.
Exit-pupil, 375. 537; reciprocity between
image and exit-pupil. 537. See also £«-
trance-pupil.
F.
Fermat, p., 33, 34. 40, 499; law of. 33.
Field of view, 549-551 ; angular measure of,
550. 551; in a projection-system of finite
aperture (i) with one entrance-port, 563-
568; and (2) with two entrance-ports
568-571.
Field-stop. 550.
Finsterw alder. S.. 471.
Flat (or plane) image. Conditions of. 440.
467. See Curvature of Image.
"Flucht*' Lines of conjugate pianes. 204;
see Focal Lines.
"Flucht" Planes of two collinear space-
systems, 208; see Focal Planes.
"Flucht" Points of projective point-ranges
(or of conjugate rays), 166. 203. 219;
see also Focal Points.
Fluorescence, 12.
Focal Lengths (/, «0 of Object-Space and
Image-Space, Definitions of, 233, 237;
also 157; relations to the image-constants
a, b, 233.
Focal Lengths (/, «0 of centered system of
spherical surfaces. 264-267, 271; of com-
pound optical system, 248. 258; of spher-
ical refracting surface. 155. 264; of thick
lens, 275; of thin lens. 284; of infinitely
thin lens. 186, 284; of a system of two
lenses, 285. Focal Length (/) of spher-
ical mirror, 140. ^
Focal Lengths (/«. Cu' and /m, fnO of the
meridian and sagittal collinear systems
of a narrow bundle of rays refracted at a
spherical surface. 354; and through a
centered system of spherical surfaces, 359.
Focal Lines, 154, 168, 206; see ''FluchV*
Lines.
Focal Planes (0. c') of Object-Space and
Image-Space, 208; of refracting sphere,
154. See also **Flucht** Planes.
Focal Planes. Sine-Condition in the. 408.
Focal Points (F. £') of Object-Space and
Image-Space, 211; of centered system of
spherical surfaces, 177, 271; of compound
optical system, 247, 256; of refracting
sphere. 150; of spherical mirror. 140; of
thick lens. 275; of Infinitely thin lens. 184.
284; of a system of two lenses, 285.
Focal Points (7, /' and 7. /') of meridian
and sagittal rays of narrow bundle of
rays in case of refraction at a spherical
surface. 340-344; in case of refraction at
a plane. 361; in case of reflexion at a
spherical mirror. 362 ; Smith's construct-
ion of the Focal Points J, /', 348.
Focal Surface. 471.
Foci. Conjugate. 41; focus, 44.
Focus-Depth of projection-systems of finite
aperture, 557-560; of systems of finite
aperture used in conjunction with the eye,
560, 561 ; lack of detail in the image due
to, 560.
Focus-Plane, 552.
FoucAULT. L.. 19.
Fraunhofer. J., 21, 83. 87, 104, 319. 415,
469. 470. 475. 477. 478. 479. 482. 502. 525,
526. 528. 529, 610; so-called Fraun-
HOFER-Condition. essentially same as
Sine-Condition and condition of abolition
of Coma. 415; FRAUNHOFBR-Lines of the
solar spectrum. 477, 478.
Fresnel. a.. I. 7. 9; explanation of the
so-called rectilinear propagation of light,
7. 8.
G.
Galileo, i, 398. 478.
Gauss. C. F.. 54. 178. 179, 198. 199. 200,
233. 237. 239. 263, 319, 367, 369. 371, 372.
373. 374. 379. 380. 385. 400, 401. 403. 416.
417. 422. 430. 431. 432. 433, 440. 446. 454,
456. 457. 458. 459. 464. 506. 526, 528.
532, 534. 588. 590. 591, 595. 603; his
famous work on Optics. 198; his defini-
tions of the focal lengths. 233 ; use of the
Principal Points. 237; GAUSsian Imagery.
263, 367. 369; GAUSsian parameters
of the incident and refracted rays. 456-
459; so-called Gauss's Condition. 528.
Gehler. G. S. T.. 128. 400.
Geometrical Optics. Its scope and plan. 2. 3.
Geometrical Theory of Optical Imagery:
see Table of Contents, Chap. VII.
Glass, Optical: Kinds of, 480; Jena Glass,
478-483: investigations of Abbe and
ScHOTT. 478.
Gleichen. a., 97, 98, 114. 358.
GoERZ, P.. 358.
Graphical Method of showing imagery by
Paraxial Rays, 142.
618
Index.
Grubb, T.. 127.
Grunert, J. a., 239.
GUENTHER, S., 262.
H.
Hall, 476.
Hamilton, Sir W. R.. 33. 36. 38. 39, 472;
Hamilton's Characteristic Function, 36-
39. 472.
Hankel, H., 199.
Harting, H., 366.
Heath, R. S., 40, 44, 55, 125, 349, 476.
Helmholtz, H. von, 44, 98, 128, 192, 197,
268, 353, 406. 493. 494, 497, 577; so-
called Helmholtz Equation, 197, 268
(see also Smith, Lagrange); proof of
Sine-Condition, 406, 577; measure of
purity of spectrum, 493.
Heppercer, J. VON, 128.
Hero of Alexandria, 33.
Herschel, J. F. W., 394. 407. 470. 487. 556;
HERSCHEL-Condition, 394, 470, 556.
Hertz, H., 2.
HocKiN, C. 407.
Homocentric Bundle of Rays, Definition,
44.
Homocentric Image-Points, with respect
to prism, 102, 133; with respect to slab,
1 09-1 11; with respect to prism-system,
122; with respect to lens, 358, 364.
Homocentric Refraction through a prism,
97-105, 128-133; through a prism-ssra-
tem. 122; across a slab, loo-iii; through
a lens, 358.
HooRWBG, J. L., 127, 128.
HuYGENS, C, 1, 3, 4, 5, 6, 7, 16, 17, 18, 522;
construction of wave-front in general, 4-
. 7; construction of reflected and refracted
wave-fronts, 16-19; Huygsns's Ocular,
522.
I.
Illumination, Intensity of, 571-575.
Illumination in the Image-Space, 578.
Image, Abbe's measure of the indistinctness
or lack of detail of the, 385; numerical
calculation, 395-397'
Image. Diffraction-, 42.
Image, Flat, Conditions of, 440, 467.
Image of extended object by astigmatic
bundles of rajrs, 349-351; see also Astig-
matism.
Image, Order of, according to Pbtzval, 370,
371.
Image, Perfect or Ideal, 41, 198; Max-
well's definition, 200.
Image, Practical, 367-369, 412; require-
ments of a good image, 367; practical
images by means of wide-angle bundles
of rays, 412; and by means of narrow
bundles, 416. 417.
Image, Projected on screen-plane, 553, 555*
Image, Real or Virtual, 41; erect or in-
verted, 228.
Image-Constants (a, b, c), Signs of the, 218,
228: their connections with the focal
lengths (J, tT), 232.
Image-EIquations: In general, 216; in case
of symmetry around the principal axes,
222; in terms of the focal lengths, 233;
referred to conjugate axial points, and,
especially, to the Principal Points {A, AO.
239. 240; in the case of Telescopic Im-
agery, 243, 244, 245; in the case of a re-
fracting sphere, 158-162, 264; in the case
of an infinitely thin lens, 187.
Image-Lines, Primary and Secondary, of
narrow astigmatic bundle of image-rays,
44-50. 331. 334-336. 348; directions of,
48-50, 335; exactly what is meant by,
48-50. 335.
Image-Lines, Primary and Secondary, of
narrow astigmatic bundle of rays re-
fracted at a plane, 65, 66; through a
prism, 94; at a spherical surface, 331,
334-336, 348.
Image-Plane, 370.
Image-Point or Point-Image, according to
Phj^ical Optics. 42. 287. 368.
Image-Points, Primary and Secondary, of
narrow astigmatic bundle of image-rays,
46, 33 1 ; see also Infinitely Narrow Bundle
of Rays.
Image-Points, Primary and Secondary, of
narrow astigmatic bundle of rays re-
fracted (i) at a plane, 65-68, 360; con-
structions of I. Image-Point, 71, 361; (a)
through a prism: constmction of, 90;
formulae for, 92; (3) across a slab: con-
struction of, 106; formulae for, 107; (4)
through a prism-system: construction oif,
115; formulae for, 117; (5) at a spherical
surface, 336-348; (6) through a centered
system of spherical surfaces. 356-35S;
(7) in case of reflexion at a spherical
mirror, 362.
Image-Points, Homocentric, X02; see Homo-
centric Image-Points.
Imagery, Ideal, by means of Paraxial Rays:
see Table of Contents, Chap. VIII; see
also Optical Imagery, GAUSsian Imagery.
Imagery in the planes of the meridian and
sagittal sections of an infinitely narrow
bundle of nys refracted at a spherical
surface, 349-356. 402-405.
Imagery, So-called Seidbl-: see L. Sbidbl.
Image-Space and Object-Space: see Object'
Space and Image-Space.
Image-Surfaces, Astigmatic, 416, 42^, 430;
curvatures of, 434-441; sec CurwUwre cf
Image.
Incidence, Angle of, 13; incidence-height,
135* 295; incidence-normal, 13; plane of
incidence, 13; incidence-point, 13.
Index.
619
Incident light, lo; incident ray, 13.
Inclined Mirrors. 54, 55.
Index of Refraction: Absolute, 20; relative,
14, 20; "artificial'* index of refraction in
case of oblique refraction, 30, 126; con-
nection between index of refraction and
wave-length of light, 474.
Indistinctness (or Lack of Detail) of image
due to spherical aberration along the axis.
Abbe's measure of, 385, 395-397-
Infinitely Narrow Bundle of Rays, 42-50,
331-336; see Table of Contents, Chaps.
II, III, IV and XI; also Chap. XII under
Astigmalism; see also Astigmatic Refract'
ion.
Infinitely Narrow Bundle of Rays, Astig-
matic Reflexion of, at a spheriod mirror,
361-363-
Infinitely Narrow Bundle of Rays, Astig-
matic Refraction of. at a spherical surface
or through a centered system of spherical
surfaces, see Table of Contents, Chaps. XI
and XII; collinear correspondence, 351-
356, 402-405; lateral magnifications (K«,
Yu) of the meridian and sagittal rays,
403-405; imagery in the planes of the
meridian and sagittal sections, 349-356,
402-405.
Infinitely Thin Lens, Chromatic Variations
in case of an, 516.
Infinitely Thin Lens, Refraction of Paraxial
Rays through an, 173, 182-190, 388; a
case of central collineation, 173; conju-
gate axial points, 183; focal points, 184,
284; focal lengths, 186, 284; power or
strength (0), 186; dispersive strength
and "dispersor", 516; lateral magnifica-
tion (F). 187; imagery, 187.
Infinitely Thin Lens, Special Notation in
case of an, 387.
Infinitely Thin Lens, The Spherical Aber-
rations in case of an: Astigmatism, 363-
366; Axial or Longitudinal Aberration,
387-392; Coma, 455; Curvature of Image,
443; Distortion-Aberration, 428.
Infinitely Thin Lenses, System of: Chro-
matic Aberration of a, 517, foil.
Infinitely Thin Lenses, System of: Refract-
ion of Paraxial Rasrs through a, 190;
lateral magnification (K), 191; Cotes's
formula for "apparent distance" of object
viewed through such a system, 191-197.
System of two infinitely thin lenses. 285.
System of infinitely thin lenses in contact,
191.
Infinitely Thin Lenses, System of: Spheri-
cal (Longitudinal) Aberration of a, 392-
394; case when lenses are in contact,
393-
Initial Values of the Ray-Parameters: in
Kerber's Refraction-Formulae, 323; in
Sbidel's Refraction-Formulae, 329; in
Seidel's Theory of Spherical Aberra-
tions, 373-375.
Interval (A) between two consecutive com-
ponents of a compound optical system,
247; in case of a lens, 274.
Invariant (c) of Central Collineation, 168;
invariant (/) of refraction at a sphere.
299; invariant (/) of refraction of par-
axial rays at a sphere, so-called "zero
invariant", 159; invariant, optical (K =
n 'sin a), 21.
Invariant-Method of £. Abbe, especially
as applied to the development of the
formulae for the curvatures of the image-
surfaces, 434. foil.; and of the formulae
for the comatic aberrations, 448, foil.
Invariants ((?. Q) of Astigmatic Refraction,
434. 450.
Iris of optical system. 537.
Isoplethic Curves of Von Rohr, 529.
J.
Jena glass, 478-483; table of some varieties
of, 480;. opticail properties of, 482.
jBTTlifAR, H. VON, 128.
K.
Kabstner, a. G., 191*
Kayser, H., 79, 114, 128, 485, 488, 493, 498.
Kepler, 15.
Kerber, a.. 305, 306, 310, 311, 312, 322,
467. 473. 528, 529, 602, 605, 606, 607;
Refraction-Formulae. 305-307, 310-312,
322-325; chromatic correction of optical
system, 528.
Kessler, F., 73, 79» 288, 336, 360, 514;
investigations of the chromatic aberra-
tions of a lens. 514.
KiRCHHOFF, G. R., 192, 477, 577.
KiRKBY, J. H., 79.
Klein, F., 38.
Klingenstierna. S., 476.
KoENiG, A., 312, 376, 398, 412, 420, 422,
430, 434. 446. 448, 455. 468, 473, 506. 509,
512. 513. 519. 523. 524. 526, 528.
kohlrausch, f., 86.
Kruess, H., 526, 528.
KUMMER, E. E., 44, 48, 335, 351, 471.
KUNDT, A., 21.
KuRZ, A., 79.
L.
Lagrange. J. L. de, 192, 195, 197, 268, 353;
so-called Lagrange-Helmholtz Equa-
tion, 197, 268; see Helmholtz, Smith.
Lambert, J. H.. 538, 573.
Lateral Aberration, 379; formula for, 385;
chromatic, 510.
Lateral Magnification (K) of two collinear
space-systems, 214-216, 221, 234; rela-
tion to the other magnification-ratioe
620
Index.
(X and Z). 234; in the case of a tdescopic
system. 244.
Lateial Magnification (K) in case of a
spherical mirror. 145.
Lateral Magnification ( K) in case of refract-
ion at a plane. 58; at a sphere. 160. 264;
through a centered system of spherical
surfaces. 178. 510; through a lens. 180;
through an infinitely thin lens. 187;
through a system of thin lenses, 191.
Lateral Magnification (K)> Chromatic
variation of, 505, foil. __
Lateral Magnifications (K«. K«) of the
meridian and sagittal sections of narrow
astigmatic bundle of rays refracted at a
spherical suriace (or through a centered
system of spherical suriaces). 403-405.
Law of independence of rays of light. 2, 8,
9; of rectilinear propagation of light 2»
3~8; of reflexion and of refraction. 2. 13-20.
Least Action. Maupbriuis's Principle of.
36.
Least Circle of Aberration. 378.
Least Confusion. Place or Circle of, 48, 349.
429. 433.
Least Time. Principle of, 33-36.
Left-screw imagery. 223, 225. 228.
Lens. Definition. 179; types of lenses. 180;
convergent and divergent, positive and
negative lenses, 180. 185; character of
the different forms of lenses, 276-283;
bending of lenses 390; rectilinear lens.
418. See also Thick Lenses, Thin Lenses,
Infinitely Thin Lenses, Lens-Systems.
Lens-Systems, 284-286; system of two
lenses, 284; two systems of lenses. 285;
system of two infinitely thin lenses,
285. See also Infinitely Thin Lenses,
Leonardi da Vinci, i.
L'HospiTAL, 348.
Lie. S.. 472.
Light, Mode of propagation of, 2. 4; see
also Rectilinear Propagation of Light.
Light, Theories of: Emission, i, 19; Wave,
I, 19; Electromagnetic, 2.
Light. Tube of, 571.
Light, Velocity of, 19.
Light-rays: see Rays of Light.
Light-Source, Equivalent, 574.
Linear Magnitudes, Symbols of, 596-604.
Lines, Designations of, in the diagrams, 593,
594-
LiPPiCH, F., 44. 199. 239, 262, 288, 332, 339,
352. 354. 360.
Listing, J. B., 238.
LOEWE. F.. 98. 99. 114. 128, 485, 492, 493.
LoMMEL, E., 79.
Longitudinal Aberration: see Spherical Ab-
erration and Chromatic Aberration.
Lucas of Liege, 482.
Luminous Surface-Element, Radiation of,
573.
Luiocnu O., 41, 349. 368. 408. 471, 527.
M.
Magnification. Angular (Z): see Comver-
gence-Ratio,
Magnification. Ajdal or Depth (X): see
Axial Aiagnificaium.
Magnification. Lateral (F): see Laieral
Magnifu4Mtion.
Magnification. Objective. 544.
Magnification-Ratio. 214; relation of the
magnification-ratios to each other. 234;
in the case of telescopic imagery. 243-245.
Magnifications, Different, of the different
zones of a spherically corrected system,
402.
Magnifying Power, Objective. 544; subject-
ive. 545-549; intrinsic magnifying power
of optical sy^em. 548.
Malik's. £. L.. 39. 40. 42; law of Malus. 39.
40.
Matthiessen. L., 48, 49, 50. 64. 239. 262.
335. 351-
Maupertuis. 36.
Maxwell. J. C. 2. 38, 199. 200. 201;
theory of periect optical instruments.
199. 200.
Measurement. Optical Instruments for pur-
pose of. 541.
Medium. Optical. 9; transparent, translu-
cent, opaque. 13.
Meridian Planes of Optical Sjrstem. 212.
214. 227.
Meridian Rajrs of narrow bundle of ra3rs:
see Astigmatism, Astig;nuUic R^r actum,
etc.; see also Table of Contents. Chaps.
II. Ill, IV. XI and XII.
Meridian Ra3rs of narrow bundle of rays
refracted at a sphere. 331; convergence
of. 333; collinear relations. 351-356; lat-
eral magnification (KOt 403; imagery,
353. 403.
Meridian Section of narrow astigmatic
bundle of rays. 46, 331.
Meridian Section of bundle of rays refracted
at a sphere. Lack of ssrmmetry in the,
445, 446-448.
Metric Relations of two collinear space-
systems, 213-217.
MiCHELSON, A. A., 19.
Micrometer- Microscope, 542.
Minimum Deviation by a prism. 78, 79,
81-83, 87, 99, 127; in case of oblique
refraction through prism, 126; minimum
deviation by a prism-system, 114.
Minimum Dispersion by a prism, 487.
Minimum Property of the Light- Path: sec
Least Time: see also Fermat. Hamilton.
Minor, 21.
Mirror: see Plane Mirror, Spherical Mirror.
MoEBius, A. F., 44, 178, 199, 20I, 262.
Index.
621
Monocentric Bundle of Rays, Definition,
44.
monoyer. f., 262.
Mueller. Fr. C. G., 79.
Muellbr-Pouillet's Lehrbuch der Physik,
349. 368. 407. 408. 527.
N.
Neumann, C, 44.
Neumann, C. C, 239:
Newton, Sir I., i, 19. 192, 347. 348. 475.
476, 478, 482 ; discoverer of astigmatism,
347; prism-experiments, 475.
Nodal Points (AT, N') and Planes of optical
system, 238; nodal points of refracting
sphere, 264; of lens, 275.
Normal Sections of a surface. 28, 42.
Notation. System of: see Appendix, 583-
612.
Numerical Aperture, 538.
Numerical Illustration of the calculation of
the path of a ray through an optical sys-
tem: (i) Paraxial Ray, 320; and (2)
Edge-Ray in principal section, 321; of
the calculation of the spherical aberra-
tion, 319-321. 394-397-
O.
Object, Projected on focus-plane, 553, 555
Objective Magnification, and Magnifying
Power, 544.
Object-Plane, 370.
Object-Space and Image-Space. 207; geo-
metrical characteristics of, 210-212; con-
jugate planes of, 210; focal points (F, £').
211; principal axes (x, x'), 212; axes of
co-ordinates. 212; positive directions of
co-ordinate axes. 220. 221. 227; relation
between conjugate rays. 229; focal lengths
if, e'), 232.
Oblique Refraction in general. 28-32; con-
struction of obliquely refracted ray. 31.
Oblique Refraction at a plane. 311. 315.
Oblique Refraction at a sphere: Parameters
of the ray. 304-310; Kerber's Refraction-
Formulae. 310-312; Seidel's Refraction-
Formulae, 313-315-
Oblique Refraction through a centered
system of spherical surfaces: Kerber's
Refraction-Formulae, 322-325; Seidel's
Refraction-Formulae, 325-330.
Oblique Refraction through a prism: Con-
struction of the path of the ray, 123; cal-
culation of the path, 124; deviation of
the ray. 125. Oblique Refraction of
narrow bundle of rays through a prism,
128-133.
Oculars of Huygens and Ramsden. 522.
Optical Axis of spherical surface, 134; of
centered system. 174. 227; positive direc-
tion of. 135. 227.
Optical Centre of Lens, 181.
Optical Image, 40,' 42; from standpoint of
Physical Optics, 42, 287, 368; brightness
of, 579-582; intensity of radiation of.
575-579-
Optical Imagery, Geometrical Theory of:
see Table of Contents, Chap. VII; Abbe's
Theory of. 198-201 ; characteristic metric
relation of. 213. 220; general character-
istics of, 218-229; different types of, 223-
229. See also Imagery.
Optical Instrument, Function of, in general,
198; Maxwell's definition of "perfect"
optical instrument, 200.
Optical Invariant {K = n • sin a), 21.
Optical Length. 35.
Optical Measuring Instruments. 541.
Optical Systems, CoUinear. 218-262; com-
bination of, 245-262; convergent and
divergent, 228.
Order of image, according to Petzval. 370,
371-
Orthoscopic (or Angle-true) image, 418.
Orthoscopic Points of Optical System, 421.
Orthoscopy, 415-429; condition of, in gen-
eral, 41 3; in case the system is spherically
corrected with respect to the pupil-cen-
tres, 420. See Distortion.
Orthotomic system of rays, 40.
P.
Parallel Plane Refracting Surfaces, Path of
ray traversing a series of. 89; see also Slab.
Parameters of incident and refracted rays
in case of refraction at a spherical surface.
296; in case of oblique refraction at a
spherical surface. 304-310; initial values
in case of oblique refraction through a
centered system of spherical surfaces.
323-329; parameters of Gauss, 456-459;
parameters used by Seidel in his theory
of spherical aberrations. 371, 459; ap-
proximate values of the SEiDEL-param-
eters and the corrections of the 3rd
order. 372. 459. foil.; relations of the
SEiDEL-parameters to those of Gauss.
459-
Paraxial Ray. Definition. 136; numerical
illustration of calculation of path of par-
axial ray through a centered system of
spherical surfaces. 320.
Paraxial Rays, Ideal Imagery by means
of: see Table of Contents. Chap. VIII;
graphical method of showing imagery
by paraxial rays. 142.
Paraxial Rays. Reflexion and Refraction of,
at a spherical surface: see Table of Con-
tents, Chap. V.
Paraxial Rays. Refraction of. at a plane. 57-
59. 161, 172; through a centered system
of spherical surfaces or through a lens or
lens-sjrstem: see Table of Contents.
Chaps. VI and VIII.
622
Index.
Path of ray reflected at a spherical mirror,
299.
Path of ray refracted at a plane, 55-56, 300,
311. 315-
Path of ray refracted at a spherical surface:
Calculation of (i) when ray lies in a prin-
cipal section, 298. 299. 302; and (2) when
ray does not lie in a principal section,
304-315; see Table of Contents. Chap.
IX. Geometrical investigation of path
of ray refracted at spherical surface, 288-
294; Young's construction of refracted
ray. 288.
Path of ray refracted through a centered
system of spherical surfaces: (i) when
ray lies in a principal section, 316-321;
and (2) when ray does not lie in a princi-
pal section. 322-330. See Table of Con-
tents, Chap. X.
Peacock. G., 339.
Pencil of rays. 41. 202.
Perspective, Centres of: see Centres of Per^
speciive.
Perspective elongation of projected object
(or image). 553.
Pktzval. J.. 370. 371. 438, 439. 440. 529.
564; order of the image, 370. 371; for-
mula for curvature of image, 439; field
of view of projection-system. 564.
Pbzenas. Le Fere. 191.
Photograph, Correct distance of viewing a,
554.
Pin-hole camera, 288.
Plane-field, Definition. 163, 202; central
collineation of two plane-fields, 162-173;
plane-fields in coUinear relation, 201-206;
projective relation. 163. 202; affinity-re-
lation of plane-fields. 206.
Plane Image. Conditions of, 440, 467; see
Cturvature of Image.
Plane Mirror, 51-55; conjugate points with
respect to, 51 ; collinear imagery, 52, 288;
image of extended objects in a, 53; uses
of. 54; number of images by successive
reflexions in a pair of plane mirrors, 54,
55.
Plane Surface, Path of ray refracted at a.
55* 56. 300; Kerber*s formulae for path
of ray refracted obliquely at a. 311; and
Sbidel's formulae for the same, 315.
Plane Surface, Reflexion at a: see Plane
Mirror.
Plane Surface, Refraction of paraxial rays
at a, 55, 56. 161, 172.
Plane Surface, Astigmatic Refraction of
narrow bundle of ra>*s at a, 64-73. 360.
361; geometrical relations between ob-
ject-points and image-points, 70; con-
struction of the I. Image- Point, 71; and
of the I. and II. Image-Points, 360, 361.
POGGENDORF. J. C, 54.
Foint-Image: see Imaic-Poini.
Point-Range, Definition. 202; projectively
similar ranges of points. 211; directly and
oppositely projective point-ranges. 219.
See Projeaive Relations, Affinity-Rela-
tions.
Points, Designations of, in the diagrams,
583-593.
Point-Source of light, 6; radiation of. 571;
candle power of, 572; brightness of. 581.
Ports. Entrance- and Exit-, 549; see En-
trance-Port.
Positive Directions of incident, reflected
and refracted rays, 22, 251; positive
direction of optical ray, 219; of a straight
line. 22; of the principal axes (x. xO of
the Object-Space and Image-Space, 220.
221. 227; of the secondary axes of co-
ordinates, 221, 227; of the optical axis,
135. 227.
Power, or Strength (0), of infinitely thin
lens. 186.
Primary and Secondary Astigmatic Image-
Suriaces, 416. 429. 430; curvatures of,
434-441. See Curvature of Image.
Primary and Secondary Focal Lengths: see
Focal Lengths.
Primary and Secondary Focal Points: see
Focal Points.
Primary and Secondary Image-Points and
Image-Lines: see Image-Points, Image-
Lines, Infinitely Narrow Bundles of Rays,
Astigmatism, etc
Principal Axes (x, xO of Object-Space and
Imaige-Space, 212; positive directions of,
220. 221; symmetry around, 221.
Principal Axes (at, mO of the two pairs_of
collinear plane-systems (**, r' and t, rQ
in the case of refraction of narrow bundle
of rays at a spherical surface, 351-354.
Principal Planes of optical system, 1 78, 237.
Principal Points (A, A') of optical system
in general. 237; image-equations lefened
to, 239, 240.
Principal Points (A, A') of a centered sys-
tem of spherical surfaces. 178. 271; of a
spherical refracting surface, 264; of a
lens, 275; of a system of two lenses, 285.
Principal Sections of a surface, 42.
Principal Sections of a prism, 74; of a re-
fracting sphere, 294.
Prism, Definition. 74; refracting angle, 74;
principal section. 74; construction of ny
refracted through prism in principal sect-
ion. 74; total reflodoo at second face of
prism. 78, 84. 87; normal emergence at
second face. 83. 87; deviatioa of ray by
prism, 78-81, 125; ray of minimnm de-
x-iation, 78, 79, 8i-«3, 87, 99, 126, 127;
path of ray refracted obUqoely through a
prism. 123-128.
Prism. Astigmatic Refractkxi of narrow
bundle of ra^-s through a. 90-97.
Index.
623
Prism, Homocentric Refraction through a,
(i) when chief ray lies in a principal
section, 97-105; (2) when chief ray is
obliquely refracted through prism, 128-
133.
Prism, Dispersion of, 487; see Dispersion,
Prism-formulae, Collection of, 87.
Prism-spectra: see Table of Contents, Chap.
XIII.
Prism-System, Achromatic, 483, 489-491.
Prism-System, Direct-vision, 484, 491, 502.
Prism-System, Dispersion of, 484-492; see
Dispersion.
Prism-System, Path of Ray through a, 1 11-
115; construction, 112; calculation, 113;
condition of minimum deviation, 114.
See Table of Contents. Chap. IV.
Prism-System, Resolving power of, 498-
503.
Prism, Thin, 83; achromatic combination
and direct-vision combination of two thin
prisms, 483, 484.
Projected Object and Image. 553; in case
of projection-systems of finite aperture,
555.
Projections of incident and refracted rays.
Theorems concerning, 30.
Projective Relation of two collinear plane-
fields, 202; in special case of Central
Collineation, 163.
Projective Relations of ranges of I. and II.
object-points and image-points lying on
chief rays of narrow bundles of incident
and refracted rays, in case of refraction
at a spherical surface, 338, 343, 345, 347.
ProLEMiBUS, C, 15.
PULFRICH. C. 86.
Pupils, Entrance- and Exit-, 532-540; see
Entrance-Pupil, Exit-Pupil, Planes of
the Pupils. 374.
Purity of the Spectrum, 492-498; Ideal
Purity of Spectrum, 497; see Spectrum,
R.
Radau, R., 75.
Radiation of point-source, 571; of luminous
surface-element, 573.
RxMSDEN-ocular. 522.
Range of points: see Point-Range.
Rays of light, 2, 3, 4, 8-20; mutual inde-
pendence of, 2, 8, 9; rays meet wave-
surface normally, 39. See also Bundle of
Rays, Pencil of Rays, etc.
Rays, Geometrical, 202 ; orthotomic system
of rays. 40.
Rays. Chief: see Chief Rays.
Ray-co-ordinates. 56, 296.
Ray-length, 295.
Ray-parameters: see Parameters of Ray.
Ray-Path, Reversibility of, 15, 207. See
also Path of Ray.
Rayleigh, Lord, 191, 192, 195, 268, 342,
497. 498, 500, 502, 581; resolving power
of prism-system, 498-503.
Real and virtual, 10; real and virtual
images, 41.
Rectilinear Propagation of Light, 2, 3-8.
Reflected Light, 10.
Reflected Ray, 13, construction of, 25; de-
viation of, 27; positive direction of, 22,
251.
Reflexion, Angle of, 13; laws of, 2, 13-20;
total, 22-25; regular and irregular (or
diffused), 11; reflexion as special case of
refraction, 22, 161, 299, 361.
Refracted Light, 10.
Refracted Ray, 13; construction of, 26;
deviation of, 27; positive direction of, 22.
Refracting Angle of Prism, 74.
Refraction, Angle of, 13; laws of, 2, 13-20;
regular and irregular. 11, 12; index of,
see Index of Refraction.
Refraction-Formulae of A. Kerbbr and L.
Sbidbl: see Kerber, Seidel; see also
Table of Contents, Chaps. IX and X.
Refraction (or Reflexion), Oblique: see
Oblique Refraction.
Refractive Index: see Index of Refraction,
Residual Dispersion: see Secondary Spec-
trum.
Resolving Power: Of eye, 369, 557. 559;
of prism-system, 498-503.
Rbusch, E., 31, 72, 75. 91. 127, 142.
Reversibility of Ray-Path, Principle of, 15.
207.
Right-screw imagery, 223, 225, 228.
RoETHiG, O., 262, 458.
Rudolph, P., 529.
S.
Sagittal or s-aberrations, 374.
Sagittal Rays of narrow bundle of rays: see
Astigmatism, Astigmatic Refraction, etc.;
see also Table of Contents, Chaps. II,
III, IV, XI and XII.
Sagittal Rays of narrow bundle of rays re-
fracted at a sphere, 333, 343-345; con-
vergence of, 334; collinear relations, 351-
356; lateral magnification (Km), 403-405;
imagery, 353, 404; symmetry in the sag-
ittal section, 445.
Sagittal Section of narrow bundle of rays,
46.
Salmon, G., 60.
schellbach, k., 288.
schellbach, r. h., 79-
schleiermacher, l., 370.
Schmidt, W., 475.
ScHOTT, O., 87, 478, 479.
Schuster, A., 501.
Screen-plane, 553.
Secondary Focal Point, Focal Length,
Image-Point, Image-Line, Image-Sur-
face: see Focal Points, Focal Lengths,
624
Index.
Image-Points, Image-Lines, Image-Sur^
faces.
Secondary Spectrum, 479, 482, 504, 523-
526; of a system of thin lenses in contact,
534.
Sbidel. L., 268, 269, 270, 271, 297, 305, 307,
308. 309, 310, 313. 314, 325, 326, 327, 328,
329. 330. 366, 369. 370, 371, 373, 376. 41S.
438. 439. 440, 456, 458. 459. 465. 467. 468,
469, 470, 471, 472, 510, 526, 607, 608; for-
mulae for refraction of paraxial rays
through a centered system. 269-273;
formulae for calculation of path of ray
through centered optical system, 305,
307-310, 313-315. 325-330; Seidkl
Imagery. 369-376; parameters of inci-
dent and refracted rays, 307-310, 371-
373. 459; theory of the spherical aberra-
tions of the 3rd order, 369-376, 456-
473; development of formulae for the y-
and 3B-aberrations, 456-468; Sbidel's
Five Sums, 467.
Series-Developments of the spherical aber-
rations of the 3rd order, 371-376, 397-
400, 456-470.
Series-Developments of the comatic aber-
rations, 446, 448, foil.
. Sheaf of Planes, Definition, 202.
Shortest Route, Principle of the, 35.
Signs of the Image-Constants a, h, c, 218;
228; of the focal lengths/, ^, 233.
Similar Ranges of Points, 71, 92, 99, 117;
projectively similar ranges of points, 211,
214, 243.
Similarity between object and image. Con-
dition of, 222; also, 418. See Distortion,
Orthoscopy.
SIMMS, W., 128.
Sine-Condition, 400-415; its derivation and
meaning. 400-407; derived from a general
law of R. Clausius, 406; development
of formulae for the sine-condition, 412-
415; identical with condition of abolition
of Coma, 455; sine-condition in the focal
planes, 408. See also Aplanatism.
Sine-Condition, Chromatic Variation of,
530.
Sine-Condition satisfied with respect to
aplanatic points (Z, Z') of refracting
sphere. 291. 301, 401.
Slab, with plane parallel faces: as special
case of prism. 86; construction of path of
ray across a. 88 ; astigmatic refraction of
narrow bundle of rays across a. 106-111;
homocentric refraction across a, 109-111.
Slit-image, as seen through prism, 94. 105;
as seen through prism-system, 120; dif-
fraction-image of slit, 495.
Slope of ray, 135, 296, 316.
Smith, R., 191, 192, 195, 196, 197, 267, 268,
347, 348, 353, 385, 401, 427, 440. 441, 461,
465. 581; his law, 196, 197, 267; proof of
CoTBS's formula and corollaries there-
from, 192-197; construction of focal
points J, I' of narrow bundle of rasrs in
case of refraction at a sphere. 348.
Smith-Hblmholtz Equation. 268.
Snell, W., 15, 19. 158.
SOUTHALL, J. p. C, 173, 218.
Space-Systems, in collinear relation, 206-
210.
Specific Intensity of Radiation of luminous
surface, 573.
Spectrum, 475; continuous spectrum, 478;
solar spectrum, 478; purity of, 492-498;
purity of spectrum of single prism, 494;
ideal purity of spectrum, 497.
Spherical Aberration in its narrow sense, so-
called Longitudinal Aberration or Aber-
ration measured along the optical axis,
292, 300, 376-400, 467; development of
general formulae for, 380-385 ; formula for
the abolition of, 384, 467; in the special
cases (i) of a single spherical suriace. 386;
(2) of an infinitely thin lens, 387-392; (3)
of a system of two or more thin lenses,
392-394; (4) of a system of two thin
lenses in contact, 393. The terms of the
higher orders in the series-developments.
397-400.
Spherical Aberrations, Theory of: see Table
of Contents, Chap. XII; see also Astigr
matism. Coma, Curvature of Image, Dis-
tortion-Aberration, Orthoscopy, Sine-Con-
dition, Seidel, etc.
Spherical Aberrations, Chromatic Varia-
tions of the, 504, 526-531; chromatic
variation of the longitudinal aberration,
526; chromatic variation of the sine-con-
dition, 530.
Spherical Mirror, Astigmatic Reflexion of
narrow bundle of rays at a, 361-363.
Spherical Mirror, Reflexion of paraxial rays
at a, 137-147, 161; a case of central col-
lineation, 173; conjugate axial points
(M, M') with respect to, 137-142; con-
struction of axial image-point M', 139;
focal point and focal length, 140; extra-
axial conjugate points (Q, QO and the
lateral magnification (F), 142-147; con-
struction of image-point Q*, 144.
Spherical Mirror, Path of ray reflected at a.
299.
Spherical Over-Correction and Under-Cor-
rection, 378.
Spherical Refracting Suriace, Aplanatic
Points of: see Aplanatic Points,
Spherical Refracting Suriace, Astigmatic
Refraction of narrow bundle of rays at a:
see Table of Contents, especially Chap.
XI.
Spherical Refracting Suriace, Path of ray
refracted at a: Construction of the ray,
288; trigonometric formulae for calculat-
Index.
625
ing path of ray: see Table of Contents,
Chap. IX; see also Kerber and Sbidel.
Spherical Refracting Surface, Refraction of
Paraxial Rays at a, 147-162. 264; a case
of Central Collineation, 171; conjugate
axial points (M, 3fO. 147-152; construc-
. tion of axial image-point M\ 149; the
focal points (F. £0t 150; construction of
extra-axial image-point Q', 153; the focal
planes, 154; the focal lengths (/, O. I55.
264; the image-equations, 158-162, 264;
the zero-invariant (7), 159; the lateral
magnification (F), 160. 264; the angular
magnification (Z), 264; principal points
and nodal points. 264.
Spherical Refracting Surface, Spherical Ab-
errations incase of a: Astigmatism, 442;
Comatic Aberration, 455; Curvature of
Image, 442; Distortion- Aberration. 427;
Spherical or Longitudinal Aberration,
380-383, 386.
Spherical Surfaces, Centered System of:
see Centered System of Spherical Surfaces;
sec also Table of Contents, Chaps. VI,
VIII, X, XI, XII, XIII and XIV.
Spherically Corrected Optical System. 377;
different magnifications of the different
zones of, 402.
Steinheil, a., 307, 313, 327, 329, 398.
Steinheil, R., 473.
Stigmatic Image, Curvature of, 439.
Stokes, Sir G. G., 127.
Stops, Effect of, 532; aperture-stop, 533;
field-stop. 550; front and rear stops. 533;
interior stop. 374. 416, 420, 533; virtual
stop, 536.
Strength or Power (^) of infinitely thin lens,
186.
Strutt, J. W.: see Rayleigh.
Sturm, J. C, 44, 46. 48. 50, 66, 335; his
theory of astigmatism, 44-50. 335.
Subjective Magnifjnng Power, 545-549.
Surfaces, Designations of, in the diagrams,
594-596.
Sutton, T., 421; Bow-Sutton Condition,
421.
Symbols of angular magnitudes, 604-608;
of linear magnitudes, 596-604; of non-
geometrical magnitudes (constants, co-
efficients, etc.), 608-612.
Symmetry around principal axes, 221.
Symmetry and Congruence, in special
senses, 223.
Symmetry, Lack of, of a pencil of meridian
rays of finite aperture, 445, 446-448.
System of Lenses and System of Prisms:
see LenS'System, Prism-System,
T.
Tait, p. G.. 36, 571.
Tangential or y-aberrations, 374.
Tangent Condition of Airy. 420; see Orthos'
copy.
Taylor, H. D.. 319. 396. 438, 445.
Telecentric Optical System, 543.
Telescopes, Oculars of, 522.
Telescopic Imagery, 210, 243-245; image-
equations, 243, 244. 245; characteristics
of. 244.
Telescopic Optical System. 162. 243-245;
produced by a combination of two non-
telescopic systems, 251; combination of
two telescopic systems, 254; combination
of telescopic with non-telescopic system,
252.
Thick Lens, Chromatic Variations in case
of a, 513-516.
Thick Lens, Refraction of Paraxial Rays
through a. 179. 273-283; lateral magni-
fication (F). 180; optical centre, 181;
focal points, principal points and focal
lengths, 275. See Lenses, Lens-Systems.
Thicknesses measured along the optical
SLxis, 177, 316; thickness of a lens, 179,
274.
Thiesen, M., 38, 472.
Thin Lens, 283; see Infinitely Thin Lens.
Thin Prism : see Prism.
Thollon, L., 488, 495.
Thompson, S. P., 41, 445, 471.
ToEPLER, A., 238.
Total Reflexion, 22-25; sit second face of
prism, 78, 84.
V.
Veillon, H., 79.
Vertex of spherical surface, 134.
Vignette-angle, 565.
Virtual and real, 10; virtual image, 41.
Virtual stop, 536.
Vorr, E., 307, 313, 327, 329, 398.
Von Hoegh, E., 522.
Von Rohr, M., 98, 114, 128, 192, 218. 223,
262, 268. 312, 348, 350, 353, 366, 376, 398,
412, 420, 421, 422, 430, 434, 439, 446, 448,
455. 468, 473, 479, 485, 492, 493. 506, 509,
51a, 513. 519. 523. 526, 528. 529. 536, 550,
551. 554. 565.
W.
Wadsworth, F. L. O., 498.
Wagner, R., 238.
Wanach, B., 310, 366.
Wandersleb, E., 218, 223, 262.
Wave-Front. Hugybns's construction of, 4-
7; in special cases. 17-19.
Wave-Front, met by rays orthogonally. 39*
Wave-length of light and refractive index,
474.
Wave-motion, i.
Wave Theory of Light, I, 19.
Weierstrass, 288.
Whewbll, W., 200.
626 Index.
Wide-angle bundles of ra]^ Practical 339* 348* 352. 360, S93> 596; construction
Images by means of, 412. of ray refracted at a sphere, 288; con-
WiLSiNG, J.. 98, 128. tributions to theory of astigmatism. 348.
WoLLASTON. W. H., 477.
Wood. R. W., 44. Z.
Y
Zero-invariant (/), 159.
Young, T.. i. 288, 289, 290, 292. 294, 336. Zinken gen. Sommer. H., 438, 440.
:1> 3fc
.«
3 2044 024 236 275
This
Library on
. should be returned to the
r before the last date stamped
A fine of five cents a day b incurred by
retaining it beyond the specified time.
Please r
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11